Let $Au=-u''$ be the minimal symmetric operator on $\cH=L^2(0,1)$ with domain $\dom(A)=H^2_0(0,1)$ so that $A^*u=-u''$ with $\dom(A^*)=\cH_+=H^2(0,1)$, set $\mathfrak{H}=\bbC^2$
and introduce the surjective trace operator $\tr=(\Gamma_0,\Gamma_1)\in\cB(\cH_+,\mathfrak{H}\times\mathfrak{H})$ using the Dirichlet and (inward) Neumann traces $\Gamma_0u=[u(0), u(1)]^\top$ and $\Gamma_1u=[u'(0), - u'(1)]^\top$. Integration by parts yields (<ref>), and thus $(\mathfrak{H},\Gamma_0,\Gamma_1)$ is a boundary triplet, cf. <cit.>. For $t\in[0,1]$ we let $\cA_t$ denote the self-adjoint extension of $A$ with the domain

\begin{equation}\label{domAT}
\dom(\cA_t)=\{u\in H^2(0,1): \cos({\pi}t/2)\Gamma_0 u-\sin({\pi}t/2)\Gamma_1 u=0\}=\ker([X_t,Y_t]),\end{equation}
where $X_t=\cos({\pi}t/2)I_2$ and $Y_t=-\sin({\pi}t/2)I_2$, cf. (<ref>).
Given a bounded real-valued potential $V$, we
let $H_tu=-u''+Vu$, $t\in[0,1]$, be the family of scalar Schrödinger operators on $L^2(0,1)$ equipped with the boundary conditions
specified in (<ref>) so that Hypothesis <ref> holds.
In particular, $H_0$ is the Dirichlet and $H_1$ is the Neumann Schrödinger operator.
To apply Theorem <ref> we first perform a standard calculation of the resolvent $R_t(\zeta)=(H_t-\zeta)^{-1}$, cf., e.g., <cit.>: For $t\in[0,1]$ and $\zeta\in\bbC$ we let $v_t(\cdot\,;\zeta)$,  $w_t(\cdot\,;\zeta)$ denote the solutions to the equation $-u''+Vu=\zeta u$ that satisfy the initial conditions

\begin{align*}
(v_t(0;\zeta), v'_t(0,\zeta))&=\big(\sin({\pi}t/2), \cos({\pi}t/2)\big),\\ (w_t(1;\zeta), w'_t(1,\zeta))
&=\big(\sin({\pi}t/2), -\cos({\pi}t/2)\big),
\end{align*}

and let $\cW_t(\zeta)=v_t(x;\zeta)w'_t(x;\zeta)-v'_t(x;\zeta)w_t(x;\zeta)$ denote their Wronskian. Then for each $u\in L^2(0,1)$ the function $R_t(\zeta)u$ is given by the formula

\[\big(R_t(\zeta)u\big)(x)=(\cW_t(\zeta))^{-1}\Big(w_t(x;\zeta)\int_0^xv_t(y;\zeta)u(y) {\rm d } y
+v_t(x;\zeta)\int_x^1w_t(y;\zeta)u(y) {\rm d} y\Big),\]

$x\in[0,1]$.Using this, it is convenient to write $\tr R_t(\zeta)=K_t(\zeta)L_t(\zeta)$ where we temporarily introduced the $(4\times2)$ matrix $K_t(\zeta)$ and the operator $L_t(\zeta)$ by the formulas

\begin{align}
&K_t(\zeta)=(\cW_t(\zeta))^{-1}\big[\sin({\pi}t/2)I_2, \cos({\pi}t/2)I_2\big]^\top,\\
&L_t(\zeta)u=\big[\langle w_t(\cdot;\zeta), \overline{u}\rangle_{L^2}, \langle v_t(\cdot;\zeta), \overline{u}\rangle_{L^2}\big]^\top, L_t(\zeta)\in\cB(L^2(0,1), \bbC^2)
\end{align}

so that $(L_t(\zeta))^*$ maps $(z_1,z_2)\in\bbC^2$ into $ w_t(\cdot\,;\overline{\zeta})z_1+ v_t(\cdot\,;\overline{\zeta})z_2\in L^2(0,1)$. Theorem <ref> and  a short calculation  now yield

\begin{align}
\big(&R_t(\zeta)-R_s(\zeta)\big)u=(\cW_t(\zeta)\cW_s(\zeta))^{-1}\sin(\pi(t-s)/2)\\
&\times\big(\langle w_s(\cdot;\zeta), \overline{u}\rangle_{L^2}w_t(\cdot;\zeta)+ \langle v_s(\cdot;\zeta), \overline{u}\rangle_{L^2}v_t(\cdot;\zeta)\big), \,\zeta\notin\spec(H_t)\cup\spec(H_s),
\end{align}

\begin{align}
\dot{R}_t(\zeta)u&=\frac{\pi}{2}(\cW_t(\zeta))^{-2}
\big(\langle w_t(\cdot;\zeta), \overline{u}\rangle_{L^2}w_t(\cdot;\zeta)+ \langle v_t(\cdot;\zeta), \overline{u}\rangle_{L^2}v_t(\cdot;\zeta)\big),\\&\hskip5cm\zeta\notin\spec(H_t), \\
\dot{\lambda}(t_0)&=-\frac{\pi}{2}
\big\|\sin(\pi t_0/2)\Gamma_0u_0+\cos(\pi t_0/2)\Gamma_1u_0\big\|^2_{\bbC^2},\,  t_0\in[0,1],
\end{align}

where $u_0$ is the normalized  eigenfunction corresponding to the eigenvalue $\lambda(t_0)\in\spec(H_{t_0})$.

§.§ Laplace operator on bounded domains via boundary triplets

The main result of this section is Theorem <ref> in which we derive the Krein-type formula, Riccati equation, and Hadamard-type formula for a family of Robin-type Laplacians. To that end, we employ abstract results of Theorem <ref> with a boundary triplet specifically defined for the Laplace operator. This triplet is discussed in [52, 54, 58] and recalled in Appendix <ref>.

Throughout this section
we assume that  $n\in\mathbb{N},n\geq2$ and  $\Om\subset\mathbb{R}^n$ is a bounded domain with $C^{1,r}$-boundary with $r>1/2$ (although this assumption can be considerably weakened, see Remark <ref> below).
We define the maximal and minimal Laplace operators as follows,

\begin{align}
-\Delta_{\max}&:\dom(-\Delta_{\max})\subset L^2(\Omega)\rightarrow L^2(\Omega),\no\\
\dom(-\Delta_{\max})&=\big\{u\in L^{2}(\Om)\big|\  \Delta u\in L^{2}(\Om)\big\},\no\\
-\Delta_{\max}u&=-\Delta u\ \  (\text{in the sence of distribudtions}),\\\no
\dom(-\Delta_{\min})&=H^2_0(\Om),\,-\Delta_{\min}u=-\Delta u,\no
\end{align}

and remark that by <cit.> one has

\begin{equation}\label{m=m*}\begin{split}
\dom(-\Delta_{\min})&=H^2_0(\Om)=\{u\in L^{2}(\Om)|\  \Delta u\in L^{2}(\Om) ,\ \gd(u)=0,\ \gn(u)=0\},\\
\end{equation}

Here and below we use the following extensions of the  Dirichlet and (weak) Neumann

\begin{equation}\label{dfnbtrDN1}\begin{split}
\gd&:\{u\in L^{2}(\Om)\,|\,\Delta u\in L^{2}(\Om)\}\to H^{-1/2}(\dOm),\\
\gn&:\{u\in L^{2}(\Om)\,|\,\Delta u\in L^{2}(\Om)\}\to H^{-3/2}(\dOm),\end{split}
\end{equation}

and consider the map

\begin{equation}\label{dfnbtrDN2}
\tN:\{u\in L^2(\Omega)| \Delta u\in L^2(\Omega)\}\rightarrow H^{1/2}(\partial\Omega),\,%\no,\\
\tN u:=\gn u+ M_{D,N}(\gd u),
\end{equation}

where $M_{D,N}$ is the Dirichlet-to-Neumann map acting by the rule $M_{D,N}:g\mapsto -\gn u$ for
$u$ being the solution of the boundary value problem

\begin{equation}\label{bvpg}
-\Delta u =0,\,
u\in L^2(\Om),\quad
\gd u =g   \text{\ on\ } \partial\Omega. \end{equation}

More details regarding the definitions of $\gd, \gn$ and $\tau_{{}_N}$ and their properties are discussed in Appendix <ref>, cf. Lemma <ref>, <ref>, <ref> taken from [54]. In the sequel we use the Reisz isomorphism given by

\begin{align}
\begin{split}\lb{aub27}
&\Phi: H^{-1/2}(\partial\Omega)\rightarrow H^{1/2}(\partial\Omega),\\
&H^{-1/2}(\partial\Omega)\ni\psi\mapsto \Phi_{\psi}\in H^{1/2}(\partial\Omega),\\
& \langle f,\psi  \rangle_{{-1/2}}:=\psi(f)=\langle f,\Phi_{\psi}  \rangle_{1/2}, f\in H^{1/2}(\partial\Omega), \psi\in H^{-1/2}(\partial\Omega),
\end{split}
\end{align}

in particular, for $f, \psi \in H^{1/2}(\partial\Omega)\hookrightarrow
L^2(\partial\Omega)\hookrightarrow H^{-1/2}(\partial\Omega)$ we have

\begin{equation}
\langle f,\psi  \rangle_{{-1/2}}=\langle f,\psi  \rangle_{L^2(\partial\Omega)}.
\end{equation}

We also note that $\Phi$ is a conjugate linear mapping. The next lemma is a well-known fact that goes back to [58].
Assume that $\Om\subset\mathbb{R}^n$ is a bounded domain with $C^{1,r}$-boundary, $r>1/2$, and the boundary traces $\gd$, $\tau_{{}_N}$ are as in (<ref>), (<ref>). Then

\begin{equation}\lb{int70}
(\mathfrak H, \Gamma_0, \Gamma_1):=(H^{1/2}(\partial\Omega), \tau_{{}_N}, \Phi\gd)
\end{equation}

is a boundary triplet for $A=-\Delta_{\min}$.

The trace operator $\tr:= [\tau_{{}_N}, \Phi\gd]^\top$ is defined on the space
\[\cH_+:=\{u\in L^2(\Omega): \Delta u\in L^2(\Omega)\}\]
with the norm
\[\|u\|_{\cH_+}=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}. \]

Recalling the Green identity (<ref>)

\begin{align}
&(-\Delta u,v)_{L^{2}(\Om)}-(u,-\Delta v)_{L^{2}(\Om)}\no\\
&=-{\lnoh \tN u, \gd v \rnohs}+\overline{\lnoh \tN v, \gd u \rnohs},
\end{align}

we rewrite it as

\begin{align}
&\langle A^* u,v\rangle_{\cH}-\langle u,A^* v\rangle_{\cH}=-\langle\Gamma_0u, \Gamma_1 v\rangle_{\mathfrak H}+\overline{\langle\Gamma_0v, \Gamma_1 u\rangle_{\mathfrak H}},\no\\
&=\langle\Gamma_1 u, \Gamma_0v\rangle_{\mathfrak H}-\langle\Gamma_0u, \Gamma_1 v\rangle_{\mathfrak H},
\end{align}

and thus check that (<ref>) satisfies the abstract Green identity. It remains to show that the map $\tr:\cH_+\to H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega)$ is onto.
We fix a vector $(f,g)\in H^{1/2}(\partial\Omega)\times H^{-1/2}(\partial\Omega)$.
By  (<ref>) there exists
$u_0\in H^2(\Omega)\cap H_0^1(\Omega)$ such that $\tN u_0=f$. By <cit.>, the boundary value problem (<ref>)
has a unique solution that we denote by $v_0$ (we note that zero is outside of the spectrum of the Dirichlet Laplacian). Employing (<ref>) and $v_0\in \ker(\tN)$  yields

\begin{align}
\tr (u_0+v_0)=(\tN(u_0+v_0),\Phi \gd(u_0+v_0))=(\tN u_0 ,\Phi \gd v_0)=(f,\Phi g)
\end{align}

since $\gd u_0=\gaD u_0 =0$.

In PDE literature, boundary value problems are often formulated in terms of the Dirichlet and Neumann traces defined by

\begin{align}
&\gaD: \{u\in H^1(\Omega): \Delta u\in L^2(\Omega) \}\rightarrow H^{1/2}(\dOm), \gaD:=\gd\upharpoonright_ {\{u\in H^1(\Omega): \Delta u\in L^2(\Omega) \}},\\
&\gaN: \{u\in H^1(\Omega): \Delta u\in L^2(\Omega) \}\rightarrow H^{-1/2}(\dOm), \gaN:=\gn\upharpoonright_ {\{u\in H^1(\Omega): \Delta u\in L^2(\Omega) \}}.
\end{align}

We note that $(-\Delta_{\max}, \gaD, \gaN)$ is not a boundary triplet. Firstly, $\tr:=(\gaD, \gaN)$ is not defined on the entire space $\dom(-\Delta_{\max})$. Secondly, $\tr$ is not onto, see <cit.>. However, Hypothesis <ref> is still satisfied with $\cD:=\{u\in H^1(\Omega): \Delta u\in L^2(\Omega) \}$ and equipped with the norm $(\|u\|_{H^1(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}$. In fact, Hypothesis <ref> is also satisfied for this choice of $\tr, \cD$. These facts serve as our main motivation for introducing Hypotheses <ref>, <ref>. We elaborate on this further in Section <ref>.

Having constructed the  boundary triplet for the Laplacian, we can now apply the abstract results from Theorem <ref>.
theorem5.4 Let $\Om\subset\mathbb{R}^n$ be a bounded domain with $C^{1,r}$-boundary, $r>1/2$, and let
$t\mapsto \Xi_t\in \cB(H^{1/2}(\partial\Omega))$, $t\in[0,1]$, be a differentiable family of self-adjoint operators. Then for $t\in[0,1]$ the linear operator

\begin{align}\lb{aub1}
&-\Delta_t: \dom(-\Delta_t)\subset L^2(\Omega)\rightarrow L^2(\Omega), -\Delta_t u=-\Delta u, \\
&u\in\dom(-\Delta_t):=\{ u\in \dom(\Delta_{\max}): \Phi\gd u+\Xi_t \tN u=0 \},
\end{align}

is self-adjoint.
The following Krein-type formula holds

\begin{align}\begin{split}\lb{deltaKrein}
&= \big(\tN(-\Delta_t-\overline{\zeta})^{-1} \big)^* (\Xi_t-\Xi_s)\big(\tN(-\Delta_s-\zeta)^{-1}\big),
\end{split}
\end{align}

for $t,s\in[0,1]$, $\zeta\not\in(\spec(-\Delta_t)\cup\spec(-\Delta_s))$. Moreover, for a fixed $t_0\in[0,1]$ the mapping

\begin{equation}\lb{deltacont}
t\mapsto (-\Delta_t-\zeta)^{-1}\in \cB(L^2(\Omega))
\end{equation}

is well defined for $t$ near $t_0$ as long as $\zeta\not\in\spec(-\Delta_{t_0})$. This mapping is differentiable at $t_0$ and satisfies the following Riccati equaiton

\begin{align}
\begin{split}\lb{deltaRic}
\frac{\bd}{\bd t}\big|_{t=t_0}\big((-\Delta_t&-\zeta)^{-1}\big)\\
&= \big(\tN(-\Delta_{t_0}-\overline{\zeta})^{-1} \big)^* \big(\frac{\bd}{\bd t}\big|_{t=t_0}\Xi_t\big)\big(\tN(-\Delta_{t_0}-\zeta)^{-1}\big).
\end{split}
\end{align}

Finally, if $\lambda(t_0)$ is an eigenvalue of $-\Delta_{t_0}$ of multiplicity $m\geq 1$ then there exists  a choice of orthonormal eigenfunctions  $\{u_j\}_{j=1}^m\subset \ker(-\Delta_{t_0}-\lambda(t_0))$
and a labeling of eigenvalues $\{\lambda_j(t)\}_{j=1}^m$  of $-\Delta_{t}$, for $t$ near $t_0$, such that

\begin{equation}\lb{thetder}
\dot{\lambda}_j(t_0)= -\langle\dot \Xi_{t_0} \tN u_j, \tN u_j\rangle_{L^2(\partial\Omega)}, 1\leq j\leq m.
\end{equation}

By Lemma <ref>, $(H^{1/2}(\partial\Omega), \tN, \Phi\gd)$ is a boundary triplet. In order to check that $-\Delta_t$ is self-adjoint, it suffices to check conditions (<ref>), (<ref>) with $X:=\Xi_t$, $Y:=I$. Indeed, (<ref>) holds since $\Xi_t$ is self-adjoint, (<ref>) holds since the operator $XX^*+YY^*$  given by $I+\Xi_t^2>0$ is invertible.  The fact that (<ref>) is well defined for $t$ near $t_0$ follows from continuity of $\Xi_t$ and Theorems <ref> and <ref> upon setting $\cA_t:=-\Delta_t$, $V_t:=0$, $\tr_t:=[\tN, \Phi\gd]^\top$. In order to prove (<ref>), (<ref>), (<ref>), we use (<ref>), (<ref>), (<ref>), respectively, with

\begin{align}
&\big(W(\Xi_t, I)\big) \tr   R_{t}(\zeta)= (I+\Xi_t^2)^{-1}( -\Gamma_0R_{t}(\zeta)+\Xi_t\Gamma_1R_{t}(\zeta))\no\\
&\quad=(I+\Xi_t^2)^{-1}( -\Gamma_0R_{t}(\zeta)-\Xi_t^2\Gamma_0R_{t}(\zeta))=-\Gamma_0R_{t}(\zeta)=-\tau_N R_{t}(\zeta)
\end{align}

and $\phi_j=-\tN u_j$.

The assumption $\partial\Omega$ being $C^{1,r}$, $r>1/2$, imposed in this section could be replaced by  $\partial\Omega$ being Lipschitz and $\Omega$ quasi-convex, see <cit.> for the definition. As proved  in [54], these weaker assumptions are sufficient for the domains of the Dirichlet and Neumann Laplacians to belong to $H^2(\Omega)$, which in turn is equivalent to (<ref>) to hold. Also, for the case of Lipschitz domains Lemma <ref> and, as demonstrated in [54], leading to it Lemmas <ref>, <ref>, <ref> hold with the Sobolev spaces $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$ replaced by $N^{1/2}(\partial\Omega)$ and its adjoint $\big(N^{1/2}(\partial\Omega)\big)^*$, respectively, where the space $N^{1/2}(\partial\Omega)$ is defined as $\{f\in L^2(\partial\Omega): f\nu_j\in H^{1/2}(\partial\Omega)\}$, $\nu=(\nu_j)_{j=1}^n$, and is equal to $H^{1/2}(\partial\Omega)$ provided $\partial\Omega$ is $C^{1,r}$,  $r>1/2$.

Our motivation to consider the boundary condition in Theorem <ref> stems from [35, 52, 73]. More generally, the boundary condition described in Theorem <ref> can be replaced by  $ X_t \gn u+Y_t \tN u=0$ for $X_t, Y_t\in \cB(H^{1/2}(\partial\Omega))$ satisfying (<ref>), (<ref>).  In this case, as in Theorem <ref>,  continuity of the mappings $t\mapsto X_t$, $t\mapsto Y_t$ yields continuity of the resolvent operator with respect to $t$. Moreover, differentiability of the mappings $t\mapsto X_t$, $t\mapsto Y_t$ yields differentiability of the resolvent operator with respect to $t$ as well as the Reccati equation and the formula for the slopes of the eigenvalue curves (both obtained by dropping the potential terms  $V_t$ in  (<ref>), (<ref>), respectively).

§.§ Quantum graphs
The main result of this section is Theorem <ref> in which we derive the Krein-type formula, Riccati equation, and Hadamard-type formula for Schrödinger operators on metric graphs. To that end, we employ the abstract results discussed in Theorem <ref> with a boundary triplet specifically defined for quantum graphs. Examples <ref>  and   <ref> give two applications of Theorem <ref>. Both examples concern monotonicity of eigenvalue curves of Schrödinger operators with respect to some natural parameter present in the boundary conditions.

We begin by discussing  differential operators on metric graphs. To set the stage, let us fix a discrete graph $(\cV,\cE)$ where $\cV$ and $\cE$ denote the set of vertices and edges respectively. We assume that $(\cV,\cE)$ consists of a finite number of vertices,  $|\cV|$, and a finite number of edges,  $|\cE|$. We assign to each edge $e\in\cE$ a positive and finite length $\ell_{e}\in(0,\infty)$. The corresponding metric graph is denoted by $\cg$. The boundary $\partial\cg$ of the metric graph is defined by

\begin{equation}
\partial\cg:=\cup_{e\in\cE} \{a_e,b_e\},
\end{equation}

where $a_e, b_e$ denote the end points of the edge $e$.  It is convenient to treat $2|\cE|$ dimensional vectors as functions on the boundary $\partial\cg$, in particular,
$L^2(\partial\cg)\cong \bbC^{2|\cE|}$,
where the space $L^2(\partial\cg)=\bigoplus_{e\in\cE}\left( L^2(\{a_e\})\times L^2(\{b_e\})\right)$  corresponds to the discrete Dirac measure with support $\cup_{e\in \cE} \{a_e, b_e\}$. In addition to the space of functions on the boundary we consider the Sobolev spaces of functions on the graph $\cg$,

\begin{align}
&L^2(\cg):=\bigoplus_{e\in\cE}L^2(e),\  \hatt{H}^2(\cg):=\bigoplus_{e\in\cE}H^2(e)\no,
\end{align}

where $H^2(e)$ is the standard $L^2$ based Sobolev space.
As in the setting of  Laplace operators on bounded domains, the spaces ${L}^2(\cg)$ and $L^2(\partial \cg)$ are related via the trace maps. We define the trace operators $(\Gamma_0, \Gamma_1)$  by the formulas

\begin{align}
&\Gamma_0: \hatt{H}^2(\cg)\rightarrow L^2(\partial \cg),
\ \Gamma_0u:=u|_{\partial \cg}, u\in \hatt{H}^2(\cg),\\%\label{vv2}\\
&\Gamma_1: \hatt{H}^2(\cg)\rightarrow L^2(\partial \cg),
\ \Gamma_1 u:=\partial_{n}u|_{\partial \cg}, u\in \hatt{H}^2(\cg),%\label{vv3}
\end{align}

where $\partial_{n} u$ denotes the derivative of $u$ taken in the inward direction. The trace operator is a bounded, linear operator given by

\begin{equation}%\lb{2.4new}
\tr:=
\Gamma_1]^\top,\, \tr: \hatt{H}^2(\cg)\rightarrow L^2(\partial \cg)\times L^2(\partial \cg)\cong \bbC^{4|\cE|}.
\end{equation}

The Sobolev space of functions vanishing on the boundary $\partial\cg$ together with their derivatives is denoted by

\begin{equation}\no
H^2_0(\cg):=\left\{u\in \hatt{H}^2(\cg): \tr u=0\right\}.
\end{equation}

Using our notation for the trace maps, the Green identity can be written as follows,

\begin{align}\label{vv5}
\int_{\cg} (-u'')\overline{v}-u\overline{(-v'')}
\partial_{n}u\overline{v}-u\overline{\partial_{n}v} \\
&=\langle [J\otimes I_{2|\cE|}]\tr u, \tr v\rangle_{\bbC^{4|\cE|}},\  u,v\in  \hatt{H}^2(\cg).\no
\end{align}

The right-hand side of the Green identity defines a symplectic form

\begin{align}
\label{vv9}
&\omega:\  \dL^2(\partial \cg)\times\,\dL^2(\partial
\cg)\rightarrow \bbC, \\
\label{eq:def_omega}
&\omega( (f_1, f_2), (g_1, g_2)):=\int_{\partial\cg}f_2 \overline {g_1}-f_1\overline{g_2},\\
& (f_1, f_2), (g_1, g_2)\in \dL^2(\partial\cg),\label{vv10}
\end{align}

where $\dL^2(\partial \cg):=L^2(\partial \cg)\times L^2(\partial \cg)$.

Next, we introduce the minimal Laplace operator $A_{min}$ and its adjoint $A_{max}$. The operator

\begin{equation}\label{b5}
A_{min}:=-\frac{\bd^2}{\bd x^2},\quad \dom(A_{min})=\hatt H^2_0(\cg),
\end{equation}

is symmetric in $L^2(\cg)$. Its adjoint $A_{max}:=A_{min}^*$ is given by

\begin{equation}\label{b6}
A_{max}:=-\frac{\bd^2}{\bd x^2},\quad \dom(A_{max})=\hatt{H}^2(\cg).
\end{equation}

The dificiency indices of $A_{min}$ are finite and equal, that is,

\begin{equation}
\end{equation}

theorem4.5Assume that

\begin{align}
&t\mapsto V_t \text{ is in } C^1([0,1], L^{\infty}(\cg)),\\
&t\mapsto X_t, Y_t \text{ is in } C^1([0,1], \bbC^{2|\cE|\times 2|\cE|}),\  \det(X_tX^*_t+Y_tY_t^*)\not=0, X_tY_t^*=Y_t^*Y_t.
\end{align}

Then the operator

\begin{align}
&\cA_t: L^2(\cg)\rightarrow L^2(\cg), \dom(\cA_t):=\{u\in H^2(\cg): [X_t, Y_t]\tr u=0 \},\\
&\cA_t u=-u'', u\in\dom(\cA_t),
\end{align}

is a self-adjoint extension of $A_{min}$. The operator-valued function

\begin{equation}
t\mapsto R_t(\zeta_0):=(\cA_t+V_t-\zeta_0)^{-1} \text{ for all } \zeta_0\not\in\spec(\cA_t)
\end{equation}

is in $C^1([0,1], \cB(L^2(\cg)))$ and for any $t_0\in[0,1]$ one has

\begin{align}\lb{derRnewnew}
\begin{split}
&\dot{R}_{t_0}(\zeta_0)=-R_{t_0}(\zeta_0)\dot V_{t_0}R_{t_0}(\zeta_0)\\
&+(\tr R_{t_0}(\overline{\zeta_0}))^*\big(W(X_{t_0}, Y_{t_0})\big)^*(\dot X_{t_0}{Y}_{t_0}^*-\dot Y_{t_0}{X}_{t_0}^*)\big(W(X_{t_0}, Y_{t_0})\big)\tr  R_{t_0}(\zeta_0),
\end{split}
\end{align}

where $W(X_{t_0}, Y_{t_0})$ is as in (<ref>).
Furthermore, if $\lambda(t_0)$ is an eigenvalue of $\cA_{t_0}+V_{t_0}$ of multiplicity $m\geq 1$ then there exist  a choice of orthonormal eigenfunctions

\begin{equation}\no
\{u_j\}_{j=1}^m\subset \ker(\cA_{t_0}+V_{t_0}-\lambda(t_0))
\end{equation}

and a labeling of eigenvalues $\{\lambda_j(t)\}_{j=1}^m$  of $\cA_t+V_t$, for $t$ near $t_0$, such that

\begin{equation}\lb{5.10newnew24}
\dot{\lambda}_j(t_0)= \langle \dot V_{t_0}u_j, u_j\rangle_{L^2(\cg)}+\big\langle (X_{t_0}\dot{Y}_{t_0}^*-Y_{t_0}\dot{X}_{t_0}^*) \phi _j, \phi_j\big\rangle_{L^2(\partial\cg)},
\end{equation}

where $\phi_{j}=W(X_{t_0}, Y_{t_0})\tr u_j$ is a unique $2|\cE|$ dimensional vector satisfying $\Gamma_0 u_j=-Y_{t_0}^*\phi_j$ and $\Gamma_1 u_j=X_{t_0}^*\phi_j$, $1\leq j\leq m$.

Since $(L^2(\partial\cg), \Gamma_0, \Gamma_1)$ is a boundary triplet,  equations (<ref>) and (<ref>) in Theorem <ref>
give (<ref>) and (<ref>) respectively.

Consider the Schrödinger operator $\displaystyle{H_{t}=-\frac{\bd^2}{\bd x^2}+V}$ on a compact star graph $\cg=(\cE,\mathcal{V})$ with a bounded real-valued potential $V$ subject to arbitrary self-adjoint vertex conditions at the vertices of degree one and the following $\delta$-type condition at the center ${\rm v_c}\in\mathcal{V}$,

\begin{equation}\label{BCnu}
\sum_{e\sim {\rm v_c}}{\partial_{n} u_e({\rm v_c})}=t u({\rm v_c}),\ t\in\bbR.
\end{equation}

In this example, we will derive an Hadamard-type formula (<ref>) for the derivative of the eigenvalues of $H_t$. Such a formula is discussed in <cit.> for simple eigenvalues. The general case can be treated using (<ref>) as follows.  The  boundary matrices describing the vertex conditions are given by $\wti X\times X_{t}$ and $\wti Y\times Y$ where

\begin{equation}\no
X_{t}= \begin{bmatrix}
1&-1&&\cdots &0\\
\end{bmatrix},\quad
Y= \begin{bmatrix}
0&&\cdots &&0\\
0&&\cdots &&0\\
\end{bmatrix},
\end{equation}

and the matrices $\wti X$ and $\wti Y$ correspond to the vertex conditions at $\cV\setminus\{{\rm v_c}\}$. A direct computation gives

\begin{equation}\lb{3.33}
X_{t}^*Y=Y^*X_{t}= \begin{bmatrix}
0&0&\cdots &&0\\
\end{bmatrix}.\
\end{equation}

For the eigenvalue $\lambda(t_0)$ of $H_{t_0}$ of multiplicity $m\in\bbN$  we use (<ref>) to get

\begin{equation}
\dot{\lambda}_j(t_0)= \big\langle (X_{t_0}\dot{Y}_{t_0}^*-Y_{t_0}\dot{X}_{t_0}^*) \phi _j, \phi_j\big\rangle_{L^2(\partial\cg)}=|\phi_j({\rm v_c})|^2,
\end{equation}

where $1\leq j\leq m$, $\phi_{j}=W(X_{t_0}, Y_{t_0})\tr u_j$, and $\{u_j\}_{j=1}^m$ are the eigenfunctions of $H_{t_0}$ corresponding to $\lambda(t_0)$. Furthermore, using (<ref>) we obtain $\phi_j({\rm v_c})=-u_j({\rm v_c})$, hence,

\begin{equation}\lb{BKHad}
\dot{\lambda}_j(t_0)=|u_j({\rm v_c})|^2,\, 1\leq j\leq m.
\end{equation}

ex47 This example concerns monotonicity of eigenvalue curves of a class of Schrödinger operators on a compact interval arising in the spectral theory of periodic Hamiltonians. Specifically, we consider the Schrödinger operators $H_\vartheta$ with a real valued potential $V\in L^{\infty}(0,1)$ which are parameterized by $\vartheta\in[0, 2\pi)$ and defined as follows,

\begin{align}
&H_\vartheta=\cA_\vartheta+V,\, \cA_{\vartheta}: L^2(0,1)\rightarrow L^2(0,1), \cA_{\vartheta} u=-u'', u\in\dom(\cA_{\vartheta}), \\
&\dom(\cA_{\vartheta}):=\{u\in H^2(0,1): e^{\bfi \vartheta}u(0)=u(1), e^{\bfi \vartheta}u'(0)=u'(1)  \}.\lb{aub14}
\end{align}

Such operators are of interest, in particular, because their eigenvalues fill up the spectral bands of the Schrödinger operator
in $L^2(\bbR)$ with  the potential given by the periodic extension of $V$, see   <cit.>. We claim that the eigenvalue curves satisfy

\begin{equation}\label{formdernew}
\dot\lambda_j(\vartheta_0)=2\Im(u_j'(0)\overline{u_j(0)}) \text{ for all } \vartheta_0\in(0, 2\pi),
\end{equation}

where, as usual, $u_j\in \ker(\cA_{\vartheta_0}-\lambda_j(\vartheta_0))$, $j=1,2$ (in fact all but, possibly, periodic and antiperiodic operators have simple spectra). We derive this formula from (<ref>) by defining trace operators appropriately.  It is well known that ordinary differential operators fit well into the scheme of boundary triplets, cf. e.g. <cit.>, however, for completeness we recall the setting. Indeed, we set

\begin{align}
&\cH:=L^2(0,1), \cH_+:=H^2(0,1), A=-\frac{\bd^2}{\bd x^2}, \dom(A)=H^2_0(0,1)\\
&\tr: H^2(0,1)\rightarrow \C^4, \Gamma_0u:=(u(0), u(1))^{\top}, \Gamma_1u:=(u'(0), -u'(1))^{\top}.
\end{align}

Next, to utilize (<ref>) we first rewrite the boundary conditions in (<ref>) as follows,

\begin{equation}
X_{\vartheta}\Gamma_0 u+Y_{\vartheta}\Gamma_1 u=0, \text{ where }
-\e^{\bfi\vartheta}& 1\\0&0
\end{bmatrix}, Y_{\vartheta}:=\begin{bmatrix}
\e^{\bfi \vartheta}&1
\end{bmatrix},
\end{equation}

and compute

\begin{align}
\phi_j=W(X_{\vartheta}, Y_{\vartheta})\tr u_j&= \frac{1}{2}(-Y_{\vartheta_0}\Gamma_0u_j+X_{\vartheta_0}\Gamma_1u_j)= -e^{\bfi \vartheta_0}(u_j'(0), u_j(0))^{\top},\\
X_{\vartheta_0} \dot Y^*_{\vartheta_0}-Y_{\vartheta_0}X^*_{\vartheta_0} &=\begin{bmatrix}
\end{bmatrix}.
\end{align}

Plugging this in (<ref>) yields (<ref>). Monotonicity of the eigenvalues follows from linear independence of $u_j, \overline{u_j}$ and the next formula involving the Wronskian,

\begin{equation}
2|\Im(u_j'(0)\overline{u_j(0)})|=|\cW(u_j, \overline{u_j})(0)|\not=0, \vartheta_0\in(0, 2\pi).
\end{equation}

§.§ Periodic Kronig–Penney model
[An alternative approach applicable to a very broad class of second-order operators is discussed in the upcoming work of D. Damanik, J. Fillman and the second author]  In  this section we give yet another application of Theorem <ref> proving a version of B. Simon's theorem [94] which states that a certain open gap property (described below) of periodic Schrödinger operators is generic in the class of periodic $C^{\infty}(\bbR)$ potentials.  The main result of this section, Theorem  <ref>, states this assertion for singular $\delta-$type potentials. Its proof is based on a perturbative argument inspired by [94] and technically made available by Theorem <ref>.

The spectrum of the Schrödinger operator with periodic potential on the line has a band-gap structure, that is, in general, it consists of closed segments, called bands, such that two adjacent bands can either have a common endpoint or be separated by an open interval, a gap, of the resolvent set; in the latter case we say that the gap is open.
We will now use Theorem <ref> to prove that all gaps of a generic periodic Kronig–Penney model are open. The operators in question are the Schrödinger operators with $\delta-$type potentials which in physics literature are written as follows,

\begin{align}
&H_{{\alpha}}:=-\frac{\bd^2}{\bd x^2}+\sum_{k\in\bbZ}\alpha_k \delta(x-k),
\end{align}

and mathematically are defined by

\begin{align}
&H_{\alpha}u:=-u'', u\in\dom(H_{\alpha}), H_{\alpha}:\dom(H_{\alpha})\subset L^2(\bbR)\rightarrow L^2(\bbR),\\
&\dom(H_{{\alpha}})= \{u\in \hatt H^2(\bbR\setminus\bbZ): u\ \text{satisfies}\  \eqref{6.8new}\text{\ for all}\ k\in\bbZ\},\\
&\hspace{3.3cm}u(k^+)=u(k^-),\ u'(k^+)-u'(k^-)=\alpha_ku(k),\lb{6.8new}
\end{align}

where $\alpha=\{\alpha_k\}_{k\in\bbZ}\in \ell^{\infty}(\bbZ; \bbR)$, $u(k^\pm)$ are the one-sided limits, and $\hatt{H}^2$ denotes the direct sum of the Sobolev spaces on respective intervals. The spectrum of $H_{\alpha}$ for the case of periodic sequence $\alpha$ has a band-gap structure, see <cit.>. This was originally proved for $1-$periodic sequences but can be directly extended to any $p-$periodic ones. Specifically, given  a $p-$periodic sequence $\alpha=\{\alpha_k\}_{k\in\bbZ}\in \ell^{\infty}(\bbZ; \bbR)$, the operator $H_{\alpha}$ is unitary equivalent to the direct integral

\begin{equation}
\int_{[0,2\pi)}^{\oplus} H_{{\alpha^{(p)}}, \vartheta}\frac{\bd\vartheta}{2\pi}, \, \text{ where we denote $\alpha^{(p)}:=\{\alpha_0, ..., \alpha_{p-1}\}\in\bbR^p$,}
\end{equation}

and $H_{{\alpha^{(p)}}, \vartheta}$ for $\vartheta\in[0,2\pi)$ is the operator defined in $ L^2(I_p)$ with $I_p:=(-1/2,p-1/2)$  by

\begin{align}
&H_{{\alpha^{(p)}}, \vartheta}u:=-u'',\ H_{{\alpha^{(p)}}, \vartheta}: \dom(H_{\alpha^{(p)}, \vartheta})\subset L^2(I_p)\rightarrow L^2(I_p),\\
&\dom(H_{\alpha^{(p)}, \vartheta})= \Big\{u\in \hatt H^2(I_p\setminus\bbZ):  u\ \text{satisfies}\  \eqref{6.8new} \text{\ for $k\in I_p\cap\bbZ$ and\ } \eqref{6.9new}\Big\},\\
&u(-1/2^+)=e^{\bfi\vartheta}u((p-1/2)^-),\, u'(-1/2^+)=e^{\bfi\vartheta}u'((p-1/2)^-),\lb{6.9new}
\end{align}

\begin{equation}
\hatt H^2(I_p\setminus\bbZ):=H^2(-1/2, 0)\oplus H^2(0, 1)\oplus...\oplus H^2(p-2, p-1)\oplus H^2(p-1, p-1/2).
\end{equation}

Denoting the eigenvalues of $H_{\alpha^{(p)}, \vartheta}$ (ordered in non-decreasing order) by
\[\lambda_j(\alpha^{(p)}, \vartheta), j=1,2,\dots,\]
we have

\begin{align}
\lambda_1(\alpha^{(p)}, 0)&\leq \lambda_1(\alpha^{(p)}, \vartheta)\leq  \lambda_1(\alpha^{(p)}, \pi)\leq \lambda_2(\alpha^{(p)}, \pi)\leq  \lambda_2(\alpha^{(p)}, \vartheta)\leq\lambda_2(\alpha^{(p)}, 0)\\
&\leq \lambda_3(\alpha^{(p)}, 0)\leq \lambda_3(\alpha^{(p)}, \vartheta) \leq\lambda_3(\alpha^{(p)}, \pi)\leq\dots \text{ for $\vartheta\in[0,\pi]$.}
\end{align}

Then the spectrum of $H_{\alpha}$ is given by

\begin{align}
\spec(H_{\alpha})&=\bigcup\limits_{\vartheta\in[0,\pi]}\spec(H_{\alpha^{(p)}, \vartheta})\\
&=[\lambda_1(\alpha^{(p)}, 0),  \lambda_1(\alpha^{(p)}, \pi)]\cup[\lambda_2(\alpha^{(p)}, \pi), \lambda_2(\alpha^{(p)}, 0)]\cup\dots.
\end{align}

The intervals $[\lambda_1(\alpha^{(p)}, 0),  \lambda_1(\alpha^{(p)}, \pi)]$, $[\lambda_2(\alpha^{(p)}, \pi), \lambda_2(\alpha^{(p)}, 0)], \dots$  are called bands. The endpoints of two adjacent bands may coincide. In this case we say that the respective gap is closed; otherwise the respective gap,  $\big(\lambda_1(\alpha^{(p)}, \pi), \lambda_2(\alpha^{(p)}, \pi)\big)$, $\big(\lambda_2(\alpha^{(p)}, 0),  \lambda_3(\alpha^{(p)}, 0)\big),\dots$ is said to be open.
In the following theorem we show that all gaps are open for a generic periodic sequence $\alpha$.
thm4.14There is a dense $G_{\delta}-$set $\cS\subset \ell^{\infty}(\bbZ; \bbR)$ of sequences $\alpha$ such that for each $\alpha\in\cS$ all gaps in the spectrum of $H_{\alpha}$ are open.

We let

\[\cS_n:=\{\alpha\in \ell^{\infty}(\bbZ;\bbR): \alpha \text{\ is $p-$periodic and the $n$-th gap of $H_{\alpha}$ is open} \}.\]
It is enough to prove that each $\cS_n$ is open and dense (then $\bigcap_{n\in\bbN}\cS_n$ gives the required dense $G_{\delta}$-set of potentials). To begin,
let us rewrite $\dom(H_{\alpha^{(p)}, \vartheta})$ in terms of Lagrangian planes in $\Lambda(\bbC^{4(p+1)})$. For $u\in \hatt H^2(I_p\setminus\bbZ)$ we introduce the traces $\Gamma_0 u, \Gamma_1 u\in\bbC^{2(p+1)}$ by

\begin{align}
&\Gamma_0 u:=\{u(-1/2^+),u((p-1/2)^-),u(0^-),u(0^+),\dots, u(k^-),u(k^+),\dots,\\
&\hspace{5cm}u((p-1)^-),u((p-1)^+) \}\in \bbC^{2(p+1)},\\
&\Gamma_1 u:=\{u'(-1/2^+),-u'((p-1/2)^-),-u'(0^-),u'(0^+),\dots, -u'(k^-),u'(k^+),\dots,\\
&\hspace{5cm}-u'((p-1)^-),u'((p-1)^+) \}\in \bbC^{2(p+1)}.
\end{align}

Also, let us introduce $2(p+1)\times 2(p+1)$ matrices

\begin{align}
&X_{\alpha^{(p)}, \vartheta}:=
\begin{bmatrix}
-e^{\bfi \vartheta}&1\\
\end{bmatrix}
\oplus
\begin{bmatrix}
\end{bmatrix}
\oplus...\oplus
\begin{bmatrix}
\end{bmatrix},\\
&Y_{\alpha^{(p)}, \vartheta}:=
\begin{bmatrix}
e^{\bfi \vartheta}&1
\end{bmatrix}
\oplus
\begin{bmatrix}
\end{bmatrix}
\oplus...\oplus
\begin{bmatrix}
\end{bmatrix}.
\end{align}

Then one has

\begin{equation}
\dom(H_{\alpha^{(p)}, \vartheta})=\{u\in \hatt H^2(I_p\setminus\bbZ): X_{\alpha^{(p)}, \vartheta}\Gamma_0 u+Y_{\alpha^{(p)}, \vartheta}\Gamma_1 u=0\}.
\end{equation}

That is, the Lagrangian plane corresponding to $H_{\alpha^{(p)}, \vartheta}$ is given by
\[\ker[X_{\alpha^{(p)}, \vartheta}, Y_{\alpha^{(p)}, \vartheta}]. \]

In order to prove that $\cS_n$ is open, let us recall that the edges of the spectral gaps are given by consecutive eigenvalues of the periodic,  $H_{\alpha^{(p)}, 0}$, or anti-periodic, $H_{\alpha^{(p)}, \pi}$, operators. Suppose that $\alpha\in\cS_n$ and that the edges of the $n-$th  gap satisfy $\lambda_n(\alpha^{(p)}, \vartheta)<\lambda_{n+1}(\alpha^{(p)}, \vartheta)$ with either $\vartheta=0$ or $\vartheta=\pi$. We claim that this strict inequality holds for all $\wti{\alpha}^{(p)}\in \bbR^p$ near $\alpha^{(p)}$, i.e. that the gap is open under small perturbations of $\alpha^{(p)}$.    Indeed, since the mapping

\begin{equation}
\bbR^p\ni\alpha^{(p)}\mapsto  [X_{\alpha^{(p)}, \vartheta}, Y_{\alpha^{(p)}, \vartheta}] \text{ for $\vartheta=0$ or $\vartheta= \pi$}
\end{equation}

is continuous, Theorem <ref> yields continuity of the mapping

\begin{equation}
\bbR^p\ni\alpha^{(p)}\mapsto  (H_{\alpha^{(p)}, \vartheta}-\bfi)^{-1}\in \cB(L^2(I_p)) \text{ for $\vartheta=0$ or $\vartheta= \pi$};
\end{equation}

hence, the mappings

\begin{equation}
\alpha^{(p)}\mapsto \lambda_j(\alpha^{(p)}, \vartheta), \alpha^{(p)}\mapsto \lambda_{j+1}(\alpha^{(p)}, \vartheta), \text{ for $\vartheta=0$ or $\vartheta= \pi$}
\end{equation}

are also continuous, which implies the asserted strict  inequality

\[\lambda_n(\wti{\alpha}^{(p)}, \vartheta)<\lambda_{n+1}(\wti{\alpha}^{(p)}, \vartheta)\]
for all $\wti{\alpha}^{(p)}$ near ${\alpha}^{(p)}$.

In order to prove that $\cS_n$ is dense we need to show that for both cases $\vartheta=0$ and $\vartheta= \pi$ the equality $\lambda_n(\alpha^{(p)}, \vartheta)=\lambda_{n+1}(\alpha^{(p)}, \vartheta)$  will not hold if $\alpha^{(p)}$ is replaced by its small perturbation. We will consider the case $\vartheta=0$, that is, we will assume that $\lambda_n(\alpha^{(p)}, 0)=\lambda_{n+1}(\alpha^{(p)}, 0)$; the case $\vartheta=\pi$ is treated analogously. For $t\in\bbR$, let us introduce the perturbation $\alpha^{(p)}(t):= \{t+\alpha_0,\alpha_1,\dots, \alpha_{p-1}\}$. We claim that for every $\varepsilon>0$ there is a $t_0\in(0, \varepsilon)$ with

\begin{equation}\lb{eigcur}
\lambda_n(\alpha^{(p)}(t_0), 0)<\lambda_{n+1}(\alpha^{(p)}(t_0), 0).
\end{equation}

When proven, this inequality shows that there exist arbitrarily close to $\alpha^{(p)}$  perturbations which open the closed gap. To prove the claim we utilize the Hadamard-type formula (<ref>) for the boundary matrices $X_{\alpha^{(p)}(t), 0}$, $Y_{\alpha^{(p)}(t), 0}$.  We recall that $\lambda:=\lambda_n(\alpha^{(p)}, 0)=\lambda_{n+1}(\alpha^{(p)}, 0)$ is an eigenvalue of $H_{\alpha^{(p)}, 0}$ of multiplicity two. By Theorem <ref>, there is a basis $\{u_1, u_2\}$ in $\ker(H_{\alpha^{(p)}, 0}-\lambda)$ such that

\begin{align}
&\frac{\rm d}{{\rm d}t}\Big|_{t=0}\lambda_n(\alpha^{(p)}(t), 0)=|u_1(0)|^2,\lb{derone}\\
&\frac{\rm d}{{\rm d}t}\Big|_{t=0}\lambda_{n+1}(\alpha^{(p)}(t), 0)=|u_2(0)|^2.\lb{dertwo}
\end{align}

Next, we will prove that the values of the derivatives in (<ref>), (<ref>) are not equal to each other. This fact  implies that the eigenvalue curves $t\mapsto\lambda_n(\alpha^{(p)}(t), 0)$ and   $t\mapsto\lambda_{n+1}(\alpha^{(p)}(t), 0)$ do not coincide for $t$ near $t=0$, which, in turn, yields (<ref>) as needed. Starting the proof of the fact, we first remark that the eigenfunctions $u_1$ and $u_2$ are real valued because the boundary conditions for $\vartheta=0$ are real. Upon multiplying the eigenfunctions by appropriate constants
we may and will assume that $u_1(0)$ and $u_2(0)$ are non-negative. If  $u_1(0)\not=u_2(0)$ then the left-hand sides of (<ref>), (<ref>) are not equal as required. If $u_1(0)=u_2(0)$ then for any $t\in\bbR$ the function $u_1-u_2$ satisfies the boundary condition at $x=0$ with $\alpha_0$ replaced by $t+\alpha_0$. Therefore,
u_1-u_2\in \ker (H_{\alpha^{(p)}(t), 0}-\lambda)\setminus\{0\}
and thus $\lambda$ is an eigenvalue of $H_{\alpha^{(p)}(t), 0}$ for all $t\in\bbR$. That is, either $\lambda_n(\alpha^{(p)}(t), 0)$ or $\lambda_{n+1}(\alpha^{(p)}(t), 0)$ should be identically equal to $\lambda$ for all $t$ near $0$. Hence, one of the derivatives in (<ref>), (<ref>) vanishes, say, the first one. Then $u_1(0)=0$. But in this case $u_2(0)\not=0$ for otherwise $u_1$ and $u_2$ would be linearly dependent. Thus, the value of the derivative in (<ref>) is equal to zero while the value of the derivative in (<ref>) is not, as required.

§.§ Maslov crossing form for abstract  boundary triplets
In this section, we discuss an infinitesimal version of the formula equating the Maslov index and the spectral flow for the family of operators $H_t=\cA_t+V_t$ satisfying Hypothesis <ref>, which is assumed throughout this section. Formulas relating these two quantities are quite classical, and we refer the reader to  the papers  [19, 20, 21, 22, 26, 35, 36, 48, 71, 72, 73, 88] and the literature therein. Employing the abstract Hadamard-type formula obtained in Theorem <ref>, we prove in Theorem <ref>  that the signature of the Maslov crossing form defined in (<ref>) at an eigenvalue $\lambda$ of the operator $H_{t_0}$ is equal to the difference between the number of  monotonically decreasing and the number of monotonically increasing eigenvalue curves for $H_t$ bifurcating from $\lambda$.

For $\lambda\in\bbR$ and $t\in[0,1]$ we introduce the following subspaces,

\begin{align}
&\cG_t:=\ran (Q_t)\subset\bndra, \\
&\Upsilon_{\lambda,t}:=\bbK_{\lambda,t}\oplus\cG_t\subset\big((\mathfrak{H}\times\mathfrak{H})\oplus(\mathfrak{H}\times\mathfrak{H})\big), \label{dfnUPS}\\
&\mathfrak{D}:=\{\bp=(p,p)^\top: p\in\mathfrak{H}\times\mathfrak{H}\}
\subset\big((\mathfrak{H}\times\mathfrak{H})\oplus(\mathfrak{H}\times\mathfrak{H})\big).
\end{align}

Since $\tr_t(\dom(\cA_t))=\ran(Q_t)$ by Hypothesis <ref>,  the following assertions are equivalent

\begin{equation}\label{equivEV}
(i)\,\, \ker(H_t-\lambda)\not=\{0\}, \quad (ii)\,\, \bbK_{\lambda,t}\cap\cG_t\neq\{0\}, \quad (iii)\,\, \Upsilon_{\lambda,t}\cap\mathfrak{D}\neq\{0\}
\end{equation}

since $\mathfrak{D}$  is the diagonal subspace in $(\mathfrak{H}\times\mathfrak{H})\oplus(\mathfrak{H}\times\mathfrak{H})$. In fact, using a fundamental Proposition 3.5 in [19], one can deduce deeper connections between the spectral information for $H_t$ and the behavior of Lagrangian planes under the following hypotheses.
i17 Given $\lambda\in \bbR$ and $t_0\in[0,1]$,  we assume that

(i) $\lambda\not\in\spec_{\rm ess}(H_{t_0})$.

Moreover, there exists an interval $\cJ\subset[0,1]$ centered at $t_0$ such that

(ii) the mappings $t\mapsto \tr_t$, $t\mapsto V_t$, $t\mapsto Q_t$ are $C^1$ on $\cJ$,

(iii) $\ker(A^*+V_t-\lambda)\cap\dom(A)=\{0\}$ for all $t\in\cJ$.

Hypothesis <ref>  will be assumed through this section.
Part (iii) of this hypothesis is an abstract version of the unique continuation principle for PDEs, and we refer to Theorems 3.2 and Hypothesis 5.9 in [71] for a discussion of this connection.
Part (i) implies that the operator $H_{t_0}-\lambda$ is Fredholm.
Since $\ker(\tr)=\dom(A)$ by Proposition <ref>(1), parts (i) and (iii) of Hypothesis <ref> imply that $\tr\big|_{\ker(H_{t_0}-\lambda)}$ is an isomorphism between $\ker(H_{t_0}-\lambda)$ and
$ \bbK_{\lambda,t}\cap\cG_t$, cf. (<ref>). Moreover,
the subspaces  $\bbK_{\lambda,t_0}$ and  $\cG_{t_0}$ form a Fredholm pair (i.e. their intersection is finite dimensional and their sum is closed and has finite co-dimension). The latter fact has been established in <cit.> in the setting of Lagrangian planes in $\dom(A^*)/\dom(A)$; using this one can readily deduce the Fredholm property of the pair in the present setting via the symplectomorphism introduced in <cit.>. The subspace   $\cG_t$ is Lagrangian by Hypothesis <ref>. The subspace $ \bbK_{\lambda,t}$ is also Lagrangian again by <cit.>.
Furthermore,  part (ii) of Hypothesis <ref> yields continuity in $t$ of the resolvent operators for $H_t$ by Theorem <ref>. This, together with part (i), shows that $\lambda\not\in\spec_{\rm ess}(H_{t})$ for $t$ near $t_0$, hence,  the subspaces $\bbK_{\lambda,t}, \cG_{t}$ form a Fredholm  pair of Lagrangian subspaces for each $t$ near $t_0$. Hence, $\big(\Upsilon_{\lambda,t}, \mathfrak{D}\big)$ is a Fredholm  pair of Lagrangian subspaces for each $t$ near $t_0$.

Let $\Pi_{\lambda,t}$ be the orthogonal projection onto $\Upsilon_{\lambda,t}$ from (<ref>) so that the mapping $t\mapsto \Pi_{\lambda,t}$ is continuously differentiable on $[0,1]$ for each $\lambda\in\bbR$, see <cit.>. Furthermore, for $\lambda\in\bbR$ and $t_0\in[0,1]$ satisfying Hypothesis <ref> there is an interval $\cI\subseteq\cJ\subset [0,1]$ centered at $t_0$ and a family of operators $t\mapsto \cM_{\lambda,t}$, $t\in\cI$, which is in $C^1\big(\cI,\cB(\Upsilon_{\lambda, t_0},(\Upsilon_{\lambda, t_0})^\bot)\big)$ with  $\cM_{\lambda,t_0}=0$ such that

\begin{equation}\label{RT}
\Upsilon_{\lambda,t}=\big\{\bq+\cM_{\lambda,t}\bq\ \big|\, \bq\in\Upsilon_{\lambda, t_0}\big\}, t\in\cI,
\end{equation}

see,  e.g., <cit.>. We call $(\lambda,t_0)$  a conjugate point if $\ker(H_{t_0}-\lambda)\not=\{0\}$, or equivalently, if assertions (ii) and (iii) in (<ref>) hold for $t=t_0$.  The Maslov crossing form $\mathfrak{m}_{t_0}$ for $\Upsilon_{\lambda,t}$ relative to $\mathfrak{D}$ at the conjugate point ${(\lambda, t_0)}$
is defined on the finite dimensional  intersection $\Upsilon_{\lambda, t_0}\cap\mathfrak{D}$ of the Lagrangian subspaces by the formula

\begin{equation}\label{dfnMcf}
\mathfrak{m}_{t_0}(\bq,\bp):=\frac{\bd}{\bd t}\big|_{t=t_0}\widehat{\omega} (\bq,\cM_{\lambda, t}\bp)=\widehat{\omega} (\bq,\dot{\cM}_{\lambda, t_0}\bp),\ \bp,\bq\in \Upsilon_{\lambda, t_0}\cap\mathfrak{D},
\end{equation}

where $\widehat{\omega} ={\omega} \oplus(-{\omega} )$ is a symplectic form on $(\mathfrak{H}\times\mathfrak{H})\oplus(\mathfrak{H}\times\mathfrak{H})$ and, as usual, we abbreviate  $\displaystyle{\dot{\cM}_{\lambda, t_0}=\frac{\bd}{\bd t}\cM_{\lambda, t}\big|_{t=t_0}}$.

Let $(\lambda, t_0)$ be a conjugate point satisfying Hypothesis <ref> and let $u\in\ker(H_{t_0}-\lambda)$. Then there exist an open interval $\cI\subseteq\cJ$ centered at $t_0$, a  family $t\mapsto w_t$  in $C^1(\cI, \cH_+)$, and
a family $t\mapsto g_t\in\ran(Q_t)$ in $C^1(\cI, \mathfrak{H}\times\mathfrak{H})$ such that

\begin{align}
w_{{t_0}}&=u,\quad g_{{t_0}}=\tr_{t_0} u,\lb{2.43}\\
\big(
\tr_{t}w_t, g_t
\big)^\top
\big(
\tr_{t_0} u, \tr_{t_0} u
\big)^\top
\cM_{\lambda, t} \big(
\tr_{t_0} u, \tr_{t_0} u
\big)^\top,\ t\in\cI,\label{propwt}
\end{align}

where $\cM_{\lambda, t}$ is as in (<ref>).

The proof is similar to that of Lemma 2.6 in <cit.>.  For brevity, we denote $N_t:=\ker(A^*+V_t-\lambda)$, $q:= \tr_{t_0}u$, $\bq:=(q,q)$ and let $P_t$ be the orthogonal projections onto $\bbK_{\lambda, t}$. Then $P_t\in C^1\big(\cI,  \cB(\mathfrak{H}\times\mathfrak{H})\big)$ for some open interval $\cI\subseteq\cJ$ centered at $t_0$ (see, e.g., <cit.>, <cit.>). We now consider the projections in $(\mathfrak{H}\times\mathfrak{H})\times(\mathfrak{H}\times\mathfrak{H})$ given by

\begin{equation*}
\widehat{P}_t:=\begin{bmatrix}P_t&0\\0&0\end{bmatrix}, \
\widehat{Q}_t:=\begin{bmatrix}0&0\\0&Q_t\end{bmatrix},
\end{equation*}

so that $\widehat{P}_t+\widehat{Q}_t=\Pi_{\lambda,t}$, $\ran (\Pi_{\lambda, t})=\Upsilon_{\lambda, t}=\bbK_{\lambda,t}\oplus\cG_t$.
Using the definition of $\Upsilon_{\lambda, t}$ and $\cM_{\lambda,t}$, see (<ref>) and (<ref>), we define

\begin{equation}
h_t\in\ran (P_t)\subset\mathfrak{H}\times\mathfrak{H},\ g_t\in\ran (Q_t)\subset\mathfrak{H}\times\mathfrak{H},
\end{equation}

such that
\begin{equation}\label{Dfngt}
(h_t,0)^\top=\widehat{P}_t(\bq+\cM_{\lambda,t}\bq)\text{ and $(0,g_t)^\top=\widehat{Q}_t(\bq+\cM_{\lambda,t}\bq)$},\end{equation}

and so $h_{{t_0}}=g_{{t_0}}=q$. Since $t\mapsto \cM_{\lambda,t}$, $t\mapsto P_t$ and $t\mapsto Q_t$ are $C^1$, we know that the maps $t\mapsto h_t$ and $t\mapsto g_t$ are $C^1$.
As above, employing Hypothesis <ref> and $\ker \tr_t= \dom(A)$, see  Proposition <ref> (1),   we conclude that the restriction

\begin{equation}
\tr_{t}\upharpoonright_{N_t}: N_t\to\ran(P_t)\subset\mathfrak{H}\times\mathfrak{H},
\end{equation}

of $\tr_{t}$ to $N_t$ is a bijection. Therefore, there is a unique vector $w_t\in N_t$ satisfying $\tr_{t}w_t=h_t$. Assertions (<ref>), (<ref>) hold with this choice of $w_t$ and $g_t$ .

It remains to show that the function $t\mapsto w_t$ is in $C^1(\cI, \cH_+)$.
Let $U_t$ denote the $C^1$ family of boundedly invertible transformation operators in $\cH_+$ that split the projections $\cP_{N_t}$ onto $N_t$ and $\cP_{N_{{t_0}}}$ onto $N_{{t_0}}$ so that the identity
$U_t\cP_{N_{{t_0}}}=\cP_{N_t}U_t$ holds, and $U_t: N_{{t_0}}\mapsto N_t$ are bijections for $t$ near $t_0$, cf. <cit.>,
<cit.>, <cit.>, <cit.>. We temporarily  introduce  $v_t\in N_{{t_0}}$ by $v_t=U_t^{-1}w_t$ so that $\tr_{t}w_t=h_t$ yields $(\tr_{t}\circ\, U_t)v_t=h_t$.
The map $\tr_{t}\circ \,U_t\big|_{N_{{t_0}}}:N_{{t_0}}\to\ran(P_t)$ is a bijection and
$t\mapsto\tr_{t}\circ\, U_t\big|_{N_{{t_0}}}$ is in $C^1\big(\cI, \cB(N_{{t_0}},\mathfrak{H}\times\mathfrak{H})\big)$  by the assumptions in the lemma. Since
$w_t=U_t\circ\big(\tr_{t}\circ\, U_t\big)^{-1}h_t$, the function $t\mapsto w_t$ is $C^1$  because each of the three terms in the composition is $C^1$.

Under Hypothesis <ref>, let $(\lambda, t_0)$ be a conjugate point satisfying Hypothesis <ref>. Let $\{\lambda_j(t)\}_{j=1}^m$, with $\lambda=\lambda(t_0)$, $\{u_j\}_{j=1}^m$ be as in Theorem <ref>, and let $\bq_j:=(\tr_{t_0}u_j, \tr_{t_0}u_j)^\top\in\Upsilon_{\lambda, t_0}\cap\mathfrak{D}$. Then the slope of the eigenvalue curves satisfies

\begin{equation}\label{formdr}
\dot\lambda_j(t_0)=\mathfrak{m}_{t_0}(\bq_j,\bq_j),\ 1\leq j\leq m,
\end{equation}

where $\mathfrak{m}_{t_0}$ is the Maslov form  introduced in (<ref>).

For a fixed  $j$, let $(w_t, g_t)$ be as in Lemma <ref> with $u:=u_j$. Differentiating

\begin{equation}\lb{2.49}
A^*w_t+V_tw_t-\lambda w_t=0,
\end{equation}

at $t_0$ and multiplying the result by  $w_{{t_0}}=u_j$ we get

\begin{equation}
\langle(A^*+V_{{t_0}}-\lambda) \dot{w}_{{t_0}},\, w_{{t_0}}\rangle_\cH+	\langle\dot V_{{t_0}} {w}_{{t_0}},\, w_{{t_0}}\rangle_\cH=0.
\end{equation}

Using the Green identity (<ref>) with $u=\dot{w}_{{t_0}}$ and $v=w_{{t_0}}$ we obtain

\begin{align}\lb{2.50}
\begin{split}
&\langle(A^*+V_{{t_0}}-\lambda) \dot{w}_{{t_0}},\, w_{{t_0}}\rangle_\cH= \langle\dot{w}_{{t_0}},\, (A^*+V_{{t_0}}-\lambda)w_{{t_0}}\rangle_\cH\\
\end{split}
\end{align}

Combining (<ref>) and (<ref>) yields

\begin{equation}\label{df5}
\omega\big(\tr_{{t_0}}\dot{w}_{{t_0}},\, \tr_{t_0}u_j\big)+\langle\dot{V}_{{t_0}}u_{j},\, u_{j}\rangle_\cH=0.
\end{equation}

Next,  (<ref>)  and (<ref>)  yield

\begin{align}\lb{2.53}
\mathfrak{m}_{{t_0}}(\bq_j,\bq_j)
=\omega\big(\tr_{t_0}u_j,\frac{\bd}{\bd t}\big|_{t={t_0}}(\tr_{t}w_t)\big)
\end{align}

Since $g_t=Q_tg_t$ we have
\[\dot{g}_{{t_0}}=\dot{Q}_{{t_0}}g_{t_0}+Q_{{t_0}}\dot{g}_{{t_0}}=\dot{Q}_{{t_0}}\tr_{t_0} u_j+Q_{{t_0}}\dot{g}_{{t_0}}.\]
Utilizing this, the fact that $\ran(Q_{{t_0}})$ is Lagrangian and $\tr u_j\in\ran(Q_{t_0})$ we get

\begin{equation}\lb{2.54}
\omega(\tr_{t_0} u_j,\,\dot{g}_{{t_0}})=	\omega(\tr_{t_0} u_j,\,\dot{Q}_{{t_0}}\tr_{t_0} u_j+Q_{{t_0}}\dot{g}_{{t_0}})=\omega(\tr_{t_0} u_j,\dot{Q}_{{t_0}}\tr_{t_0}u_j).
\end{equation}

Then (<ref>), (<ref>), and (<ref>) yield

\begin{align}
\begin{split}\lb{aub34}
\mathfrak{m}_{{t_0}}(\bq_j,\bq_j)
+\langle\dot{V}_{{t_0}}u_{j},\, u_{j}\rangle_\cH\\
\end{split}
\end{align}

where we used $\omega(\dot{Q}_{{t_0}}\tr_{t_0}u_j,\tr_{t_0}u_j)\in\bbR$, see (<ref>). Comparing (<ref>) and (<ref>) one infers (<ref>) as required.

Formula (<ref>) in Theorem <ref> yields a fundamental relation between the Maslov index and the spectral flow of the family of operators $H_t=\cA_t+V_t$ satisfying the condition $\tr_t(\dom(H_t))=\cG_t$ for a given family of Lagrangian subspaces $\cG_t$, $t\in[0,1]$. This relation goes back to the celebrated Atiyah–Patodi–Singer Theorem and it has been a subject of intensive research ever since, see, e.g.,  [19, 20, 21, 22, 26, 35, 88] and many more references therein. We will briefly comment on the equality of the Maslov index and the spectral flow. First, we recall the definition of the Maslov index via crossing forms.  For a fixed $\lambda=\lambda_0$ from now on we assume that Hypothesis <ref> is satisfied for all $t=t_0\in[0,1]$. Then, given the subspaces defined in (<ref>), and assuming that all conjugate points $(\lambda, t_0)$ for $t_0\in[0,1]$ are non-degenerate (in the sense that the quadratic form $\mathfrak{m}_{t_0}$ from (<ref>) is non-degenerate), one defines the Maslov index by the formula

\begin{equation}\label{defMasInd}
{\rm Mas } \big(\Upsilon_{\lambda_0,t}: t\in[0,1]\big)=
\end{equation}

where the summation is taken over all $t_0$ such that $(\lambda, t_0)$ is a conjugate point and we denote by $m_+(t_0)$, respectively, $m_-(t_0)$ the number of positive, respectively, negative squares of the quadratic form $\mathfrak{m}_{t_0}$ at the conjugate point. Next, we recall the definition of the spectral flow: The spectral flow ${\rm SpF }_{\lambda_0}(H_t: t\in[0,1])$ for the family of operators $H_t$ is the net count of the eigenvalues of $H_t$ passing through $\lambda_0$ as $t$ changes from $t=0$ to $t=1$ and is defined as follows, cf., e.g., <cit.>. Take a partition $0=t_0<t_1<\dots<t_N=1$ and $N$ intervals $[a_\ell, b_\ell]$ such that $a_\ell<\lambda_0<b_\ell$ and
$a_\ell, b_\ell\notin\spec(H_t)$ for all $t\in[t_{\ell-1},t_\ell]$, $1\le\ell\le N$. Then the spectral flow is defined by

\begin{equation}\label{dfnspfl}
{\rm SpF }_{\lambda_0}(H_t: t\in[0,1])=\sum_{\ell=1}^N\sum_{a_\ell\le\lambda<\lambda_0}\big(\dim\ker(H_{t_{\ell-1}}-\lambda)-\dim\ker(H_{t_{\ell}}-\lambda)\big).
\end{equation}

By our assumptions, due to part (i) in Hypothesis <ref>, $\lambda_0$ does not belong to the essential spectrum of the operator $H_t$ for all $t\in[0,1]$. Moreover, let us assume, in addition, that for each $t_0\in[0,1]$ such that $\lambda_0\in\spec_{\rm disc}(H_{t_0})$
the inequality $\dot{\lambda}_j(t_0)\neq0$ holds for all $j=1,\dots,m$.
Here,  $m=m(t_0)$ is the multiplicity of the  isolated eigenvalue $\lambda_0$ of $H_{t_0}$, and
$\{\lambda_j(t)\}$ are the eigenvalues of $H_t$ as in Theorem <ref>(2) and Theorem <ref>(2) for $t\in[t_0',t_0'']$ near $t_0$. With no loss of generality $t=t_0$ could be assumed to be the only point in $[t_0',t_0'']$ such that $\lambda_0\in\spec(H_{t})$. By our assumptions and formula (<ref>) in Theorem <ref> the quadratic
form $\mathfrak{m}_{t_0}$ defined in (<ref>) is non-degenerate and $m_+(t_0)$, respectively, $m_-(t_0)$ is equal to the number of $j$'s such that the eigenvalue $\lambda_j(t)$ moves through $\lambda_0$ in the positive, respectively, negative direction as $t$ changes from $t_0'$ to $t_0''$. Formulas (<ref>) and (<ref>) now show that
${\rm Mas } \big(\Upsilon_{\lambda_0,t}: t\in[t_0',t_0'']\big)={\rm SpF }_{\lambda_0}(H_t: t\in[t_0',t_0''])$.
Passing to a partition of $[0,1]$ then gives

\begin{equation}\label{MISF}
{\rm Mas } \big(\Upsilon_{\lambda_0,t}: t\in[0,1]\big)={\rm SpF }_{\lambda_0}(H_t: t\in[0,1]),\end{equation}

the desired equality of the Maslov index and the spectral flow.

§ HADAMARD-TYPE FORMULA FOR ELLIPTIC OPERATORS VIA DIRICHLET AND NEUMANN TRACES

In this section concerns self-adjoint realizations of second order elliptic operators on bounded domains. We begin by discussing a Krein-type formula, see Proposition <ref>, an Hadamard-type formula, (<ref>), and asymptotic resolvent expansions, Theorem <ref>,  for the elliptic operators (<ref>) posted on bounded domains with smooth boundary. We deduce all these results from Theorem <ref> by appropriately choosing the trace maps. The main technical issue is to validate Hypotheses <ref> and <ref>, which is done in Proposition <ref>. Next, these results are utilized to give simple and unified proofs of Friedlander's Theorem <cit.>, see Example <ref>, and Rohleder's Theorem <cit.>, see Example <ref>. Furthermore, in Section <ref> we consider the heat equation with space-dependent diffusion coefficient equipped with Robin boundary conditions so that both the equation and the boundary conditions contain a physically relevant parameter, the thermal conductivity. The results in this section provide, in particular, a new proof of the fact  that the temperature of a non-homogeneous material immersed into a surrounding medium of constant temperature depends continuously on the thermal conductivity of the material.

§.§ Elliptic operators
We consider the following differential expression,

\begin{align}
\begin{split}
\cL:&=-\sum_{j,k=1}^{n} \partial_j \mathtt{a}_{jk}\partial_k + \sum_{j=1}^{n}\mathtt{a}_j\partial_j -\partial_j {\mathtt{a}_j}+\mathtt{q},\lb{1.15b} \\
&=-\div(\mathtt{A}\nabla)+ \mathtt{a}\cdot \nabla -\nabla \cdot \mathtt{a} +\mathtt{q},
\end{split}
\end{align}

with coefficients $\mathtt{A}=\{\mathtt{a}_{ij}\}_{1\leq i, j\leq n}$, $\mathtt{a}:=\{\mathtt{a}_i\}_{1\leq i\leq n}$ satisfying, for some $c=c(\cL)>0$,

\begin{align}
& \sum\limits_{j,k=1}^n\mathtt{a}_{jk}(x)\xi_k\overline{\xi_j} \geq c \sum\limits_{j=1}^n|\xi_j|^2, x\in\overline{\Omega}, \xi=\{\xi_j\}_{j=1}^n\in\C^n,\lb{unifel}\\
&\mathtt{a}_{jk}, \mathtt{a}_j\in C^{\infty}(\overline{\Omega};\bbR), \mathtt{q}\in L^{\infty}(\Omega; \bbR), \mathtt{a}_{jk}(x)={\mathtt{a}_{kj}(x)}, 1\leq j,k\leq n.
\end{align}

Associated with $\cL$ is the following space of distributions,

\begin{align}\lb{1.15c}
& \cD^{s}(\Omega):=\{u\in H^s(\Omega): \cL u\in L^2(\Omega)\},\ s\geq 0,
\end{align}

equipped with the norm

\begin{equation} \lb{1.15d}
\|u\|_{s}:=\left(\|u\|_{H^s({\Omega} )}^2+\|\cL u\|_{L^2({\Omega })}^2\right)^{1/2},
\end{equation}

where $\cL u$ should be understood in the sense of distributions. Let us introduce two operators acting in $L^2(\Omega)$,

\begin{align}
&\cL_{min}f:=\cL f,\ f\in\dom(\cL_{min}):=H^2_0(\Omega), \\
&\cL_{max}f:=\cL f,\ f\in\dom(\cL_{max}):=\cD^0(\Omega).
\end{align}

The operator $\cL_{min}$ is closed, symmetric, and  $(\cL_{min})^*=\cL_{max}$. Associated with $\cL$  is a first order trace operator $\gamma_{{}_{N, \cL}} \in\cB(\cD^1(\Omega), H^{-1/2}(\partial\Omega))$ which is a unique extension of the co-normal derivative

\begin{equation}
{\gamma_{{}_{N,\cL}}}u:=\sum_{j,k=1}^{n} \mathtt{a}_{jk}\nu_j\gaD(\partial_k u)+\sum_{j=1}^{n}{\mathtt{a}_j}\nu_j \gaD u, u\in H^2(\Omega)\label{dfncnd}
\end{equation}

to the space $\cD^{1}(\Omega)$ (here, $(\nu_1,\cdots,\, \nu_n)$ is the outward unit normal on $\partial\Omega$). Then the following Green identity holds,

\begin{align}
\langle\cL u, v\rangle_{L^2(\Omega )}- \langle u,\cL v\rangle_{L^2(\Omega )}={\langle{\gaD u, \gamma_{{}_{N, \cL}}} v  \rangle}_{{-1/2}}-\overline{\langle{\gaD v, \gamma_{{}_{N, \cL}}} u \rangle_{{-1/2}}},\lb{1.2}
\end{align}

for all $u,v\in\cD^1(\Omega)$. In order to rewrite this identity in a form compatible with (<ref>) let $\Phi$ denote the Riesz isomorphism $\Phi\in\cB (H^{-1/2}(\partial \Omega), H^{1/2}(\partial \Omega))$ as in (<ref>)
and define

\begin{equation}\lb{1.60}
\Gamma_0:=\gaD\in \cB (\cD^{1}( \Omega), H^{1/2}(\partial \Omega)),\ \Gamma_1:=-\Phi\gamma_{N,\cL} \in \cB (\cD^{1}( \Omega), H^{1/2}(\partial \Omega)).
\end{equation}

Then we have, for all $u,v\in\cD^1(\Omega)$,

\begin{align}
\begin{split}\lb{1.23o}
\langle\cL_{max} u, v\rangle_{L^2(\Omega )}&- \langle u,\cL_{max} v\rangle_{L^2(\Omega )}\\
&={\langle\Gamma_1u, \Gamma_0v \rangle_{H^{1/2}(\partial \Omega)}}-\langle\Gamma_0u, \Gamma_1v \rangle_{H^{1/2}(\partial \Omega)}.
\end{split}
\end{align}

We claim that Hypotheses <ref> and  <ref> are satisfied for

\begin{equation}\lb{aub11}
A=\cL_{min}, \cH_+=\cD^0(\Omega), \cD=\cD^1(\Omega), \Gamma_0=\gaD, \Gamma_1=-\Phi\gamma_{N,\cL}.
\end{equation}

Since we already checked the Green identity, (<ref>),
to justify the claim it remains to show that $\tr(\cD)$ is dense in $H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega)$ and that $\cD^1(\Omega)$ is dense in $\cD^0(\Omega)$. By <cit.>, <cit.> one has

\begin{equation}\no
(\gaD, \gamma_{{}_{N,\cL}})(H^2(\Omega))=H^{3/2}(\partial\Omega)\times H^{1/2}(\partial\Omega),
\end{equation}

and the right-hand side is dense in $H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega)$. By <cit.>, $H^2(\Omega)$ is dense in $\cD^s(\Omega), s<2$, hence $\cD^1(\Omega)$ is dense in $\cD^0(\Omega)$.
prop1.7 Under the assumptions on $\cL$ imposed in this section, for any two self-adjoint extensions $\cL_1, \cL_2$ of $\cL_{min}$ with domains containing in $\cD^1(\Omega)$ and $\zeta\not\in (\spec(\cL_1)\cup\spec(\cL_2))$, the following Krein-type formula holds,

\begin{align}
(\cL_2-\zeta)^{-1}-(\cL_1-\zeta)^{-1}&=(\tr(\cL_2-\overline{\zeta})^{-1})^*\, J\tr (\cL_1-\zeta)^{-1},
\end{align}

where $\tr=[\Gamma_0, \Gamma_1]^\top$ is defined in (<ref>), and
\[(\tr(\cL_2-\overline{\zeta})^{-1})^*\in\cB( H^{1/2}(\partial\Omega)\times  H^{1/2}(\partial\Omega), L^2(\Omega)).\]

The results follows directly from (<ref>).

§.§ Hadamard-type formulas for Robin elliptic operators, L. Friedlander's and J. Rohleder's inequalities

In this section we obtain an Hadamard-type formula for a one-parameter family of differential operators $\cL_tu=\cL u$ as in (<ref>) for which the dependence on the parameter $t$ enters through the Robin boundary condition $\gamma_{{}_{N,\cL}} u=\Theta_t \gaD u$, see Theorem <ref>. We will utilize Theorem <ref> by choosing the symmetric operator $A$, the function spaces $\cH, \cH_+, \mathfrak H$, and the trace operator $\tr$ as indicated in (<ref>). The main challenge is to check Hypothesis <ref> which in this setting reads as follows,

\begin{equation}
\big\|(\cL_t-\bfi)^{-1}-(\cL_{t_0}-\bfi)^{-1}\big\|_{\cB(L^2(\Omega), \cD^1(\Omega))}=o(1),\ t\rightarrow t_0,
\end{equation}

and can be reduced to showing that for some constant $c>0$ one has
the inequality

\begin{align}
&\|\nabla u\|_{L^2(\Omega)}^2\leq c\big( \|  \cL u\|^2_{L^2(\Omega)}+  \|u\|_{L^2(\Omega)}^2\big), u\in\dom(\cL_t),
\end{align}

for $t$ near $t_0$. We discuss the reduction and give the proof of this inequality in Proposition <ref>.

theorem6.2 Suppose that, in addition to  the assumptions on $\cL$ listed in Subsection <ref>, we are given a mapping $t\mapsto\Theta_t$  belonging to $C^1 ([0,1], L^{\infty}( \partial\Omega, \bbR))$. Then for $t\in[0,1]$ the Robbin elliptic operator $\cL_t$ defined by

\begin{align}
\begin{split}\lb{elt}
&\cL_t: \dom(\cL_t)\subset L^2(\Omega)\rightarrow L^2(\Omega),\quad \cL_t u=\cL u, \\
&u\in\dom(\cL_t)=\{ u\in \cD^1(\Omega): \Phi\gamma_{{}_{N,\cL}} u=\Theta_t \gaD u \},
\end{split}
\end{align}

is self-adjoint.  The following Krein-type formula holds,

\begin{align}\lb{cLKrein}
(\cL_t-\zeta)^{-1}-(\cL_s-\zeta)^{-1}= \big(\gaD(\cL_t-\overline{\zeta})^{-1} \big)^* (\Theta_t-\Theta_s)\big(\gaD(\cL_s-\zeta)^{-1}\big),
\end{align}

for $t,s\in[0,1]$, $\zeta\not\in(\spec(\cL_t)\cup\spec(\cL_s))$. Moreover, the mapping

\begin{equation}\label{cLcont}
t\mapsto (\cL_t-\zeta)^{-1}\in \cB(L^2(\Omega))
\end{equation}

is well defined for $t$ near $t_0$ as long as $\zeta\not\in\spec(\cL_{t_0})$. This mapping is differentiable at $t_0$ and satisfies the following Riccati equaiton,

\begin{align}\lb{cLRic}
\frac{\bd}{\bd t}|_{t=t_0}\big((\cL_t-\zeta)^{-1}\big)=\big(\gaD(\cL_{t_0}-\overline{\zeta})^{-1}\big)^* \big(\frac{\bd}{\bd t}|_{t=t_0}\Theta_t\big)\big(\gaD(\cL_{t_0}-\zeta)^{-1}\big).
\end{align}

Finally, if $\lambda(t_0)$ is an isolated eigenvalue of $\cL_{t_0}$ of multiplicity $m\geq 1$ then there exist  a choice of orthonormal eigenfunctions  $\{u_j\}_{j=1}^m\subset \ker(\cL_{t_0}-\lambda(t_0))$
and a labeling of eigenvalues $\{\lambda_j(t)\}_{j=1}^m$  of $\cL_{t}$, for $t$ near $t_0$, such that

\begin{equation}\lb{cLthetder}
\dot{\lambda}_j(t_0)=- \langle\dot \Theta_{t_0} \gaD u_j, \gaD u_j\rangle_{L^2(\partial\Omega)}, 1\leq j\leq m.
\end{equation}

We will employ Theorem <ref>. The proof consists of two steps. First, we derive (<ref>) from (<ref>). We can use (<ref>) because Hypothesis <ref> is trivially satisfied. Second,  we derive
(<ref>) and (<ref>) from (<ref>) and (<ref>). To apply (<ref>) and (<ref>) we need to verify Hypotheses <ref> and <ref>. They are satisfied by Proposition <ref> given next; the proof of this proposition uses formula (<ref>) proved in the first step.

To proceed, we choose $ \cH_+, \cD, A$ as in (<ref>) and rewrite the Robin condition $\gamma_{{}_{N, {\cL}}}u=\Theta_t\gaD u$ in the definition of $\cL_t$ as

\begin{equation}
X_t\Gamma_0u+Y_t\Gamma_1u=0, \text{ where we set }\, X_t:=\Theta_t, Y_t:=I.
\end{equation}

Continuity of $\Theta_t$ with respect to $t$ and Theorem <ref> with $\cA_t:=\cL_t$, $V_t:=0$, $\tr_t:=[\gaD, -\Phi \gamma_{N,\cL}]^\top$ yield that the map $t\mapsto R_t(\zeta):=(\cL_t-\zeta)^{-1}$ is well defined for $t$ near $t_0$.
Next, with $W$ defined in (<ref>), we observe that  $R_t(\zeta)u\in\dom(\cA_t)$ yields

\begin{equation}
\big(W(\Theta_t, I)\big) \tr  R_t(\zeta)u = -\Gamma_0 R_t(\zeta)u=-\gaD R_t(\zeta)u \text{ for all $u\in L^2(\Omega)$}.
\end{equation}

This can be checked directly or by noting that $\phi=\big(W(\Theta_t, I)\big) \tr    R_t(\zeta)u$ is the unique vector satisfying the relations
$\Gamma_0R_t(\zeta)u=-\phi$, $\Gamma_1R_t(\zeta)u=\Theta_t\phi$, cf. (<ref>).
This observation together with (<ref>) yield (<ref>).
We can now involve Proposition <ref> given next and verify
Hypotheses <ref> and <ref> in the present setting.
Thus, Theorem <ref> applies and therefore
(<ref>) and (<ref>) follow from  (<ref>) and (<ref>) with $\phi_j=-\Gamma_0 u_j$.

It is worth comparing Theorems <ref>  and <ref> for the case $\cL=-\Delta$ where both theorems apply. The major difference is in the type of trace operators utilized in each theorem. In Theorem  <ref> we use $\tr=[-\tau_{{}_N}, \Phi \gd]^\top$ which is defined on the entire space $\cH_+=\dom(-\Delta_{max})$ and is surjective, while in Theorem <ref> we have $\tr=[\gaD, -\Phi \gamma_{{}_{N, \cL}}]^\top$ which is defined only on a dense subset $\cD=\cD^1(\Omega)$ of $\cH_+=\cD^0(\Omega)$. We note that the latter trace operator is local while the former is not. In addition, these trace maps do not match even on smooth functions on $\Omega$. Another major technical difference is that Hypotheses <ref> and  <ref> are automatically satisfied in one case but not in the other.

prop5.2 Under assumptions of Theorem <ref> one has

\begin{align}
&\|(\cL_t-\bfi)^{-1}\|_{\cB(L^2(\Omega), \cD^1(\Omega))}=\cO(1),\ t\rightarrow t_0,\lb{5.7}\\
&\big\|(\cL_t-\bfi)^{-1}-(\cL_{t_0}-\bfi)^{-1}\big\|_{\cB(L^2(\Omega), \cD^1(\Omega))}=o(1),\ t\rightarrow t_0, \lb{5.8}
\end{align}

for all $t_0\in[0,1]$. In other words, Hypotheses <ref> and <ref> hold for $\cA_t:=\cL_t$.

To prove (<ref>) it is enough to show that there exists a constant $c>0$ such that

\begin{equation}
\|u\|_{\cD^1(\Omega)}^2\leq c\|\cL u-\bfi u\|^2_{L^2(\Omega)}, u\in\dom(\cL_t),
\end{equation}

for all $t\in[0,1]$. By the definition of $\cD^1(\Omega)$-norm, see (<ref>), we need to prove that

\begin{equation}\lb{5.7n}
\|\nabla u\|_{L^2(\Omega)}^2\leq c (\|\cL u\|^2_{L^2(\Omega)}+\| u\|^2_{L^2(\Omega)}), u\in\dom(\cL_t).
\end{equation}

To show this, we first notice that for $u\in\dom(\cL_t)$ one has

\begin{equation}
\langle \mathtt{A}\nabla u,\nabla u\rangle_{L^2(\Omega)}=\langle\cL u,u\rangle_{L^2(\Omega)}-\langle \mathtt{q} u, u\rangle_{L^2(\Omega)} -\langle \Theta_t \gaD u,\gaD u\rangle_{L^2(\partial\Omega)}.
\end{equation}

Using the Cauchy–Schwartz inequality and (<ref>) we get

\begin{equation}\lb{gradest}
\| \nabla u\|_{L^2(\Omega)}^2\leq c( \|\cL u\|^2_{L^2(\Omega)}+\|u\|^2_{L^2(\Omega)}+\|\Theta_t\|_{L^{\infty}(\partial\Omega)} \|\gaD u\|_{L^2(\partial\Omega)}^2),
\end{equation}

for $c>0$ (which is $t-$ and $u-$independent). Let us recall from  <cit.> the inequality

\begin{equation}
\|\gaD u\|^2_{L^2(\Omega)}\leq \varepsilon \|\nabla u\|_{L^2(\Omega)}^2+\beta(\varepsilon)\|u\|^2_{L^2(\Omega)}, \text{ where $\varepsilon>0$ and $\beta(\varepsilon)\underset{\varepsilon\rightarrow 0}{=}\cO(\varepsilon^{-1})$}.
\end{equation}

Thus, continuing (<ref>) we infer

\begin{align}
&\|\nabla u\|_{L^2(\Omega)}^2%\leq c\big( \|  \cL u\|^2_{L^2(\Omega)}+  \|u\|_{L^2(\Omega)}^2+\|\Theta_t\|_{L^{\infty}(\partial\Omega)}\|\gaD u\|^2_{L^2(\Omega)}\big)\\&\quad
\leq  c \Big(\|  \cL u\|^2_{L^2(\Omega)}+  \|u\|_{L^2(\Omega)}^2+\varepsilon\|\Theta_t\|_{L^{\infty}(\partial\Omega)}\|\nabla u\|^2_{L^2(\Omega)}\\
\end{align}

for some $c>0$.  Taking $\varepsilon>0$ sufficiently small

yields (<ref>) and thus (<ref>).

Starting the proof of (<ref>), we first show that

\begin{equation}\lb{L2H1}
\big\|(\cL_t-\bfi)^{-1}-(\cL_{t_0}-\bfi)^{-1}\big\|_{\cB(L^2(\Omega), H^1(\Omega))}=o(1),\ t\rightarrow t_0.
\end{equation}

We denote $R(t):=(\cL_t-\bfi)^{-1}$ and
recall that we may use Krein's formula (<ref>) already established in the first part of the proof of Theorem <ref>. It yields

\begin{align}\lb{thetagreen}
&\langle R(t)u-R(t_0)u,v\rangle_{L^2(\Omega)}=\langle(\Theta_{t_0}-\Theta_{t})\gaD R(t)u, \gaD R(t_0)v\rangle_{L^2(\partial\Omega)}
\end{align}

for all  $u, v\in L^2(\Omega)$.
For $v\in(H^1(\Omega))^*=H^{-1}(\Omega)$ we view $w:=R(t_0)v\in H^1(\Omega)$ as the solution to the boundary value problem $(\cL-\bfi)w=v$, $ \gamma_{{}_{N,\cL}}w=\Theta_{t_0}\gaD w$. Using a well-known elliptic estimate  $\|w\|_{H^1(\Omega)}\le c\|v\|_{H^{-1}(\Omega)}$ from <cit.>,
the operator $R(t_0)$ can be extended to an operator in $\cB((H^1(\Omega))^*, H^1(\Omega))$.
So, (<ref>) can be extended as follows,

\begin{equation}
_{H^1(\Omega))}\langle R(t)u-R(t_0)u,v\rangle_{(H^1(\Omega))^*}=\langle(\Theta_{t_0}-\Theta_{t})\gaD R(t)u, \gaD R(t_0)v\rangle_{L^2(\partial\Omega)},
\end{equation}

now for all $u\in L^2(\Omega)$ and  $v\in (H^1(\Omega))^*$. Hence,

\begin{align}
| _{H^1(\Omega))}\langle R(t)u&-R(t_0)u,v\rangle_{(H^1(\Omega))^*}|\leq \|\Theta_{t_0}-\Theta_{t}\|_{L^{\infty}(\partial\Omega)}\|\gaD\|^2_{\cB(\cD^1(\Omega), H^{1/2}(\partial\Omega))}\\
& \times \|R(t)\|_{\cB(L^2(\Omega), \cD^1(\Omega))}\|u\|_{L^2(\Omega)} \|R(t_0)\|_{\cB((H^1(\Omega))^*, H^1(\Omega))}\|v\|_{(H^1(\Omega))^*}.
\end{align}

Since $\|R(t)\|_{\cB(L^2(\Omega), \cD^1(\Omega))}=\cO(1)$ by (<ref>), and $\|\Theta_{t_0}-\Theta_{t}\|_{L^{\infty}(\partial\Omega)}=o(1)$, $t\rightarrow t_0,$ the above inequality gives (<ref>).
We now combine (<ref>) with the estimate

\begin{align}
\big\|(\cL_t-\bfi)^{-1}u&-(\cL_{t_0}-\bfi)^{-1}u\big\|_{\cD^1(\Omega)}^2=\big\|(\cL_t-\bfi)^{-1}u-(\cL_{t_0}-\bfi)^{-1}u\big\|_{H^1(\Omega)}^2\\
&\qquad + \big\|\cL(\cL_t-\bfi)^{-1}u-\cL(\cL_{t_0}-\bfi)^{-1}u\big\|_{L^2(\Omega)}^2\\
&\le2\big\|(\cL_t-\bfi)^{-1}u-(\cL_{t_0}-\bfi)^{-1}u\big\|_{H^1(\Omega)}^2, u\in L^2(\Omega),
\end{align}

finishing the proof of (<ref>).

Theorem <ref> can be used in proving the celebrated  Friedlander Inequalities $\lambda_{D,k}\geq \lambda_{N, k+1}$, $k=1,2,\dots$, for the eigenvalues of the Dirichlet and Neumann Laplacians, see [47], which was improved in [45] to state  that $\lambda_{D,k}>\lambda_{N, k+1}$, see also [15, 46, 53, 90] for further advances, detailed bibliography and a historical account of this beautiful subject. Also, we refer to Example <ref> for connections to the Maslov index.
The proof of the Friedlander Inequalities consists of two major steps. First,  one proves that the counting functions of the
Dirichlet and Neumann boundary problems differ by a number of negative eigenvalues of the Dirichlet-to-Neumann operator, see (<ref>) below.
Second, one proves the existence of a nonnegative eigenvalue of the latter.
The first step involves  a one-parameter family of Robin boundary value problems giving a homotopy of the Dirichlet to the Neumann boundary problem.
The critical issue here is to show monotonicity of the eigenvalues of the Robin  problems
with respect to the parameter, and this is where the results of the current papers help. (In fact, monotonicity holds not merely for the Laplacian but for  general elliptic operators as described in Subsection <ref>).
Indeed, formula (<ref>) in Theorem <ref>  with
$\cL=-\Delta$ and $\Theta_t=-\cot(\frac{\pi}{2}t)$  shows that the eigenvalues $\lambda=\lambda(t)$ of the Robin problem

\begin{equation}\label{eq:RobEV}
\begin{cases}
\cL u=\lambda u \text{\ in\ }\ \Omega,\\
\sin(\frac{\pi}{2}t)\gaN u+\cos(\frac{\pi}{2}t)\gaD u=0\text{\ on $\partial\Omega$ for $t\in[0,1]$,}
\end{cases}
\end{equation}

are monotonically decreasing with respect to $t\in[0,1]$. We note that

\begin{align*}
&\lambda_k(0)=\lambda_{D,k}\le\lambda_{D,k+1} =\lambda_{k+1}(0)\text{ and }\\
&\lambda_k(1)=\lambda_{N,k}\le\lambda_{N,k+1}=\lambda_{k+1}(1), k=1, 2, \ldots,
\end{align*}

are the Dirichlet and  Neumann eigenvalues. From this point on the arguments given in [47] and [45] are as follows. Monotonicity in $t$ of the Robin eigenvalues $\lambda_k(t)$ just proved, and the standard inequalities $\lambda_{D,k}\ge\lambda_{N,k}$ show the strict inequalities $\lambda_{D,k}>\lambda_{N, k+1}$ provided we know the fact, cf. <cit.>, that for each $\lambda$ there is a $t\in[0,1]$ such that (<ref>)  has a nontrivial solution. This fact is equivalent to the existence of a positive eigenvalue $\cot(\frac{\pi}{2}t)$ of the Dirichlet-to-Neumann operator when $\lambda\notin\Sp(-\Delta_D)$, and its proof  has been carried out in [47] and  [45] for the Laplacian using the minimax principle and infinitely many linearly independent explicit functions $e^{i\eta\cdot x}$, with $\eta\in\bbR^n$ such that $\|\eta\|^2_{\bbR^n}=\lambda$, that satisfy $-\Delta(e^{i\eta\cdot x})=\lambda e^{i\eta\cdot x}$.

We will now derive from Theorem  <ref> an elegant result in <cit.> regarding monotonicity of Robin eigenvalues. Given $\Theta^{(\ell)}\in L^\infty(\Omega; \bbR)$,  $\ell=0,1$, we define the Robin operators $\cL^{(\ell)}u=\cL u$ such that
\[\dom(\cL^{(\ell)})=\{u\in\cD^1(\Omega): \gamma_{{}_{N,\cL}}u=\Theta^{(\ell)}\gaD u\}\]
for the elliptic  differential expression in (<ref>). We let $\lambda_1(\cL^{(\ell)})\le\lambda_2(\cL^{(\ell)})\le\dots$ denote the eigenvalues of
$\cL^{(\ell)}$ counting multiplicities. Assume that $\Theta^{(0)}\le\Theta^{(1)}$. We will give a new proof of J. Rohleder's result stating that

\begin{equation}\label{RohT}
\text{if $\Theta^{(0)}<\Theta^{(1)}$ on a set of positive measure then $\lambda_k(\cL^{(0)})>\lambda_k(\cL^{(1)})$}
\end{equation}

for $k=1,2,\dots$. Denote $\Theta_t=\Theta^{(0)}+t(\Theta^{(1)}-\Theta^{(0)})$ for $t\in[0,1]$ and introduce operators $\cL_t$ as in Theorem <ref> such that $\cL_0=\cL^{(0)}$ and $\cL_1=\cL^{(1)}$. Denoting by $\lambda_k(t):=\lambda_k(\cL_t)$ the eigenvalues of $\cL_t$ counting multiplicities and by $u_k$ the respective eigenfunctions, formula (<ref>) implies

\begin{equation}\lb{aub20}
\frac{\bd\lambda_k(t)}{\bd t}=-\langle(\Theta^{(1)}-\Theta^{(0)})\gaD u_k,\gaD u_k\rangle_{L^2(\Omega)}<0 \text{, $k=1,2,\dots$, $t\in[0,1]$}
\end{equation}

because $\Theta^{(0)}<\Theta^{(1)}$ on a set of positive measure, thus proving (<ref>). Let us elaborate on some additional consequences of monotonicity of eigenvalues.
As the eigenvalue curves $t\mapsto \lambda_k(t)$ are strictly monotone and continuous we obtain the following count for the eigenvalues, see Figure <ref>,

\begin{align}\lb{aub15}
\begin{split}
(\#\{k: \lambda_k(\cL^{(1)})<\lambda\})-(\#\{k: \lambda_k&(\cL^{(0)})<\lambda\})\\
&=\sum_{t\in [0,1]}\dim\ker(\cL_t-\lambda).
\end{split}
\end{align}

A weaker version of this counting formula

\begin{align}
(\#\{k: \lambda_k(\cL^{(1)})<\lambda\})-(\#\{k: \lambda_k(\cL^{(0)})<\lambda\})\geq \dim\ker(\cL^{(0)}-\lambda),
\end{align}

was obtained by J. Rohleder <cit.> by variational methods. This is a key estimate in [89] leading to (<ref>) in the original proof. Now, (<ref>) can be viewed as a prequel to Section <ref>, where the left-hand side of (<ref>) is treated as the spectral flow of the family $\{\cL_t\}_{t\in[0,1]}$ through $\lambda$ and the write-hand side is viewed as the Maslov index of a certain path of Lagrangian planes. The equality between the Maslov index and the spectral flow in a very general setting has been recently investigated in, for example, [35, 36, 37, 71, 72] and the wast literature cited therein.

(100,12)$ \lambda$
(15,105)$ t$
(50, 12)${\ }_{\lambda_1(0)}$
(8,87)${\ }_{\lambda_1(1)}$
(31,87)${\ }_{\lambda_2(1)}$
(55,87)${\ }_{\lambda_3(1)}$

Illustration of (<ref>), (<ref>)fig1

§.§ Continuous dependence of solutions to heat equation on thermal conductivity
In this section we apply our general results to give a new proof that solutions to the linear homogeneous heat equation depend continuously on a certain physically relevant parameter present in both the operator and the boundary condition. The assertions of this type  have a long and distinguished history, and have been resolved even for quite general Wentzell boundary conditions. We refer the reader to [33, 34] where one can also find further literature. We did not attempt to cover the case of Wentzell boundary conditions anywhere in this paper but remark parenthetically
that it is an interesting open area to develop a version of the asymptotic  perturbation theory for operators equipped with this type of dynamical boundary conditions. At the moment, as in [55], we consider the following heat equation,

\begin{equation}\lb{aub2}
\begin{cases}
&u_\mathtt{t}(\mathtt{t},x)=\kappa \rho(x) \Delta_x u (\mathtt{t},x), x\in\Omega,
\mathtt{t}\ge0,\\
&-\kappa \frac{\partial  u}{\partial n}= u,\text{\ on\ }\partial \Omega,
\end{cases}
\end{equation}

describing the temperature $u$ of a material in the region $\Omega\subset \bbR^3$ with thermal conductivity $\kappa$ immersed in a surrounding medium of zero temperature. Here, $1/\rho(x)$ is the product of the density of the material times its heat capacity.  The continuous dependence of the temperature $u$ on the thermal conductivity $\kappa$ with respect to $L^2(\Omega)$ norm follows from Theorem <ref> proved below, which is a version of Theorem <ref>. To sketch the argument, we consider the self-adjoint operator  $\cL_{\kappa}:=-\kappa\Delta$, $\cL_{\kappa}:\dom(\cL_{\kappa})\subset L^2(\Omega)\rightarrow L^2(\Omega)$ with $\dom(\cL_{\kappa})=\{u\in\cD^1(\Omega): -\kappa \gaN u= \gaD u\}$. Then by Trotter–Kato Approximation Theorem <cit.>, the family of semigroups $\{e^{-\mathtt{t} \rho \cL_{\kappa}}\}_{\mathtt{t}\ge0}$ is strongly continuous in $\kappa $ uniformly for $\mathtt{t}$ from compact subsets whenever $\kappa\mapsto         (\rho \cL_{\kappa}-\zeta)^{-1}$ is continuous as a mapping from $(0,+\infty)$  to $\cB(L^2(\Omega))$ for some $\zeta\not\in \spec(\cL_{\kappa})$ (we note that $\rho \cL_{\kappa}$ is not necessarily self-adjoint). The next theorem gives a rigorous argument for the required continuity of the resolvent in a slightly more general form. (In the next theorem, to keep up with notation used in the rest of the paper, we denote the parameter with respect to which the continuity is established by $t$, not by $\kappa$; this is not to be confused with notation $\mathtt{t}$ for time used in (<ref>)).

Assume that $\Omega\subset \bbR^d$ is a bounded open set with $ C^{\infty}$-smooth boundary $\partial\Omega$. Let $t\mapsto \alpha_t$, $t\mapsto \beta_t$ be mappings in $C([0,1], L^{\infty}(\partial\Omega; \bbR))$ such that $\alpha_t^2(x)+\beta_t^2(x)\not=0$ for  $x\in\partial\Omega$, $t\in[0,1]$, and  $t\mapsto\rho_t\in C([0,1], C(\overline{\Omega};\bbR))$ be such that $\inf\{\rho_t(x): t\in[0,1], x\in\overline{ \Omega}\}>0$. Recall the differential expression $\cL$ from (<ref>) and define the following operator acting in $L^2(\Omega)$,

\begin{align}
&\cL_{t,\rho}u:=\rho_t\cL u, u\in\dom(\cL_{t, \rho}), \\
&\dom(\cL_{t,\rho}):=\{u\in \cD^1(\Omega): \alpha_t\gaD u+\beta_t \gamma_{{}_{N,\cL}} u=0\}.
\end{align}

Then the operator $\cL_{t,\rho}$ is sectorial and the mapping $t\mapsto (\cL_{t, \rho}-\zeta)^{-1}$ lies in $C([0,1], \cB(L^2(\Omega)))$ for all $\zeta\in\bbC\setminus\spec(\cL_{t, \rho})$.

To prove that $\cL_{t,\rho}$ is sectorial we have to show the existence of such $\theta\in(0,\frac{\pi}{2})$ and $M=M(\theta)>0$ that
\[\zeta\in\bbC\setminus\spec(\cL_{t, \rho})\,\text{ and }\, \|(\cL_{t,\rho}-\zeta)^{-1}\|_{\cB(L^2(\Omega))}\le M|\zeta|^{-1}\]
provided $\zeta\neq0$ and $|\arg\zeta|\in(\theta,\pi]$.
First, we introduce a self-adjoint operator $\cL_t$ acting in $L^2(\Omega)$ and defined by
$\cL_{t}u:=\cL u$ for $u\in\dom(\cL_{t}):=\dom(\cL_{t, \rho})$
so that $\cL_{t,\rho}=\rho_t\cL_t$.
Since $\cL_{t}$ is bounded from below we may assume without loss of generality that $\cL_{t}\ge0$ and, given a $\theta\in(0,\frac{\pi}{2})$, use the estimate

\begin{equation}\label{estrxi}
\|(\cL_t-\xi)^{-1}\|_{\cB(L^2(\Omega))}\leq (|\xi|\sin\theta)^{-1}
\text{ for all $\xi\in\bbC\setminus\{0\}$ such that $|\arg\xi|\in(\theta,\pi]$.}
\end{equation}

Indeed,  (<ref>) follows from the estimate
\[\|(\cL_t-\xi)^{-1}\|_{\cB(L^2(\Omega))}\leq |\Im \xi|^{-1}\le(|\xi|\sin\theta)^{-1}\]

provided $|\arg\xi|\in(\theta,\frac{\pi}{2}]$ and

\[\|(\cL_t-\xi)^{-1}\|_{\cB(L^2(\Omega))}=\big(\dist(\xi,\spec(\cL_{t}))\big)^{-1}\le|\xi|^{-1}\le(|\xi|\sin\theta)^{-1}\]
provided $|\arg\xi|\in(\frac{\pi}{2},\pi]$.

Throughout the rest of this proof we take all $\inf$'s and $\sup$'s over $(t,x)\in[0,1]\times\overline{\Omega}$.  We pick $\theta\in(0,\frac{\pi}{2})$ such that

\begin{equation}\label{chofth}
(1-\sin^2\theta)\sup\rho_t(x)<\inf\rho_t(x)% 0\le 1-\frac{\inf\rho_t(x)}{\sup\rho_t(x)}<\sin^2\theta<1,
\end{equation}

and fix any $\zeta\in\bbC\setminus\{0\}$ such that $|\arg\zeta|\in(\theta,\pi]$.
Using (<ref>), we can choose $\xi\in\bbC$
such that $\arg\xi=\arg\zeta$ with $|\xi|$ that satisfies the inequality

\begin{equation}\label{xizeta}
(1-\sin^2\theta)\sup\rho_t(x)<|\zeta| |\xi|^{-1}<\inf\rho_t(x).
\end{equation}

Dividing this by $\rho_t(x)$ we infer

\begin{equation}\label{xizeta2}
\sup\big|(|\zeta| (|\xi|\rho_t(x))^{-1}-1)\big|\le\sin^2\theta.
\end{equation}

Since  $\xi\in\bbC\setminus\spec(\cL_{t})$ we have

\begin{equation}\lb{aub3}
\rho_t\cL_t-\zeta=\rho_t(\cL_t-\xi)\big(I-(\cL_t-\xi)^{-1}(\zeta\rho_t^{-1}-\xi)\big). \end{equation}

Combining (<ref>) and (<ref>)
we infer

\begin{align}
\begin{split}
\|(\cL_t-\xi)^{-1}(\zeta\rho_t^{-1}-\xi)\|_{\cB(L^2(\Omega))}&\leq (|\xi|\sin\theta)^{-1} \sup\big|e^{\bfi\arg\zeta}(|\zeta|\rho_t(x)^{-1}-|\xi|)\big|\\
\end{split}
\end{align}

which by (<ref>) gives $\lambda\in\bbC\setminus \spec(\rho_t\cL_t)$ and, using  the second inequality in (<ref>), the required resolvent estimate $\|(\cL_{t,\rho}-\zeta)^{-1}\|_{\cB(L^2(\Omega))}\le M|\zeta|^{-1}$. Thus, $\cL_{t,\rho}$ is sectorial.

It is enough to prove continuity of the resolvent mapping at any $\zeta\in\bbR$ in the resolvent set of $\cL_{t,\rho}$. We note that if
$\zeta\in\bbR\setminus\spec(\cL_{t,\rho})$ then $0\in\bbC\setminus\spec(\cL_t-\rho_t^{-1}\zeta)$ and the identity $(\rho_t\cL_t-\zeta)^{-1}=(\cL_t-\rho_t^{-1}\zeta)^{-1}\rho_t^{-1}$ holds. Since the map $t\mapsto\rho_t^{-1}$ is continuous, it remains to prove continuity of the map $t\mapsto(\cL_t-\rho_t^{-1}\zeta)^{-1}$, that is, of the resolvent of the operator $H_t=\cL_t-\rho_t^{-1}\zeta$ at zero. This follows from Theorem <ref> with $\cA_t=\cL_t$, $V_t=-\rho_t^{-1}\zeta$,
$\tr:=(\gaD, \gamma_{{}_{N,\cL}})\in \cB (\cD^1(\Omega), \bndr)$ and

\begin{align}
Z_{t,s}:=[W(\alpha_{t}, \beta_{t})]^*( \alpha_{t}\beta_{s}-\beta_{t}\alpha_{s})[W(\alpha_{s}, \beta_{s})]\rightarrow 0, s\rightarrow t.
\end{align}

To justify the use of Theorem <ref>, we note that Hypothesis <ref> in the theorem is satisfied, that is, $(\cL_t-\bfi)^{-1}=\cO(1)$ as $t\rightarrow s$ in $\cB(L^2(\Omega), \cD^1(\Omega))$. The proof of this assertion is similar to that of (<ref>) (one imposes Robin boundary condition with $\Theta_t(x):=-\alpha_t(x)\beta_t^{-1}(x)$ on the portion of the boundary where $\beta_t^{-1}(x)\not=0$ and the Dirichlet condition elsewhere).

§.§ The Hadamard formula for star-shaped domains
In this section we show how
to use Theorem <ref>
to derive the classical Hadamard formula for the  Schrödinger operators subject to the Dirichlet boundary condition on variable star-shaped domains.

Let $\Omega\subset\bbR^n$ be a smooth star-shaped domain centered at zero and $\Omega_t=\{tx: x\in\Omega\}$ be its variation for $t\in(0,1]$. We consider a smooth ($N\times N$)-matrix potential $V=V(x)$ for $x\in\overline{\Omega}$ taking symmetric values. Suppose that $\mu\in\bbR$ is such that $\dim\ker(-\Delta_{D, \Omega}+V-\mu)=m\geq 1$, where $-\Delta_{D, \Omega}$ denotes the Dirichlet Laplacian acting in $L^2(\Omega)$. We claim that there exists a choice of orthonormal eigenfunctions $\{u_j\}_{j=1}^m\subset (-\Delta_{D,\Omega}+V-\mu)$ and a labeling of the eigenvalues $\{\mu_j(t)\}_{j=1}^m$ of $-\Delta_{D, \Omega_t}+V\upharpoonright_{\Omega_t}$, for $t$ near $1$, such that $\mu_j(1)=\mu$ for each $j$, and that the following classical Rayleigh–Hadamard–Rellich formula holds,
cf. <cit.>,

\begin{equation}\label{clHF}
\dot \mu_j(1)=-\int_{\partial\Omega}(\nu\cdot x)(\nu\cdot\nabla u_j)^2 {\rm d} x, 1\leq j\leq m.
\end{equation}

Rescaling $\Omega\ni t\mapsto tx\in \Omega_t$
of the operator $\big(-\Delta_{D,\Omega_t}+V\big)\big|_{\Omega_t}$  back to $\Omega$ yields a one-parameter family of self-adjoint operators  $H_t=-\Delta_{D,\Omega}+t^2V(tx)$, $t\in(0,1]$  acting in the fixed space $L^2(\Omega)$. This family of operators fits the framework of Theorem <ref> with
$\cA_t\equiv-\Delta_{\Omega}$, $V_t(x)=t^2V(tx)$, $\tr_t=[\gaD, -t^{-1}\Phi\gaN]^\top$, cf (<ref>), $t_0=1$, $\lambda(t_0)=\mu$ and $Q_t$ given by the $t$-independent projection onto the Dirichlet subspace   $\{(0,g): g\in H^{1/2}(\partial\Omega)\}$ for all $t$. All assumptions of Theorem <ref> are clearly satisfied in the present setting. By the theorem there exists a choice of orthonormal eigenfunctions $\{u_j\}_{j=1}^m\subset \ker (-\Delta_{D,\Omega}+V-\mu)$ and a labeling of the eigenvalues $\{\lambda_j(t)\}_{j=1}^m$ of $H_t$, for $t$ near $1$, such that

\begin{align}
\begin{split}
\label{l1}
\dot\lambda_j(1)&=\left\langle \frac{{\rm d} (t^2V(tx))}{{\rm d} t}\Big|_{t=1}u_j,u_j\right\rangle_{L^2(\Omega)}\\
&=2\langle Vu_j,u_j\rangle_{L^2(\Omega)}+\langle(\nabla V\cdot x)u_j,u_j \rangle_{L^2(\Omega)}, 1\leq j\leq m.
\end{split}
\end{align}

By the same rescaling as above, the eigenvalues $\lambda_j(t)$ uniquely determine the eigenvalues $\mu_j(t)$ for $t$ near $1$, and one has $\lambda_j(t)=t^2\mu_j(t)$. Our next objective is to use this identity together with (<ref>) to derive (<ref>).

We pause to consider the case of the Laplace operator with no potential. If $V\equiv 0$ then the proof is essentially completed as $H_t$ does not depend on $t$ and $0=\dot\lambda_j(1)=2\mu_j(1)+\dot\mu_j(1)$. This yields (<ref>) by the celebrated Rellich formula [86] expressing the eigenvalues $\lambda_j(1)=\mu_j(1)$ of the Dirichlet Laplacian via the Neumann boundary values of the respective eigenfunctions (this formula in turn easily follows from the Pokhozaev–Rellich identity, see, e.g., <cit.>, <cit.>, and formula (<ref>) below).

Returning to the general case of nonzero potential, to derive (<ref>) from (<ref>) we will follow the strategy of <cit.>. Let us fix $j$ and denote, for brevity, $u:=u_j$ and $\lambda(t):=\lambda_j(t)$, $\mu(t)=\mu_j(t)$.  First, integration by parts for $\Omega\subseteq\bbR^n$ yields

\begin{equation}\label{l3}
\langle(\nabla V\cdot x)u,u \rangle_{L^2(\Omega)}
=-\langle Vu ,2(\nabla u\cdot x)+nu\rangle_{L^2(\Omega)}
\text{ and } \langle u,\nabla
u\cdot x\rangle_{L^2(\Omega)}=-{n}/{2}.
\end{equation}

Using $-\Delta u+Vu=\lambda(1)u$ and replacing $Vu$ by $\Delta u+\lambda(1)u$ in (<ref>) and (<ref>),
a short calculation gives

\begin{equation}\label{l2}
\dot{\mu}(1)=\dot\lambda(1)-2\lambda(1)=(2-n)\langle\Delta u,u\rangle_{L^2(\Omega)}-2\langle\Delta u,\nabla u\cdot x\rangle_{L^2(\Omega)}.
\end{equation}

The standard Rellich's identity, see, e.g., <cit.>,  yields

\begin{align}\label{l4}
\langle\Delta u,\nabla u\cdot x\rangle_{L^2(\Omega)}
&=\int_{\partial\Omega}\big((\nu\cdot\nabla u)(x\cdot\nabla u)-\frac12(x\cdot\nu)\|\nabla u\|^2\big){\rm d}x\\
&\hskip4cm+\frac{n-2}{2}\int_\Omega\|\nabla u\|^2{\rm d}x.\nonumber
\end{align}

Since $u$ satisfies the Dirichlet condition, $\partial\Omega$ is a level curve, and thus $\nabla u$ and $\nu$ are parallel, that is, $\nabla u=(\nu\cdot \nabla u)\nu$. Using all this in (<ref>) yields (<ref>) because

\[\dot\mu(1)=\int_{\partial\Omega}\big(-2(\nu\cdot\nabla u)(x\cdot\nabla u)
+(x\cdot\nu)\|\nabla u\|^2\big){\rm d}x=
-\int_{\partial\Omega}(\nu\cdot\nabla u)^2(\nu\cdot x){\rm d} x.\]

§.§ Maslov crossing form for elliptic operators
section5.4 In this section  we continue the discussion began in Section <ref> on the relation between the Maslov crossing form and the slopes of the eigenvalue curves bifurcating from a multiple eigenvalue of the unperturbed elliptic operator. Here,
we assume the setting of Theorem <ref> and obtain a version of formula (<ref>) for the Robin-type elliptic operators $\cL_t$, see Proposition <ref> below.
For $\lambda\in\bbR$ we let

\begin{align}
\begin{split}
&{\cK}_{\lambda}:=\tr\Big(\Big\{u\in H^1(\Omega):\sum_{j,k=1}^{n}\langle \mathtt{a}_{jk}\partial_ku,\partial_j \varphi\rangle_{L^2(\Omega)}+\sum_{j=1}^{n}\langle \mathtt{a}_j\partial_ju,\varphi\rangle_{L^2(\Omega)}\\
&\quad+\sum_{j=1}^{n}\langle u,\mathtt{a}_j\partial_j\varphi\rangle_{L^2(\Omega)}+\langle vu-\lambda u,\varphi\rangle_{L^2(\Omega)}=0,\ \varphi\in H^1_0(\Omega)\Big\}\Big), \no
\end{split}
\end{align}

where the trace  operator $\tr=[\Gamma_0,\Gamma_1]^\top$is as in (<ref>). This is a “weak” version of the set $\bbK_{\lambda,t}$ from Section <ref>. The mapping $\lambda\mapsto \cK_{\lambda}$ is in $C^1(\bbR, \Lambda(H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega)))$ by <cit.>.

Let $t\mapsto \cG_t:=\{(f, -\Theta_t f): f\in H^{1/2}(\partial\Omega)\}$, then for $t_0\in[0,1]$ there is an interval $\cI\subset [0,1]$ centered at $t_0$ and a family of operators $t\mapsto \cM_{t}, t\in\cI$, which is in $C^1\big(\cI,\cB(\cG_{t_0},\cG_{t_0}^\bot)\big)$ with  $\cM_{t_0}=0$ and

\begin{equation}
\cG_t=\big\{\bq+\cM_{t}\bq\ \big|\, \bq\in\cG_{t_0}\big\}, t\in\cI,
\end{equation}

see,  e.g., <cit.>. In other words, $\cG_t$ can be written locally as the graph of the operator $\cM_t$, which is a replacement of $\cM_{\lambda,t}$ from Section <ref>. We say that $(\lambda,t_0)$ is a conjugate point if $\cK_{\lambda}\cap \cG_{t_0}\not=\{0\}$ or, equivalently,  if $\ker(\cL_{t_0}-\lambda)\not=\{0\}$.

We recall $\lambda(t_0)\in\spec_{\rm disc}(\cL_{t_0})$ from Theorem <ref> and let $\lambda:=\lambda(t_0)$. Then $(\lambda, t_0)$ is a conjugate point at which the Maslov crossing form $\mathfrak{m}_{t_0}$ for the path $t\mapsto \cK_{\lambda}\oplus\cG_{t}$ relative to the diagonal subspace $\mathfrak{D}=\{\bp=(p,p): p\in H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega))\}$
is defined by the formula

\begin{equation}\label{dfnMcfnew}
\mathfrak{m}_{t_0}(\bq,\bp):=\frac{\bd}{\bd t}\big|_{t=t_0}\widehat{\omega} (\bq,\cM_{t}\bp)=\widehat{\omega} (\bq,\dot{\cM}_{t_0}\bp),\ \bp,\bq\in (\cK_{\lambda}\oplus\cG_{t_0})\cap\mathfrak{D},
\end{equation}

where $\widehat{\omega} ={\omega} \oplus(-{\omega} )$ and
$\displaystyle{\dot{\cM}_{t_0}=\frac{\bd}{\bd t}\cM_{t}\big|_{t=t_0}}$. We stress that the pair of Lagrangian subspaces  $\big(\cK_{\lambda}, \cG_{t_0}\big)$ is Fredholm since $\lambda=\lambda(t_0)\not\in \spec_{\rm{ess}}(\cL_{t_0})$, see <cit.>. Hence, $\dim\big((\cK_{\lambda}\oplus\cG_{t_0})\cap\mathfrak{D}\big)<\infty$ and $\mathfrak{m}_{t_0}$ is a finite dimensional bilinear form. In fact,
the pair of Lagrangian subspaces $\big(\cK_{\lambda}, \cG_{t}\big)$ is Fredholm for $t$ near $t_0$ due to continuity of the path of the resolvent operators $t\mapsto (\cL_t-\bfi)^{-1}$.

Let $\lambda(t_0)$, $\{\lambda_j(t)\}_{j=1}^m$ and $\{u_j\}_{j=1}^m$ be as in Theorem <ref>, and denote $\bq_j:=(\tr u_j, \tr u_j)$. Then $\bq_j\in (\cK_{\lambda(t_0)}\oplus\cG_{t_0})\cap\mathfrak{D}$ and

\begin{equation}\label{formdr5}
\dot\lambda_j(t_0)=\mathfrak{m}_{t_0}(\bq_j,\bq_j),\ 1\leq j\leq m,
\end{equation}

where $\mathfrak{m}_{t_0}$ is the Maslov crossing form introduced in (<ref>).

The inclusion $\bq_j\in (\cK_{\lambda(t_0)}\oplus\cG_{t_0})\cap\mathfrak{D}$  holds since $u_j$ is an eigenfunction of $\cL_{t_0}$ corresponding to the eigenvalue $\lambda(t_0)$. For a fixed $j$
we abbreviate $\bq:=\bq_j=\tr u_j$ and introduce $g_t\in H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega)$ as in (<ref>)
but with $\cM_{\lambda,t}$ replaced by $\cM_t$. In particular, $g_{t_0}=\tr u_j$ because $\cM_{t_0}=0$.
Since $g_t=Q_tg_t$ where $Q_t$ is the orthogonal projection onto $\cG_t$, we have
\[\dot{g}_{{t_0}}=\dot{Q}_{{t_0}}g_{t_0}+Q_{{t_0}}\dot{g}_{{t_0}}=\dot{Q}_{{t_0}}\tr u_j+Q_{{t_0}}\dot{g}_{{t_0}}.\]
This and  that $\ran(Q_{{t_0}})$ is Lagrangian yields, as in (<ref>),

\begin{equation}
\omega(\tr u_j,\,\dot{g}_{{t_0}})=\omega(\tr u_j,\dot{Q}_{{t_0}}\tr_{t_0}u_j).
\end{equation}

As in (<ref>),  by definition of $\mathfrak{m}_{t_0}$ this implies

\begin{equation}
\mathfrak{m}_{{t_0}}(\bq_j,\bq_j)=-\omega(\tr u_j,\,\dot{g}_{{t_0}})=-\omega(\tr u_j,\dot{Q}_{{t_0}}\tr u_j)=\omega(\dot{Q}_{{t_0}}\tr u_j, \tr u_j).
\end{equation}

By formula (<ref>) in Theorem <ref> we have
$\dot{\lambda}_j(t_0)=-\langle\dot{\Theta}_{t_0}\gaD u_j,\gaD u_j\rangle_{L^2(\partial\Omega)}$. Thus,
it remains to show that
\[\omega(\dot{Q}_{t_0}\tr u_j, \tr u_j )=-\langle\dot{\Theta}_{t_0} \gaD u_j, \gaD u_j\rangle_{L^2(\partial\Omega)}.\]

The latter assertion follows from (<ref>) with $\phi_j=-\gaD u_j$ and $X_t=\Theta_t$, $Y_t=I$ as $\cG_t=\gr(-\Theta_t)=\ker([X_t, Y_t])$ with this choice of $X_t$ and $Y_t$.

As discussed in Remark <ref>,  formula (<ref>)
relating the derivative of the eigenvalues of the elliptic operators $\cL_t$ with respect to the parameter $t$ and the value of the (Maslov) crossing form for the flow $t\mapsto\cK_{\lambda(t)}\oplus\cG_t$ of  Lagrangian planes could be viewed as an infinitesimal version of the fundamental relation between the spectral flow and the Maslov index. Indeed, as in Remark <ref>,  formula (<ref>) implies relation (<ref>) with $H_t$ replaced by $\cL_t$ and $\Upsilon_{\lambda,t}$ replaced by $\cK_{\lambda(t)}\oplus\cG_t$.

We will now briefly  return to the Robin eigenvalue problem (<ref>) related to the Friedlander Inequalities but at once for the general elliptic operator $\cL$  described in Subsection <ref>.
We recall that for $\lambda\notin\spec(\cL_D)$ the Dirichlet-to-Neumann operator $M_{D,N}(\lambda)$ is defined by $f\mapsto-\gaN u$ (in the relevant papers [37, 47], $M_{D,N}$ is defined by $f\mapsto\gaN u$) where $u$ is the solution to $\cL u=\lambda u$, $\gaD u=f$. It is easy to see that (<ref>) has a nontrivial solution if and only if $\mu=\cot(\frac{\pi}{2}t)$ is an eigenvalue of  $M_{DN}(\lambda)$. Combining Remarks <ref>, <ref>
and Example <ref> with Proposition <ref> can be used to show the following formula relating the spectral counting functions
of the Dirichlet and Neumann realizations $\cL_D$ and $\cL_N$ and the Dirichlet-to-Neuman map $M_{D,N}(\lambda)$,

\begin{equation}\label{rem:FrMorse1}\begin{split}
\#\{\lambda\in\Sp(\cL_N): \lambda<0\}&-\#\{\lambda\in\Sp(\cL_D): \lambda<0\}\\&=\#\{\mu\in\Sp(M_{D,N}(\lambda)): \mu\geq0\},
\end{split}
\end{equation}

see [47] and,
specifically,  <cit.> and the literature therein (in [37, 47] the RHS of (<ref>) is given by the number of negative eigenvalues of $M_{D, N}(\lambda)$, this is due to sign discrepancy in the definition of $M_{D, N}(\lambda)$).
We omit details and just mention that the monotonicity of the eigenvalue curves $\lambda_k(t)$, $k=1,2,\dots$, established in Example <ref> and formula (<ref>) show that the Maslov crossing form is sign definite at each conjugate point on the vertical line through $\lambda$
when $t$ changes from $0$ to $1$ (Figure <ref> serves as a schematic illustration of this assertion). By a standard calculation, see, e.g.,  Step 1 in the proof of <cit.>, the Maslov crossing form is also sign definite at each conjugate point on the horizontal lines through $t=0$ and $t=1$ when $\lambda$ is changing.
These two properties are sometimes referred to as the monotonicity of the Maslov index. Thus, cf. Remark <ref>, the spectral flow through zero given by the LHS of (<ref>) is equal to the Maslov index along the vertical line through $\lambda$ which, in turn, is equal to the RHS.

§ KREIN-TYPE FORMULAS FOR DUAL PAIRS

In this section we give a  generalization of the Krein-type resolvent formula (<ref>) to the case of boundary triplets for an adjoint pair $A, \tA$, see, e.g., [1, 23, 25] and the literature cited therein. The theory of adjoint pairs goes back to [75], see also [7, 24, 76]. It allows one to describe non-selfadjoint extensions for an adjoint pair of densely defined closed (but not neceserely symmetric) operators. A typical example of the adjoint pair, see, e.g., [23, 25], is furnished by a non-symmetric elliptic second order partial differential operator and its formal adjoint; this example is discussed in detail in the end of this section.

We follow [25] to recall the definition of the adjoint pair and its boundary triplet. Let $A,\tA$ be closed densely defined operators on a Hilbert space $\cH$ forming an adjoint pair, that is, we assume that $\tA\subseteq A^*$ and $A\subseteq(\tA)^*$. We denote by $\cH_+$, respectively, $\wti{\cH}_+$ the domain $\dom(A^*)$, respectively, $\dom((\tA)^*)$ equipped with the graph-scalar product and graph norm for $A^*$, respectively, $(\tA)^*$, cf. Section <ref>. Let $\mathfrak{H}$ and $\mathfrak{K}$ be some “boundary” Hilbert spaces and

\begin{equation}\label{Gammas}
\Gamma_0:\wti{\cH}_+\to\mathfrak{H},\quad \Gamma_1:\wti{\cH}_+\to\mathfrak{K}, \quad \wti{\Gamma}_0: {\cH}_+\to\mathfrak{K}, \quad \wti{\Gamma}_1: {\cH}_+\to\mathfrak{H}
\end{equation}

be some “boundary trace operators". The collection $\{\mathfrak{H},\mathfrak{K}, \Gamma_0, \Gamma_1,\wti{\Gamma}_0, \wti{\Gamma}_1\}$ is called a
boundary triplet for the adjoint pair $A,\tA$ when the following hypothesis is satisfied.

Suppose that $A,\tA$ is an adjoint pair of densely defined closed operators such that $\tA\subseteq A^*$ and $A\subseteq(\tA)^*$. Consider linear operators, called the trace operators,

\begin{equation}\label{APeq1}
\tr:=[\Gamma_0,\Gamma_1]^\top:\wti{\cH}_+\to\mathfrak{H}\times\mathfrak{K},\, %\dom(\tr)=\wti{\cH}_+, \,
\wti{\tr}:=[\wti{\Gamma}_0,\wti{\Gamma}_1]^\top:\cH_+\to\mathfrak{K}\times\mathfrak{H}.\, %\dom(\tr)=\cH_+.
\end{equation}

Assume that the operators $\tr$ and $\wti{\tr}$ are surjective and satisfy

\begin{equation}\label{asgf}
\langle(\tA)^* u, v\rangle_\cH-\langle u,A^*v\rangle_\cH=
\langle\Gamma_1u,\wti{\Gamma}_0v\rangle_\mathfrak{K}
\end{equation}

for all $u\in\wti{\cH}_+$ and $v\in\cH_+$.

The existence of a boundary triplet for every adjoint pair $A,\tA$ was proved in [75], where, in addition, it was shown that

\begin{equation}
\dom(A)=\dom((\tA)^*)\cap\ker\Gamma_0\cap\ker\Gamma_1,\,
\dom(\tA)=\dom(A^*)\cap\ker\wti{\Gamma}_0\cap\ker\wti{\Gamma}_1.
\end{equation}

It is standard that the operators $\tr$ and $\wti{\tr}$ in (<ref>) are bounded, cf. <cit.>.
Indeed, it suffices to show that they are closed. To prove this, take a sequence $u_n\to0$ in $\wti{\cH}_+$ as $n\to\infty$ and assume that $\tr u_n\to(f,g)\in\mathfrak{H}\times\mathfrak{K}$.
For each $v\in\cH_+$ formula (<ref>) yields

\begin{align*}
\langle g,\wti{\Gamma}_0v\rangle_\mathfrak{K}-\langle f,\wti{\Gamma}_1v\rangle_\mathfrak{H}&=
\lim\big(\langle\Gamma_1u_n,\wti{\Gamma}_0v\rangle_\mathfrak{K}
-\langle\Gamma_0u_n,\wti{\Gamma}_1v\rangle_\mathfrak{H}\big)\\&=\lim\big(\langle(\tA)^* u_n, v\rangle_\cH-\langle u_n,A^*v\rangle_\cH\big)=0.
\end{align*}

By surjectivity, choosing $v$ such that $\wti{\tr}\,v=(0,f)$ gives $f=0$, and such that  $\wti{\tr}\,v=(g,0)$ gives $g=0$, as required to check that $\tr$ is closed. The proof for $\wti{\tr}$ is analogous.

The following Krein's-type formula is a direct generalization of Theorem <ref>. It gives the difference of the resolvent operators of any two (not necessarily sel-adjoint) extensions of the operator $A$ that are parts of $(\tA)^*$.

$\{\mathfrak{H},\mathfrak{K}, \Gamma_0, \Gamma_1,\wti{\Gamma}_0, \wti{\Gamma}_1\}$ be a boundary triplet for an adjoint pair $A,\tA$, and let $\cA_j$ for $j=1,2$ be any two closed extensions of $A$ acting in $\cH$ and satisfying $A\subseteq\cA_j\subseteq(\tA)^*$. Suppose that $\zeta\in\bbC\setminus(\spec(\cA_1)\cup\spec(\cA_2))$ and denote $R_j(\zeta):=(\cA_j-\zeta)^{-1}$ for $j=1,2$. Then one has

\begin{align}\label{AP5.14}
\big(\wti{\Gamma}_1R_2^*(\zeta)\big)^*\Gamma_0R_1(\zeta),\\
\label{AP5.14a}
R_2(\zeta)-R_1(\zeta)&=\big(\wti{\tr}R_2^*(\zeta)\big)^*Q_2JQ_1\big(\tr R_1(\zeta)\big),
\end{align}

where $R_2^*(\zeta)=((\cA_2)^*-\overline{\zeta})^{-1}$,
the operator $\wti{\tr}R_2^*(\zeta)=\big(\wti{\Gamma}_0R_2^*(\zeta),\wti{\Gamma}_1R_2^*(\zeta)\big)$ is considered as an operator in $\cB(\cH,\mathfrak{K}\times\mathfrak{H})$ and the adjoint operators in (<ref>), (<ref>) are defined correspondingly, $Q_1$, respectively, $Q_2$ denotes the orthogonal projection onto $\overline{\tr(\dom(\cA_1))}$ in the space $\mathfrak{H}\times\mathfrak{K}$, respectively, onto $\overline{\wti{\tr}(\dom((\cA_2)^*))}$ in the space $\mathfrak{K}\times\mathfrak{H}$, and the operator $J$ maps a pair $(f, g)$ from $\mathfrak{H}\times\mathfrak{K}$ into the pair
$(g,-f)$ from $\mathfrak{K}\times\mathfrak{H}$.

The inclusion $A\subseteq\cA_j\subseteq(\tA)^*$  yields
$\tA\subseteq(\cA_j)^*\subseteq A^*$ for $j=1,2$ <cit.>. The operator $R_2^*(\zeta)\in\cB(\cH)$ is also  bounded from $\cH$ onto $\dom((\cA_2)^*)\subseteq\cH_+=\dom(A^*)$, cf. (<ref>). Thus, the operator $\wti{\tr}R_2^*(\zeta)$ is well defined, and, analogously, the operator
$\tr R_1(\zeta)$ is well defined. Moreover, for all $u,v\in\cH$ one has

\begin{equation}\label{domaa}
(A^*-\overline{\zeta})R_2^*(\zeta)v=(\cA_2-\zeta)^*R_2^*(\zeta)v=v, \,
\big((\tA)^*-\zeta\big)R_1(\zeta)u=(\cA_1-\zeta)R_1(\zeta)u=u.\end{equation}

We also have $Q_2 \wti{\tr}R_2^*(\zeta)=\wti{\tr}R_2^*(\zeta)$ and $Q_1 \tr R_1(\zeta)=\tr R_1(\zeta)$ by the definition of the orthogonal projections $Q_2$ and $Q_1$. Thus, (<ref>) is just a reformulation of (<ref>). For the proof of (<ref>), we use (<ref>) and (<ref>) to write

\begin{align*}
\langle\big(R_2(\zeta)&-R_1(\zeta)\big)u,v\rangle_{\cH}=
\langle R_2(\zeta)u-R_1(\zeta)u,(\cA_2-\overline{\zeta})^*R_2^*(\zeta)v\rangle_{\cH}\\
&=\langle (\cA_2-\zeta)R_2(\zeta)u,R_2^*(\zeta)v\rangle_{\cH}-
\langle R_1(\zeta)u,(A^*-\overline{\zeta})R_2^*(\zeta)v\rangle_{\cH}\\
&=\langle u,R_2^*(\zeta)v\rangle_{\cH}-
\langle \big((\tA)^*-\zeta\big)R_1(\zeta)u,R_2^*(\zeta)v\rangle_{\cH}\\
\end{align*}

for all $u,v\in\cH$, yielding (<ref>).

In particular, for $j=1,2$,  given an operator
$\Psi_j\in\cB(\mathfrak{H},\mathfrak{K})$ (not necessarily self-adjoint), we consider in $\cH$ the extension $\cA_j$ of $A$ satisfying $A\subseteq\cA_j\subseteq(\tA)^*$ and defined by the formulas

\[\cA_ju=(\tA)^*u\, \text{ for }\, u\in\dom(\cA_j):=\{u\in\wti{\cH}_+: \Gamma_1u=\Psi_j\Gamma_0u\}, \quad j=1,2.\]

Under assumptions in Theorem <ref> one has

\begin{equation}
\end{equation}

We claim that $\tA\subset(\cA_j)^*\subset A^*$ satisfies
$\dom((\cA_j)^*)=\{v\in \cH_+: \wti{\Gamma}_1v=(\Psi_j)^*\wti{\Gamma}_0v\}$. Indeed, to see the inclusion “$\subseteq$” we note that for $u\in\dom(\cA_j)$ and $v\in\dom((\cA_j)^*)$ by (<ref>)

\begin{align*}
0&=\langle\cA_ju,v\rangle_\cH-\langle u, \cA_j^*v\rangle_\cH=\langle(\tA)^* u, v\rangle_\cH-\langle u,A^*v\rangle_\cH\\&=
\langle\Gamma_1u,\wti{\Gamma}_0v\rangle_\mathfrak{K}
\end{align*}

yielding $\wti{\Gamma}_1v=(\Psi_j)^*\wti{\Gamma}_0v$ as $\Gamma_0$ is surjective, while the opposite inclusion follows by running this chain of equalities backwards, thus proving the claim. Next, we note that $\Gamma_1R_1(\zeta)=\Psi_1\Gamma_0R_1(\zeta)$ and
$\wti{\Gamma}_1R_2^*(\zeta)=(\Psi_2)^*\wti{\Gamma}_0R_2^*(\zeta)$ since the resolvents map $\cH$ into  the domains of respective operators.
Now (<ref>) yelds

\begin{align*}
\big(\wti{\Gamma}_1R_2^*(\zeta)\big)^*\Gamma_0R_1(\zeta)\\&=\big(\wti{\Gamma}_0R_2^*(\zeta)\big)^*\Psi_1\Gamma_0R_1(\zeta)-
\big((\Psi_2)^*\wti{\Gamma}_0R_2^*(\zeta)\big)^*\Gamma_0R_1(\zeta)\\&=
\big(\wti{\Gamma}_0R_2^*(\zeta)\big)^*\big(\Psi_1-\Psi_2\big)\Gamma_0R_1(\zeta),
\end{align*}

as required.

We conclude this section with an example of the abstract boundary triplet for the adjoint pair formed by an elliptic second order partial differential operator and its formal adjoint, cf.  [12, 13, 14, 23, 25]. In particular, Krein-type formulas in Theorem <ref> and Corollary <ref> hold for these operators. The discussion below regarding the boundary traces is based on the paper [58], see also [25], dealing with general elliptic operators on smooth domains and is related to the material in Appendix <ref> taken from
[52, 54] where the case of the Laplacian on Lipschitz domains was considered.
So, we assume throughout that $\Omega\subset\bbR^n$ is a bounded domain with $C^\infty$-boundary, and consider the following
mutually formally adjoint differential expressions, cf. (<ref>),

\begin{align}
\begin{split}
\cL:&=-\sum_{j,k=1}^{n} \partial_j \mathtt{a}_{jk}\partial_k + \sum_{j=1}^{n}\big(\mathtt{a}_j\partial_j -\partial_j {\mathtt{b}_j}\big)+\mathtt{q}\\
&\hspace{3cm}= -\div(\mathtt{A}\nabla)+ \mathtt{a}\cdot \nabla -\nabla \cdot \mathtt{b} +\mathtt{q},\lb{1.15b-new}\\
\end{split}
\end{align}

\begin{align}
\begin{split}
\wti{\cL}:&=-\sum_{j,k=1}^{n} \partial_k\overline{\mathtt{a}}_{jk}\partial_j + \sum_{j=1}^{n}\big(\overline{\mathtt{b}}_j\partial_j -\partial_j \overline{\mathtt{a}}_j\big)+\overline{\mathtt{q}}\\
&\hspace{3cm}= -\div(\mathtt{A}^*\nabla)+ \mathtt{b}^*\cdot \nabla -\nabla \cdot \mathtt{a}^* +\overline{\mathtt{q}},\lb{1.15c-new}
\end{split}
\end{align}

where the bar stands for complex conjugation, with coefficients $\mathtt{A}=\{\mathtt{a}_{ij}\}_{1\leq i, j\leq n}$, $\mathtt{a}:=\{\mathtt{a}_j\}_{1\leq j\leq n}$,
$\mathtt{b}:=\{\mathtt{b}_j\}_{1\leq j\leq n}$ and $\mathtt{q}$ satisfying
$\mathtt{a}_{jk}, \mathtt{a}_j, \mathtt{b}_j, \mathtt{q}\in C^{\infty}(\overline{\Omega};\bbC)$, $1\leq j,k\leq n$. In addition,  we assume the ellipticity condition: For $c>0$ we have

\begin{align}
& \Re\Big(\sum\limits_{j,k=1}^n\mathtt{a}_{jk}(x)\xi_k\overline{\xi_j}\Big) \geq c \sum\limits_{j=1}^n|\xi_j|^2, x\in\overline{\Omega}, \xi=\{\xi_j\}_{j=1}^n\in\C^n.
\end{align}

As in (<ref>), we associate with $\cL$ from
(<ref>) the space $\cD^s(\Omega)$ equipped with the $\cL$-graph norm (<ref>), and with $\wti{\cL}$ from
the space
$\wti{\cD}^{s}(\Omega):=\{u\in H^s(\Omega): \wti{\cL} u\in L^2(\Omega)\}$
equipped with the  $\wti{\cL}$-graph norm
$\|u\|_{\wti{s}}:=\big(\|u\|_{H^s({\Omega} )}^2+\|\wti{\cL} u\|_{L^2({\Omega })}^2\big)^{1/2}$, $s\geq 0$,
where $\wti{\cL}u$ should be understood in the sense of distributions.
We introduce operators acting in $L^2(\Omega)$ by
$\cL_0u=\cL u$ and $\wti{\cL}_0=\wti{\cL}u$ for $u\in\dom(\cL_0)=\dom(\wti{\cL}_0):=C^\infty_0(\Omega)$, the space of functions compactly supported in $\Omega$, and recall from <cit.> that these operators are closable. We will denote by $\cL_{min}$ the closure of $\cL_0$ and by $\wti{\cL}_{min}$ the closure of $\wti{\cL}_0$.
Also, we will consider the maximal operators, $\cL_{max}u:=\cL u$ for $u\in\dom(\cL_{max}):=\cD^0(\Omega)$ and $\wti{\cL}_{max}u:=\wti{\cL} u$ for $u\in\dom(\wti{\cL}_{max}):=\wti{\cD}^0(\Omega)$. It is known, see, e.g.,
<cit.>, that

\begin{equation}\label{incladj}
\cL_{min}\subset(\wti{\cL}_{min})^*=\cL_{max},
\quad \wti{\cL}_{min}\subset(\cL_{min})^*=\wti{\cL}_{max}.
\end{equation}

Thus, $A=\cL_{min}$ and $\wti{A}=\wti{\cL}_{min}$ is an adjoint pair of operators. We will now construct the boundary traces so that the Green identity (<ref>) holds.

It will be convenient to use the notation
\[\gamma_{{}_{D,\cL}}=\gamma_{{}_{D,\wti{\cL}}}:=\gaD\in\cB(H^1(\Omega), H^{1/2}(\partial\Omega))\]
for the usual Dirichlet trace. As in Subsection <ref>, we will associate with $\cL$ from (<ref>) the first order trace operator $\gamma_{{}_{N, \cL}} \in\cB(\cD^1(\Omega), H^{-1/2}(\partial\Omega))$
which is a unique extension of the co-normal derivative $\gamma_{{}_{N, \cL}}\in\cB(H^2(\Omega), H^{1/2}(\Omega))$ defined in (<ref>). Analogously, associated with $\wti{\cL}$ from (<ref>) is the first order trace operator $\gamma_{{}_{N, \wti{\cL}}} \in\cB(\wti{\cD}^1(\Omega), H^{-1/2}(\partial\Omega))$
which is a unique extension of the co-normal derivative $\gamma_{{}_{N, \wti{\cL}}}\in\cB(H^2(\Omega), H^{1/2}(\Omega))$ defined by the formula

\begin{align}
&{\gamma_{{}_{N,\wti{\cL}}}}u:=\sum_{j,k=1}^{n} \overline{\mathtt{a}}_{kj}\nu_j\gamma_{{}_{D,\wti{\cL}}}(\partial_k u)+\sum_{j=1}^{n}{\overline{\mathtt{b}}_j}\nu_j \gamma_{{}_{D,\wti{\cL}}} u,\quad u\in H^2(\Omega).
\end{align}

Then the following Green identity,

\begin{align}
\langle\cL u, v\rangle_{L^2(\Omega )}
&- \langle u,\wti{\cL} v\rangle_{L^2(\Omega )} \lb{1.2.6}\\&={\langle\gamma_{{}_{D,\cL}} u, \gamma_{{}_{N, \wti{\cL}}} v  \rangle_{{-1/2}}}-\overline{\langle\gamma_{{}_{D,\wti{\cL}}} v, \gamma_{{}_{N, {\cL}}} u  \rangle_{{-1/2}}},\no
\end{align}

holds for all $u\in\cD^1(\Omega)$ and $v\in\wti{\cD}^1(\Omega)$. In order to rewrite this identity in a form compatible with (<ref>) we will need to take four more steps.

First, we will extend further the usual Dirichlet and weak Neumann trace operators
\begin{align*}
&\gamma_{{}_{D,\cL}} =\gamma_{{}_{D,\wti{\cL}}}\in\cB(H^1(\Omega), H^{1/2}(\partial\Omega)),\\
&\gamma_{{}_{N, \cL}} \in\cB(\cD^1(\Omega), H^{-1/2}(\partial\Omega)),\, \gamma_{{}_{N, \wti{\cL}}} \in\cB(\wti{\cD}^1(\Omega), H^{-1/2}(\partial\Omega))\end{align*}

defined on the spaces $H^1(\Omega)$, $\cD^1(\Omega)$, $\wti{\cD}^1(\Omega)$, respectively, to $\cD^0(\Omega)$ and $\wti{\cD}^0(\Omega)$, the domains of the maximal operators $\cL_{max}$ and $\wti{\cL}_{max}$, and obtain the bounded and surjective traces

\begin{align*}
\widehat{\gamma}_{{}_{D,\wti{\cL}}}\in\cB(\wti{\cD}^0(\Omega), H^{-1/2}(\partial\Omega)),\\
& \widehat{\gamma}_{{}_{N,{\cL}}}\in\cB(\cD^0(\Omega), H^{-3/2}(\partial\Omega)),\,\widehat{\gamma}_{{}_{N,\wti{\cL}}}\in\cB(\wti{\cD}^0(\Omega), H^{-3/2}(\partial\Omega)),\end{align*}

cf. Lemmas <ref> and <ref>,
that agree with the usual Dirichlet and Neumann trace operators on their respective domains. The existence of such extensions for smooth domains and general elliptic operators is given in <cit.>, and for Lipschitz domains and the Laplacian is given in [54].

Armed with the trace operators defined on the domains $\cD^0(\Omega)$ and $\wti{\cD}^0(\Omega)$ of the maximal operators, we proceed, following <cit.>, with the second step that involves the Dirichlet-to-Neumann operators, $M_{D,N}$ and $\wti{M}_{D,N}$, associated with the operators $\cL$ from (<ref>) and $\wti{\cL}$ from (<ref>), respectively. We define  $M_{D,N}$ next; one deals with $\wti{M}_{D,N}$ analogously. By adding to $\mathtt{q}$ a constant, if needed, we may and will assume that zero is not in the spectrum of the Dirichlet realizations of $\cL$ as defined in <cit.> or <cit.>. Thus, for each $f\in H^{-1/2}(\partial\Omega)$ there is a unique solution $u=u_f\in\cD^0(\Omega)$ to the
boundary value problem $\cL u=0$, $\widehat{\gamma}_{{}_{D,\cL}}u=f$. We may now define $M_{D,N}f:=-\widehat{\gamma}_{{}_{N,{\cL}}}u_f$ as an operator acting from $H^{-1/2}(\partial\Omega)$ to $H^{-3/2}(\partial\Omega)$.

Our third step is to introduce yet another Neumann trace,
$\tau_{{}_{N,\cL}}$, defined by the formula
$\tau_{{}_{N,\cL}}u:=\widehat{\gamma}_{{}_{N,{\cL}}}u+M_{D,N}\widehat{\gamma}_{{}_{D,\cL}}u$ for $u\in\cD^0(\Omega)$. The following remarkable property of $\tau_{{}_{N,\cL}}$ is a consequence of elliptic regularity of solutions to the Dirichlet problem, see <cit.> and <cit.> (or Lemma  <ref> below): Although both distributions $\widehat{\gamma}_{{}_{N,{\cL}}}u$ and $M_{D,N}\widehat{\gamma}_{{}_{D,\cL}}u$ belong to $H^{-3/2}(\partial\Omega)$, we claim that the sum $\tau_{{}_{N,\cL}}u$ of these two distributions is, in fact, a function from $H^{1/2}(\partial\Omega)$. Indeed, given a $u\in\cD^0(\Omega)$ and letting $f=\widehat{\gamma}_{{}_{D,\cL}}u$ we observe that
$\tau_{{}_{N,\cL}}u=\widehat{\gamma}_{{}_{N,{\cL}}}(u-u_f)$ with $u_f$ as in the definition of $M_{D,N}$ in step two above. But then
$\widehat{\gamma}_{{}_{D,\cL}}(u-u_f)=0$, which shows that $u-u_f$ is in the domain of the Dririclet realization of $\cL$. By elliptic regularity then
$u-u_f\in H^2(\Omega)$, see, e.g., <cit.>, and thus $\tau_{{}_{N,\cL}}u=\widehat{\gamma}_{{}_{N,{\cL}}}(u-u_f)={\gamma}_{{}_{N,{\cL}}}(u-u_f)\in H^{1/2}(\partial\Omega)$ as claimed. Analogously, we define
$\tau_{{}_{N,\wti{\cL}}}u:=\widehat{\gamma}_{{}_{N,\wti{\cL}}}u+\wti{M}_{D,N}\widehat{\gamma}_{{}_{D,\wti{\cL}}}u$ for $u\in\wti{\cD}^0(\Omega)$.
Moreover, the operators $\tau_{{}_{N,\cL}}\in\cB(\cD^0(\Omega), H^{1/2}(\partial\Omega))$ and $\tau_{{}_{N,\wti{\cL}}}\in\cB(\wti{\cD}^0(\Omega), H^{1/2}(\partial\Omega))$ are surjective, and the following Green identity,

\begin{align}\label{GF3}
(\cL u,v)_{L^{2}(\Om)}&-(u,\wti{\cL} v)_{L^{2}(\Om)}\\
\overline{\langle \tau_{{}_{N,\wti{\cL}}} v, \widehat{\gamma}_{{}_{{D,\cL}}} u\rangle_{{-1/2}}}-{{\langle \tau_{{}_{N, {\cL}}}
u, \widehat{\gamma}_{{}_{D,\wti{\cL}}} v \rangle_{{-1/2}}}},\no
\end{align}

holds for all $u\in\cD^0(\Omega)$, $v\in\wti{\cD}^0(\Omega)$ by <cit.>, see also <cit.> and <cit.> or Lemma <ref> below.

Finally, in the last step we use the Riesz isomorphism $\Phi: H^{-1/2}(\partial\Omega)\rightarrow H^{1/2}(\partial\Omega)$ defined in (<ref>). We are ready to define the boundary triplet in question:  the function spaces are given by

\begin{equation}
\cH_+=\dom((\cL_{min})^*)=\wti{\cD}^0(\Omega),
\wti{\cH}_+=\dom((\wti{\cL}_{min})^*)={\cD}^0(\Omega),
\mathfrak{H}=\mathfrak{K}=H^{1/2}(\partial\Omega)
\end{equation}

and the trace operators are given by

\begin{equation}
\Gamma_0=-\Phi\widehat{\gamma}_{{}_{D,\cL}}, \Gamma_1=\tau_{{}_{N,\cL}}, \wti{\Gamma}_0=-\Phi\widehat{\gamma}_{{}_{D,\wti{\cL}}}, \wti{\Gamma}_1=\tau_{{}_{N,\wti{\cL}}}.
\end{equation}

Now (<ref>) shows that this collection is indeed a boundary triplet for $A=\cL_{min}$ and $\wti{A}=\wti{\cL}_{min}$ as (<ref>) readily holds.

§ THE KREIN–NAIMARK RESOLVENT FORMULA REVISITED
In this appendix, we revisit the classical Krein–Naimark (<ref>) and Krein (<ref>) formulas for the difference of resolvents of two self-adjoint extensions of an abstract symmetric operator, see, e.g, <cit.>. As we demonstrate in the proof of Proposition <ref> the Krein–Naimark formula (<ref>) can be naturally derived from formula (<ref>) in Theorem <ref> by specializing it to the case of  boundary triplets.  Conversely,  in Remark <ref> we show how to derive (<ref>) from (<ref>). The poof of Krein's resolvent formula for the case of finite deficiency indices is given in Proposition <ref>.

Let $(\mathfrak{H}, \Gamma_0, \Gamma_1)$ be a boundary triplet as described in Definition <ref>. Following common convention we define  one of the two self-adjoint extensions of $A$ in the Krein-Naimark formula  by

\begin{equation}\lb{1.35a}
\cA_0:=A^*\upharpoonright_{\ker(\Gamma_0)},
\end{equation}

and subtract from its resolvent the resolvent of yet another, arbitrary, self-adjoint extension.

First, we recall some known facts, see, e.g., <cit.>. Since $\dom(A^*)=\dom(A_0)\dot{+}\ker(A^*-\zeta)$ for $\zeta\in\bbC\setminus\bbR$ the map
$\Gamma_0\upharpoonright_{\ker(A^*-\zeta)}: \ker(A^*-\zeta)\to\mathfrak{H}$ is bijective
and thus we define $\gamma(\zeta):=(\Gamma_0\upharpoonright_{\ker(A^*-\zeta)})^{-1}$ and notice that $\gamma(\zeta)\in\cB(\mathfrak H, \cH)$ and $\Gamma_0\gamma(\zeta)h=h$ for any $h\in\mathfrak{H}$. In particular, $\gamma(\zeta)$ is injective.  We will use the well-known Derkach-Malamud Lemma saying that $\gamma^*(\overline{\zeta})=\Gamma_1(\cA_0-\zeta)^{-1}$, see
<cit.> or <cit.>.  The operator-valued function $\gamma(\cdot)$ can be extended analytically to $\bbC\setminus\spec (\cA_0)$ giving rise to the abstract Weyl function $M(\zeta):=\Gamma_1\gamma(\zeta)$, $\zeta\in \bbC\setminus\spec(\cA_0)$.

Next, let $\cA$ be an arbitrary self-adjoint extension of $A$, and let
$\cG\in\Lambda(\mathfrak{H}\times\mathfrak{H})$ be the Lagrangian subspace such that $\cG=\tr(\dom(\cA))$, cf. Theorems <ref>, <ref> and Remark <ref>. We will treat $\cG$ as a linear relation,
see, e.g., <cit.>.
Slightly abusing notation we do not distinguish between the operator $M(\zeta)$ and its graph, in particular, we write $\cG-M(\zeta):=\cG-\gr(M(\zeta))$ and treat both terms in the right-hand side as linear relations. The linear relation $\cG-M(\zeta)$ is called invertible whenever

\begin{align}\label{inj}
\ker(\cG-M(\zeta))&:=\{f\in\mathfrak{H}: (f,0)\in(\cG-M(\zeta))\}=\{0\},\text{ and }\\
\ran(\cG-M(\zeta))&:=\{g\in\mathfrak H:\exists f\in\mathfrak{H}\  {s.t.}\  (f,g)\in(\cG-M(\zeta)) \}=\mathfrak H.\label{surj}\end{align}

In this case there exists an operator in $\cB(\mathfrak{H})$ whose graph is given by

\begin{equation}
\{(g,f) \in \bndra: (f,g)\in(\cG-M(\zeta)) \};
\end{equation}

this operator is denoted by   $(\cG-M(\zeta))^{-1}$.
Let  $(\mathfrak{H}, \Gamma_0, \Gamma_1)$ be a boundary triplet for the symmetric operator $A$, see Definition <ref>, let $\cA_0$ be the self-adjoint extension of $A$ from (<ref>), let $\cA$ be an arbitrary self-adjoint extension of $A$ and $\cG=\tr(\dom(\cA))$.
Then $\cG-M(\zeta)$ is invertible and

\begin{equation}\label{1.23}
\text{ for $\zeta\not\in\Sp(\cA_0)\cup \Sp(\cA)$. }
\end{equation}

We denote $R_0(\zeta):=(\cA_0-\zeta)^{-1}$ and
$R(\zeta)=(\cA-\zeta)^{-1}$. Since $\Gamma_0 R_0(\overline{\zeta})=0$ by (<ref>), the Krein-type formula from Theorem <ref> and the Derkach-Malamud Lemma above yield

\begin{equation*}
\label{1.25new}
R_{0}(\zeta)- R(\zeta)= (\Gamma_0R_0(\overline{\zeta}))^*\Gamma_1 R(\zeta)-(\Gamma_1R_0(\overline{\zeta}))^*\Gamma_0 R(\zeta)=-\gamma(\zeta)\Gamma_0 R(\zeta).
\end{equation*}

It remains to prove (<ref>), (<ref>), and that

\begin{equation}\label{g0R}
\Gamma_0R(\zeta)=(\cG-M(\zeta))^{-1}\gamma^*(\overline{\zeta}).
\end{equation}

The main identity needed for the proofs is that

\begin{equation}\label{1.32}
\gamma^*(\overline{\zeta})u=\Gamma_1R_0(\zeta)u= \Gamma_1R(\zeta)u-M(\zeta) \Gamma_0 R(\zeta)u \,
\text{ for all  $u\in\cH$}.
\end{equation}

To justify the second equality in (<ref>), we use  $(A^*-\zeta)\gamma(\zeta)=0$
and $\Gamma_0(I_\cH-\gamma(\zeta)\Gamma_0)=0$, yielding
$\ran(I_\cH-\gamma(\zeta)\Gamma_0)\subset\dom(\cA_0)$, and write

\begin{align*}
\Gamma_1R_0(\zeta)&=\Gamma_1R_0(\zeta)(\cA-\zeta)R(\zeta)=\Gamma_1R_0(\zeta)(A^*-\zeta)R(\zeta)\\
\Gamma_1R(\zeta)-M(\zeta) \Gamma_0 R(\zeta).
\end{align*}

Since $R(\zeta)$ is a bijection of $\cH$ onto $\dom(\cA)$, we have $\cG=\{(\Gamma_0R(\zeta)u, \Gamma_1R(\zeta)u): u\in\cH\}$. This and (<ref>) yield

\begin{align}\label{linre}
\cG-M(\zeta)&=\{(f,g-M(\zeta)f): (f,g)\in\cG\}\\&=\big\{\big(\Gamma_0R(\zeta)u, \Gamma_1R(\zeta)u-M(\zeta) \Gamma_0 R(\zeta)u\big):u\in\cH\big\}\\
&=\big\{\big(\Gamma_0R(\zeta)u, \gamma^*(\overline{\zeta})u\big):u\in\cH\big\}.\nonumber
\end{align}

Since $\tr$ is surjective, (<ref>) follows from (<ref>). Indeed, for any $g\in\mathfrak{H}$ there is some $v\in\dom(A^*)$ such that $\Gamma_0v=0$ and $\Gamma_1v=g$. Since $v\in\dom(\cA_0)$, there is some $u\in\cH$ such that $v=R_0(\zeta)u$ and so $g=\Gamma_1R_0(\zeta)u\in\ran(\cG-M(\zeta))$ by (<ref>) and (<ref>). To begin the proof of (<ref>), we first notice that
$\gamma(\zeta)\ker(\cG-M(\zeta))\subset\dom(\cA)$. Indeed,
by (<ref>) and (<ref>) we have $\ker(\cG-M(\zeta))=\big\{\Gamma_0R(\zeta)u: \Gamma_1R(\zeta)u=M(\zeta)\Gamma_0R(\zeta)u, u\in\cH\big\}$ and thus

\begin{align}
\tr&\gamma(\zeta)\ker(\cG-M(\zeta))%=\{(\Gamma_0\gamma(\zeta)x,\Gamma_1\gamma(\zeta)x):x\in\ker(\cG-M(\zeta))\}\\&
\\&=\{\big(\Gamma_0\gamma(\zeta)\Gamma_0R(\zeta)u,
\Gamma_1\gamma(\zeta)\Gamma_0R(\zeta)u\big): \Gamma_1R(\zeta)u=M(\zeta)\Gamma_0R(\zeta)u, u\in\cH\}
\\&=\{\big(\Gamma_0R(\zeta)u,
M(\zeta)\Gamma_0R(\zeta)u\big): \Gamma_1R(\zeta)u=M(\zeta)\Gamma_0R(\zeta)u, u\in\cH\}\\
%\\&\subseteq\{\big(\Gamma_0R(\zeta), \Gamma_1R(\zeta)f\big): f\in\cH\big\}\cap {\rm graph } M(\zeta)
&=\cG\cap {\rm graph } (M(\zeta)).
\end{align}

Therefore, $(\cA-\zeta)\gamma(\zeta)\ker(\cG-M(\zeta))=
(A^*-\zeta)\gamma(\zeta)\ker(\cG-M(\zeta))=\{0\}$ yields
the inclusion $\gamma(\zeta)\ker(\cG-M(\zeta))\subset\ker(\cA-\zeta)=\{0\}$
and thus $\ker(\cG-M(\zeta))=\{0\}$ because $\gamma(\zeta)$ is injective, thus finishing the proof of (<ref>).
Finally, using (<ref>) again,

\begin{align*}
{\rm graph } (\cG-M(\zeta))^{-1}&=\big\{ (g,f)\in\mathfrak{H}\times\mathfrak{H}: (f,g)\in(\cG-M(\zeta))\big\}
\\&=\big\{\big(\gamma^*(\overline{\zeta})u,\Gamma_0R(\zeta)u\big): u\in\cH \big\}
\end{align*}

yielding $(\cG-M(\zeta))^{-1}\gamma^*(\overline{\zeta})
as required to finish the proof of (<ref>).

In the course of proof of the Krein-Naimark formula (<ref>) we established relation (<ref>). Using this relation we now show how to derive formula (<ref>) in Theorem <ref> from formula (<ref>). For any two self-adjoint extensions $\cA_1$ and $\cA_2$ and the extension $\cA_0$ given by (<ref>) we denote
$R_j(\zeta)=(\cA_j-\zeta)^{-1}$ for any $\zeta$ which is not in the spectrum of $\cA_j$, $j=0,1,2$. Applying (<ref>) and using (<ref>) for $\cA_1$ and $\cA_2$ yields

\begin{equation}\label{firstres}
\end{equation}

Multiplying (<ref>) by $\Gamma_1$ and using formulas $\gamma^*(\overline{\zeta})=\Gamma_1R_0(\zeta)$ and $M(\zeta)=\Gamma_1\gamma(\zeta)$ gives

\[\Gamma_1R_1(\zeta)=\gamma^*(\overline{\zeta})+M(\zeta)\Gamma_0 R_1(\zeta), \quad
\Gamma_1R_2(\overline{\zeta})=
\gamma^*({\zeta})+M(\overline{\zeta})\Gamma_0 R_2(\overline{\zeta}).\]

Plugging this in the RHS of formula (<ref>) and using
the property  $M^*(\zeta)=M(\overline{\zeta})$ of the Weyl function, see, e.g., <cit.>, yileds

\begin{align*}
\big(&\Gamma_0 R_2(\overline{\zeta})\big)^*\Gamma_1 R_1({\zeta})-\big(\Gamma_1 R_2(\overline{\zeta})\big)^*\Gamma_0 R_1({\zeta})\\&=\big(\Gamma_0 R_2(\overline{\zeta})\big)^*\big(\gamma^*(\overline{\zeta})+M(\zeta)\Gamma_0 R_1(\zeta)\big)
-\big(\gamma^*({\zeta})+M(\overline{\zeta})\Gamma_0 R_2(\overline{\zeta})\big)^*\Gamma_0 R_1({\zeta})\\
&=\big(\gamma(\overline{\zeta})\Gamma_0 R_2(\overline{\zeta})\big)^*-\big(\gamma(\zeta)\Gamma_0R_1(\zeta)\big)+\big(\Gamma_0 R_2(\overline{\zeta})\big)^*\big(M(\zeta)-M^*(\overline{\zeta})\big)\Gamma_0 R_1({\zeta})\\&=\big(R_2(\overline{\zeta})-R_0(\overline{\zeta})\big)^*-\big(R_1(\zeta)-R_0(\zeta)\big)=
\end{align*}

where, to pass to the last line, we used (<ref>) again.
This proves (<ref>) as required.

We will conclude this section by deriving from formula (<ref>) in Theorem <ref> yet another classical Krein's resolvent formula
(<ref>)  valid under the temporary assumption that the equal deficiency indices of $A$ are finite and for which we refer to the classical text
<cit.> and a very nice newer exposition in <cit.>.

Let  $(\mathfrak{H}, \Gamma_0, \Gamma_1)$ be a boundary triplet for the symmetric operator $A$ with equal and finite deficiency indices, let $\cA_1$ and $\cA_2$ be any two self-adjoint extensions of $A$, let $(m,m)$ denote the deficiency indices of the operator  $A_0=A^*\big|_{\dom(A_0)}$  defined by the equality $\dom(A_0):=\dom(\cA_1)\cap\dom(\cA_2)$, and let $\{u_k(\zeta)\}_{1\le k\le m}$ be any basis in the subspace $\ker(A_0^*-\zeta)$.  Then there exists a unique non-singular matrix $\cP(\zeta)=\big(p_{lj}(\zeta)\big)_{1\le l,j\le m}$ such that the resolvents of the operators $\cA_1$ and $\cA_2$ for each $u\in\cH$ satisfy

\begin{equation}\label{fdi-krf}
\big(R_1(\zeta)-R_2(\zeta)\big)u=\sum_{l,j}p_{lj}(\zeta)\langle  u, u_j(\overline{\zeta})\rangle_{{}_\cH} u_l(\zeta)\,
\end{equation}

for all $\zeta\in\bbC\setminus(\spec(\cA_1)\cup\spec(\cA_2))$.

We temporarily denote by $R(\zeta)$ the RHS of  (<ref>) in Theorem <ref>, that is, we set $R(\zeta)=(\tr R_2(\overline{\zeta}))^*J (\tr R_1(\zeta))$. Thus, our objective is to prove that $R(\zeta)$ is equal to the RHS of (<ref>).
First, we will use the fact that the subspaces
$\cG_1=\tr(\dom(\cA_1))$ and $\cG_2=\tr(\dom(\cA_2))$ are Lagrangian in $\mathfrak{H}\times\mathfrak{H}$, cf. Remark <ref>, and prove the following elementary assertions:

\begin{equation}\label{assrts}
{\rm (i)}\,\ \ker(R(\zeta))=\ran(A_0-\zeta);\quad {\rm (ii)}\,\
\ran(R(\zeta))=\ker(A_0^*-\zeta).
\end{equation}

To begin the proof we notice that for any $u,v\in\cH$ by (<ref>) one has

\begin{equation}\label{Romega}\begin{split}
\langle R(\zeta)u,v\rangle_{{}_\cH}&= \langle  (\tr R_2(\overline{\zeta}))^*J (\tr R_1(\zeta))u,v\rangle_{{}_\cH}\\
&=\langle  J (\tr R_1(\zeta))u, \tr R_2(\overline{\zeta})v\rangle_{{}_{\mathfrak{H}\times\mathfrak{H}}}=\omega\big(\tr R_1(\zeta)u, \tr R_2(\overline{\zeta})v \big).
\end{split}\end{equation}

If $u=(A_0-\zeta)w$ for some $w\in\dom(A_0)=\dom(\cA_1)\cap\dom(\cA_2)$ then
\[R_1(\zeta)u=R_1(\zeta)(A_0-\zeta)w=R_1(\zeta)(\cA_1-\zeta)w=w\]

because $w\in\dom(\cA_1)$ and thus $\tr R_1(\zeta)u=\tr w\in\cG_2$ because  $w\in\dom(\cA_2)$. Since $\cG_2$ is isotropic, (<ref>) yields $R(\zeta)u=0$ and thus $\overline{\ran(A_0-z)}\subseteq\ker(R(\zeta))$. On the other hand, if $R(\zeta)u=0$ then $\tr R_1(\zeta)u\in\cG_2$ by (<ref>) since $\cG_2=\tr(\dom(\cA_2))$ is co-isotropic. Then $w:=R_1(\zeta)u\in\dom(\cA_1)\cap\dom(\cA_2)$ and thus $u=(A_0-\zeta)w$ yielding $\ker(R(\zeta))\subseteq\ran(A_0-z)$. This proves (i) in (<ref>). In particular, $\ran(A_0-\zeta)$ is closed and thus $\cH=\ran(A_0-\zeta)\oplus\ker(A_0^*-\overline{\zeta})$ where the sum is orthogonal. To show that $R(\zeta)u\in\ker(A_0^*-\zeta)=\big(\ran(A_0-\overline{\zeta})\big)^\perp$ for each $u\in\cH$, we pick any vector $v=(A_0-\overline{\zeta})w\in\ran(A_0-\overline{\zeta})$ with some $w\in\dom(A_0):=\dom(\cA_1)\cap\dom(\cA_2)$. As above,

\[R_2(\overline{\zeta})v=R_2(\overline{\zeta})(A_0-\overline{\zeta})w=R_2(\overline{\zeta})(\cA_2-\overline{\zeta})w=w\]

because $w\in\dom(\cA_2)$ and thus $\tr R_2(\overline{\zeta})v=\tr w\in\cG_1$ because  $w\in\dom(\cA_1)$. Since $\cG_1$ is isotropic, (<ref>) yields $\langle R(\zeta)u, v\rangle_{{}_\cH}=0$ and so
the inclusion $\ran(R(\zeta))\subseteq\ker(A_0^*-\zeta)$ in assertion (ii) of (<ref>) does hold. What we have proved so far shows that the finite dimensional operator

\begin{equation}\label{Risom}
\ker(A_0^*-\overline{\zeta})\to \ker(A_0^*-{\zeta})
\text{ is an isomorphism}
\end{equation}

as it is injective by
assertion (i) of (<ref>) and $\ker(A_0^*-\overline{\zeta})\cap\ran(A_0-\zeta)=\{0\}$. This implies assertion (ii)
and finishes the proof of  (<ref>).

The rest easily follows by representing the isomorphism in (<ref>) as an $(m\times m)$ matrix ${\mathbf r}(\zeta)$ using the bases $(u_k(\overline{\zeta}))_{1\le k\le m}$ and $(u_k(\zeta))_{1\le k\le m}$ in $\ker(A_0^*-\overline{\zeta})$ and $\ker(A_0^*-\zeta)$, respectively. Indeed, let ${\mathbf r}(\zeta)=\big(r_{lk}(\zeta)\big)_{1\le l,k\le m}$ be chosen such that $R(\zeta)u_k(\overline{\zeta})=\sum_{l=1}^mr_{lk}(\zeta)u_l(\zeta)$, and let
$G(\zeta)=\big(\langle u_k(\zeta), u_l(\zeta)\rangle_{{}_\cH}\big)_{1\le k,l\le m}$ denote the Gramm matrix so that $G(\zeta)^\top=\overline{G(z)}$ while its inverse will be written as $G(\zeta)^{-1}=\big(G^{-1}_{kj}(\zeta)\big)_{1\le k,j\le m}$. If $u=\sum_{k=1}^mc_ku_k(\overline{\zeta})\in\ker(A_0^*-\overline{\zeta})$ then $\overline{G(\overline{\zeta})}(c_k)_{1\le k\le m}=\big(\langle u, u_j(\overline{\zeta})\rangle_{{}_\cH}\big)_{1\le j\le m}$ as vectors  in $\bbC^m$ and therefore

\begin{equation}\begin{split}
R(\zeta)u&=\sum_{k=1}^mc_kR(\zeta)u_k(\overline{\zeta})=\sum_{k=1}^m\big(\sum_{j=1}^m\overline{G^{-1}_{kj}(\overline{\zeta})}\langle u, u_j(\overline{\zeta})\rangle_{{}_\cH}\big)\big(\sum_{l=1}^mr_{lk}(\zeta)u_l(\zeta)\big)\\
&=\sum_{l,j}\big(\sum_{k=1}^mr_{lk}(\zeta)\overline{G^{-1}_{kj}(\overline{\zeta})}\big)\langle u, u_j(\overline{\zeta})\rangle_{{}_\cH}u_l(\zeta).
\end{split}\end{equation}

We now define ${\mathcal P}(\zeta)=(p_{lj}(\zeta))_{1\le l,j\le m}$ by the formula ${\mathcal P}(\zeta):={\mathbf r}(\zeta)\overline{G(\overline{\zeta})^{-1}}$
and obtain equation
(<ref>)  for
$u\in\ker(A_0^*-\overline{\zeta})$. By $\cH=\ran(A_0-\zeta)\oplus\ker(A_0^*-\overline{\zeta})$ and assertion (i) in (<ref>)
it also holds for all $u\in\cH$.

§ DIRICHLET AND NEUMANN TRACE OPERATORS

In this appendix we recall  definitions and some facts about
various types of Dirichlet and Neumann trace operators which are discussed in detail in [52], [54].
Let $n\in\mathbb{N},n\geq2$, and  $\Om\subset\mathbb{R}^n$ be a bounded domain with $C^{1,r}$, $r>1/2$, boundary.

First, we define the strong trace operators.
Let us introduce the boundary trace operator $\gaD^0$ (the Dirichlet
trace) by

\begin{equation}\lb{2.4}
\gaD^0\colon C^0(\ol{\Om})\to C^0(\dOm), \quad
\gaD  ^0 u = u|_\dOm .
\end{equation}

By the standard trace theorem, see, e.g., <cit.>, there exists a bounded, surjective Dirichlet
trace operator

\begin{equation}
\gaD  \colon H^{s}(\Om)\to H^{s-1/2}(\dOm) \hookrightarrow
L^2(\dOm), \quad 1/2<s<3/2.
\lb{6.1}
\end{equation}

Next, retaining Hypothesis <ref>, we introduce the Neumann trace operator $\gaN$ by

\begin{equation}\lb{Nstrong}
\gaN=\nu\cdot\gaD\nabla  \colon H^{s+1}(\Om)\to L^2(\dOm), \quad 1/2<s<3/2,
\end{equation}

where $\nu$ denotes the outward pointing normal unit vector to $\dOm$.
Furthermore, one can extend $\gaN$ to the weak Neumann trace operator still denoted by $\gaN$ such that

\begin{equation}\lb{weaknt}
\gaN:\{u\in H^{1}(\Om)\,|\,\Delta u\in L^2(\Om)\}\to H^{-1/2}(\dOm).
\end{equation}

Assume Hypothesis <ref>. Then the Neumann trace operator $\gaN$ considered in the context

\begin{equation}\lb{gaN0}
\gaN:H^2(\Om)\cap H^1_0(\Om)\to H^{1/2}(\dOm),
\end{equation}

is well-defined, linear, bounded, onto, and with a linear bounded right-inverse.
In addition, the null space of $\gaN$ in (<ref>)
is $H^2_0(\Om)$, the closure of $C^{\infty}_0(\Om)$ in $H^2(\Om)$.

Assume Hypothesis <ref>. Then there exists a unique linear bounded operator

\begin{equation}
\gd:\{u\in L^{2}(\Om)\,|\,\Delta u\in L^{2}(\Om)\}\to H^{-1/2}(\dOm),
\end{equation}

which is compatible with the Dirichlet
trace introduced in (<ref>). This extension of the Dirichlet trace operator has dense range and allows for the following integration by
parts formula,

\begin{equation}
\lnoh \gaN w, \gd u\rnohs= (\Delta w,u)_{L^{2}(\Om)}-(w,\Delta u)_{L^{2}(\Om)},
\end{equation}

valid for every $u\in L^{2}(\Om)$ with $\Delta u\in L^{2}(\Om)$ and every $w\in H^{2}(\Om)\cap H^{1}_0(\Om)$.

Assume Hypothesis <ref>. Then there exists a unique linear bounded operator

\begin{equation}
\gn:\{u\in L^{2}(\Om)\,|\,\Delta u\in L^{2}(\Om)\}\to H^{-3/2}(\dOm),
\end{equation}

which is compatable with the Neumann
trace, introduced in (<ref>). This extension of the Neumann trace operator has dense range and allows for the following integration by parts formula,

\begin{equation}
{}_{H^{3/2}(\dOm)}\langle \gaD w, \gn u\rangle_{H^{-3/2}(\dOm)}= (w,\Delta u)_{L^{2}(\Om)}-(\Delta w,u)_{L^{2}(\Om)},
\end{equation}

valid for every $u\in L^{2}(\Om)$ with $\Delta u\in L^{2}(\Om)$ and every $w\in H^{2}(\Om)$ with $\gaN w=0$.

Next, we introduce the Dirichet-to-Neumann map $M_{D,N}$ associated with $-\Delta$ on $\Om$ as

\begin{equation}
H^{-1/2}(\dOm)\to H^{-3/2}(\dOm):
\end{equation}

where $u_D$ is the unique solution of the boundary value problem

\begin{equation}
-\Delta u=0\,\,\hbox{in}\,\,\Om,\,\,\,\,u\in L^{2}(\Om),\,\,\gd u=g\,\,\hbox{in}\,\,\dOm.
\end{equation}

Assume Hypothesis <ref>. Then the map

\begin{align}
\tN:\{u\in L^2(\Omega)| \Delta u\in L^2(\Omega)\}\rightarrow H^{1/2}(\partial\Omega),\,%\no,\\
\tN u:=\gn u+ M_{D,N}(\gd u),
\end{align}

is bounded when the space
$\{u\in L^2(\Omega)| \Delta u\in L^2(\Omega)\}=\dom(-\Delta_{\max})$ is equipped with the natural graph norm
$(\|u\|_{L^{2}(\Om)}^2+\|\Delta u\|_{L^{2}(\Om)}^2)^{1/2}$. Moreover, this operator is onto. In fact,

\begin{equation}\lb{onto}
\tN(H^2(\Om)\cap H^1_0(\Om))=H^{1/2}(\partial\Omega).
\end{equation}

Also, the null space of the map $\tN$ is given by

\begin{equation}\lb{kernel}
\ker(\tN)=H_0^2(\Omega)\dot{+}\{u\in L^2(\Om),\,\, -\Delta u=0 \}.
\end{equation}

Finally, the following Green formula holds for every $u,v\in\dom(-\Delta_{\max})$,

\begin{align}\lb{Green}
&(-\Delta u,v)_{L^{2}(\Om)}-(u,-\Delta v)_{L^{2}(\Om)}\no\\