# Resolvent expansions for self-adjoint operators via boundary triplets

Yuri Latushkin  Department of Mathematics, The University of Missouri,
Columbia, MO 65211, USA<EMAIL_ADDRESS>and  Selim Sukhtaiev
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849,
USA<EMAIL_ADDRESS>
###### Abstract.

In this paper we develop certain aspects of perturbation theory for self-
adjoint operators subject to small variations of their domains. We use the
abstract theory of boundary triplets to quantify such perturbations and give
the second order asymptotic analysis for resolvents, spectral projections,
discrete eigenvalues of the corresponding self-adjoint operators. In
particular, we derive explicit formulas for the first variation and the
Hessian of the eigenvalue curves bifurcating from a discrete eigenvalue of an
unperturbed operator. An application is given to a matrix valued Robin
Laplacian and more general Robin-type self-adjoint extensions.

This work naturally emerged from [19] and [5] as the result numerous
stimulating discussions with Prof. G. Berkolaiko. We thank him for generously
sharing his ideas as well as his notes on related topics. We wish to thank
Prof. P. Kuchment for pointing out many references. Y.L. was supported by the
NSF grants DMS-1710989 and DMS-2106157, and would like to thank the Courant
Institute of Mathematical Sciences and especially Prof. Lai-Sang Young for the
opportunity to visit CIMS

##  1\. Introduction

This paper concerns asymptotic perturbation theory for self-adjoint extensions
of symmetric operators. Our main goal is to derive explicit second order
Taylor expansions for resolvents, spectral projections, and eigenvalue curves
for one-parameter families of self-ajdoint extensions of a given symmetric
operator. The type of admissible perturbations covers, in particular,
variation of boundary conditions for quantum graph operators and, more
generally, variation of domain of self-adjointness for extensions of abstract
symmetric operators. Such non-additive perturbations arise, for example, in
the geometric spectral theory, in which case ad-hoc methods are used to
relate, say, the Hessian of parameter-dependent eigenvalues to the reduced
Dirichlet-to-Neumann map, see, for example, recent manuscripts [3], [4], [5],
[15]. We provide a complete operator theoretic treatment for the second order
expansion problem which, in particular, yields an abstract version of the
reduced Dirichlet-to-Neumann map and its relation to the Hessian of eigenvalue
curves. The first and second order expansion formulas go back to Hadamard,
Rayleigh, and Rellich and have been closely studied subsequently, cf. e.g. [1,
7, 14, 16, 17]

The first and second order asymptotic expansions discussed in this paper are
also central in the Maslov index theory, cf. [6, 23], for partial differential
operators on varying domains or subject to varying boundary conditions, a
subject that recently attracted much attention, cf. [8, 9, 10, 11, 19, 20] and
the bibliography therein. For example, Deng–Jones’s [11] foundational approach
to the spectral count via Maslov index for Robin Laplacian essentially reduces
to finding the sign of the derivative of the eigenvalue curves, see recent
papers [8, 20]. Further developments of this approach, see [10], led to
elaborate spectral flow formulas and provided new tools in geometric analysis
of Laplace eigenfunctions.

We stress that the perturbation theory developed here covers not merely
“additive” perturbations of a given self-adjoint operator but rather a
“structural” perturbation of the dense subspace on which the perturbed
operator is defined. More specifically, we focus on the second order Taylor
expansion for resolvents, spectral projections, and eigenvalue curves of
$C^{2}$-regular one parameter families of self-adjoint extensions of a given
symmetric operator in abstract Hilbert spaces. We use the abstract theory of
boundary triples and Krein–Naimark-type resolvent formulas, a subject having
rich and distinguished history [2, 13, 24]. Among applications is an explicit
formula for the Hessian and the first derivative of an eigenvalue curve of a
one dimensional Schrödinger operator with matrix-valued potential subject to
varying Robin boundary conditions on a bounded interval.

To give this work yet another perspective, let us mention some closely related
papers. In [9, 10], G. Cox, C. Jones, J. Marzuola utilized explicit formulas
for the first derivative of eigenvalue curves of parameter dependent Robin-
type elliptic operators to evaluate the spectral shift between the Dirichlet
and Neumann realizations in terms of the Morse index of the Dirichlet-to-
Neumann map, which yields a plethora of interesting insights in spectral
geometry, e.g., a novel formula for the number of nodal domains of Dirichlet
eigenfunctions, and a geometric proof Friedlander’s inequality, [12]. Two more
interesting connections are to a recent paper by G. Berkolaiko, P. Kuchment
[5], where the Hessian is computed for simple eigenvalues in the case of
“additive” perturbations, and to the work of G. Berkolaiko, Y. Canzani, G.
Cox, J. L. Marzuola [3], where the Hessian is computed for dispersion
relations (i.e. eigenvalue surfaces parameterized by quasimomenta which arise
in the Floquet-Bloch decomposition) of some periodic discrete graphs. It is
worth noting that in some of these works, e.g. in [3] the parameter is higher-
dimensional while we focus on one-parameter families. That said, we believe
that the methods developed in the present paper could be applicable in multi-
parameter setting.

This work should be considered as a sequel to [19] where a symplectic form of
the Krein resolvent formula was derived in order to establish the first order
expansion for resolvents and eigenvalues of self-adjoint extensions. We stress
that the abstract setting considered in [19] is significantly more general
than the one assumed in the current work but the expansion carried out in [19]
is only to the first order. The major difference is in the type of trace maps
used in [19] where they are assumed to be merely densely defined and with
dense range while in the current work the traces are bounded (in particular,
have full domain) and surjective. In short, in [19],
$(\mathfrak{H},\Gamma_{0},\Gamma_{1})$ is not required to be a classical
boundary triplet, [2, 13, 24], but it is in the current manuscript.

Let ${\mathcal{H}},\mathfrak{H}$ be complex Hilbert spaces. Let $A$ be a
densely defined, closed, symmetric operator acting in ${\mathcal{H}}$ and
having equal (possibly infinite) deficiency indices. Let
$(\Gamma_{0},\Gamma_{1},\mathfrak{H})$ be a boundary triplet associated with
$A$, see Hypothesis 2.1. It is well known that self-adjoint extensions of $A$
are parametrized by Lagrangian planes (also known as self-adjoint linear
relations) in $\mathfrak{H}\times\mathfrak{H}$, cf., e.g., [2, 13, 24], by
means of the identity
$[\Gamma_{0},\Gamma_{1}]^{\top}(\operatorname{dom}({\mathcal{A}}))={\mathcal{F}}$,
where $A\subset{\mathcal{A}}={\mathcal{A}}^{*}\subset A^{*}$ and
${\mathcal{F}}\subset\mathfrak{H}\times\mathfrak{H}$ is a Lagrangian plane.
Firstly, we show that for a $C^{2}$ path of Lagrangain planes
$t\mapsto{\mathcal{F}}_{t}$ and a family of self-adjoint extensions of $A$
with
$[\Gamma_{0},\Gamma_{1}]^{\top}(\operatorname{dom}({\mathcal{A}}_{t}))={\mathcal{F}}_{t}$,
the path $t\mapsto({\mathcal{A}}_{t}-{\bf i})^{-1}$ is also $C^{2}$-regular in
uniform operator topology. Our argument is based on a Krein-type formula from
[19], cf. (2.8), expressing resolvent difference in terms of the difference of
orthogonal projections onto the respective planes. Secondly, in our main
Theorems 2.5, 2.9, 2.10, we derive explicit expressions for the coefficients
of the second order Taylor expansions for the resolvents of $t-$dependent
operators ${\mathcal{A}}_{t}$, the operators per se (more precisely, their
conjugated restrictions onto spectral subspaces), and their eigenvalue curves.
Next, in Theorem 2.11 we specialized to the case when the Lagrangian planes
${\mathcal{F}}_{t}$ are given as kernels of the $1\times 2$ block operator
matrices $Z_{t}=[X_{t},Y_{t}]$ in $\mathfrak{H}\times\mathfrak{H}$, and derive
explicit formulas for the first and second derivatives of the eigenvalues of
${\mathcal{A}}_{t}$ in terms of $Z_{t}$ and its derivatives. In particular,
Corollary 2.12 gives an expansion of the eigenvalue curves for the abstract
Robin-type self-adjoint extensions of $A$ up to the second order and
demonstrates that the Hessian in this case features a remarkable similarity
with that derived by G. Berkolaiko and P. Kuchement in [5, Eq. 40] for
additive perturbations of a fixed self-adjoint operator with a simple
eigenvalue.

##  2\. Resolvent expansions

Throughout, $A$ denotes a densely defined, symmetric operator acting in a
complex Hilbert space ${\mathcal{H}}$ and having equal deficiency indices,
i.e., $\dim\operatorname{ker}(A^{*}-{\bf i})=\dim\operatorname{ker}(A^{*}+{\bf
i})\leq\infty$. We equip the vector space
${\mathcal{H}}_{+}=\operatorname{dom}(A^{*})$ with the graph scalar product

$\langle u,v\rangle_{{\mathcal{H}}_{+}}:=\langle
u,v\rangle_{{\mathcal{H}}}+\langle A^{*}u,A^{*}u\rangle_{{\mathcal{H}}},\
u,v\in\operatorname{dom}(A^{*}).$ (2.1)

Let ${\mathcal{H}}_{-}:=({\mathcal{H}}_{+})^{*}$ be the dual to
${\mathcal{H}}_{+}$ space consisting of linear bounded functionals. We notice
a chain of natural embeddings,
${\mathcal{H}}_{+}\hookrightarrow{\mathcal{H}}\hookrightarrow{\mathcal{H}}_{-}$,
where the first embedding is given by ${\mathcal{H}}_{+}\ni u\mapsto
u\in{\mathcal{H}}$, and the second embedding is given by ${\mathcal{H}}\ni
v\mapsto\langle\cdot,v\rangle_{{\mathcal{H}}}$. Let
$\Phi^{-1}:{\mathcal{H}}_{+}\to{\mathcal{H}}_{-}$ be the conjugate linear
Riesz isomorphism such that

${}_{{\mathcal{H}}_{+}}\langle u,\Phi^{-1}w\rangle_{{\mathcal{H}}_{-}}=\langle
u,w\rangle_{{\mathcal{H}}_{+}}=\langle u,w\rangle_{{\mathcal{H}}}+\langle
A^{*}u,A^{*}w\rangle_{{\mathcal{H}}},u,w\in{\mathcal{H}}_{+}.$

Next, we fix the standard setting of boundary triplets, see [2, 24] for more
details, the history of the subject and a vast bibliography. Below, we write
$[a,b]^{\top}$ for a $2\times 1$ column vector with entries $a,b$.

######  Hypothesis 2.1.

Assume that $(\Gamma_{0},\Gamma_{1},\mathfrak{H})$ is a boundary triplet. That
is, we assume that we are given an auxiliary (boundary) Hilbert space
$\mathfrak{H}$ and a trace operator

$\mathrm{T}:=[\Gamma_{0},\Gamma_{1}]^{\top}\in{\mathcal{B}}({\mathcal{H}}_{+},\mathfrak{H}\times\mathfrak{H}),$

which is bounded, surjective and satisfies the following abstract Green
identity,

$\langle A^{*}u,v\rangle_{{\mathcal{H}}}-\langle
u,A^{*}v\rangle_{{\mathcal{H}}}=\langle\Gamma_{1}u,\Gamma_{0}v\rangle_{\mathfrak{H}}-\langle\Gamma_{0}u,\Gamma_{1}v\rangle_{\mathfrak{H}}\text{
for all $u,v\in{\mathcal{H}}_{+}$}.$ (2.2)

Let us introduced a bounded symplectic form $\omega$ defined by the formulas

$\displaystyle\begin{split}&\omega\big{(}(f_{1},f_{2})^{\top},(g_{1},g_{2})^{\top}\big{)}:=\langle
f_{2},g_{1}\rangle_{\mathfrak{H}}-\langle f_{1},g_{2}\rangle_{\mathfrak{H}}\\\
&\quad=\left\langle
J(f_{1},f_{2})^{\top},(g_{1},g_{2})^{\top}\right\rangle_{\mathfrak{H}\times\mathfrak{H}},\
J:=\begin{bmatrix}0&I_{\mathfrak{H}}\\\
-I_{\mathfrak{H}}&0\end{bmatrix},f_{k},g_{k}\in\mathfrak{H},k=1,2,\end{split}$
(2.3)

so that the right-hand side of (2.2) becomes
$\omega(\mathrm{T}u,\mathrm{T}v)$. We denote the annihilator of a subspace
${\mathcal{F}}\subset\mathfrak{H}\times\mathfrak{H}$ by

${\mathcal{F}}^{\circ}:=\\{(f_{1},f_{2})^{\top}\in\mathfrak{H}\times\mathfrak{H}:\omega\big{(}(f_{1},f_{2})^{\top},(g_{1},g_{2})^{\top}\big{)}=0\text{\
for all}\ (g_{1},g_{2})^{\top}\in{\mathcal{F}}\\},$ (2.4)

and recall that the subspace ${\mathcal{F}}$ is called Lagrangian if
${\mathcal{F}}={\mathcal{F}}^{\circ}$. Further, let
$\Lambda(\mathfrak{H}\times\mathfrak{H})$ denote the metric space of
Lagrangian subspaces of $\mathfrak{H}\times\mathfrak{H}$ equipped with the
metric

$d({\mathcal{F}}_{1},{\mathcal{F}}_{2}):=\|Q_{1}-Q_{2}\|_{{\mathcal{B}}(\mathfrak{H}\times\mathfrak{H})},\
{\mathcal{F}}_{1},{\mathcal{F}}_{2}\in\Lambda(\mathfrak{H}\times\mathfrak{H}),$
(2.5)

where $Q_{j}$ is the orthogonal projection onto ${\mathcal{F}}_{j}$ acting in
$\mathfrak{H}\times\mathfrak{H}$, $j=1,2$. Let us also recall the classical,
and central for this paper, fact that the Lagrangian planes in
$\Lambda(\mathfrak{H}\times\mathfrak{H})$ are in one-to-one correspondence
with self-adjoint extensions of $A$, see [24]. That is, for every self-adjoint
extension ${\mathcal{A}}$ of $A$ the subspace
$\mathrm{T}(\operatorname{dom}({\mathcal{A}}))\subset\mathfrak{H}\times\mathfrak{H}$
is Lagrangian with respect to $\omega$, and, conversely, every Lagrangian
plane ${\mathcal{F}}\in\Lambda(\mathfrak{H}\times\mathfrak{H})$ gives rise to
a self-adjoint extension of $A$ with domain
$\mathrm{T}^{-1}({\mathcal{F}})\subset\operatorname{dom}(A^{*})$, see [19] and
the references therein for more details.

In this paper we are interested in explicit second order Taylor expansions for
resolvents and eigenvalues of a family of self-adjoint extensions
${\mathcal{A}}_{t}$ of $A$ defined by a given family of Lagrangian planes
$t\mapsto{\mathcal{F}}_{t}$ for $t$ near some $t_{0}$. We note that the
regularity of a path of Lagrangian planes ${\mathcal{F}}_{t}$ is determined by
the regularity of the path $t\mapsto Q_{t}$ of orthogonal projections $Q_{t}$
acting in $\mathfrak{H}\times\mathfrak{H}$ with
$\text{\rm{ran}}(Q_{t})={\mathcal{F}}_{t}$, in other words, the topology on
$\Lambda(\mathfrak{H}\times\mathfrak{H})$ is determined by the metric in
(2.5). Throughout, we use “dot” to denote $t$-derivative so that, in
particular, $\dot{Q}_{t_{0}}=\frac{\rm d}{{\rm d}t}Q_{t}\big{|}_{t=t_{0}}$.
Let us now record our assumptions.

######  Hypothesis 2.2.

Let $Q:[0,1]\rightarrow{\mathcal{B}}(\mathfrak{H}\times\mathfrak{H}),t\mapsto
Q_{t}$ be a one-parameter family of orthogonal projections and assume that
$\text{\rm{ran}}(Q_{t})\in\Lambda(\mathfrak{H}\times\mathfrak{H})$ is a
Lagrangian plane. Fix $t_{0}\in[0,1]$ and assume that

$\displaystyle\begin{split}&Q_{t}\underset{t\rightarrow
t_{0}}{=}Q_{t_{0}}+\dot{Q}_{t_{0}}(t-t_{0})+\frac{1}{2}\ddot{Q}_{t_{0}}(t-t_{0})^{2}+o(t-t_{0})^{2}\text{\
in\ }{\mathcal{B}}(\mathfrak{H}\times\mathfrak{H}).\end{split}$ (2.6)

Let ${\mathcal{A}}_{t}$ be a family of self-adjoint extensions of $A$
satisfying

$\displaystyle{\mathrm{T}\big{(}\operatorname{dom}({\mathcal{A}}_{t})\big{)}}=\text{\rm{ran}}(Q_{t}).$
(2.7)

Finally, denote
$R_{t}(\zeta):=({\mathcal{A}}_{t}-\zeta)^{-1}\in{\mathcal{B}}({\mathcal{H}})$
for $\zeta\not\in\operatorname{Spec}({\mathcal{A}}_{t})$.

We recall from [19, Theorem 3.11] the following Krein-type resolvent formula.

######  Theorem 2.3.

Assume Hypothesis 2.2 and let $t,t_{0}\in[0,1],$
$\zeta\not\in\operatorname{Spec}({\mathcal{A}}_{t})\cup\operatorname{Spec}({\mathcal{A}}_{t_{0}})$.
Then one has

$\displaystyle
R_{t}(\zeta)-R_{t_{0}}(\zeta)=(\mathrm{T}R_{t}(\zeta))^{*}(Q_{t}-Q_{t_{0}})JQ_{t_{0}}\mathrm{T}R_{t_{0}}(\zeta).$
(2.8)

######  Remark 2.4.

1) Due to Lagrangian properties of $Q_{t}$ we have $Q_{t_{0}}JQ_{t_{0}}=0$; in
addition, since
$\mathrm{T}(\text{\rm{ran}}(R_{t_{0}}(\zeta)))=\mathrm{T}(\operatorname{dom}({\mathcal{A}}_{t_{0}}))=\text{\rm{ran}}(Q_{t_{0}})$,
we have $Q_{t_{0}}\mathrm{T}R_{t_{0}}(\zeta)=\mathrm{T}R_{t_{0}}(\zeta)$ and
formula (2.8) can be written as follows,

$\displaystyle
R_{t}(\zeta)-R_{t_{0}}(\zeta)=(\mathrm{T}R_{t}(\zeta))^{*}Q_{t}J\mathrm{T}R_{t_{0}}(\zeta).$
(2.9)

2) The adjoint operator $(\mathrm{T}R_{t}(\zeta))^{*}$ in the right-hand side
of (2.8) is a rather subtle object. It should be viewed as an operator in
${\mathcal{B}}(\mathfrak{H}\times\mathfrak{H},{\mathcal{H}})$. Indeed, one has
$\mathrm{T}\in{\mathcal{B}}({\mathcal{H}}_{+},\mathfrak{H}\times\mathfrak{H})$
and $\text{\rm{ran}}(R_{t}(\zeta))\subset{\mathcal{H}}_{+}$, so that the
product $\mathrm{T}(R_{t}(\zeta))$ is well defined as a mapping from
${\mathcal{H}}$ to $\mathfrak{H}\times\mathfrak{H}$. Moreover, we claim that
$R_{t}(\zeta)\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$, hence, in
particular
$\mathrm{T}(R_{t}(\zeta))\in{\mathcal{B}}({\mathcal{H}},\mathfrak{H}\times\mathfrak{H})$
so its adjoint is indeed in
${\mathcal{B}}(\mathfrak{H}\times\mathfrak{H},{\mathcal{H}})$ as claimed
above. To check the inclusion
$R_{t}(\zeta)\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$ we notice that
for some $c>0$ and every $u\in{\mathcal{H}}$ one has

$\displaystyle\|R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}_{+}}$
$\displaystyle=\|R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}+\|A^{*}R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}$
(2.10)
$\displaystyle\leq\|R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}+2\|({\mathcal{A}}_{t_{0}}-\zeta)R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}+2\|\zeta
R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}$ (2.11)
$\displaystyle\leq\|R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}+2\|u\|^{2}_{{\mathcal{H}}}+2\|\zeta
R_{t_{0}}(\zeta)u\|^{2}_{{\mathcal{H}}}\leq c\|u\|^{2}_{{\mathcal{H}}}.$
(2.12)

3) It is possible to represent $(\mathrm{T}R_{t}(\zeta))^{*}$ as the product
of two adjoints, $(R_{t}(\zeta))^{*}$ and $\mathrm{T}^{*}$. In fact, the
adjoint of the resolvent
$(R_{t}({\zeta}))^{*}\in{\mathcal{B}}({\mathcal{H}}_{-},{\mathcal{H}})$ is a
natural extension of $R_{t}(\overline{\zeta})$, originally defied as a mapping
from ${\mathcal{H}}$ to itself, to a mapping from ${\mathcal{H}}_{-}$ to
${\mathcal{H}}$, see [19, Proposition 2.4]. In this work, however, we chose
not to unfold the product $(\mathrm{T}R_{t}(\zeta))^{*}$ and shall view
$R_{t}(\zeta)$ as an element of
${\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$ whenever it precedes
$\mathrm{T}$.

4) From the outset we only have
$\text{\rm{ran}}((\mathrm{T}R_{t}(\zeta))^{*})\subset{\mathcal{H}}$. However,
(2.9) suggests that
$\text{\rm{ran}}((\mathrm{T}R_{t}(\zeta))^{*})\subset{\mathcal{H}}_{+}$ as
both resolvents in the left-hand side of (2.9) have ranges in
${\mathcal{H}}_{+}$. It turns out, cf. Theorem 2.5 (3), that Krein’s-type
formula (2.9) produces a similar effect for the derivative
$\dot{R}_{t_{0}}(\zeta)$ that satisfies
$\text{\rm{ran}}(\dot{R}_{t_{0}}(\zeta))\subset{\mathcal{H}}_{+}$ although a
priori one only has
$\text{\rm{ran}}(\dot{R}_{t_{0}}(\zeta))\subset{\mathcal{H}}$.

Our first principal result gives Taylor expansions for the resolvent of
$R_{t}(\zeta)$

######  Theorem 2.5.

Assume Hypothesis 2.2. Suppose that
$\zeta_{0}\not\in\operatorname{Spec}({\mathcal{A}}_{t_{0}})$ for some
$t_{0}\in[0,1]$ and $\zeta_{0}\in{\mathbb{C}}$, and define

$\displaystyle{\mathcal{U}}_{\varepsilon}=\\{(t,\zeta)\in[0,1]\times{\mathbb{C}}:|t-t_{0}|\leq\epsilon,|\zeta-\zeta_{0}|\leq\epsilon\\}\text{
for $\epsilon>0$}.$ (2.13)

  1. (1)

There exists an $\varepsilon>0$ such that if
$(t,\zeta)\in{\mathcal{U}}_{\varepsilon}$ then
$\zeta\not\in\operatorname{Spec}({\mathcal{A}}_{t})$.

  2. (2)

For $\varepsilon>0$, $(t,\zeta)\in{\mathcal{U}}_{\varepsilon}$ as in part (1)
above one has

$(\mathrm{T}R_{t}(\zeta))^{*}(Q_{t}-Q_{t_{0}})JQ_{t_{0}}\mathrm{T}R_{t_{0}}(\zeta)\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+}).$
(2.14)

  3. (3)

For $\varepsilon>0$, $\zeta\in{\mathbb{C}}$ as in part (1) above we define

$\displaystyle\dot{R}_{t_{0}}(\zeta):=(\mathrm{T}R_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}R_{t_{0}}(\zeta).$
(2.15)

Then one has

$\displaystyle\dot{R}_{t_{0}}(\zeta)\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+}).$
(2.16)

Moreover, the following asymptotic expansion holds uniformly for
$|\zeta-\zeta_{0}|<\varepsilon$,

$\displaystyle
R_{t}(\zeta)=R_{t_{0}}(\zeta)+\dot{R}_{t_{0}}(\zeta)(t-t_{0})+o(t-t_{0})\text{\
in\ }{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+}).$ (2.17)

  4. (4)

For $\varepsilon>0$ as in part (1) the following asymptotic expansion holds
uniformly for $|\zeta-\zeta_{0}|<\varepsilon$,

$\displaystyle R_{t}(\zeta)\underset{t\rightarrow
t_{0}}{=}R_{t_{0}}(\zeta)+\dot{R}_{t_{0}}(\zeta)(t-t_{0})+\frac{1}{2}\ddot{R}_{t_{0}}(\zeta)(t-t_{0})^{2}+o(t-t_{0})^{2}\
\text{\ in\ }{\mathcal{B}}({\mathcal{H}}),$ (2.18)

where $\dot{R}_{t_{0}}(\zeta)$ is given by (2.15) and

$\displaystyle\ddot{R}_{t_{0}}(\zeta):=$
$\displaystyle(\mathrm{T}R_{t_{0}})^{*}\ddot{Q}_{t_{0}}JTR_{t_{0}}+2(\mathrm{T}\dot{R}_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}R_{t_{0}}(\zeta).$
(2.19)

###### Proof.

Proof of (1). First, we show that for $\zeta={\bf i}$ we have

$\|R_{t}({\bf
i})\|_{{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})}\underset{t\rightarrow
t_{0}}{=}{\mathcal{O}}(1).$ (2.20)

Of course, such asymptotic formula holds in ${\mathcal{B}}({\mathcal{H}})$ by
self-adjointness of ${\mathcal{A}}_{t}$. A stronger version which we are after
holds due to the fact that $R_{t}({\bf i})$ is the resolvent of a self-adjoint
restriction of $A^{*}$; this observation is crucial throughout the proof.

Since $\|R_{t}({\bf i})\|_{{\mathcal{B}}({\mathcal{H}})}\leq 1$ there exists
$c>0$ such that for all $u\in{\mathcal{H}}_{+}$ and $t$ near $t_{0}$ one has

$\displaystyle\begin{split}\|R_{t}({\bf
i})u\|_{{\mathcal{H}}_{+}}^{2}&=\|R_{t}({\bf
i})u\|_{{\mathcal{H}}}^{2}+\|A^{*}R_{t}({\bf i})u\|_{{\mathcal{H}}}^{2}\\\
&=\|R_{t}({\bf i})u\|_{{\mathcal{H}}}^{2}+2\|(A_{t}-{\bf i})R_{t}({\bf
i})u\|_{{\mathcal{H}}}^{2}+2\|{\bf i}R_{t}({\bf i})u\|_{{\mathcal{H}}}^{2}\leq
c\|u\|_{{\mathcal{H}}}^{2}\end{split}$ (2.21)

which confirms (2.20) (we used $({\mathcal{A}}_{t}-{\bf i})R_{t}({\bf i})=I$).
Next, by Krein’s formula (2.9) with $\zeta={\bf i}$, we have

$R_{t}({\bf i})=R_{t_{0}}({\bf i})+(\mathrm{T}R_{t}({\bf
i}))^{*}(Q_{t}-Q_{t_{0}})J\mathrm{T}R_{t_{0}}({\bf i}).$

Combining this with $\|\mathrm{T}R_{t}({\bf
i})\|_{{\mathcal{B}}({\mathcal{H}},\mathfrak{H}\times\mathfrak{H})}\underset{t\rightarrow
t_{0}}{=}{\mathcal{O}}(1)$ we get

$\|R_{t}({\bf i})-R_{t_{0}}({\bf
i})\|_{{\mathcal{B}}({\mathcal{H}})}\underset{t\rightarrow
t_{0}}{=}{\mathcal{O}}(t-t_{0}),$ (2.22)

so that ${\mathcal{A}}_{t}\rightarrow{\mathcal{A}}_{t_{0}}$, $t\rightarrow
t_{0}$ in the norm resolvent sense. Then [21, Theorem VIII.23(a)] yields the
assertion.

Proof of (2). By Krein’s formula the operator in (2.14) is equal to the
difference of the resolvents of two self-adjoint restrictions of $A^{*}$, each
of which is an element of ${\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$ as
discussed in Remark 2.4 part 2).

Proof of (3). We begin by obtaining an improved version of (2.22) given by

$\|R_{t}({\bf i})-R_{t_{0}}({\bf
i})\|_{{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})}\underset{t\rightarrow
t_{0}}{=}{\mathcal{O}}(t-t_{0}).$ (2.23)

To that end we write

$\displaystyle\|R_{t}({\bf i})u-R_{t_{0}}({\bf
i})u\|_{{\mathcal{H}}_{+}}^{2}=\|R_{t}({\bf i})u-R_{t_{0}}({\bf
i})u\|_{{\mathcal{H}}}^{2}+\|A^{*}R_{t}({\bf i})u-A^{*}R_{t_{0}}({\bf
i})u\|^{2}_{{\mathcal{H}}}$ (2.24) $\displaystyle\quad\leq\|R_{t}({\bf
i})u-R_{t_{0}}({\bf i})u\|_{{\mathcal{H}}}^{2}+2\|({\mathcal{A}}_{t}-{\bf
i})R_{t}({\bf i})u-({\mathcal{A}}_{t_{0}}-{\bf i})R_{t_{0}}({\bf
i})u\|^{2}_{{\mathcal{H}}}$ (2.25) $\displaystyle\hskip 145.10922pt+2\|-{\bf
i}R_{t}({\bf i})u+{\bf i}R_{t_{0}}({\bf i})u\|^{2}_{{\mathcal{H}}}$ (2.26)
$\displaystyle\quad\leq 3\|R_{t}({\bf i})u-R_{t_{0}}({\bf
i})u\|_{{\mathcal{H}}}^{2}\underset{t\rightarrow
t_{0}}{=}{\mathcal{O}}(t-t_{0}),\text{\ uniformly for\
}\|u\|_{{\mathcal{H}}}\leq 1.$ (2.27)

We will need a slightly more general version of (2.23) in which ${\bf i}$ is
replaced by $\zeta\in{\mathbb{C}}$ near $\zeta_{0}$. Let us fix
$\varepsilon_{0}>0$ such that
$\mathbb{B}_{\varepsilon_{0}}(\zeta_{0})\subset{\mathbb{C}}\setminus\operatorname{Spec}({\mathcal{A}}_{t_{0}})$
for the $\varepsilon_{0}$-disc centered at $\zeta_{0}$. Then by (2.22) and
[21, Theorem VIII.23] we have
$\mathbb{B}_{\varepsilon_{0}}(\zeta_{0})\cap\operatorname{Spec}({\mathcal{A}}_{t})=\emptyset$
for $t$ sufficiently close to $t_{0}$. Hence,

$\sup\\{\|R_{t}(\zeta)\|_{{\mathcal{B}}({\mathcal{H}})}:(t,\zeta)\in{\mathcal{U}}_{\varepsilon}\\}<\infty$
(2.28)

for a sufficiently small $\varepsilon\in(0,\varepsilon_{0})$. Then combining
the resolvent identity

$R_{t}(\zeta)=R_{t}({\bf i})-({\bf i}-\zeta)R_{t}({\bf i})R_{t}(\zeta),$
(2.29)

and (2.28) we infer that the asymptotic formula

$\displaystyle\|R_{t}(\zeta)-R_{t_{0}}(\zeta)\|_{{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})}\underset{t\rightarrow
t_{0}}{=}{\mathcal{O}}(t-t_{0}),$ (2.30)

holds uniformly for $|\zeta-\zeta_{0}|<\varepsilon$.

We are now ready to turn to the proof of (2.16), (2.17). To that end we will
first establish the same expansion but in the space
${\mathcal{B}}({\mathcal{H}})$ possessing weaker topology. Employing Krein’s
formula (2.4) ones again we obtain

$\displaystyle\begin{split}&R_{t}(\zeta)\underset{t\rightarrow
t_{0}}{=}R_{t_{0}}(\zeta)+\big{(}TR_{t_{0}}(\zeta)+{\mathcal{O}}_{{\mathcal{B}}({\mathcal{H}},\mathfrak{H}\times\mathfrak{H})}(t-t_{0})\big{)}^{*}\big{(}\dot{Q}_{t_{0}}(t-t_{0})+o(t-t_{0})\big{)}J\mathrm{T}R_{t_{0}}(\zeta)\\\
&=R_{t_{0}}(\zeta)+\dot{R}_{t_{0}}(\zeta)(t-t_{0})+o(t-t_{0})\text{\ in\
}{\mathcal{B}}({\mathcal{H}}).\end{split}$ (2.31)

Our next objective is to derive this asymptotic expansion in
${\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$ and to prove (2.16). To that
end we pick any sequence $t_{n}\rightarrow t_{0}$, $n\rightarrow\infty$ and
introduce a sequence of operators

$\displaystyle{\mathcal{D}}_{n}(\zeta):=\frac{R_{t_{n}}(\zeta)-R_{t_{0}}(\zeta)}{{t_{n}}-t_{0}},n\in{\mathbb{N}}.$
(2.32)

First we notice that
$\text{\rm{ran}}({\mathcal{D}}_{n})\subset\operatorname{dom}(A^{*})={\mathcal{H}}_{+}$
and, in fact, by Remark 2.4 (2) one has
${\mathcal{D}}_{n}\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$. We claim
that the sequence $\\{{\mathcal{D}}_{n}(\zeta)\\}$ is Cauchy in
${\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$. Let us observe that

$\displaystyle
A^{*}(R_{t_{n}}(\zeta)-R_{t_{m}}(\zeta))=\zeta(R_{t_{m}}(\zeta)-R_{t_{n}}(\zeta)),$
(2.33)

that is
$A^{*}({\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta))=\zeta({\mathcal{D}}_{m}(\zeta)-{\mathcal{D}}_{n}(\zeta))$,
so for arbitrary $u\in{\mathcal{H}}$ we have

$\displaystyle\|({\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta))u\|_{{\mathcal{H}}_{+}}^{2}=\left\|({\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta))u\right\|^{2}_{{\mathcal{H}}}+\left\|A^{*}({\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta))u\right\|^{2}_{{\mathcal{H}}}$
(2.34)
$\displaystyle=(1+|\zeta|^{2})\|({\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta))u\|^{2}_{{\mathcal{H}}},$
(2.35)

thus, one has

$\displaystyle\|{\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta)\|_{{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})}^{2}\leq(1+|\zeta|^{2})\|{\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{m}(\zeta)\|^{2}_{{\mathcal{B}}({\mathcal{H}})}\underset{m,n\rightarrow\infty}{\rightarrow}0,$
(2.36)

where we used that $\\{{\mathcal{D}}_{n}\\}_{n\in{\mathbb{N}}}$ is Cauchy in
${\mathcal{B}}({\mathcal{H}})$ by (LABEL:frstord), (2.22). Thus, the sequence
${\mathcal{D}}_{n}$ converges in
${\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$ to an operator
$D_{\infty}\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$. By definition
of $\|\cdot\|_{{\mathcal{H}}_{+}}$, we automatically also infer

$\|{\mathcal{D}}_{n}(\zeta)-{\mathcal{D}}_{\infty}(\zeta)\|_{{\mathcal{B}}({\mathcal{H}})}\rightarrow
0,n\rightarrow\infty.$ (2.37)

At the same time, (LABEL:frstord) show that

$\|{\mathcal{D}}_{n}(\zeta)-\dot{R}_{t_{0}}(\zeta)\|_{{\mathcal{B}}({\mathcal{H}})}\rightarrow
0,n\rightarrow\infty.$ (2.38)

Hence
$\dot{R}_{t_{0}}(\zeta)={\mathcal{D}}_{\infty}(\zeta)\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$
as asserted in(2.16). Also, ${\mathcal{D}}_{\infty}(\zeta)$ is independent of
the choice of $t_{n}\rightarrow t_{0}$. Thus

$R_{t}(\zeta)-R_{t_{0}}(\zeta)={\mathcal{D}}_{\infty}(\zeta)(t-t_{0})+o(t-t_{0}),\
t\rightarrow t_{0},$ (2.39)

and (2.17) holds.

Proof of (4). The asymptotic formula (2.18) is obtained by substituting
resolvent expansion (2.17) into Krein’s formula (2.8) and grouping powers of
$(t-t_{0})$. This formal expansion is indeed justified since the operator
$\mathrm{T}\dot{R}_{t_{0}}(\zeta)$ is well defined and, by (2.17), we have

$\displaystyle\mathrm{T}R_{t}(\zeta)=\mathrm{T}R_{t_{0}}(\zeta)+\mathrm{T}\dot{R}_{t_{0}}(\zeta)(t-t_{0})+o(t-t_{0})\text{\
in\ }{\mathcal{B}}({\mathcal{H}},\mathfrak{H}\times\mathfrak{H}).$ (2.40)

∎

In what follows we will describe how a multiple isolated eigenvalue $\lambda$
of ${\mathcal{A}}_{t_{0}}$ bifurcates for $t$ near $t_{0}$.

######  Hypothesis 2.6.

Fix $t_{0}\in[0,1]$, suppose that $\lambda=\lambda(t_{0})$ is an isolated
eigenvalue of ${\mathcal{A}}_{t_{0}}$ with finite multiplicity
$m\in{\mathbb{N}}$. Let

$\gamma:=\big{\\{}z\in{\mathbb{C}}:2|z-\lambda|=\operatorname{dist}\big{(}\lambda,\operatorname{Spec}({\mathcal{A}}_{t_{0}})\setminus\\{\lambda\\}\big{)}\big{\\}}.$

We continue to assume Hypothesis 2.2. Then, by Theorem 2.5, there exists
$\varepsilon>0$ such that $\gamma$ encloses the $\lambda$-group of $m$
eigenvalues (not necessarily distinct) of the operator ${\mathcal{A}}_{t}$
whenever $|t-t_{0}|<\varepsilon$ and $\varepsilon>0$ is sufficiently small.
For such $t$ we let $P_{t}$ denote the Riesz projection for
${\mathcal{A}}_{t}$ corresponding to the $\lambda$-group,

$P_{t}:=\frac{-1}{2\pi{\bf i}}\int_{\gamma}R_{t}(\zeta)d\zeta,\,\text{ where
$R_{t}(\zeta)=({\mathcal{A}}_{t}-\zeta)^{-1}$}.$ (2.41)

Let us recall from [18, Section III.6.5] the following Laurent expansion for
the resolvent,

$R_{t_{0}}(\zeta)=(\lambda-\zeta)^{-1}P_{t_{0}}+\sum_{n=0}^{\infty}(\zeta-\lambda)^{n}S^{n+1}_{t_{0}},$
(2.42)

where
$S_{t_{0}}=\lim_{\zeta\to\lambda}R_{t_{0}}(\zeta)(I_{\mathcal{H}}-P_{t_{0}})$
is the reduced resolvent of the operator ${\mathcal{A}}_{t_{0}}$ at the
eigenvalue $\lambda$ given by the formula

$S_{t_{0}}:=\frac{-1}{2\pi{\bf
i}}\int_{\gamma}(\lambda-\zeta)^{-1}R_{t_{0}}(\zeta)d\zeta.$ (2.43)

######  Remark 2.7.

We note that by Theorem 2.5, part (3) one has

$(\mathrm{T}R_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}R_{t_{0}}(\zeta)\in{\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+}).$
(2.44)

Therefore, in particular

$\displaystyle\mathrm{T}(\mathrm{T}R_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}R_{t_{0}}(\zeta)\in{\mathcal{B}}({\mathcal{H}},\mathfrak{H}\times\mathfrak{H}).$
(2.45)

Furthermore, since the integrals in (2.41) and (2.43) converge with respect to
the norm of the space ${\mathcal{B}}({\mathcal{H}},{\mathcal{H}}_{+})$, the
two inclusions above also hold with the two copies of $R_{t_{0}}(\zeta)$
replaced by $S_{t_{0}}^{2}$, $S_{t_{0}}$, $P_{t_{0}}$ in any combination. In
particular, the operators in (2.47), (2.48), (2.51), (2.52), (2.93), and in
Remark 2.14 used below are all well defined and bounded in the respective
spaces.

The following result gives the second order Taylor expansion for the Riesz
projection just introduced.

######  Proposition 2.8.

Under Hypotheses 2.2 and 2.6, for the Riesz projection introduced in (2.41)
one has

$\displaystyle P_{t}\underset{t\rightarrow
t_{0}}{=}P_{t_{0}}+\dot{P}_{t_{0}}(t-t_{0})+\frac{1}{2}\ddot{P}_{t_{0}}(t-t_{0})^{2}+o(t-t_{0})^{2},\
\text{\ in\ }{\mathcal{B}}({\mathcal{H}}),$ (2.46)

with

$\displaystyle\dot{P}_{t_{0}}$
$\displaystyle:=(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}+(\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}},$
(2.47)

and

$\displaystyle\begin{split}\ddot{P}_{t_{0}}:=&(\mathrm{T}S_{t_{0}})^{*}\ddot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}+(\mathrm{T}P_{t_{0}})^{*}\ddot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}\\\
&+2\big{(}\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}+2\big{(}\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}\\\
&+2\big{(}\mathrm{T}(\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}-2\big{(}\mathrm{T}(\mathrm{T}S_{t_{0}}^{2})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\\\
&-2\big{(}\mathrm{T}(\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}^{2}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}-2\big{(}\mathrm{T}(\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}^{2}.\end{split}$
(2.48)

###### Proof.

We plug (2.42) in (2.15), (2.19) and observe that the coefficients of the
terms containing $(\lambda-\zeta)^{-1}$ in the Laurent expansions of
$\dot{R}_{t_{0}}(\zeta)$, $\ddot{R}_{t_{0}}(\zeta)$ are given by the right-
hand sides of (2.47), (2.48) respectively. Upon using the right hand sides of
(2.47), (2.48), in the Laurent expansions of $\dot{R}_{t_{0}}(\zeta)$,
$\ddot{R}_{t_{0}}(\zeta)$, we multiply both sides of (2.18) by
$\frac{-1}{2\pi{\bf i}}$, integrate over $\gamma$, and employ (2.41) together
with Cauchy’s integral formula to arrive at (2.46). In other words,
$\dot{P}_{t_{0}}$, $\ddot{P}_{t_{0}}$ are residues of the operator-valued
functions $\zeta\mapsto-\dot{R}_{t_{0}}(\zeta)$,
$\zeta\mapsto-\ddot{R}_{t_{0}}(\zeta)$; the residues are given by (2.47),
(2.48). ∎

Our next objective is to derive an asymptotic expansion for the operator that
is similar to $P_{t}{\mathcal{A}}_{t}P_{t}$ for $t$ near $t_{0}$ but acts on a
$t$-independent $m-$dimensional space $\text{\rm{ran}}(P_{t_{0}})$. To that
end, we introduce the operator $U_{t}$ mapping $\text{\rm{ran}}(P_{t_{0}})$
onto $\text{\rm{ran}}(P_{t})$ unitarily, for $t$ near $t_{0}$, and defined by
the formulas

$\displaystyle\begin{split}&U_{t}:=(I-D^{2}_{t})^{-1/2}\big{(}(I-P_{t})(I-P_{t_{0}})+P_{t}P_{t_{0}}\big{)},\\\
&U_{t}^{-1}=\big{(}(I-P_{t_{0}})(I-P_{t})+P_{t_{0}}P_{t}\big{)}(I-D^{2}_{t})^{-1/2},\end{split}$
(2.49)

where $D_{t}:=P_{t}-P_{t_{0}}$ is invertible (for $t$ near $t_{0}$) since
$\|D_{t}\|_{{\mathcal{B}}({\mathcal{H}})}\underset{t\rightarrow t_{0}}{=}o(1)$
by (2.46). In particular, one has $U_{t}P_{t_{0}}=P_{t}U_{t}$. Given the
auxiliary operators $U_{t}$ we consider an $m-$dimensional operator
$W_{t}:=P_{t_{0}}U_{t}^{-1}{\mathcal{A}}_{t}U_{t}P_{t_{0}}$ and obtain its
second order Taylor expansion next. Here and in what follows we identify
$W_{t}$ and its restriction to $\text{\rm{ran}}(P_{t_{0}})$.

######  Theorem 2.9.

Assume Hypotheses 2.2 and 2.6 and let
$W_{t}:=P_{t_{0}}U_{t}^{-1}{\mathcal{A}}_{t}U_{t}P_{t_{0}}$ for $t$ near
$t_{0}$. Then the following second order Taylor expansion holds,

$\displaystyle\begin{split}W_{t}\underset{t\rightarrow
t_{0}}{=}W_{t_{0}}+\dot{W}_{t_{0}}(t-t_{0})+\frac{1}{2}\ddot{W}_{t_{0}}(t-t_{0})^{2}+o(t-t_{0})^{2},\end{split}$
(2.50)

where $W_{t_{0}}:=\lambda P_{t_{0}}$ and

$\displaystyle\dot{W}_{t_{0}}:=(TP_{t_{0}})^{*}J\dot{Q}_{t_{0}}TP_{t_{0}},$
(2.51)
$\displaystyle\ddot{W}_{t_{0}}:=(TP_{t_{0}})^{*}J\ddot{Q}_{t_{0}}TP_{t_{0}}-2\big{(}\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}.$
(2.52)

###### Proof.

Our strategy is to expand the function

$t\mapsto P_{t_{0}}U_{t}^{-1}R_{t}(\zeta)U_{t}P_{t_{0}},$ (2.53)

uniformly for $\zeta$ near $\lambda$, to the second order in $(t-t_{0})$, then
multiply both sides of the obtained expansion by $\frac{-1}{2\pi{\bf
i}}\zeta$, integrate over $\gamma$, and use the formula

${\mathcal{A}}_{t}P_{t}=\frac{-1}{2\pi{\bf i}}\int_{\gamma}\zeta
R_{t}(\zeta)d\zeta.$ (2.54)

In order to obtain the expansion for the operator in (2.53), we split it into
four terms,

$\displaystyle\begin{split}&P_{t_{0}}U_{t}^{-1}R_{t}(\zeta)U_{t}P_{t_{0}}=\\\
\hfil\displaystyle\begin{split}&=P_{t_{0}}U_{t}^{-1}P_{t_{0}}R_{t}(\zeta)P_{t_{0}}U_{t}P_{t_{0}}+P_{t_{0}}U_{t}^{-1}(I-P_{t_{0}})R_{t}(\zeta)P_{t_{0}}U_{t}P_{t_{0}}\\\
&\quad+P_{t_{0}}U_{t}^{-1}P_{t_{0}}R_{t}(\zeta)(I-P_{t_{0}})U_{t}P_{t_{0}}+P_{t_{0}}U_{t}^{-1}(I-P_{t_{0}})R_{t}(\zeta)(I-P_{t_{0}})U_{t}P_{t_{0}}\end{split}\\\
&=\tau_{1}(\zeta)+\tau_{2}(\zeta)+\tau_{3}(\zeta)+\tau_{4}(\zeta),\end{split}$
(2.55)

where $\tau_{k}(\zeta)$ denotes the respective term in (2.55). The remaining
part of the argument is divided into three steps. In Steps 1 and 2, we derive
auxiliary expansions in $t-t_{0}$. Step 3 gives asymptotic expansion for
integrated versions of $\tau_{k}(\zeta)$.

Step 1. Let us denote

$\displaystyle\begin{split}&\mathfrak{S}_{0}:=(\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}},\\\
\mathfrak{S}_{1}:=\big{(}\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\big{)}^{*}&\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}},\mathfrak{S}_{2}:=\big{(}\mathrm{T}(\mathrm{T}S_{t_{0}}^{2})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}\big{)}^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}.\end{split}$
(2.56)

Then one has

$\displaystyle P_{t_{0}}P_{t}P_{t_{0}}$
$\displaystyle=P_{t_{0}}-\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2},$ (2.57)
$\displaystyle P_{t_{0}}P_{t}$
$\displaystyle=P_{t_{0}}+\mathfrak{S}_{0}(t-t_{0})+o(t-t_{0}).$ (2.58)

###### Proof of (2.57), (2.58)..

To prove (2.57), we multiply both sides of (2.46) by $P_{t_{0}}$ from the left
and right sides and use (2.47), (2.48), and the identities
$S_{t_{0}}P_{t_{0}}=0$ and
$P_{t_{0}}(\mathrm{T}S_{t_{0}})^{*}=(\mathrm{T}S_{t_{0}}P_{t_{0}})^{*}$.
Similarly, one obtains (2.58). ∎

Step 2. One has

$\displaystyle
P_{t_{0}}U_{t}P_{t_{0}}=P_{t_{0}}-\frac{1}{2}\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2},$
(2.59) $\displaystyle
P_{t_{0}}U^{-1}_{t}(I-P_{t_{0}})=\mathfrak{S}_{0}(t-t_{0})+o(t-t_{0}),$ (2.60)

where we used notation (2.56).

###### Proof of (2.59), (2.60).

Let us derive (2.59). First, recalling $U_{t}$ from (LABEL:int27) we get

$\displaystyle P_{t_{0}}U_{t}P_{t_{0}}$
$\displaystyle=P_{t_{0}}(I-D^{2}_{t})^{-1/2}P_{t}P_{t_{0}}=P_{t_{0}}(I+\frac{1}{2}D^{2}_{t}+o(t-t_{0})^{2})P_{t}P_{t_{0}}$
(2.61)
$\displaystyle=P_{t_{0}}P_{t}P_{t_{0}}+\frac{1}{2}P_{t_{0}}D^{2}_{t}P_{t}P_{t_{0}}+o(t-t_{0})^{2}.$
(2.62)

Next, since $D_{t}=P_{t}-P_{t_{0}}=\dot{P}_{t_{0}}(t-t_{0})+o(t-t_{0})$, we
have

$\displaystyle\begin{split}&P_{t_{0}}D^{2}_{t}P_{t}P_{t_{0}}=P_{t_{0}}(\dot{P}_{t_{0}}^{2}(t-t_{0})^{2}+o(t-t_{0})^{2})(P_{t_{0}}+\dot{P}_{t_{0}}(t-t_{0})+o(t-t_{0}))P_{t_{0}}\\\
&=P_{t_{0}}\dot{P}_{t_{0}}^{2}P_{t_{0}}(t-t_{0})^{2}+o(t-t_{0})^{2}=\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2},\end{split}$
(2.63)

where in the last equality we utilized (2.47) and $S_{t_{0}}P_{t_{0}}=0$. Then
(2.59) can be inferred from (2.57), (2.62), (LABEL:pdsp).

To prove (2.60), we use (LABEL:int27), (2.58) and get

$\displaystyle P_{t_{0}}U^{-1}_{t}(I-P_{t_{0}})$
$\displaystyle=P_{t_{0}}(I-P_{t_{0}}-P_{t}+2P_{t_{0}}P_{t}+{\mathcal{O}}(t-t_{0})^{2})(I-P_{t_{0}})$
(2.64)
$\displaystyle=P_{t_{0}}P_{t}(I-P_{t_{0}})+o(t-t_{0})=\mathfrak{S}_{0}(t-t_{0})+o(t-t_{0}),$
(2.65)

where we used $\mathfrak{S}_{0}(1-P_{t_{0}})=\mathfrak{S}_{0}$. ∎

Step 3. Let us denote

$\displaystyle\mathfrak{Q}_{1}:=(\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}},\
\mathfrak{Q}_{2}:=(\mathrm{T}P_{t_{0}})^{*}\ddot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}},$
(2.66)

recall $\tau_{k}(\zeta)$ introduced in (2.55), and $\mathfrak{S}_{0}$,
$\mathfrak{S}_{1}$, $\mathfrak{S}_{2}$ from (2.56). Then we have

$\displaystyle\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta\tau_{1}(\zeta)d\zeta=\lambda
P_{t_{0}}-\mathfrak{Q}_{1}(t-t_{0})-\Big{(}\frac{1}{2}\mathfrak{Q}_{2}+2\lambda\mathfrak{S}_{2}+\mathfrak{S}_{1}\Big{)}(t-t_{0})^{2}+o(t-t_{0})^{2},$
(2.67) $\displaystyle\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta\tau_{k}(\zeta)d\zeta=\lambda\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2},\
k=2,3,$ (2.68) $\displaystyle\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta\tau_{4}(\zeta)d\zeta=o(t-t_{0})^{2}.$ (2.69)

###### Proof of (2.67), (2.68).

To prove (2.67) we will combine the second order Taylor expansions (2.59) and
(2.18) with respect to parameter $t$, collect powers of $(\lambda-\zeta)$, and
contour integrate with respect to $\zeta$ as follows.

Introducing
$\tilde{\mathfrak{R}}(\zeta):=(\mathrm{T}(\mathrm{T}R_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}$
and noting that by conjugation $P_{t_{0}}U_{t}^{-1}P_{t_{0}}$ admits the same
asymptotic as $P_{t_{0}}U_{t}P_{t_{0}}$), we infer

$\displaystyle\tau_{1}(\zeta)=P_{t_{0}}\Big{(}P_{t_{0}}-\frac{1}{2}\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2}\Big{)}P_{t_{0}}\times$
(2.70) $\displaystyle\times
P_{t_{0}}\Big{(}(\lambda-\zeta)^{-1}P_{t_{0}}+(\lambda-\zeta)^{-2}\mathfrak{Q}_{1}(t-t_{0})+\big{(}\frac{1}{2}(\lambda-\zeta)^{-2}\mathfrak{Q}_{2}+(\lambda-\zeta)^{-2}\tilde{\mathfrak{R}}(\zeta)\big{)}(t-t_{0})^{2}\Big{)}P_{t_{0}}\times$
(2.71)
$\displaystyle\times\Big{(}P_{t_{0}}-\frac{1}{2}\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2}\Big{)}P_{t_{0}},$
(2.72)

hence,

$\displaystyle\begin{split}&\tau_{1}(\zeta)=(\lambda-\zeta)^{-1}P_{t_{0}}+(\lambda-\zeta)^{-2}\mathfrak{Q}_{1}(t-t_{0})\\\
&+\Big{(}\frac{1}{2}(\lambda-\zeta)^{-2}\mathfrak{Q}_{2}-(\lambda-\zeta)^{-1}\mathfrak{S}_{2}+(\lambda-\zeta)^{-2}\tilde{\mathfrak{R}}(\zeta)\Big{)}(t-t_{0})^{2}+o(t-t_{0})^{2}.\end{split}$
(2.73)

The Laurent expansion (2.42) yields

$\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta(\lambda-\zeta)^{-2}R_{t_{0}}(\zeta)d\zeta=-S_{t_{0}}-\lambda
S_{t_{0}}^{2},$ (2.74)

and, consequently,

$\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta(\lambda-\zeta)^{-2}\tilde{\mathfrak{R}}(\zeta)d\zeta=-\mathfrak{S}_{1}-\lambda\mathfrak{S}_{2}.$
(2.75)

Multiplying both sides of (LABEL:tau1) by $\frac{-1}{2\pi{\bf i}}\zeta$,
integrating over $\gamma$, and using (2.75) we obtain (2.67).

Let us derive (2.68) for $k=2$ (the case $k=3$ follows by conjugation). Using
Taylor expansions (2.18), (2.60) and denoting
$\widehat{\mathfrak{R}}(\zeta):=(\mathrm{T}R_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}S_{t_{0}}$
we arrive at

$\displaystyle\tau_{2}(\zeta)=\big{(}\mathfrak{S}_{0}(t-t_{0})+o(t-t_{0})\big{)}(I-P_{t_{0}})\big{(}(\lambda-\zeta)^{-1}\widehat{\mathfrak{R}}(\zeta)(t-t_{0})+o(t-t_{0})\big{)}P_{t_{0}}$
(2.76) $\displaystyle\hskip
28.45274pt\times\big{(}P_{t_{0}}-\frac{1}{2}\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2}\big{)},$
(2.77)

and, consequently, infer

$\displaystyle\tau_{2}(\zeta)$
$\displaystyle=(\lambda-\zeta)^{-1}\mathfrak{S}_{0}\widehat{\mathfrak{R}}(\zeta)(t-t_{0})^{2}+o(t-t_{0})^{2}.$
(2.78)

Let us observe that

$\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta(\lambda-\zeta)^{-1}R_{t_{0}}(\zeta)d\zeta=-P_{t_{0}}+\lambda
S_{t_{0}},\ S_{t_{0}}P_{t_{0}}=0,$ (2.79)

thus

$\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta(\lambda-\zeta)^{-1}\mathfrak{S}_{0}\widehat{\mathfrak{R}}(\zeta)d\zeta=\lambda\mathfrak{S}_{2},$
(2.80)

where we used $S_{t_{0}}P_{t_{0}}=0$. Multiplying both sides of (2.78) by
$\frac{-1}{2\pi{\bf i}}\zeta$, integrating over $\gamma$ and using (2.80)we
obtain

$\displaystyle\frac{-1}{2\pi{\bf
i}}\int_{\gamma}\zeta\tau_{2}(\zeta)d\zeta=\lambda\mathfrak{S}_{2}(t-t_{0})^{2}+o(t-t_{0})^{2}.$
(2.81)

Let us prove (2.69). Using the expansion of the resolvent (2.18) to the first
order, (2.60) and its conjugate version, denoting
$\mathfrak{R}(\zeta)=(\mathrm{T}R_{t_{0}}(\zeta))^{*}\dot{Q}_{t_{0}}J\mathrm{T}R_{t_{0}}(\zeta)$
we obtain

$\displaystyle\tau_{4}(\zeta)=\big{(}\mathfrak{S}_{0}(t-t_{0})+o(t-t_{0})\big{)}((I-P_{t_{0}})R_{t_{0}}(\zeta)(I-P_{t_{0}})+\mathfrak{R}(\zeta)(t-t_{0})+o(t-t_{0}))$
(2.82)
$\displaystyle\quad\times\big{(}\mathfrak{S}_{0}^{*}(t-t_{0})+o(t-t_{0})\big{)}=\mathfrak{S}_{0}(I-P_{t_{0}})R_{t_{0}}(\zeta)(I-P_{t_{0}})\mathfrak{S}_{0}^{*}(t-t_{0})^{2}+o(t-t_{0})^{2}$
(2.83)

Multiplying both sides of the above equation by $\frac{-1}{2\pi{\bf i}}\zeta$,
integrating over $\gamma$ and noticing that the function
$\zeta\mapsto(I-P_{t_{0}})R_{t_{0}}(\zeta)(I-P_{t_{0}})$ is analytic with
respect to $\zeta$ near $\lambda$ we obtain (2.69).∎

Finally, to prove the main assertion, (2.50), one combines (2.67)–(2.69) (add
up all equations) together with (2.55), (2.54). ∎

We are now ready to present our next principal result, a version of what is
sometimes called the Kato Selection Theorem, cf. [18, Theorem II.5.11], that
allows one to compute the second order Taylor expansion of the appropriately
chosen eigenvalue curves bifurcating from a multiple eigenvalue.

######  Theorem 2.10.

Assuming Hypotheses 2.2 and 2.6, we let
$W_{t}=P_{t_{0}}U_{t}^{-1}{\mathcal{A}}_{t}U_{t}P_{t_{0}}$ and suppose that
$\dot{W}_{t_{0}}$, considered as an operator in $\text{\rm{ran}}(P_{t_{0}})$,
has $m^{\prime}\leq m$ distinct eigenvalues denoted by
$\mu_{1},...,\mu_{m^{\prime}}$. We denote the corresponding to $\mu_{i}$
eigenprojection of $\dot{W}_{t_{0}}$ in $\text{\rm{ran}}(P_{t_{0}})$ by
$P^{(i)}_{t_{0}}$, $1\leq i\leq m^{\prime}$. Furthermore, for each $i\leq
m^{\prime}$ we suppose that $P^{(i)}_{t_{0}}\ddot{W}_{t_{0}}P^{(i)}_{t_{0}}$
considered as an operator in $\text{\rm{ran}}(P^{(i)}_{t_{0}})$ has
$m^{\prime}_{i}\leq m$ distinct eigenvalues denoted by
$\nu_{i,1},...,\nu_{i,m^{\prime}_{i}}$. Then for $t$ near $t_{0}$ there is a
labeling $\lambda_{i,k}(t)$ of the eigenvalue curves of the operator
${\mathcal{A}}_{t}P_{t}$ or $W_{t}$ such that

$\displaystyle\lambda_{i,k}(t)=\lambda+\mu_{i}(t-t_{0})+\frac{1}{2}\nu_{i,k}(t-t_{0})^{2}+o(t-t_{0})^{2},\
k=1,..,m^{\prime}_{i},i=1,...,m^{\prime}.$ (2.84)

Moreover, there is an orthonormal set of eigenvectors $u_{i,k}$ of the
operators $P^{(i)}_{t_{0}}\ddot{W}_{t_{0}}P^{(i)}_{t_{0}}$ (and therefore of
the operators $\dot{W}_{t_{0}}$, $W_{t_{0}}$ and ${\mathcal{A}}_{t_{0}}$),

$\\{u_{i,k}\in\text{\rm{ran}}(P^{(i)}_{t_{0}})):k=1,..,m^{\prime}_{i},\,i=1,...,m^{\prime}\\}\subset\text{\rm{ran}}(P_{t_{0}}),$

such that

$\displaystyle\begin{split}&\lambda_{i,k}(t)=\lambda+\omega(\dot{Q}_{t_{0}}\mathrm{T}u_{i,k},\mathrm{T}u_{i,k})(t-t_{0})\\\
&\quad+\frac{1}{2}\Big{(}\omega(\ddot{Q}_{t_{0}}\mathrm{T}u_{i,k},\mathrm{T}u_{i,k})-2\langle\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}u_{i,k},\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}u_{i,k}\rangle_{\mathfrak{H}\times\mathfrak{H}}\Big{)}(t-t_{0})^{2}\\\
&\quad\quad+o(t-t_{0})^{2},\
k=1,..,m^{\prime}_{i},\,i=1,...,m^{\prime},\end{split}$ (2.85)

where
$\mathrm{T}\in{\mathcal{B}}({\mathcal{H}}_{+},\mathfrak{H}\times\mathfrak{H})$
is the trace operator, the symplectic form $\omega$ is defined in (LABEL:5.3),
$S_{t_{0}}$ is the reduced resolvent of the operator ${\mathcal{A}}_{t_{0}}$
at $\lambda$.

###### Proof.

Taylor’s expansion (2.50) yields

$\displaystyle\frac{1}{(t-t_{0})}\Big{(}\frac{1}{(t-t_{0})}\big{(}W_{t}-W_{t_{0}}\big{)}-\dot{W}_{t_{0}}\Big{)}\underset{t\rightarrow
t_{0}}{=}\frac{1}{2}\ddot{W}_{t_{0}}+o(1).$ (2.86)

Multiplying this by $P^{(i)}_{t_{0}}$ from both sides and using the identities

$P^{(i)}_{t_{0}}W_{t_{0}}P^{(i)}_{t_{0}}=\lambda
I_{\text{\rm{ran}}P_{t_{0}}},\quad
P^{(i)}_{t_{0}}\dot{W}_{t_{0}}P^{(i)}_{t_{0}}=\mu_{i}I_{\text{\rm{ran}}P^{(i)}_{t_{0}}},$
(2.87)

we obtain

$\displaystyle\frac{1}{(t-t_{0})}\Big{(}\frac{1}{(t-t_{0})}\big{(}P^{(i)}_{t_{0}}W_{t}P^{(i)}_{t_{0}}-\lambda\big{)}-\mu_{i}\Big{)}\underset{t\rightarrow
t_{0}}{=}\frac{1}{2}P^{(i)}_{t_{0}}\ddot{W}_{t_{0}}P^{(i)}_{t_{0}}+o(1).$
(2.88)

By continuity of the spectrum, see, e.g., [22, Theorem VIII.24], every point
in the set

$\operatorname{Spec}\left(\frac{1}{2}P^{(i)}_{t_{0}}\ddot{W}_{t_{0}}P^{(i)}_{t_{0}}\right)=\left\\{\frac{1}{2}\nu_{i,1},...,\frac{1}{2}\nu_{i,m^{\prime}_{i}}\right\\}$

can be approximated by an eigenvalue of the operator in the left-hand side of
(2.88) for $t$ near $t_{0}$. Such an eigenvalue is of the form

$\frac{1}{(t-t_{0})}\Big{(}\frac{1}{(t-t_{0})}\big{(}\lambda_{i,k}(t)-\lambda\big{)}-\mu_{i}\Big{)},\text{
for some choice of
$\lambda_{i,k}(t)\in\operatorname{Spec}({\mathcal{A}}_{t}P_{t}|_{\text{\rm{ran}}(P_{t})})=\operatorname{Spec}(W_{t})$.}$
(2.89)

That is, we have

$\displaystyle\frac{1}{(t-t_{0})}\Big{(}\frac{1}{(t-t_{0})}\big{(}\lambda_{i,k}(t)-\lambda\big{)}-\mu_{i}\Big{)}\underset{t\rightarrow
t_{0}}{=}\frac{1}{2}\nu_{i,k}+o(1),$ (2.90)

hence, (2.84) holds as asserted.

To prove (LABEL:razl), let $u_{i,k}\in\text{\rm{ran}}(P^{(i)}_{t_{0}})$ be the
normalized eigenvectors of $\ddot{W}_{t_{0}}$ such that
$\ddot{W}_{t_{0}}u_{i,k}=\nu_{i,k}u_{i,k}$. Since
$u_{i,k}\in\text{\rm{ran}}(P^{(i)}_{t_{0}})$ we have
$\dot{W}_{t_{0}}u_{i,k}=\mu_{i}u_{i,k}$. Using these identities and (2.51),
(2.52) we arrive at

$\displaystyle\begin{split}&\mu_{i}=\langle\dot{W}_{t_{0}}u_{i,k},u_{i,k}\rangle_{{\mathcal{H}}}=\omega(\dot{Q}_{t_{0}}\mathrm{T}u_{i,k},\mathrm{T}u_{i,k}),\\\
&\nu_{i,k}=\langle\ddot{W}_{t_{0}}u_{i,k},u_{i,k}\rangle_{{\mathcal{H}}}=\omega(\ddot{Q}_{t_{0}}\mathrm{T}u_{i,k},\mathrm{T}u_{i,k})-2\langle\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}u_{i,k},\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}u_{i,k}\rangle_{{\mathcal{H}}}.\end{split}$
(2.91)

Finally, combining (LABEL:munu) and (2.84) we obtain (LABEL:razl) ∎

In applications, e.g. such as quantum graphs, spectral theory of differential
operators, one often describes a Lagrangian plane
${\mathcal{F}}\in\Lambda(\mathfrak{H}\times\mathfrak{H})$ by means of two
bounded operators $X,Y\in{\mathcal{B}}(\mathfrak{H})$, satisfying
$XY^{*}=YX^{*}$, $0\not\in\operatorname{Spec}(XX^{*}+YY^{*})$, via

${\mathcal{F}}=\\{(u,v)\in\mathfrak{H}\times\mathfrak{H}:Xu+Yv=0\\}=\operatorname{ker}(Z),\,\text{
where
$Z=[X,Y]\in{\mathcal{B}}(\mathfrak{H}\times\mathfrak{H},\mathfrak{H})$}.$

It is therefore natural to derive the second order Taylor expansion in this
setting, which is done in the following theorem.

######  Theorem 2.11.

Under hypotheses of Theorem 2.10, suppose that the Lagrangian planes
$\text{\rm{ran}}(Q_{t})=\mathrm{T}(\operatorname{dom}({\mathcal{A}}_{t}))$ are
given by $\text{\rm{ran}}(Q_{t})=\operatorname{ker}Z_{t}$ where
$Z_{t}:=[X_{t},Y_{t}]$ and $X_{t},Y_{t}\in{\mathcal{B}}(\mathfrak{H})$,
$X_{t}Y_{t}^{*}=Y_{t}X_{t}^{*}$,
$0\not\in\operatorname{Spec}(X_{t}X^{*}_{t}+Y_{t}Y^{*}_{t})$, satisfy

$\displaystyle Z_{t}\underset{t\rightarrow
t_{0}}{=}Z_{t_{0}}+\dot{Z}_{t_{0}}(t-t_{0})+\frac{1}{2}\ddot{Z}_{t_{0}}(t-t_{0})^{2}+o(t-t_{0})^{2}\text{\
in\ }{\mathcal{B}}(\mathfrak{H}\times\mathfrak{H}).$ (2.92)

Then the eigenvalue curves $\lambda_{i,k}(t)$, defined as in Theorem 2.10,
bifurcating from the isolated eigenvalue
$\lambda\in\operatorname{Spec}({\mathcal{A}}_{t_{0}})$ admit the second order
Taylor expansion

$\displaystyle\begin{split}\lambda_{i,k}(t)\underset{t\rightarrow
t_{0}}{=}&\lambda+\langle
Z_{t_{0}}J\dot{Z}^{*}_{t_{0}}\phi_{i,k},\phi_{i,k}\rangle_{\mathfrak{H}}(t-t_{0})\\\
&\quad+\frac{1}{2}\left(\langle
Z_{t_{0}}J\ddot{Z}^{*}_{t_{0}}\phi_{i,k},\phi_{i,k}\rangle_{\mathfrak{H}}-2\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Z}_{t_{0}}^{*}\phi_{i,k},\dot{Z}_{t_{0}}^{*}\phi_{i,k}\rangle_{\mathfrak{H}\times\mathfrak{H}}\right)(t-t_{0})^{2}\\\
&\quad\qquad+o(t-t_{0})^{2},\end{split}$ (2.93)

where $S_{t_{0}}$ is the reduced resolvent of ${\mathcal{A}}_{t_{0}}$ at
$\lambda$, and we introduce notation

$\phi_{i,k}=(Z_{t_{0}}Z_{t_{0}}^{*})^{-1}Z_{t_{0}}J\mathrm{T}u_{i,k}$

for $u_{i,k}$ given in Theorem 2.10.

###### Proof.

For brevity, we denote $u=u_{i,k}$, $\phi:=\phi_{i,k}$ and suppress dependence
of $P_{t_{0}}$, $Z_{t_{0}}$, $Q_{t_{0}}$ and their derivatives on $t_{0}$,
e.g., $\ddot{Z}_{t_{0}}$ will be denoted by $\ddot{Z}$. Let us record some
relevant identities. First, the orthogonal projection $Q$ onto the kernel of
$Z$, i.e. onto the Lagrangian plane
$\mathrm{T}(\operatorname{dom}({\mathcal{A}}))$, can be written as

$Q=-JZ^{*}(ZZ^{*})^{-1}ZJ,$ (2.94)

thus we have $\mathrm{T}u=Q\mathrm{T}u$, and, by
$\phi=(ZZ^{*})^{-1}ZJ\mathrm{T}u$, one also has

$\displaystyle\begin{split}&\mathrm{T}u=-JZ^{*}\phi.\end{split}$ (2.95)

In addition, the identity $XY^{*}=YX^{*}$ yields $ZJZ^{*}=0,$ therefore, by
differentiating we arrive at

$\displaystyle\dot{Z}JZ^{*}=-ZJ\dot{Z}^{*},$ (2.96)
$\displaystyle\ddot{Z}JZ^{*}=-2\dot{Z}J\dot{Z}^{*}-ZJ\ddot{Z}^{*}.$ (2.97)

Furthermore, since $\text{\rm{ran}}(Q)=\operatorname{ker}(Z)$ we have $ZQ=0$,
hence,

$\displaystyle\dot{Z}Q+Z\dot{Q}=0,$ (2.98) $\displaystyle
Q\dot{Z}^{*}+\dot{Q}Z^{*}=0.$ (2.99)

Also, since $QJ+JQ=J$ we get

$\dot{Q}J=-J\dot{Q},\ \ddot{Q}J=-J\ddot{Q}.$ (2.100)

Let us rewrite the projection $Q$ onto the Lagrangian plane as
$Q=\alpha\beta\gamma$ with

$\displaystyle\alpha:=-JZ^{*}\in{\mathcal{B}}(\mathfrak{H},\mathfrak{H}\times\mathfrak{H}),\
\beta:=(ZZ^{*})^{-1}\in{\mathcal{B}}(\mathfrak{H}),\
\gamma:=ZJ\in{\mathcal{B}}(\mathfrak{H}\times\mathfrak{H},\mathfrak{H}).$
(2.101)

We proceed by introducing three more auxiliary identities (2.102), (2.104),
(2.105) below. One has

$\dot{Q}J\mathrm{T}u=-Q\dot{Z}^{*}\phi.$ (2.102)

To prove this, we note that $\gamma J\mathrm{T}u=0$, therefore

$\displaystyle\dot{Q}J\mathrm{T}u$
$\displaystyle=\alpha\beta\dot{\gamma}J\mathrm{T}u=-\alpha\beta\dot{Z}\mathrm{T}u\underset{\eqref{tujzs}}{=}\alpha\beta\dot{Z}JZ^{*}\phi\underset{\eqref{dz}}{=}-\alpha\beta
ZJ\dot{Z}^{*}\phi=-Q\dot{Z}^{*}\phi.$ (2.103)

We also have

$\dot{Q}\mathrm{T}u=(1-Q)\mathrm{T}\dot{P}u.$ (2.104)

One arrives at this identity by differentiating both sides of the equation
$QTP=TP$ and noting that $Pu=u$. Next, we record the identity

$\dot{Q}\mathrm{T}u=(1-Q)\mathrm{T}(\mathrm{T}S)^{*}\dot{Q}J\mathrm{T}u,$
(2.105)

which follows from (2.104), explicit formula for $\dot{P}$ in (2.47), and the
fact that the second term in the RHS of (2.47) vanishes since $S_{t_{0}}u=0$.

We are now ready to prove (2.93). To that end we employ (LABEL:razl) together
with the auxiliary identities (2.95)-(2.105). It suffices to show that

$\displaystyle\omega(\dot{Q}\mathrm{T}u,\mathrm{T}u)=\langle
ZJ\dot{Z}^{*}\phi,\phi\rangle_{\mathfrak{H}},$ (2.106)
$\displaystyle\omega(\ddot{Q}\mathrm{T}u,\mathrm{T}u)-2\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Q}J\mathrm{T}u,\dot{Q}J\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}=$
(2.107) $\displaystyle\hskip 56.9055pt=\langle
ZJ\ddot{Z}^{*}\phi,\phi\rangle_{\mathfrak{H}}-2\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Z}^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}.$
(2.108)

Let us first derive (2.106). One has

$\displaystyle\omega(\dot{Q}\mathrm{T}u,\mathrm{T}u)$ $\displaystyle=\langle
J\dot{Q}\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{dqj}}{=}-\langle\dot{Q}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{dqjtu}}{=}\langle
Q\dot{Z}^{*}\phi,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}$ (2.109)
$\displaystyle=\langle\dot{Z}^{*}\phi,Q\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}=\langle\dot{Z}^{*}\phi,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{tujzs}}{=}\langle\dot{Z}^{*}\phi,-JZ^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}=\langle
ZJ\dot{Z}^{*}\phi,\phi\rangle_{\mathfrak{H}}.$ (2.110)

The derivation of (2.108) is more subtle. Our immediate objective is to derive
the following,

$\displaystyle\omega(\ddot{Q}\mathrm{T}u,\mathrm{T}u)=\langle
ZJ\ddot{Z}^{*}\phi,\phi\rangle_{\mathfrak{H}}-2\omega(Q\dot{Z}^{*}\phi,\dot{Z}^{*}\phi),$
(2.111)
$\displaystyle\omega(Q\dot{Z}^{*}\phi,\dot{Z}^{*}\phi)=\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Z}^{*}\phi,(1-Q)\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}},$
(2.112)
$\displaystyle\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Q}J\mathrm{T}u,\dot{Q}J\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}=\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Z}^{*}\phi,Q\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}},.$
(2.113)

since (2.108) readily follows from (2.111)–(2.113). To to begin the proof of
(2.111) we differentiate $Q=\alpha\beta\gamma$ twice. Since $\gamma
J\mathrm{T}u=0$ and infer

$\ddot{Q}J\mathrm{T}u=\alpha\beta\ddot{\gamma}J\mathrm{T}u+2(\dot{\alpha}\beta+\alpha\dot{\beta})\dot{\gamma}J\mathrm{T}u.$
(2.114)

Plugging this into the left-hand side of (2.111) we arrive at

$\displaystyle\omega(\ddot{Q}\mathrm{T}u,\mathrm{T}u)=-\langle\ddot{Q}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}$
(2.115)
$\displaystyle\quad=-\langle\alpha\beta\ddot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-2\langle(\dot{\alpha}\beta+\alpha\dot{\beta})\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}.$
(2.116)

Let us re-write both terms in the right-hand side of (2.115). The first term
is

$\displaystyle\begin{split}&-\langle\alpha\beta\ddot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{tujzs}}{=}-\langle\alpha\beta\ddot{Z}JZ^{*}\phi,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&\underset{\eqref{ddz}}{=}2\langle\alpha\beta\dot{Z}J\dot{Z}^{*}\phi,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}+\langle\alpha\beta
ZJ\ddot{Z}^{*}\phi,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&=2\langle(ZZ^{*})^{-1}\dot{Z}J\dot{Z}^{*}\phi,ZZ^{*}\phi\rangle_{\mathfrak{H}}+\langle\alpha\beta\gamma\ddot{Z}^{*}\phi,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&=2\langle
J\dot{Z}^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}+\langle
Q\ddot{Z}^{*}\phi,Tu\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{tujzs}}{=}\langle
ZJ\ddot{Z}^{*}\phi,\phi\rangle_{\mathfrak{H}},\end{split}$ (2.117)

where we used $\langle
J\dot{Z}^{*}\phi,\dot{Z}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}=0$. In
order to re-write the second term in the right-hand side of (2.115) we first
notice that $\beta=(ZZ^{*})^{-1}$ yields

$\dot{\beta}=-\beta(\dot{Z}Z^{*}+Z\dot{Z}^{*})\beta,$

thus

$\displaystyle\begin{split}&\langle(\dot{\alpha}\beta+\alpha\dot{\beta})\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&\quad=\langle\dot{\alpha}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\alpha\beta\dot{Z}Z^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\alpha\beta
Z\dot{Z}^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&\quad=\langle\dot{\alpha}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\alpha\beta\dot{Z}Z^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}+\langle\alpha\beta
ZJJ\dot{Z}^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&\quad=\langle\dot{\alpha}\beta\dot{\gamma}JTu,Tu\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\alpha\beta\dot{Z}Z^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\alpha\beta\gamma\dot{\alpha}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&\quad=\langle\dot{\alpha}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\alpha\beta\dot{Z}Z^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\dot{\alpha}\beta\dot{\gamma}J\mathrm{T}u,Q\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\\\
&=\quad-\langle\alpha\beta\dot{Z}Z^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}},\end{split}$
(2.118)

where we used $Q=\alpha\beta\gamma$, $Q\mathrm{T}u=\mathrm{T}u$. Now, let us
further simplify the last expression. One has

$\displaystyle\begin{split}-\langle\alpha\beta&\dot{Z}Z^{*}\beta\dot{\gamma}J\mathrm{T}u,\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}=-\langle\dot{Z}Z^{*}(ZZ^{*})^{-1}\dot{Z}JJ\mathrm{T}u,(ZZ^{*})^{-1}ZJ\mathrm{T}u\rangle_{\mathfrak{H}}\\\
&\quad\underset{\eqref{tujzs}}{=}-\langle\dot{Z}Z^{*}(ZZ^{*})^{-1}\dot{Z}JZ^{*}\phi,\phi\rangle_{\mathfrak{H}}\underset{\eqref{dz}}{=}\langle\dot{Z}Z^{*}(ZZ^{*})^{-1}ZJ\dot{Z}^{*}\phi,\phi\rangle_{\mathfrak{H}}\\\
&\quad=\langle
J(-JZ)^{*}(ZZ^{*})^{-1}ZJ\dot{Z}^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}=\omega(Q\dot{Z}^{*}\phi,\dot{Z}^{*}\phi).\end{split}$
(2.119)

Combining (2.115), (LABEL:abgdots) and (2.119) one infers (2.111) as asserted.

Next, we prove (2.112). One has

$\displaystyle\omega(Q\dot{Z}^{*}\phi,\dot{Z}^{*}\phi)=\langle
JQ\dot{Z}^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{qdz}}{=}-\langle
J\dot{Q}Z^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{dqj}}{=}\langle\dot{Q}JZ^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}$
(2.120)
$\displaystyle\quad=-\langle\dot{Q}\mathrm{T}u,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{ident3}}{=}-\langle(1-Q)\mathrm{T}(\mathrm{T}S)^{*}\dot{Q}J\mathrm{T}u,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}$
(2.121)
$\displaystyle\quad\underset{\eqref{dqjtu}}{=}\langle(1-Q)\mathrm{T}(\mathrm{T}S)^{*}Q\dot{Z}^{*}\phi,\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}$
(2.122)
$\displaystyle\quad=\langle\mathrm{T}(Q\mathrm{T}S)^{*}\dot{Z}^{*}\phi,(1-Q)\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}=\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Z}^{*}\phi,(1-Q)\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}},$
(2.123)

where we used $Q\mathrm{T}S=\mathrm{T}S$ which holds since
$\text{\rm{ran}}(S)\subset\text{\rm{ran}}(Q)$.

To complete the proof of the theorem we derive (2.113). To that end, we have

$\displaystyle\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Q}J\mathrm{T}u,\dot{Q}J\mathrm{T}u\rangle_{\mathfrak{H}\times\mathfrak{H}}\underset{\eqref{dqjtu}}{=}\langle\mathrm{T}(\mathrm{T}S)^{*}Q\dot{Z}^{*}\phi,Q\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}$
(2.124)
$\displaystyle\quad=\langle\mathrm{T}(Q\mathrm{T}S)^{*}\dot{Z}^{*}\phi,Q\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}}=\langle\mathrm{T}(\mathrm{T}S)^{*}\dot{Z}^{*}\phi,Q\dot{Z}^{*}\phi\rangle_{\mathfrak{H}\times\mathfrak{H}},$
(2.125)

where, as before, we used $Q\mathrm{T}S=\mathrm{T}S$. ∎

######  Corollary 2.12.

Under the hypotheses and general setting of Theorem 2.11 we assume that
$X_{t}=\Theta_{t}$, $Y_{t}=-I$, that is, we assume that ${\mathcal{A}}_{t}$ is
a Robin-type self-adjoint extension of $A$ with
$\operatorname{dom}({\mathcal{A}}_{t})=\\{u\in{\mathcal{H}}_{+}:\Gamma_{1}u=\Theta_{t}\Gamma_{0}u\\}$,
and that $\Theta_{t}=\Theta_{t}^{*}$. Then one has

$\displaystyle\begin{split}\lambda_{i,k}(t)\underset{t\rightarrow
t_{0}}{=}&\lambda+\langle\dot{\Theta}\Gamma_{0}u_{0},\Gamma_{0}u_{0}\rangle_{{\mathfrak{H}}}(t-t_{0})\\\
&\quad+\frac{1}{2}\left(\langle\big{(}\ddot{\Theta}_{t_{0}}-2\dot{\Theta}_{t_{0}}\Gamma_{0}(\Gamma_{0}S_{t_{0}})^{*}\dot{\Theta}_{t_{0}}\big{)}\Gamma_{0}u_{i,k},\Gamma_{0}u_{i,k}\rangle_{\mathfrak{H}}\right)(t-t_{0})^{2}\\\
&\quad+o(t-t_{0})^{2}.\end{split}$ (2.126)

###### Proof.

Formula (2.126) follows directly from (2.93) with $Z_{t}=[\Theta_{t},-I]$. ∎

######  Example 2.13 (Matrix Robin Laplacian).

Let $V\in L^{\infty}([a,b],{\mathbb{C}}^{n\times n})$ be a matrix valued
bounded function with $V(x)=(V(x))^{*}$, $x\in[a,b]$, and let
$A=-\frac{d^{2}}{dx^{2}}+V$ be the minimal operator with
$\operatorname{dom}(A)=H^{2}_{0}([a,b];{\mathbb{C}}^{n})$. Given $u\in
H^{2}([a,b],{\mathbb{C}}^{n})$ we denote
$\gamma_{{}_{D}}u=[u(a),u(b)]^{\top}$,
$\gamma_{{}_{N}}u=[u(a),-u^{\prime}(b)]^{\top}$ and view
$\gamma_{{}_{D}},\gamma_{{}_{N}}$ as bounded linear operators in
${\mathcal{B}}(H^{2}([a,b],{\mathbb{C}}^{2n});{\mathbb{C}}^{2n})$. Let
$\Theta\in C^{2}([0,1],{\mathbb{C}}^{2n\times 2n})$ be a matrix-valued
function with $\Theta_{t}=\Theta_{t}^{*}$, $t\in[0,1]$, and define a one-
parameter family of Robin extensions of $A$ by the formula

$\operatorname{dom}({\mathcal{A}}_{t})=\\{u\in
H^{2}([a,b];{\mathbb{C}}^{n}):\gamma_{{}_{N}}u=\Theta_{t}\gamma_{{}_{D}}u\\}.$
(2.127)

Suppose that $\lambda\in{\mathbb{R}}$ is a simple eigenvalue of
${\mathcal{A}}_{t_{0}}$ for some $t_{0}$ and suppose that
$u_{0}\in\operatorname{ker}({\mathcal{A}}_{t_{0}}-\lambda)$ has unit norm.
Then there is an eigenvalue curve bifurcating from $\lambda$ and obeying the
following asymptotic expansion,

$\displaystyle\begin{split}\lambda(t)=\lambda&+\left\langle\dot{\Theta}_{t_{0}}\gamma_{{}_{D}}u,\gamma_{{}_{D}}u\right\rangle_{{\mathbb{C}}^{2n}}(t-t_{0})\\\
&+\frac{1}{2}\left\langle\big{(}\ddot{\Theta}_{t_{0}}-2\dot{\Theta}_{t_{0}}\gamma_{{}_{D}}(\gamma_{{}_{D}}S)^{*}\dot{\Theta}_{t_{0}}\big{)}\gamma_{{}_{D}}u,\gamma_{{}_{D}}u\right\rangle_{{\mathbb{C}}^{2n}}(t-t_{0})^{2}+o(t-t_{0})^{2},t\rightarrow
t_{0}.\end{split}$ (2.128)

######  Remark 2.14.

Our main results, Theorem 2.9,2.10 and Theorem 2.11 can be extended to a more
general setting in which the self-adjoint extensions are subject to further
additive perturbations ${\mathcal{A}}_{t}+V_{t}$. To elaborate on this case,
we augment Hypothesis 2.2 by fixing a one-parameter family of bounded self-
adjoint operators $t\mapsto V_{t}\in{\mathcal{B}}({\mathcal{H}})$ with

$V_{t}\underset{t\rightarrow
t_{0}}{=}V_{t_{0}}+\dot{V}_{t_{0}}(t-t_{0})+\frac{1}{2}\ddot{V}_{t_{0}}(t-t_{0})^{2}+o(t-t_{0})^{2}\text{\
in\ }{\mathcal{B}}({\mathcal{H}}).$ (2.129)

Let us introduce $H_{t}:={\mathcal{A}}_{t}+V_{t}$, denote its resolvent by
$R_{t}(\zeta):=(H_{t}-\zeta)^{-1}\in{\mathcal{B}}({\mathcal{H}})$ for
$\zeta\not\in\operatorname{Spec}(H_{t})$, and its reduced resolvent at the
eigenvalue $\lambda$ by $S_{t_{0}}$. Keeping general setting of Theorem 2.10
with ${\mathcal{A}}_{t}$ replaced by $H_{t}$ we record the first and second
order terms in the Taylor expansion of the eigenvalue curves. The first order
term is given by

$\displaystyle\dot{\lambda}_{i,k}=\langle\dot{V}_{t_{0}}u_{i,k},u_{i,k}\rangle_{{\mathcal{H}}}+\omega(\dot{Q}_{t_{0}}\mathrm{T}u_{i,k},\mathrm{T}u_{i,k}).$
(2.130)

The second order term is given by

$\displaystyle\ddot{\lambda}_{i,k}$
$\displaystyle=\frac{1}{2}\Big{(}\langle\ddot{V}_{t_{0}}u_{i,k},u_{i,k}\rangle_{{\mathcal{H}}}+\omega(\ddot{Q}_{t_{0}}\mathrm{T}u_{i,k},\mathrm{T}u_{i,k})\Big{)}-\langle\mathrm{T}(\mathrm{T}S_{t_{0}})^{*}\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}u_{i,k},\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}u_{i,k}\rangle_{\mathfrak{H}\times\mathfrak{H}}$
(2.131)
$\displaystyle+2\langle\mathrm{T}S\dot{V}_{t_{0}}Pu_{i,k},\dot{Q}_{t_{0}}J\mathrm{T}P_{t_{0}}u_{i,k}\rangle_{\mathfrak{H}\times\mathfrak{H}}-\langle\dot{V}_{t_{0}}S_{t_{0}}\dot{V}_{t_{0}}u_{i,k},u_{i,k}\rangle_{{\mathcal{H}}}.$
(2.132)

The derivation relies on Krein’s formula from [19],

$\displaystyle
R_{t}(\zeta)-R_{t_{0}}(\zeta)=R_{t}({\zeta})(V_{t_{0}}-V_{t})R_{t_{0}}(\zeta)+(TR_{t}(\zeta))^{*}(Q_{t}-Q_{t_{0}})JQ_{t_{0}}TR_{t_{0}}(\zeta),$
(2.133)

and resolvent expansions arguments presented above.

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