# Tunneling effect induced by a curved magnetic edge

Søren Fournais  ,  Bernard Helffer  and  Ayman Kachmar  Department of
Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
<EMAIL_ADDRESS>Université de Nantes, Laboratoire Jean Leray, Nantes,
France<EMAIL_ADDRESS>Lebanese University, Department of
mathematics, 1700 Nabatieh, Lebanon<EMAIL_ADDRESS>
###### Abstract.

Experimentally observed magnetic fields with nanoscale variations are
theoretically modeled by a piece-wise constant function with jump
discontinuity along a smooth curve, the magnetic edge. Assuming the edge is a
closed curve with an axis of symmetry and the field is sign changing and with
exactly two distinct values, we prove that semi-classical tunneling occurs and
calculate the magnitude of this tunneling effect.

This paper is dedicated to Elliott H. Lieb on the occasion of his 90th
birthday.

##  1\. Introduction

The purpose of this paper is to study the magnetic Laplacian in
$\mathbb{R}^{2}$,

$\mathcal{P}_{h}:=(ih\nabla+\mathbf{A})^{2}=\sum_{j=1}^{2}(ih\partial_{x_{j}}+A_{j})^{2},$
(1.1)

where $\mathbf{A}:=(A_{1},A_{2})\in
H^{1}_{\textrm{loc}}(\mathbb{R}^{2};\mathbb{R}^{2})$, is a magnetic potential
generating the magnetic field $B=\textrm{
curl}\mathbf{A}:=\partial_{x_{1}}A_{2}-\partial_{x_{2}}A_{1}\in
L^{2}_{\textrm{ loc}}(\mathbb{R}^{2};\mathbb{R})$. We will also discuss the
case of the Neumann or Dirichlet realizations of $\mathcal{P}_{h}$ in smooth
bounded planar domains.

Here $h$ is a positive parameter that tends to $0$, which can be interpreted
as the semi-classical parameter. By writing
$h^{-2}\mathcal{P}_{h}=(i\nabla+h^{-1}\mathbf{A})^{2}$, we observe that the
semi-classical limit, $h\to 0_{+}$, is equivalent to the strong magnetic field
limit, $h^{-1}|B|\to+\infty$.

The spectrum of the operator $\mathcal{P}_{h}$ has been the subject of an
intense study in the past decades, particularly in the context of
superconductivity where the magnetic field $B$ is typically a constant
function [6, 11, 17, 21, 23, 32].

There is an interesting analogy between the results for the Neumann
realization of $\mathcal{P}_{h}$ in a bounded smooth domain and those for the
Schrödinger operator, $-h^{2}\Delta+V$, with an electric potential $V$, in the
full plane. The Schrödinger operator was intensively studied by Helffer-
Sjöstrand [24, 25] and Simon [36], notably in the context of quantum
tunneling. Bound states of $-h^{2}\Delta+V$ concentrate near the ‘well’
$\Gamma_{V}:=\\{x\in\mathbb{R}^{2}\,|\,V(x)=\min_{\mathbb{R}^{2}}V\\}$; if
furthermore $\Gamma_{V}$ is a regular manifold (i.e. we have a degenerate
well), bound states could concentrate near some points of $\Gamma_{V}$, the
‘mini-wells’. We have the same picture in the purely magnetic case with a
Neumann boundary condition: bound states concentrate near the boundary of the
domain, whereby the boundary plays the role of a (degenerate) well, and the
set of points of maximum curvature plays the role of mini-wells, where bound
states decay away from them. Optimal estimates describing the concentration of
bound states are very important, since they lead to accurate asymptotics for
the low lying eigenvalues. The proof of the decay away from the mini-wells
(points of maximum curvature), is more delicate compared to that of the decay
away from the well (boundary).

In this paper, our main focus will be on magnetic fields having a jump-
discontinuity. Magnetic fields that vary on very short scales (nanoscales)
have been observed experimentally, see e.g. [15]. Their theoretical
investigations, in the context of quantum mechanics [34, 35] or graphene [18],
involve the operator $\mathcal{P}_{h}$ but with the magnetic field $B$ being a
step function having a discontinuity along a curve, that we will refer to as
the _magnetic edge_.

Earlier rigorous results were devoted to the case of a flat edge [28, 29, 30].
More recently, non-flat edges have been considered in the context of spectral
asymptotics [1, 3] and in the context of superconductivity [4]. The magnetic
edge will play the role of the ‘well’, while the ‘mini-wells’ are the points
of maximum curvature of the magnetic edge [5], which is interestingly in
analogy with the setting of the Neumann realization with a constant magnetic
field in a bounded smooth domain.

The case of a single mini-well, where the curvature of the edge has a unique
and non-degenerate maximum, was analyzed by Assaad-Helffer-Kachmar [3]. The
present paper investigates the situation of a symmetric edge with several
mini-wells, the simplest case being when there are two non-degenerate maxima
of the boundary curvature. We establish a sharp asymptotics of the splitting
between the energies of the ground and first excited state, which measures a
tunneling effect induced by the geometry of the edge, see Theorem 3.4 below
which is our main result.

Let us recall how the general strategy of Helffer-Sjöstrand [24, 25] has been
applied recently to understand the tunneling effect for the Neumann
realization in a bounded domain with the breakthrough paper [7] by Bonnaillie-
Noël–Hérau–Raymond as the crowning achievement. The first step, already
performed in [21] and [16], was the analysis of a model with a flat boundary
(de Gennes model), which yields localization of bound states near the boundary
of the domain (the well), and consequently, leads to a full asymptotics of the
low-lying eigenvalues. The second step is a formal WKB expansion of bound
states [8]. The third step consists of optimal decay estimates of bound states
recently achieved in [7]. The importance of this step is that it allows one to
rigorously approximate the bound states by the formal WKB expansions, which
eventually paves the way for the analysis of an interaction matrix whose
eigenvalues measure the tunneling effect. The same approach has been
successfully applied in the context of thin domains [31] and the Robin
Laplacian [19, 20], where the proof of the tangential estimates was less
technical.

We will follow the same approach outlined above in the case of our
discontinuous magnetic field. The model problem with a flat edge was analyzed
in [5] (see also [4, 28]), while the full asymptotics for the low lying
eigenvalues are obtained in [3]. So we still need WKB expansions and optimal
tangential estimates of bound states, which we do in the present contribution.
Finally, after establishing the WKB approximation, the analysis of the
interaction matrix is quite standard.

Let us give an informal statement of our main result (Theorem 3.4 below).
Suppose that $\Gamma$ is a smooth, closed curve in $\mathbb{R}^{2}$, symmetric
with respect to an axis, and with two points of maximum curvature, denoted by
$s_{\ell}$ and $s_{r}$ ($\ell$ refers to “left” and $r$ to “right”, see Fig.
1). Consider the magnetic field satisfying $B=1$ in the interior of $\Gamma$,
and $B=a\in(-1,0)$ in the exterior of $\Gamma$. Under these assumptions, we
prove that, as $h\to 0_{+}$, the spectral gap of the operator
$\mathcal{P}_{h}$ is of exponential order,

$\lambda_{2}(h)-\lambda_{1}(h)\approx\exp\Big{(}-\frac{\mathsf{S}^{a}}{h^{1/4}}\Big{)},$
(1.2)

where $\mathsf{S}^{a}$ is the Agmon distance between the “wells” $s_{\ell}$
and $s_{r}$ defined by an appropriate potential that depends on the magnetic
field (through the parameter $a$) and the geometry of $\Gamma$ (through the
curvature).

The asymptotics in (1.2) (more precisely that in Theorem 3.4) is a consequence
of quantum tunneling. It is important to note that it is induced purely by the
magnetic field, thereby providing an example of a purely magnetic quantum
tunneling - where the case of [7] also required the interaction with the
boundary. If we look at earlier results on the tunneling effect, with or
without magnetic field, we observe that the tunneling is induced by an
external potential [24, 14] or by confinement to a bounded/thin domain [20,
31, 9]. For the Neumann realization of $\mathcal{P}_{h}$, the presence of the
magnetic field adds a challenging difficulty in the estimate of the magnitude
of the tunneling which was recently solved in [7]. Our proof of (1.2) is very
close to that of [7], but it relies on new elements that follow from a deep
investigation of magnetic steps [5, 3].

Let us give some of the heuristics behind the computations leading to (1.2).
We can construct two quasi-modes having the following structure

$\displaystyle\Psi_{h,\ell}(s,t)$ $\displaystyle\approx
e^{i\theta_{h,\ell}(s)}\,e^{-\Phi_{\ell}(s)/h^{1/4}}\,f_{0,\ell}(s)\phi_{a}(h^{1/2}t),$
$\displaystyle\Psi_{h,r}(s,t)$ $\displaystyle\approx
e^{i\theta_{h,r}(s)}e^{-\Phi_{r}(s)/h^{1/4}}\,f_{0,r}(s)\phi_{a}(h^{1/2}t),$

where $s$ denotes the arc-length parameter along $\Gamma$, $t$ denotes the
normal distance to $\Gamma$ with the convention that $t>0$ in the interior of
$\Gamma$ and $t<0$ in the exterior of $\Gamma$. The functions $\Phi_{\ell}$
and $\Phi_{r}$ are non-negative and satisfy $\Phi_{\ell}(s_{\ell})=0$ and
$\Phi_{r}(s_{r})=0$, so that $\Psi_{h,\ell}$ (resp. $\Psi_{h,r}$) is localized
near $s_{\ell}$ (resp. near $s_{r}$). The phase functions $\theta_{h,\ell}$
and $\theta_{h,r}$ involve the topology of the discontinuity curve and a
spectral constant. The function $\phi_{a}$ is the ground state eigenfunction
of a model operator related to the discontinuity of the magnetic field (see
Sec. 3). Finally, $f_{0,\ell}$ and $f_{0,r}$ are solutions of appropriate
transport equations (see Theorem 5.1).

Up to truncation, the quasi-modes $\Psi_{h,\ell}$ and $\Psi_{h,r}$ are
approximations of actual bound states

$g_{h,\ell}(s,t)\approx\Psi_{h,\ell}(s,t),\quad
g_{h,r}(s,t)\approx\Psi_{h,r}(s,t),$ (1.3)

where the bound states $g_{h,\ell}$ and $g_{h,r}$ are defined via the
orthogonal projection $\Pi$ on $V:=\oplus_{i=1}^{2}\textrm{
Ker}\left(\mathcal{P}_{h}-\lambda_{i}(h)\right)$ as follows

$g_{h,\ell}(s,t):=\Pi\Psi_{h,\ell}(s,t),\quad
g_{h,r}(s,t):=\Pi\Psi_{h,r}(s,t).$

By the Gram-Schmidt process, we transform $\\{g_{h,\ell},g_{h,r}\\}$ to an
orthonormal basis $\mathcal{B}$ of $V$ and we denote by $\mathsf{M}_{h}$ the
matrix relative to $\mathcal{B}$ of the restriction of $\mathcal{P}_{h}$ to
$V$. The spectral gap for the operator $\mathcal{P}_{h}$ is the same as that
for the matrix $\mathsf{M}_{h}$,

$\lambda_{2}(h)-\lambda_{1}(h)=\lambda_{2}(\mathsf{M}_{h})-\lambda_{1}(\mathsf{M}_{h})\,.$

Using the approximation in (1.3), we get an approximate matrix
$\widehat{\mathsf{M}}_{h}$ of $\mathsf{M}_{h}$ whose spectral gap can be
explicitly estimated (compare with (1.2))

$\lambda_{2}(\widehat{\mathsf{M}}_{h})-\lambda_{1}(\widehat{\mathsf{M}}_{h})\approx\exp\Big{(}-\frac{\mathsf{S}^{a}}{h^{1/4}}\Big{)}\,.$

We have then to show that the spectral gap for the matrix
$\widehat{\mathsf{M}}_{h}$ is a good approximation of that of
$\mathcal{P}_{h}$ up to an appropriate remainder, more precisely

$\lambda_{2}(\mathsf{M}_{h})-\lambda_{1}(\mathsf{M}_{h})=\big{(}\lambda_{2}(\widehat{\mathsf{M}}_{h})-\lambda_{1}(\widehat{\mathsf{M}}_{h})\big{)}\big{(}1+o(1)\big{)}\,.$

Such an estimate is closely related to optimal decay estimates of bound states
(resp. approximate bound states) of the operator $\mathcal{P}_{h}$, which
yield accurate errors for the approximation in (1.3).

### Organisation

In Section 2, we review the recent result of [7] for the Neumann magnetic
Laplacian with a constant magnetic field, and introduce the related de Gennes
model for a flat boundary. In Section 3, we introduce the magnetic edge along
with the related flat edge model, and state our main result, Theorem 3.4, for
the operator $\mathcal{P}_{h}$ with a magnetic step. In Section 4, we express
$\mathcal{P}_{h}$ in a Frénet frame and reduce the spectral analysis to an
operator defined near the edge. In Section 5, we introduce operators with a
single well (with ground states localized near a single point of maximum
curvature), and perform a WKB expansion for an approximate ground state (see
Theorem 5.1). In Section 6, we explain how optimal tangential estimates can be
derived along the lines of the proof of the similar statement in [7]. Finally,
in Section 7, we introduce the interaction matrix and finish the proof of
Theorem 3.4, by referring to [7] for the detailed computations, which are
essentially the same in our setting.

##  2\. Uniform magnetic fields

In this section, we review some results on the Neumann realization of the
operator $\mathcal{P}_{h}$ with a constant magnetic field. We assume that

$\left\\{\,\begin{aligned} \Omega\subset\mathbb{R}^{2}&\text{ is a
}\text{connected and simply}\text{ connected open set,}\\\
&\Sigma:=\partial\Omega\text{ is a }C^{\infty}\text{ smooth closed curve.}\\\
\end{aligned}\,\right\\}$ (2.1)

and

$\mathbf{A}=\frac{1}{2}(-x_{2},x_{1})\quad\textrm{ and}\quad B=\textrm{
curl}\mathbf{A}\equiv 1\,,$ (2.2)

We consider $\mathcal{P}_{h}$ introduced in (1.1), as a self-adjoint operator
in $L^{2}(\Omega)$, with domain,

$\textsf{ Dom}(\mathcal{P}_{h})=\\{u\in H^{2}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\
{\mathbf{n}}\cdot(h\nabla-i\mathbf{A})u|_{\partial\Omega}=0\\},$

where $H^{2}(\Omega)$ denotes the Sobolev space $W^{2,2}(\Omega)$, and
$\mathbf{n}$ the unit inward normal vector of $\Sigma$.

###  2.1. Full asymptotics and decay of bound states

The conditions in (2.1) ensure that $\Omega$ is bounded and that
$\mathcal{P}_{h}$ has compact resolvent. Let $(\lambda_{n}(h))_{n\geqslant 1}$
be the sequence of eigenvalues of $\mathcal{P}_{h}$. In generic situations,
that will be explained precisely later on, there exist complete expansions of
the eigenvalues of $\mathcal{P}_{h}$, in the form [16],

$\lambda_{n}(h)\sim\Theta_{0}h-k_{\max}C_{1}h^{\frac{3}{2}}+C_{1}\Theta_{0}^{\frac{1}{4}}(2n-1)\sqrt{-\frac{3}{2}k_{2}}h^{\frac{7}{4}}+\sum_{j\geqslant
15}\zeta_{j,n}h^{j/8}\,.$ (2.3)

The coefficients $\Theta_{0}$ and $C_{1}$ appearing in (2.3) are universal
positive constants related to the de Gennes model in the half-plane (see Sec.
2.2). The coefficients $k_{\max}$ and $k_{2}$ are related to the curvature on
the boundary. Let $\Sigma$ be parameterized by arc-length $s$ and denote by
$k(s)$ the curvature of $\Sigma$ at $s$ (see Sec. 4 for the precise definition
of $k$; in particular $k\geqslant 0$ on the convex parts of
$\overline{\Omega}$). The asymptotics in (2.3) holds provided the curvature
$k$ attains its maximum value non-degenerately and at a unique point, i.e.

$k_{\max}:=\max_{\Sigma}k(s)=k(0)\quad\textrm{ with}\quad
k_{2}:=k^{\prime\prime}(0)<0\,.$ (2.4)

The sequence $(\zeta_{j,n})_{j\geqslant 15}$ is constructed recursively, and
it can be shown that $\zeta_{j,1}=0$ for odd $j^{\prime}s$ [8].

The derivation of (2.3) is related to the decay of bound states. Assume that
$n$ is fixed and for all $h>0$ that $u_{h,n}$ is an eigenfunction of
$\mathcal{P}_{h}$, normalized in $L^{2}(\Omega)$ and with eigenvalue
$\lambda_{n}(h)$. There exist constants $\alpha_{1},C_{1,n}>0$ such that

$\int_{\Omega}|u_{h}|^{2}\exp\Big{(}\frac{\alpha_{1}\textrm{
dist}(x,\Sigma)}{h^{1/2}}\Big{)}dx\leqslant C_{1,n}\,.$

This estimate says that the bound state $u_{h,n}$ concentrates near the
boundary $\Sigma$ and is valid even when (2.4) is not satisfied [21]. If
moreover (2.4) holds, then $u_{h,n}$ concentrates near the point of maximal
curvature as follows: There exist constants
$\epsilon_{0},\alpha_{2},C_{2,n}>0$ such that [16]

$\int_{\textrm{
dist}(x,\Sigma)<\epsilon_{0}}|u_{h}|^{2}\exp\Big{(}\frac{\alpha_{2}|s(x)|^{2}}{h^{1/4}}\Big{)}dx\leqslant
C_{2,n}\,,$ (2.5)

where $s(x)$ denotes the arc-length coordinate of the point
$p(x)\in\partial\Omega$ defined by $\textrm{dist}(x,p(x))=\textrm{
dist}(x,\Sigma)$. The decay estimate (2.5) is a key ingredient in the
derivation of the asymptotics in (2.3), but is not sufficient to handle the
case of symmetries that we shall discuss below.

Let us examine the case where the curvature attains its maximum at several
points $s_{1},\cdots,s_{N}$. For all $j\in\\{1,\cdots,N\\}$ and
$m\in\mathbb{N}$, we introduce

$\lambda_{m,j}^{\textrm{
app}}(h)=\Theta_{0}h-k_{\max}C_{1}h^{\frac{3}{2}}+C_{1}\Theta_{0}^{\frac{1}{4}}(2m-1)\sqrt{-\frac{3}{2}k^{\prime\prime}(s_{j})}\,h^{\frac{7}{4}}\,.$

Consider a relabeling $(m_{n},j_{n})_{n\geqslant 1}$ of $(m,j)_{m\geqslant
1,1\leqslant j\leqslant N}$ such that

$\lambda_{m_{1},j_{1}}^{\textrm{app}}\leqslant\lambda_{m_{2},j_{2}}^{\textrm{app}}\leqslant\cdots\,.$

Then, (2.3) is replaced with

$\lambda_{n}(h)=\lambda^{\textrm{app}}_{m_{n},j_{n}}+o(h^{\frac{7}{4}})\,.$
(2.6)

If additionally $k^{\prime\prime}(s_{j_{1}})=k^{\prime\prime}(s_{j_{2}})$,
then $\lambda_{2}(h)-\lambda_{1}(h)=o(h^{\frac{7}{4}})$ and we loose the
information on the simplicity of the eigenvalues. Consequently, we need a more
detailed analysis in the case of symmetries, which will rely on an optimal
tangential decay estimate improving the one given in (2.5). We will discuss
these decay estimates later in Sec. 6. Our next step is the review of an
important model with a flat boundary.

###  2.2. The de Gennes model: flat boundary

The analysis of the model case where $\Omega=\mathbb{R}\times\mathbb{R}_{+}$
and $B=\textrm{ curl\,}\mathbf{A}=1$ leads us naturally to the family
(parametrized by $\xi\in\mathbb{R}$) of harmonic oscillators (de Gennes model)

$\mathfrak{h}^{N}[\xi]=-\frac{d^{2}}{d\tau^{2}}+(\xi+\tau)^{2},$ (2.7)

on the semi-axis $\mathbb{R}_{+}$ with Neumann boundary condition at $\tau=0$.
Let us denote by $(\mu_{j}^{N}(\xi))_{j\geqslant 1}$ the sequence of
eigenvalues of $\mathfrak{h}^{N}[\xi]$. The de Gennes constant is then defined
as follows

$\Theta_{0}=\inf_{\xi\in\mathbb{R}}\mu^{N}_{1}(\xi)\,.$ (2.8)

There exists a unique minimum $\xi_{0}<0$ such that

$\Theta_{0}=\mu^{N}_{1}(\xi_{0})\,.$

Furthermore, $\xi_{0}=-\sqrt{\Theta_{0}}$,
$(\mu^{N}_{1})^{\prime\prime}(\xi)>0$ and $\frac{1}{2}<\Theta_{0}<1$. Denoting
by $u_{0}$ the positive and normalized ground state of
$\mathfrak{h}^{N}[\xi_{0}]$, we can introduce the constant $C_{1}$ appearing
in (2.3),

$C_{1}=\frac{|u_{0}(0)|^{2}}{3}\,.$ (2.9)

###  2.3. Symmetric domains and tunneling

We continue to work under the conditions in (2.1) but we assume furthermore
that the domain $\Omega$ is symmetric with respect to an axis and the
curvature of its boundary $\Gamma$ has exactly two non-degenerate maxima. More
precisely, the hypotheses are (see Fig 1):

######  Assumption 2.1.

  1. i)

$\Omega$ is symmetric with respect to the $y$-axis.

  2. ii)

The curvature $k$ on $\Sigma$ attains its maximum at exactly two symmetric
points $a_{1}=(a_{1,1},a_{1,2})$ and $a_{2}=(a_{2,1},a_{2,2})$ with
$a_{1,1}>0$ and $a_{2,1}<0$.

  3. iii)

Denoting by $s_{r}$ and $s_{\ell}$ the arc-length coordinates of $a_{1}$ and
$a_{2}$ respectively, we have
$k^{\prime\prime}(s_{r})=k^{\prime\prime}(s_{\ell})<0$.

Figure 1. A symmetric domain with respect to the $y$-axis (dashed line). The
curvature along the boundary has two non-degenerate maxima at the points with
arc-length coordinates $s_{r}$ and $s_{\ell}$, connected by upward and
downward geodesics.

This situation induces a tunneling effect where the energy difference between
the ground and first excited states is exponentially small. The magnitude of
this splitting has been rigorously computed recently in [7].

Let us introduce the following effective quantities:

$V(s)=\frac{2C_{1}(k_{\max}-k(s))}{\mu^{\prime\prime}(\xi_{0})}\,,$ (2.10)

and

$\displaystyle\mathsf{A}_{\mathsf{u}}$
$\displaystyle=\exp\left(-\int_{[s_{r},0]}\frac{(V^{\frac{1}{2}})^{\prime}(s)+g}{\sqrt{V(s)}}ds\right)\,,$
(2.11) $\displaystyle\mathsf{A}_{\mathsf{d}}$
$\displaystyle=\exp\left(-\int_{[s_{\mathsf{\ell}},L]}\frac{(V^{\frac{1}{2}})^{\prime}(s)-g}{\sqrt{V(s)}}ds\right)\,,$
$\displaystyle g$
$\displaystyle=\left(V^{\prime\prime}(s_{r})/2\right)^{\frac{1}{2}}=\left(V^{\prime\prime}(s_{\mathsf{\ell}})/2\right)^{\frac{1}{2}}\,.$

In the above formulae, $0$ and $L$ are the arc-length coordinates of the
points of intersection between the $y$-axis and the curve $\Sigma$, with the
convention that $0$ represents the point on the upper part of $\Sigma$ (see
Fig. 1).

######  Theorem 2.2 (Bonnaillie-Noël–Hérau–Raymond [7]).

Suppose that (2.1), (2.2) and Assumption 2.1 hold. Then the first and second
eigenvalues of $\mathcal{P}_{h}$ satisfy, as $h\to 0_{+}$,

$\lambda_{2}(h)-\lambda_{1}(h)=2|\tilde{w}(h)|+o(h^{\frac{13}{8}}e^{-\mathsf{S}/h^{\frac{1}{4}}})\,,$

where

$\displaystyle\tilde{w}(h)$
$\displaystyle=\mu^{\prime\prime}(\xi_{0})h^{\frac{13}{8}}\pi^{-\frac{1}{2}}g^{\frac{1}{2}}$
$\displaystyle\quad\times\left(\mathsf{A}_{\mathsf{u}}\sqrt{V(0)}e^{-\mathsf{S}_{\mathsf{u}}/h^{1/4}}e^{iLf(h)}+\mathsf{A}_{\mathsf{d}}\sqrt{V(L)}e^{-\mathsf{S}_{\mathsf{d}}/h^{1/4}}e^{-iLf(h)}\right)\,,$

and

  1. i.

The potential $V$ is introduced in (2.10);

  2. ii.

$\mathsf{S}$ is the Agmon distance between the wells,

$\mathsf{S}=\min\left(\mathsf{S}_{\mathsf{u}},\mathsf{S}_{\mathsf{d}}\right),\leavevmode\nobreak\
\mathsf{S}_{\mathsf{u}}=\int_{[s_{r},s_{\mathsf{\ell}}]}\sqrt{V(s)}\,ds,\leavevmode\nobreak\
\mathsf{S}_{\mathsf{d}}=\int_{[s_{\mathsf{\ell}},s_{\mathsf{r}}]}\sqrt{V(s)}\,ds\,;$
(2.12)

  3. iii.

$\mathsf{A}_{\mathsf{u}}$, $\mathsf{A}_{\mathsf{d}}$ and $g$ are defined in
(2.11);

  4. iv.

$f(h)=\gamma_{0}/h+\xi_{0}/h^{1/2}-\alpha_{0}$ with

$\gamma_{0}=\frac{|\Omega|}{|\Sigma|},$

where $|\Sigma|$ is the length of $\Sigma$, and $\alpha_{0}$ is a constant
dependent on $\Omega$.

Theorem 2.2 can be extended to the situation of $N\geqslant 3$ wells, which
corresponds to a domain having symmetry by rotation of angle $2\pi/N$ and $N$
points of maximum curvature (see Fig. 2).

Figure 2. A symmetric domain with respect to the origin with $N=4$ points of
maximum curvature.

##  3\. Magnetic steps

The tunneling effect in Theorem 2.2 is a consequence of imposing the Neumann
boundary condition (if a magnetic field were not present, the first eigenvalue
would be simple and equal to $0$). The present contribution is concerned with
the following question:
Can we observe a tunneling effect, similar to the one in Theorem 2.2, but
induced purely by the magnetic field ?
That is, we would like to construct an example where the tunneling is not a
consequence of imposing a boundary condition, but rather a consequence of the
nature of the magnetic field. We will give an affirmative answer by working in
the full plane $\mathbb{R}^{2}$ and considering a magnetic field with a
discontinuity along a smooth curve111 From a technical perspective, the
magnetic discontinuity curve plays the same role in our case as the boundary
does in Theorem 2.2. (the magnetic edge). In the case of a flat edge, we get a
model in the full plane which plays the role of the de Gennes model for
uniform magnetic fields. When the edge is non-flat and has symmetries, we
observe an interesting tunneling effect.

###  3.1. A new model: flat edge

Let us recall the model in $\mathbb{R}^{2}$ where $B=\textrm{
curl\,}\mathbf{A}=\mathbf{1}_{\mathbb{R}_{+}\times\mathbb{R}}+a\mathbf{1}_{\mathbb{R}_{-}\times\mathbb{R}}$
and $a\in[-1,0)$ is a fixed constant222It is important for us to have $a<0$,
because in the opposite case, $a\in(0,1)$, $\mu_{a}(\xi)$ defined in (3.3)
becomes a monotone increasing function with
$\inf_{\xi{\mathbb{R}}}\mu_{a}(\xi)=a$. This implies that the magnetic step
will no longer attract the ground state, i.e. we do not expect localization
near the magnetic step in this case. . We get in this case a family of
Schrödinger operators [28]

$\mathfrak{h}_{a}[\xi]=-\frac{d^{2}}{d\tau^{2}}+V_{a}(\xi,\tau),$ (3.1)

on $L^{2}(\mathbb{R})$, where $\xi\in\mathbb{R}$ is a parameter and

$V_{a}(\xi,\tau)=\big{(}\xi+b_{a}(\tau)\tau\big{)}^{2},\quad
b_{a}(\tau)=\mathbf{1}_{\mathbb{R}_{+}}(\tau)+a\mathbf{1}_{\mathbb{R}_{-}}(\tau)\,.$
(3.2)

We introduce the ground state energy of $\mathfrak{h}_{a}[\xi]$,

$\mu_{a}(\xi)=\inf_{u\in B^{1}(\mathbb{R}),u\neq
0}\frac{q_{a}[\xi](u)}{\|u\|^{2}_{L^{2}(\mathbb{R})}}\,,$ (3.3)

along with the following constant

$\beta_{a}:=\inf_{\xi\in\mathbb{R}}\mu_{a}(\xi)=\mu_{a}(\zeta_{a})\,,$ (3.4)

where $\zeta_{a}<0$, is the unique minimum of $\mu_{a}(\cdot)$. Let $\phi_{a}$
be the _positive_ and $L^{2}$-normalized ground state of $\mathfrak{h}_{a}$.
We have [5]

$c_{2}(a):=\frac{1}{2}\mu_{a}^{\prime\prime}(\zeta_{a})>0$ (3.5)

and

$|a|\Theta_{0}<\beta_{a}<\min(|a|,\Theta_{0}),\quad\phi_{a}^{\prime}(0)<0\quad(-1<a<0)\,.$
(3.6)

For $a=-1$, we have by a symmetry argument

$\beta_{-1}=\Theta_{0}\,,\quad\zeta_{-1}=\xi_{0}\,,\quad\phi_{-1}(\tau)=u_{0}(|\tau|)\,,$
(3.7)

thereby returning to the de Gennes model introduced in Sec. 2.2.

Later on, the following negative constant will be of particular interest,

$M_{3}(a)=\frac{1}{3}\Big{(}\frac{1}{a}-1\Big{)}\zeta_{a}\phi_{a}(0)\phi_{a}^{\prime}(0)<0.$
(3.8)

###  3.2. Curved edge and single well

We return to the operator $\mathcal{P}_{h}$ in (1.1). Here and in the rest of
the paper, we will work under the following assumption

$\left\\{\,\begin{aligned} \Omega_{1}\subset\mathbb{R}^{2}\text{ is a
}&\text{connected and simply connected open set,}\leavevmode\nobreak\
\Omega_{2}=\mathbb{R}^{2}\setminus\overline{\Omega}_{1},\\\
&\Gamma:=\partial\Omega_{1}\text{ is a }C^{\infty}\text{ smooth closed
curve.}\\\ \end{aligned}\,\right\\}$ (3.9)

and that the magnetic field is a step function (see Fig. 3)

Figure 3. The plane $\mathbb{R}^{2}=\Omega_{1}\cup\Omega_{2}\cup\Gamma$ with
the edge $\Gamma=\partial\Omega_{1}$ dashed.

$B=\mathbf{1}_{\Omega_{1}}+a\mathbf{1}_{\Omega_{2}}\quad\textrm{
where\leavevmode\nobreak\ }-1<a<0\,.$ (3.10)

The operator $\mathcal{P}_{h}$ is then self-adjoint in $L^{2}(\mathbb{R}^{2})$
with domain

$\textsf{ Dom}(\mathcal{P}_{h})=\\{u\in
L^{2}(\mathbb{R}^{2})\leavevmode\nobreak\ :\leavevmode\nobreak\
(h\nabla-i\mathbf{A})^{j}u\in L^{2}(\mathbb{R}^{2}),\leavevmode\nobreak\
j=1,2\\}.$ (3.11)

By Persson’s lemma [33], the essential spectrum of $\mathcal{P}_{h}$ is
determined by the magnetic field at infinity (in our case it is equal to $a$),
so

$\sigma_{\textrm{ ess}}(\mathcal{P}_{h})=[|a|h,+\infty)\,.$

Since $\beta_{a}<\min(|a|,\Theta_{0})$, bound states of $\mathcal{P}_{h}$ are
localized near the edge [5]. More precisely, for every $n\in\mathbb{N}$, there
exist constants $\alpha,h_{0},C_{n}>0$ such that,

$\int_{\mathbb{R}^{2}}\big{(}|u_{h,n}|^{2}+h^{-1}|(h\nabla-i\mathbf{A})u_{h,n}|^{2}\big{)}\exp\Big{(}\frac{\alpha\,\textrm{
dist}(x,\Gamma)}{h^{1/2}}\Big{)}dx\leqslant C_{n},$ (3.12)

for all $h\in(0,h_{0}]$, where $u_{h,n}$ is a normalized eigenfunction
associated to the $n$’th eigenvalue of $\mathcal{P}_{h}$.

######  Remark 3.1 (The case of bounded domains).

We can also consider the Dirichlet or Neumann realizations of
$\mathcal{P}_{h}$ in a bounded smooth domain $\Omega$, in which case the
spectrum is purely discrete. Related to our setting is [3, Thm. 1.2] dealing
with a somehow different geometric condition, where the operator
$\mathcal{P}_{h}$ is considered in $L^{2}(\Omega)$ with Dirichlet boundary
condition, $\Omega_{1}\subset\Omega$ and $\Gamma$ a smooth curve that meets
$\partial\Omega$ transversely, see Fig. 4. However, the proofs are not altered
by considering the new setting of $\mathcal{P}_{h}$ above ($\mathcal{P}_{h}$
in the full plane and closed curve $\Gamma$). The main reason is that the
property $\beta_{a}<|a|$ for $-1<a<0$ ensures the localization of the bound
states near the edge $\Gamma$.

So the following result essentially follows from [3, Thm. 1.2]:

######  Theorem 3.2.

Assume that (3.9) and (3.10) hold and that the curvature $k$ of $\Gamma$ has a
unique non-degenerate maximum, i.e.

$k_{\max}:=\max_{\Gamma}k(s)=k(0)\quad\textrm{ with}\quad
k_{2}:=k^{\prime\prime}(0)<0$

Then, for all $n\in\mathbb{N}^{*}$ the $n$’th eigenvalue $\lambda_{n}(h)$ of
$\mathcal{P}_{h}$, defined in (1.1), satisfies as $h\rightarrow 0$,

$\lambda_{n}(h)=\beta_{a}h+k_{\textrm{
max}}M_{3}(a)h^{\frac{3}{2}}+(2n-1)\sqrt{\frac{{k_{2}}M_{3}(a)c_{2}(a)}{2}}h^{\frac{7}{4}}+\mathcal{O}(h^{\frac{15}{8}}),$

where $\beta_{a}$, $c_{2}(a)$ and $M_{3}(a)$ are introduced in (3.4), (3.5)
and (3.8) respectively.

Looking more closely at Theorem 3.2, we observe that the third term in the
expansion of $\lambda_{n}(h)$ is effectively given (up to the factor of
$h^{3/2}$) by the $n$’th eigenvalue of the following 1D operator on
$L^{2}\big{(}\mathbb{R}/(2L\mathbb{Z})\big{)}$,

$\mathfrak{L}_{h}^{\textrm{
eff}}=\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}\left(-h^{\frac{1}{2}}\partial_{s}^{2}+V_{a}(s)\right)\,,\quad
V_{a}(s)=\frac{2M_{3}(a)(k(s)-k_{\max})}{\mu^{\prime\prime}_{a}(\zeta_{a})},$
(3.13)

where $L=|\Gamma|/2$ and $|\Gamma|$ denotes the arc-length of $\Gamma$.
Notice, that $V_{a}\geqslant 0$, due to the sign of $M_{3}(a)$ (see (3.8)).
This point of view is important in order to discuss the case where $\Gamma$
has symmetries and the splitting of the eigenvalues is no more of polynomial
order in $h$.

In the presence of several points of maximal curvature, a variant of Theorem
3.2 continues to hold but we may loose the information on the simplicity of
the eigenvalues, exactly in the same manner observed for the Neumann problem
(see (2.6)).

###  3.3. Symmetric edge and tunneling

Suppose that, in addition to (3.9) and (3.10), the following holds (see Fig
1):

######  Assumption 3.3.

  1. i)

$\Omega_{1}$ is symmetric with respect to the $y$-axis.

  2. ii)

The curvature $k$ on $\Gamma$ attains its maximum at exactly two symmetric
points $a_{1}=(a_{1,1},a_{1,2})$ and $a_{2}=(a_{2,1},a_{2,2})$ with
$a_{1,1}>0$ and $a_{2,1}<0$.

  3. iii)

Denoting by $s_{r}$ and $s_{\ell}$ the arc-length coordinates of $a_{1}$ and
$a_{2}$ respectively, we have
$k^{\prime\prime}(s_{r})=k^{\prime\prime}(s_{\ell})<0$.

This is exactly the same geometric assumption on $\Omega$ as Assumption 2.1
for the Neumann realization in $L^{2}(\Omega)$, with the edge $\Gamma$ playing
the role of $\Sigma$, the boundary of $\Omega$.

The presence of a symmetric edge yields a symmetric potential, and
consequently two wells, in the effective operator introduced in (3.13), which
in turn will induce a tunneling effect whose order of magnitude can be
measured by the following quantities (similarly to what we have seen in
Theorem 2.2):

$\displaystyle\mathsf{A}_{\mathsf{u}}^{a}$
$\displaystyle=\exp\left(-\int_{[s_{r},0]}\frac{(V_{a}^{\frac{1}{2}})^{\prime}(s)+g_{a}}{\sqrt{V_{a}(s)}}ds\right)\,,$
(3.14) $\displaystyle\mathsf{A}_{\mathsf{d}}^{a}$
$\displaystyle=\exp\left(-\int_{[s_{\mathsf{\ell}},L]}\frac{(V_{a}^{\frac{1}{2}})^{\prime}(s)-g_{a}}{\sqrt{V(s)}}ds\right)\,,$
$\displaystyle g_{a}$
$\displaystyle=\left(V_{a}^{\prime\prime}(s_{r})/2\right)^{\frac{1}{2}}=\left(V_{a}^{\prime\prime}(s_{\mathsf{\ell}})/2\right)^{\frac{1}{2}}\,.$

Up to leading order, the operator in (3.13) continues to be effective under
the new assumptions on the edge, modulo additional terms related to the
circulation of the magnetic field and the geometry.

######  Theorem 3.4.

Suppose that Assumption 3.3 holds in addition to (3.9) and (3.10). The first
and second eigenvalues of $\mathcal{P}_{h}$ satisfy as $h\to 0_{+}$,

$\lambda_{2}(h)-\lambda_{1}(h)=2|\tilde{w}_{a}(h)|+o(h^{\frac{13}{8}}e^{-\mathsf{S}^{a}/h^{\frac{1}{4}}})\,,$

where:

$\displaystyle\tilde{w}_{a}(h)$
$\displaystyle=\mu_{a}^{\prime\prime}(\zeta_{a})h^{\frac{13}{8}}\pi^{-\frac{1}{2}}g^{\frac{1}{2}}_{a}$
$\displaystyle\quad\times\left(\mathsf{A}_{\mathsf{u}}^{a}\sqrt{V_{a}(0)}e^{-\mathsf{S}_{\mathsf{u}}^{a}/h^{1/4}}e^{iLf_{a}(h)}+\mathsf{A}_{\mathsf{d}}^{a}\sqrt{V_{a}(L)}e^{-\mathsf{S}_{\mathsf{d}}^{a}/h^{1/4}}e^{-iLf_{a}(h)}\right)\,,$

with $\mu_{a}$ and $\zeta_{a}$ introduced in Section 3.1, and

  1. i.

The potential $V_{a}$ is introduced in (3.13);

  2. ii.

$\mathsf{S}^{a}$ is the Agmon distance between the wells,

$\mathsf{S}^{a}=\min\left(\mathsf{S}_{\mathsf{u}}^{a},\mathsf{S}_{\mathsf{d}}^{a}\right),\leavevmode\nobreak\
\mathsf{S}_{\mathsf{u}}^{a}=\int_{[s_{r},s_{\mathsf{\ell}}]}\sqrt{V_{a}(s)}\,ds,\leavevmode\nobreak\
\mathsf{S}_{\mathsf{d}}^{a}=\int_{[s_{\mathsf{\ell}},s_{\mathsf{r}}]}\sqrt{V_{a}(s)}\,ds\,;$
(3.15)

  3. iii.

$\mathsf{A}_{\mathsf{u}}^{a}$, $\mathsf{A}_{\mathsf{d}}^{a}$ and $g_{a}$ are
defined in (3.14);

  4. iv.

$f_{a}(h)=\gamma_{0}/h+\zeta_{a}/h^{1/2}-\alpha_{0}(a)$ with

$\gamma_{0}=\frac{|\Omega_{1}|}{|\Gamma|},$ (3.16)

and $\alpha_{0}(a)$ is a constant dependent on $a$ and $\Omega_{1}$.

Theorem 3.4 is the analogue of Theorem 2.2 but for the situation where
tunneling is due to the discontinuity of the magnetic field (without the need
for imposing a Neumann boundary condition). As in the proof of Theorem 2.2 in
[7], the proof of Theorem 3.4 relies on an optimal tangential decay estimate
of ground states.

###  3.4. Bounded domains

Theorem 3.4 continues to hold if we consider the Dirichlet or Neumann
realization of the operator $\mathcal{P}_{h}$ in $L^{2}(\Omega)$, where
$\Omega$ is a domain with a $C^{2}$ boundary such that
$\overline{\Omega}_{1}\subset\Omega$ (see Fig 4). Thanks to (3.6), bound
states of $\mathcal{P}_{h}$ are localized near $\Gamma=\partial\Omega_{1}$,
and the proof of Theorem 3.4 is not altered.

Figure 4. The domain $\Omega$ is split into two parts with the edge $\Gamma$
(dashed) is a closed curve. Figure 5. The edge $\Gamma$ (dashed) splits the
domain $\Omega$ into two parts and intersects the boundary $\partial\Omega$
transversely.

We can also modify the configuration of our domains $\Omega_{1}$ and
$\Omega_{2}$ in (3.9) and still get the tunneling effect but without
oscillatory terms. Let $\Omega$ be a domain with a $C^{1}$ boundary such that
$\overline{\Omega}=\overline{\Omega}_{1}\cup\overline{\Omega}_{2}$, where
$\Omega_{1}$ and $\Omega_{2}$ are disjoint simply connected open sets. We
consider a magnetic field as in (3.10) and notice that the edge
$\Gamma=\Omega\cap\partial\Omega_{1}=\Omega\cap\partial\Omega_{2}$ (see Fig
5). We assume that $\Gamma$ is a smooth curve and consider the Dirichlet333The
Neumann realization leads to a completely different behavior, reminiscent of
domains with corners [1]. realization of $\mathcal{P}_{h}$ in $L^{2}(\Omega)$.
This is the situation considered in [3]. Now we assume that the curvature $k$
along $\Gamma$ has a non-degenerate maximum attained at two points, with arc-
length coordinates $s_{\ell}<0$ and $s_{r}=-s_{\ell}$, and that it is an even
function in a neighborhood of $[s_{\ell},s_{r}]$. In this situation, the
splitting between the first eigenvalues is given as follows:

$\lambda_{2}(h)-\lambda_{1}(h)=2|w_{a}(h)|+o(h^{\frac{13}{8}}e^{-\mathsf{S}^{a}/h^{\frac{1}{4}}}),$

where

$w_{a}(h)=2\mu_{a}^{\prime\prime}(\zeta_{a})h^{\frac{13}{8}}\pi^{-\frac{1}{2}}g^{\frac{1}{2}}_{a}\mathsf{A}_{a}\sqrt{V_{a}(0)}\,e^{-\mathsf{S}_{a}/h^{1/4}},$

and

$\mathsf{A}_{a}=2\exp\left(-\int_{[s_{\ell},s_{r}]}\frac{(V_{a}^{\frac{1}{2}})^{\prime}(s)-g_{a}}{\sqrt{V_{a}(s)}}ds\right),\quad\mathsf{S}_{a}=\int_{[s_{\ell},s_{r}]}\sqrt{V(s)}ds.$

##  4\. Reduction to a neighborhood of the edge

It will be convenient to work in Frénet coordinates, $(s,t)$, along the edge
$\Gamma$, valid in a neighborhood of $\Gamma$ of the form

$\Gamma(\epsilon)=\\{x\in\mathbb{R}^{2}\leavevmode\nobreak\
:\leavevmode\nobreak\ \textrm{
dist}(x,\Gamma)<\epsilon\\}\quad(\epsilon>0)\,.$ (4.1)

Let us briefly recall these coordinates. Consider an arc-length
parameterization of $\Gamma$ , $M:(-L,L]\to\Gamma$, so that (see Assumption
3.3)

$M(s_{\ell})=a_{1},\quad M(s_{r})=a_{2},\quad-L<s_{\ell}<0,\quad 0<s_{r}<L\,,$

and

$\Gamma\cap\\{(x,y)\in\mathbb{R}^{2}\,|\,\leavevmode\nobreak\
x=0\\}=\\{M(0)=:(0,y_{0}),M(L)=:(0,y_{1})\\}\quad\textrm{
with\leavevmode\nobreak\ }y_{0}<y_{1}\,.$

Let $\mathbf{n}(s)$ be the unit normal to $\Gamma$ pointing inward to
$\Omega_{1}$ (see 1), $\mathbf{t}(s)=\dot{\mathbf{n}}(s)$ the unit oriented
tangent, so that $\textrm{ det}(\mathbf{t}(s),\mathbf{n}(s))=1$. Let us
represent the torus $\mathbb{R}/2L\mathbb{Z}$ by the interval $(-L,L]$. We can
pick $\epsilon_{0}>0$ such that

$\Phi:\mathbb{R}/(2L\mathbb{Z})\times(-\epsilon_{0},\epsilon_{0})\ni(s,t)\mapsto
M(s)+t\mathbf{n}(s)\in\Gamma(\epsilon_{0})$

is a diffeomorphism whose Jacobian is

$\mathfrak{a}(s,t)=1-tk(s)\,,$

with $k(s)$ the curvature at $M(s)$, defined by
$\ddot{\mathbf{n}}(s)=k(s)\mathbf{n}(s)$. The Hilbert space
$L^{2}(\Gamma(\epsilon_{0}))$ is transformed to the weighted space

$L^{2}\big{(}\mathbb{R}/2L\mathbb{Z}\times(-\epsilon_{0},\epsilon_{0});\mathfrak{a}\,dsdt\big{)}$

and the operator $\mathcal{P}_{h}$ is transformed into the following operator
(after a gauge transformation
$(u,\mathbf{A})\to(v=e^{i\phi/h}u,\mathbf{A}^{\prime}=\mathbf{A}-\nabla\phi)$
to eliminate the normal component of $\mathbf{A}$, see [17, App. F]):

$\tilde{\mathcal{P}}_{h}:=-h^{2}\mathfrak{a}^{-1}\partial_{t}\mathfrak{a}\partial_{t}\\\
+\mathfrak{a}^{-1}\left(-ih\partial_{s}+\gamma_{0}-b_{a}(t)t+\frac{k}{2}b_{a}(t)t^{2}\right)\mathfrak{a}^{-1}\left(-ih\partial_{s}+\gamma_{0}-b_{a}(t)t+\frac{k}{2}b_{a}(t)t^{2}\right)$

where $b_{a}(t)$ is introduced in (3.2) and $\gamma_{0}$ is the circulation
introduced in (3.16).

Following [7], it is convenient to introduce a truncated version of the
operator $\tilde{P}_{h}$ so that it can be defined on
$\mathbb{R}/2L\mathbb{Z}\times\mathbb{R}$ instead of
$\mathbb{R}/2L\mathbb{Z}\times(-\epsilon_{0},\epsilon_{0})$. This will be
useful when rescaling the $t$ variable. What is handy in this situation is
that the actual bound states of the operator $\mathcal{P}_{h}$ decay
exponentially away from the edge, at the length scale $\hbar:=h^{1/2}$, see
(3.12). This motivates the change of variables, $t=\hbar\tau$ and $s=\sigma$,
that will allow the same spectral reduction as in [7, Prop. 2.7]. We will skip
the details which are the same as in [7].

From now on we set

$\mu=h^{\frac{1}{4}+\eta}\text{ for a fixed }\eta\in(0,\frac{1}{4})$ (4.2)

and we introduce the function

$c_{\mu}(\tau)=c(\mu\tau)\,,$ (4.3)

where $c\in C_{c}^{\infty}(\mathbb{R})$ satisfies $c=1$ on $[-1,1]$ and $c=0$
on $\mathbb{R}\setminus(-2,2)$. Consider the new weight term

$\tilde{\mathfrak{a}}_{h}(\sigma,\tau)=1-h^{1/2}c_{\mu}(\tau)\tau
k(\sigma)\,,$

and the self-adjoint operator $\tilde{\mathcal{N}}_{h}$ on the Hilbert space
$L^{2}(\mathbb{R}/2L\mathbb{Z}\times\mathbb{R};\tilde{\mathfrak{a}}_{h}d\sigma
d\tau)$,

$\displaystyle\tilde{\mathcal{N}}_{h}$
$\displaystyle=-\tilde{\mathfrak{a}}^{-1}_{h}\partial_{\tau}\mathfrak{a}_{\hbar}\partial_{\tau}$
$\displaystyle\quad+\tilde{\mathfrak{a}}^{-1}_{h}\left(-ih^{1/2}\partial_{\sigma}+\gamma_{0}-b_{a}\tau+h^{1/2}c_{\mu}\frac{k}{2}b_{a}\tau^{2}\right)$
$\displaystyle\quad\quad\times\tilde{\mathfrak{a}}^{-1}_{h}\left(-ih^{1/2}\partial_{\sigma}+\gamma_{0}-b_{a}\tau+h^{1/2}c_{\mu}\frac{k}{2}b_{a}\tau^{2}\right),$
(4.4)

with domain

$\displaystyle\textsf{ Dom}(\tilde{\mathcal{N}}_{\hbar})=\\{u\in
L^{2}(\mathbb{R}/2L\mathbb{Z}\times\mathbb{R})$
$\displaystyle\leavevmode\nobreak\ |\leavevmode\nobreak\
\partial_{\tau}^{2}u\in L^{2}(\Gamma\times\mathbb{R}),$
$\displaystyle(-ih^{1/2}\partial_{\sigma}+\gamma_{0}-b_{a}\tau)^{2}u\in
L^{2}(\mathbb{R}/2L\mathbb{Z}\times\mathbb{R})\\}.$

We have now the following spectral reduction:

######  Proposition 4.1.

Let $a\in(-1,0)$ and $\mathsf{S}^{a}$ be the Agmon distance introduced in
(3.15). There exist $K>\mathsf{S}^{a}$, $C,h_{0}>0$ such that, for all
$h\in(0,h_{0})$, we have

$\lambda_{n}(h)-Ce^{-K/h^{\frac{1}{4}}}\,\leqslant
h\lambda_{n}(\tilde{\mathcal{N}}_{h})\leqslant\lambda_{n}(h)+Ce^{-K/h^{\frac{1}{4}}}\,,$

where $\lambda_{n}(h)$ and $\lambda_{n}(\tilde{\mathcal{N}}_{h})$ are the
$n$’th (min-max) eigenvalues of the operators $\mathcal{P}_{h}$ and
$\tilde{\mathcal{N}}_{h}$ respectively.

Looking at the operator in (4), the effective semi-classical parameter is
$\hbar=h^{1/2}$ (this is the parameter appearing in front of
$\partial_{\sigma}$). So with

$\hbar=h^{\frac{1}{2}},\quad\mu=\hbar^{\frac{1}{2}+2\eta}\text{ for a fixed
}\eta\in(0,\frac{1}{4})\,,$ (4.5)

we introduce the new weight term

$\mathfrak{a}_{\hbar}(\sigma,\tau)=1-\hbar c_{\mu}(\tau)\tau k(\sigma)\,,$

and the self-adjoint operator $\mathcal{N}_{\hbar}$ on the Hilbert space
$L^{2}(\mathbb{R}/2L\mathbb{Z}\times\mathbb{R};\mathfrak{a}_{\hbar}d\sigma
d\tau)$, which is nothing but the operator in (4) but with a change of
parameter according to (4.5),

$\displaystyle\mathcal{N}_{\hbar}$
$\displaystyle=-\mathfrak{a}^{-1}_{\hbar}\partial_{\tau}\mathfrak{a}_{\hbar}\partial_{\tau}$
(4.6)
$\displaystyle\quad+\mathfrak{a}^{-1}_{\hbar}\left(-i\hbar\partial_{\sigma}+\gamma_{0}-b_{a}\tau+\hbar
c_{\mu}\frac{k}{2}b_{a}\tau^{2}\right)\mathfrak{a}^{-1}_{\hbar}$
$\displaystyle\qquad\times\left(-i\hbar\partial_{\sigma}+\gamma_{0}-b_{a}\tau+\hbar
c_{\mu}\frac{k}{2}b_{a}\tau^{2}\right)\,.$

The domain of the operator $\mathcal{N}_{\hbar}$ is

$\displaystyle\textsf{ Dom}(\mathcal{N}_{\hbar})=\\{u\in
L^{2}(\Gamma\times\mathbb{R})\leavevmode\nobreak\ |\leavevmode\nobreak\ $
$\displaystyle\partial_{\tau}^{2}u\in
L^{2}(\mathbb{R}/2L\mathbb{Z}\times\mathbb{R}),$
$\displaystyle(-i\hbar\partial_{\sigma}+\gamma_{0}-b_{a}\tau)^{2}u\in
L^{2}(\mathbb{R}/2L\mathbb{Z}\times\mathbb{R})\\}.$

With Proposition 4.1 in hand, it is enough to compute the leading order term
of $\nu_{2}(\hbar)-\nu_{1}(\hbar)$ to prove Theorem 3.4, where, for
$n\geqslant 1$, we denote by $\nu_{n}(\hbar)$ the $n$’th min-max eigenvalue of
$\mathcal{N}_{\hbar}$.

##  5\. Single well and WKB construction

We will adjust the edge $\Gamma$ so that we only have a single point of
maximum curvature, $s_{r}$ or $s_{\ell}$. This procedure will give us two new
operators, the “right well” and “left well” operators,
$\mathcal{N}_{\hbar,r,\gamma_{0}}$ and $\mathcal{N}_{\hbar,\ell,\gamma_{0}}$
respectively. The same procedure appears, for similar problems in the context
of geometrically induced tunneling effects [20, 31], but we follow here [7,
Sec. 2.4] which is slightly different, but more convenient for dealing with
the symbol of the operator later on.

###  5.1. Right well operator

We present the construction for the right well operator,
$\mathcal{N}_{\hbar,r,\gamma_{0}}$ and deduce the other one by symmetry. Let
us fix $\hat{\eta}$ as follows

$0<\hat{\eta}<\min\Big{(}\frac{1}{4},\frac{L}{4}\Big{)}\quad\textrm{
where\leavevmode\nobreak\ }L=\frac{|\Gamma|}{2}.$ (5.1)

First, we identify $\Gamma$ with $(s_{\ell}-2L,s_{\ell}]$ (by periodicity and
translation of the $s$ variable), then we extend the curvature $k$ to a
function $k_{r}$ on $\mathbb{R}$ as follows:

$\displaystyle k_{r}$ $\displaystyle=k\qquad\text{on}\qquad
I_{2\hat{\eta},r}:=(s_{\ell}-2L+\hat{\eta},s_{\ell}-\hat{\eta}),$
$\displaystyle k_{r}$
$\displaystyle=0\qquad\text{on}\qquad(-\infty,s_{\ell}-2L]\cup[s_{\ell}-\hat{\eta},+\infty)\,,$
(5.2)

and $k_{r}$ has a unique non-degenerate maximum at $s_{r}$. Consequently,
$k_{r}$ satisfies (2.4).

We consider now the operator in
$L^{2}(\mathbb{R}^{2};\mathfrak{a}_{\hbar,r}d\sigma d\tau)$,

$\displaystyle\mathcal{N}_{\hbar,r,\gamma_{0}}$
$\displaystyle=-\mathfrak{a}^{-1}_{\hbar,r}\partial_{\tau}\mathfrak{a}_{\hbar,r}\partial_{\tau}$
$\displaystyle\quad+\mathfrak{a}^{-1}_{\hbar,r}\left(-i\hbar\partial_{\sigma}+\gamma_{0}-b_{a}\tau+\hbar
c_{\mu}\frac{k_{r}}{2}b_{a}\tau^{2}\right)\mathfrak{a}^{-1}_{\hbar,r}$
$\displaystyle\qquad\times\left(-i\hbar\partial_{\sigma}+\gamma_{0}-b_{a}\tau+\hbar
c_{\mu}\frac{k_{r}}{2}b_{a}\tau^{2}\right)$ (5.3)

where

$\mathfrak{a}_{\hbar,r}(\sigma,\tau)=1-\hbar c_{\mu}(\tau)\tau
k_{r}(\sigma)\,.$ (5.4)

Since, $k_{r}$ satisfies (2.4), we have, for an arbitrarily fixed
$n\in\mathbb{N}$ (with $\beta_{a}$, $c_{2}(a)$ and $M_{3}(a)$ from (3.4),
(3.5) and (3.8)),

$\lambda_{n}(\mathcal{N}_{\hbar,r,\gamma_{0}})=\beta_{a}h+k_{max}M_{3}(a)h^{\frac{3}{2}}+(2n-1)\sqrt{\frac{{k_{2}}M_{3}(a)c_{2}(a)}{2}}h^{\frac{7}{4}}+\mathcal{O}(h^{\frac{15}{8}})\,.$
(5.5)

We are now in a simply connected domain, so the operators
$\mathcal{N}_{\hbar,r,\gamma_{0}}$ and $\mathcal{N}_{\hbar,r,0}$ are unitarily
equivalent: Denote by $u_{\hbar,r}$ a normalized ground state of
$\mathcal{N}_{\hbar,r,0}$ (the operator without flux term), a corresponding
normalized ground state of $\mathcal{N}_{\hbar,r,\gamma_{0}}$ is given by:

$\widecheck{\phi}_{\hbar,r}(\sigma,\tau)=e^{-i\gamma_{0}\sigma/\hbar^{2}}u_{\hbar,r}(\sigma,\tau).$
(5.6)

###  5.2. Left well operator

Using the symmetry operator

$Uf(\sigma,\tau):=\overline{f(-\sigma,\tau)}\,,$

we can define the left well operator on
$L^{2}(\mathbb{R}^{2};\mathfrak{a}_{\hbar,\ell}(\sigma,\tau)$ by :

$\mathcal{N}_{\hbar,\ell,\gamma_{0}}=U^{-1}\mathcal{N}_{\hbar,r,\gamma_{0}}U\,,$
(5.7)

where

$\mathfrak{a}_{\hbar,\ell}(\sigma,\tau)=\mathfrak{a}_{\hbar,r}(-\sigma,\tau)\,.$

The left and right operators have the same spectrum, and a normalized ground
state of $\mathcal{N}_{\hbar,\ell,\gamma_{0}}$ is

$\widecheck{\phi}_{\hbar,\ell}:=U\widecheck{\phi}_{\hbar,r}=e^{-i\gamma_{0}\sigma/\hbar^{2}}u_{\hbar,\ell}(\sigma,\tau)$
(5.8)

where $u_{\hbar,\ell}=Uu_{\hbar,r}$.

###  5.3. WKB expansions

We focus on the right well operator and construct an approximate eigenvalue
and an approximate ground state by WKB expansions, involving _formal series_
in the sense of [8, Notation 1.13]. The construction can be translated to the
left operator by symmetry.

Let us introduce the Agmon distance

$\Phi_{r}(\sigma)=\int_{[s_{r},\sigma]}\sqrt{V_{a,r}(s)}ds$ (5.9)

related to the “right well” potential444Recall from (3.8) that $M_{3}(a)<0$,
so $V_{a,r}\geqslant 0$.

$V_{a,r}(\sigma)=\frac{2M_{3}(a)(k_{r}(s)-k_{\max})}{\mu^{\prime\prime}_{a}(\zeta_{a})}$
(5.10)

######  Theorem 5.1.

There exist two sequences $(b_{j})_{j\geqslant 0}\subset\textsf{
Dom}(\mathcal{N}_{\hbar,r})$, $(\delta_{j})_{j\geqslant 0}\subset\mathbb{R}$,
a family of functions $(\Psi_{\hbar,r})_{\hbar\in(0,\hbar_{0}]}\subset
L^{2}(\mathbb{R}^{2})$ and a family of real numbers
$(\delta(\hbar))_{\hbar\in(0,\hbar_{0}]}$ such that

$e^{\Phi_{r}(\sigma)/\hbar^{\frac{1}{2}}}e^{-i\sigma\zeta_{a}/\hbar}\Psi_{\hbar,r}(\sigma,\tau)\underset{\hbar\to
0}{\sim}\hbar^{-\frac{1}{8}}\sum_{j\geqslant
0}b_{j}(\sigma,\tau)\hbar^{\frac{j}{2}},$ (5.11)
$\delta(\hbar)\underset{\hbar\to 0}{\sim}\sum_{j\geqslant
0}\delta_{j}\hbar^{\frac{j}{2}}\,,$

and

$e^{\Phi_{r}(\sigma)/\hbar^{\frac{1}{2}}}\left(\mathcal{N}_{\hbar,r}-\delta(\hbar)\right)\Psi_{\hbar,r}=\mathcal{O}(\hbar^{\infty})\,.$
(5.12)

Furthermore

$\delta_{0}=\beta_{a}\,,\quad\delta_{1}=0,\quad\delta_{2}=M_{3}(a)k_{\max},\quad\delta_{3}=\sqrt{\frac{{k_{2}}M_{3}(a)c_{2}(a)}{2}},$
$b_{0}(\sigma,\tau)=f_{0}(\sigma)\phi_{a}(\tau),$ (5.13)

and $f_{0}$ solves the effective transport equation

$\frac{\mu^{\prime\prime}_{a}(\zeta_{a})}{2}(\Phi^{\prime}_{r}\partial_{\sigma}+\partial_{\sigma}\Phi^{\prime}_{r})f_{0}+iF(\sigma)f_{0}=\sqrt{\frac{{k_{2}}M_{3}(a)c_{2}(a)}{2}}f_{0}\,,$
(5.14)

where $F$ is a smooth real-valued function, introduced in (5.15), such that
$F(s_{r})=0$.

######  Remark 5.2.

Let us explain precisely how the asymptotics in Theorem 5.1 are interpreted.
For every $N\geqslant 1$ we introduce the function
$\psi_{\hbar,r}^{N}(\sigma,\tau)$ and the real number $\delta^{N}(\hbar)$ as
follows:

$\Psi^{N}_{\hbar,r}(\sigma,\tau):=e^{-\Phi_{r}(\sigma)/\hbar^{\frac{1}{2}}}e^{i\sigma\zeta_{a}/\hbar}\,\hbar^{-\frac{1}{8}}\sum_{j=0}^{N}b_{j}(\sigma,\tau)\hbar^{\frac{j}{2}},\quad\delta^{N}(\hbar)=\sum_{j=0}^{N}\delta_{j}\hbar^{\frac{j}{2}}\,.$

Then (5.12) means

$e^{\Phi_{r}(\sigma)/\hbar^{\frac{1}{2}}}\left\|\left(\mathcal{N}_{\hbar,r}-\delta(\hbar)\right)\Psi_{\hbar,r}^{N}\right\|_{L^{2}(\mathbb{R}_{\tau})}=\mathcal{O}(\hbar^{N})$

locally uniformly with respect to $\sigma$.

###### Proof of Theorem 5.1.

We work in an arbitrary bounded set of $\mathbb{R}^{2}$ (see [8, Notation
1.13]), so, in the below computations, we take $c_{\mu}=1$ in (5.1).

Let us introduce the operator

$\widehat{\mathcal{N}}_{\hbar,r}:=e^{\Phi_{r}(\sigma)/\hbar^{\frac{1}{2}}}e^{i\sigma\zeta_{a}/\hbar}\mathcal{N}_{\hbar,r}e^{-i\sigma\zeta_{a}/\hbar}e^{-\Phi_{r}(\sigma)/\hbar^{\frac{1}{2}}}\,.$

It admits the formal expansion

$\widehat{\mathcal{N}}_{\hbar,r}=\mathcal{L}_{0}+\hbar^{1/2}\mathcal{L}_{1}+\hbar\mathcal{L}_{2}+\hbar^{3/2}\mathcal{L}_{3}+\hbar^{2}\mathcal{L}_{4}+\cdots$

where

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=-\partial_{\tau}^{2}+(\zeta_{a}+b_{a}\tau)^{2}$
$\displaystyle\mathcal{L}_{1}$
$\displaystyle=-2(\zeta_{a}+b_{a}\tau)i\Phi^{\prime}_{r}(\sigma)$
$\displaystyle\mathcal{L}_{2}$
$\displaystyle=k_{r}\partial_{\tau}-2(\zeta_{a}+b_{a}\tau)\Big{(}-i\partial_{\sigma}+\frac{k_{r}}{2}b_{a}\tau^{2}\Big{)}-\Phi^{\prime}_{r}(\sigma)^{2}+2k_{r}\tau(\zeta_{a}+b_{a}\tau)^{2}$
$\displaystyle\mathcal{L}_{3}$
$\displaystyle=\Big{(}-i\partial_{\sigma}+\frac{k_{r}}{2}b_{a}\tau^{2}\Big{)}i\Phi^{\prime}(\sigma)+i\Phi^{\prime}_{r}(\sigma)\Big{(}-i\partial_{\sigma}+\frac{k_{r}}{2}b_{a}\tau^{2}\Big{)}$
$\displaystyle\qquad-4\Phi^{\prime}_{r}(\sigma)\tau
k_{r}(\zeta_{a}+b_{a}\tau)$ $\displaystyle\mathcal{L}_{4}$
$\displaystyle=-\partial_{\sigma}^{2}+2k_{r}^{2}\tau^{2}(\zeta_{a}+b_{a}\tau)^{2}$
$\displaystyle\qquad-(\zeta_{a}+b_{a}\tau)\Big{[}\Big{(}-i\partial_{\sigma}+\frac{k_{r}}{2}b_{a}\tau^{2}\Big{)}k_{r}+k_{r}\Big{(}-i\partial_{\sigma}+\frac{k_{r}}{2}b_{a}\tau^{2}\Big{)}\Big{]}$
$\displaystyle\,\,\vdots$

Let $b(\sigma,\tau;\hbar):=\sum_{j\geqslant
0}b_{j}(\sigma,\tau)\hbar^{\frac{j}{2}}$ and let us formally solve the
equation

$\big{(}\widehat{\mathcal{N}}_{\hbar,r}-\delta(\hbar)\big{)}b(\sigma,\tau;\hbar)=\mathcal{O}(\hbar^{\infty}).$

Expanding the foregoing equation in powers of $\hbar^{1/2}$, the vanishing of
the coefficient of each $\hbar^{j/2}$, $j\geqslant 0$, yields the following
equations

$\displaystyle(\mathcal{L}_{0}-\delta_{0})b_{0}$ $\displaystyle=0$
$\displaystyle(\mathcal{L}_{0}-\delta_{0})b_{1}$
$\displaystyle=(\delta_{1}-\mathcal{L}_{1})b_{0}$
$\displaystyle(\mathcal{L}_{0}-\delta_{0})b_{2}$
$\displaystyle=(\delta_{2}-\mathcal{L}_{2})b_{0}+(\delta_{1}-\mathcal{L}_{1})b_{1}$
$\displaystyle(\mathcal{L}_{0}-\delta_{0})b_{3}$
$\displaystyle=(\delta_{3}-\mathcal{L}_{3})b_{0}+(\delta_{2}-\mathcal{L}_{2})b_{1}+(\delta_{1}-\mathcal{L}_{1})b_{2}$
$\displaystyle\,\,\vdots$

We will find solutions to these equations one by one. The first equation leads
us to choose $\delta_{0}=\zeta_{a}$ and
$b_{0}(\sigma,\tau)=f_{0}(\sigma)\phi_{a}(\tau)$, where $f_{0}(\sigma)$ is to
be determined at a later stage. The function $f_{0}$ will actually be free
untill the first equation involving $\mathcal{L}_{3}$.

For the equation for $b_{1}$, we determine $\delta_{1}$ by assuming that
$(\delta_{1}-\mathcal{L}_{1})b_{0}$ is orthogonal to $\phi_{a}$ in
$L^{2}(\mathbb{R})$. Then, we take the inner product with $\phi_{a}$ in
$L^{2}(\mathbb{R})$, use (9.2) and get

$\delta_{1}=0,\quad
b_{1}(\sigma,\tau)=2i\Phi^{\prime}_{r}(\sigma)f_{0}(\sigma){\mathcal{R}_{a}}\big{(}(\zeta_{a}+b_{a}\tau)\phi_{a}\big{)},$

where $\mathcal{R}_{a}$ the regularized resolvent introduced in (9.1). Since
we are applying $\mathcal{R}_{a}$ on functions orthogonal to $\phi_{a}$, we
can slightly abuse notation and say that it is equal to
$(\mathcal{L}_{0}-\delta_{0})^{-1}$.

The equation for $b_{2}$ will determine $\delta_{2}$. This equation can be
solved if $(\delta_{2}-\mathcal{L}_{2})b_{0}-\mathcal{L}_{1}b_{1}$ is
orthogonal to $\phi_{a}$ in $L^{2}(\mathbb{R})$, which we assume henceforth.
Taking the inner product with $\phi_{a}$ in $L^{2}(\mathbb{R})$, using (9.3)
and Remark 9.1, we get

$\delta_{2}f_{0}(\sigma)+\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}\Phi^{\prime}_{r}(\sigma)^{2}f_{0}(\sigma)-M_{3}(a)k_{r}(\sigma)f_{0}(\sigma)=0,$

since

$2\int_{\mathbb{R}}\tau(\zeta_{a}+b_{a}(\tau)\tau)^{2}|\phi_{a}(\tau)|^{2}\,d\tau-\int_{\mathbb{R}}b_{a}(\tau)\tau^{2}(\zeta_{a}+b_{a}(\tau)\tau)|\phi_{a}(\tau)|^{2}\,d\tau=M_{3}(a)\,.$

So, we choose $\delta_{2}=k_{r}(0)=k_{\max}$, and the foregoing equation
involving $f_{0}$ is valid everywhere in light of (5.9), independently of the
choice of $f_{0}$. At the same time, we choose $b_{2}$ as follows:

$b_{2}(\sigma,\tau)=(\mathcal{L}_{0}-\delta_{0})^{-1}\Big{(}(\delta_{2}-\mathcal{L}_{2})b_{0}-\mathcal{L}_{1}b_{1}\Big{)}.$

From the equation of $b_{3}$, we will determine $\delta_{3}$ and
$f_{0}(\sigma)$. Taking the inner product with $\phi_{a}$ in
$L^{2}(\mathbb{R})$ and using (9.3), we get

$\Big{\langle}(\delta_{3}-\mathcal{L}_{3})b_{0}+(\delta_{2}-\mathcal{L}_{2})b_{1}+(\delta_{1}-\mathcal{L}_{1})b_{2},\phi_{a}\Big{\rangle}_{L^{2}(\mathbb{R})}=\\\
\Big{(}\delta_{3}-\frac{\mu^{\prime\prime}(\zeta_{a})}{2}\big{(}\Phi^{\prime}_{r}\partial_{\sigma}+\partial_{\sigma}\Phi^{\prime}_{r}\big{)}\Big{)}f_{0}(\sigma)+iF(\sigma)f_{0}(\sigma),$

where $F(\sigma)$ is the real-valued function

$F(\sigma)=|\Phi^{\prime}_{r}(\sigma)|^{2}\int_{\mathbb{R}}g(\sigma,\tau)\phi_{a}(\tau)d\tau,$
(5.15)

where

$\displaystyle g(\sigma,\tau)$
$\displaystyle=(\mathcal{L}_{0}-\delta_{0})^{-1}\big{(}g_{1}(\sigma,\tau)+g_{2}(\sigma,\tau)\big{)}$
$\displaystyle g_{1}(\sigma,\tau)$
$\displaystyle=-\Big{(}k_{\max}+k_{r}(\sigma)\big{(}\zeta_{a}+b_{a}\tau)^{2}-b_{a}(\zeta_{a}+b_{a}\tau)\tau^{2}\big{)}-|\Phi(\sigma)|^{2}\Big{)}\phi_{a}(\tau)$
$\displaystyle\qquad+k_{r}(\sigma)\phi_{a}^{\prime}(\tau)$ $\displaystyle
g_{2}(\sigma,\tau)$
$\displaystyle=-4|\Phi_{r}(\sigma)|^{2}(\mathcal{L}_{0}-\delta_{0})^{-1}\big{(}(\zeta_{a}+b_{a}\tau)\phi_{a}(\tau)\big{)}.$

Since $\Phi_{r}^{\prime}(s_{r})=0$, we observe that $F(s_{r})=0$. We can solve
the equation of $b_{3}$ if
$(\delta_{3}-\mathcal{L}_{3})b_{0}+(\delta_{2}-\mathcal{L}_{2})b_{1}+(\delta_{1}-\mathcal{L}_{1})b_{2}$
is orthogonal to $\phi_{a}$ in $L^{2}(\mathbb{R})$, which yields the following
equation for $f_{0}$:

$\Big{(}\delta_{3}-\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}\big{(}\Phi^{\prime}_{r}\partial_{\sigma}+\partial_{\sigma}\Phi^{\prime}_{r}\big{)}\Big{)}f_{0}(\sigma)+F(\sigma)f_{0}(\sigma)=0.$

Since $F(s_{r})=\Phi^{\prime}_{r}(s_{r})=0$, the foregoing equation has a
solution satisfying $f_{0}(s_{r})\not=0$ if
$\delta_{3}=\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}\Phi^{\prime\prime}_{r}(s_{r})$,
thereby determining $\delta_{3}$ (from (5.10)) and $f_{0}$. The procedure can
be continued to any preassigned order. ∎

######  Remark 5.3 (Solving (5.14) & normalization of $\Psi_{\hbar,r}$).

In (5.14), we make the ansatz
$f_{0}(\sigma)=e^{i\alpha_{0}(\sigma)}\tilde{f}_{0}(\sigma)$, with
$\tilde{f}_{0}$ and $\alpha_{0}$ are real-valued functions such that
$\tilde{f}_{0}(0)>0$. Then we get from (5.14):

$\frac{\mu^{\prime\prime}_{a}(\zeta_{a})}{2}\big{(}\Phi^{\prime}_{r}\partial_{\sigma}+\partial_{\sigma}\Phi^{\prime}_{r}\big{)}\tilde{f}_{0}(\sigma)=\sqrt{\frac{{k_{2}}M_{3}(a)c_{2}(a)}{2}}f_{0}(\sigma)$

and

$\mu^{\prime\prime}(\zeta_{a})\alpha_{0}^{\prime}(\sigma)\Phi_{r}^{\prime}(\sigma)+F(\sigma)=0\,.$

This will determine $\tilde{f}_{0}(\sigma)$ uniquely up to the choice of
$\tilde{f}_{0}(0)$, and also $\alpha_{0}(\sigma)$ uniquely up to an additive
constant (see [7, Eq. (2.14) & Rem. 2.9]). We choose
$\tilde{f}(0)=\left(\frac{g_{a}}{\pi}\right)^{1/4}\left(\mathsf{A}_{\textsf{
u}}^{a}\right)^{1/2}$ which yields that the WKB solution $\Psi_{\hbar,r}$ in
Theorem 5.1 is almost normalized, $\|\Psi_{\hbar,r}\|\sim 1$. The constant
$\alpha_{a}$ appearing in Theorem 3.4 is

$\alpha_{a}=\frac{\alpha_{0}(0)-\alpha_{0}(-L)}{L}.$ (5.16)

##  6\. Optimal tangential Agmon estimates

The challenge of obtaining optimal decay estimates of bound states of the
Neumann magnetic Laplacian matching with the WKB solutions was recently taken
up in [7] by introducing pseudo-differential calculus with operator-valued
symbols. Fortunately, the method is quite general and can handle our situation
of magnetic steps.

###  6.1. A tangential elliptic estimate

We work with the single ‘right well’ flux free operator,
$\mathcal{N}_{\hbar,r}:=\mathcal{N}_{\hbar,r,0}$, introduced in (5.1). For the
sake of simplicity, we will omit the reference to ‘right well’ in the notation
and write $\mathcal{N}_{\hbar},k$ instead of $\mathcal{N}_{\hbar,r}$ and
$k_{r}$.

The optimal estimates, on the bound states of $\mathcal{N}_{\hbar}$, will hold
in spaces with an exponential weight, defined via a sub-solution of an
effective eikonal equation. More precisely, we consider a family of Lipschitz
functions $(\varphi_{\hbar})_{h\in(0,1]}\subset
C(\mathbb{R};\overline{\mathbb{R}_{+}})$ satisfying the following hypothesis:

######  Assumption 6.1.

For all $M>0$ there exist $\hbar_{0},C,R>0$ such that, for all
$\hbar\in(0,\hbar_{0})$, the function $\varphi:=\varphi_{\hbar}$ satisfies

  1. (i)

for all $\sigma\in\mathbb{R}$,
$\mathfrak{v}(\sigma)-\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}\varphi^{\prime}(\sigma)^{2}\geqslant
0$,

  2. (ii)

for all $\sigma$ such that $|\sigma-s_{r}|\geqslant R\hbar^{\frac{1}{2}}$,
$\mathfrak{v}(\sigma)-\frac{\mu_{1}^{\prime\prime}(\xi_{0})}{2}\varphi^{\prime}(\sigma)^{2}\geqslant
M\hbar$,

where $\mathfrak{v}(\sigma)=M_{3}(a)(k(\sigma)-k_{\max})$.

In the sequel, to lighten the notation, we write $\varphi$ instead of
$\varphi_{\hbar}$. We consider the conjugate operator, with the same domain as
$\mathcal{N}_{\hbar}$, and defined by:

$\mathcal{N}_{\hbar}^{\varphi}=e^{\varphi/\hbar^{\frac{1}{2}}}\mathcal{N}_{\hbar}e^{-\varphi/\hbar^{\frac{1}{2}}}=-a_{\hbar}^{-1}\partial_{\tau}a_{\hbar}\partial_{\tau}+a_{\hbar}^{-1}\mathcal{T}_{\hbar}^{\varphi}a_{\hbar}^{-1}\mathcal{T}_{\hbar}^{\varphi}$
(6.1)

where

$\mathcal{T}_{\hbar}^{\varphi}:=\left(-i\hbar\partial_{\sigma}-b_{a}\tau+i\hbar^{\frac{1}{2}}\varphi^{\prime}+\hbar
c_{\mu}\frac{\kappa_{r}}{2}b_{a}\tau^{2}\right)$

######  Theorem 6.2 (Bonnaillie-Noël–Hérau–Raymond).

Let $c_{0}>0$ and $\chi_{0}\in C_{c}^{\infty}(\mathbb{R})$ be $1$ in a
neighborhood of $0$. Under Assumption 6.1, there exist $c,\hbar_{0}>0$ such
that for all $\hbar\in(0,\hbar_{0})$, $z\in[\beta_{a}+M_{3}(a)k_{\max}\hbar-
c_{0}\hbar^{2},\beta_{a}+M_{3}(a)k_{\max}\hbar+c_{0}\hbar^{2}]$ and all
$\psi\in\textsf{ Dom}(\mathcal{N}_{\hbar}^{\varphi})$,

$c\hbar^{2}\|\psi\|\leqslant\|\langle\tau\rangle^{6}(\mathcal{N}^{\varphi}_{\hbar}-z)\psi\|+\hbar^{2}\|\chi_{0}(\hbar^{-\frac{1}{2}}R^{-1}(\sigma-
s_{r}))\psi\|\,,$

and

$c\hbar^{2}\|\hbar^{2}\partial_{\sigma}^{2}\psi\|\leqslant\|\langle\tau\rangle^{6}(\mathcal{N}^{\varphi}_{\hbar}-z)\psi\|+\hbar^{2}\|\chi_{0}(\hbar^{-\frac{1}{2}}R^{-1}(\sigma-
s_{r}))\psi\|\,,$

where $\langle\tau\rangle=(1+\tau^{2})^{1/2}$.

Modulo the decomposition of the symbol of the operator
$\mathcal{N}_{\hbar}^{\varphi}$ and its parametrix, the proof of Theorem 6.2
is the same as that of [7, Thm. 5.1]. In the sequel, we give only the new
ingredients.

Let us write

$\mathcal{N}_{\hbar}^{\varphi}u=\textrm{ Op}^{\textrm{
W}}_{\hbar}(n_{\hbar})\,u=\frac{1}{(2\pi\hbar)}\iint_{\mathbb{R}^{2}}e^{i(\sigma-s)\cdot\xi/\hbar}n_{\hbar}\left(\frac{\sigma+s}{2},\xi\right)u(s)dsd\xi$
(6.2)

where the foregoing quantization formula is formal, unless we consider it on,
say, for $u$ in the space
$\mathcal{S}\big{(}\mathbb{R};\widehat{\mathcal{S}}(\mathbb{R})\big{)}$ where

$\widehat{\mathcal{S}}(\mathbb{R})=\\{v\in
H^{2}(\mathbb{R}))\,|\leavevmode\nobreak\
v|_{\overline{\mathbb{R}_{\pm}}}\in\mathcal{S}(\overline{\mathbb{R}_{\pm}})\\}.$
(6.3)

The operator-valued symbol $n_{\hbar}$ can be decomposed as follows

$n_{\hbar}=n_{0}+\hbar^{\frac{1}{2}}n_{1}+\hbar
n_{2}+\hbar^{\frac{3}{2}}n_{3}+\hbar^{2}\tilde{r}_{\hbar},$

where

$\displaystyle n_{0}(\sigma,\xi)$ $\displaystyle=-\partial_{\tau}^{2}+(\xi-
b_{a}(\tau)\tau)^{2}\,,$ (6.4) $\displaystyle n_{1}(\sigma,\xi)$
$\displaystyle=2i(\xi-b_{a}(\tau)\tau)\varphi^{\prime}(\sigma)\,,$
$\displaystyle n_{2}(\sigma,\xi)$
$\displaystyle=-\varphi^{\prime}(\sigma)^{2}+\kappa
c_{\mu}(\tau)\partial_{\tau}+c_{\mu}\kappa(\sigma)(\xi-\tau)b_{a}(\tau)\tau^{2}$
$\displaystyle\quad+2\kappa(\sigma)\tau c_{\mu}(\tau)(\xi-
b_{a}\tau)^{2}+\kappa(\sigma)\tau c_{\mu}^{\prime}(\tau)\,,$
$\displaystyle\mathrm{Re}\,\,n_{3}(\sigma,\xi)$ $\displaystyle=0\,,$
$\displaystyle\tilde{r}_{\hbar}(\sigma,\xi)$
$\displaystyle={\mathcal{O}}(\tau^{4},(\xi-b_{a}(\tau)\tau)^{2}\tau^{2},(\xi-
b_{a}\tau)\tau,\tau^{2}\partial_{\tau}).$

The notation ${\mathcal{O}}$ is defined in [7, Notation 3.1]:
For differential operators $A,B,C,\cdots$ on $\mathbb{R}_{\tau}$, writing
$A=\mathcal{O}(B,C,\cdots)$ means the following:

$\exists\,c>0,\leavevmode\nobreak\
\forall\,u\in\widehat{\mathcal{S}}(\mathbb{R}),\quad\|Au\|_{L^{2}(\mathbb{R})}\leqslant
c\big{(}\|Bu\|_{L^{2}(\mathbb{R})}+\|Cu\|_{L^{2}(\mathbb{R})}+\cdots\big{)},$
(6.5)

where $\widehat{\mathcal{S}}(\mathbb{R})$ is the space introduced in (6.3) and
the constant $c$ is independent of $A,B,C,\cdots$ (in particular, in (6.4),
the estimate is uniform with respect to $(\sigma,\xi)$).

Let us introduce a modified symbol by truncating the frequency variable.
Recall that $\zeta_{a}<0$ is the unique minimum of the model operator with a
flat edge (see (3.4)). It will be convenient to introduce

$\hat{\zeta}_{a}:=-\zeta_{a}>0\,.$ (6.6)

Pick a smooth bounded and increasing function $\chi\in C^{\infty}(\mathbb{R})$
such that $\chi(\xi)=\xi$ for $\xi\in(-\hat{\zeta}_{a}/2,\hat{\zeta}_{a}/2)$,
and $\eta_{+}:=\lim_{\xi\to+\infty}\chi(\xi)\in(0,\hat{\zeta}_{a})$. We
introduce the function

$\chi_{1}(\xi)=\hat{\zeta}_{a}+\chi(\xi-\hat{\zeta}_{a}),$ (6.7)

and the operator

$\textrm{ Op}^{\textrm{ W}}_{\hbar}(p_{\hbar})\quad\textrm{
where\leavevmode\nobreak\
}p_{\hbar}(\sigma,\xi):=n_{\hbar}(\sigma,\chi_{1}(\xi)).$ (6.8)

Now consider the Grushin problem defined by the matrix operator

$\mathcal{P}_{z}(\sigma,\xi)=\left(\begin{array}[]{ll}p_{\hbar}-z&\cdot
v_{\xi}\\\
\langle\cdot,v_{\xi}\rangle&0\end{array}\right)\in\mathcal{S}\big{(}\mathbb{R}_{\sigma,\xi}^{2},\mathcal{L}(\textsf{
Dom}(p_{0})\times\mathbb{C},L^{2}(\mathbb{R})\times\mathbb{C})\big{)}$ (6.9)

where $\mathcal{S}\big{(}\mathbb{R}_{\sigma,\xi}^{2},\mathcal{L}(\textsf{
Dom}p_{0}\times\mathbb{C},L^{2}(\mathbb{R})\times\mathbb{C})\big{)}$ is
defined in [7, Notation 3.1],

$p_{0}(\sigma,\xi)=-\partial_{\tau}^{2}+(\chi_{1}(\xi)-b_{a}\tau)^{2}$ (6.10)

is the principal symbol of $p_{\hbar}$ and $v_{\xi}$ is the positive
normalized ground state of $p_{0}$, with corresponding eigenvalue
$\mu_{1}(\chi_{1}(\xi))=\mu_{a}(-\chi_{1}(\xi))$ (see (3.3)).

From the decomposition of $n_{\hbar}$, we can decompose $\mathcal{P}_{z}$ as
follows:

$\mathcal{P}_{z}=\mathcal{P}_{z}^{[3]}+\hbar^{2}\mathcal{R}_{\hbar},\quad\mathcal{P}_{z}^{[3]}=\mathcal{P}_{0,z}+\hbar^{1/2}\mathcal{P}_{1}+\hbar\mathcal{P}_{2}+\hbar^{3/2}\mathcal{P}_{3},$

where

$\mathcal{P}_{0,z}=\left(\begin{array}[]{ll}p_{0}-z&\cdot v_{\xi}\\\
\langle\cdot,v_{\xi}\rangle&0\end{array}\right),\quad\forall j\geqslant
1,\leavevmode\nobreak\ \mathcal{P}_{j}=\left(\begin{array}[]{ll}p_{j}&0\\\
0&0\end{array}\right),\quad\mathcal{R}_{\hbar}=\left(\begin{array}[]{ll}r_{\hbar}&0\\\
0&0\end{array}\right),$

and

$\displaystyle p_{1}$
$\displaystyle=2i(\chi_{1}(\xi)-b_{a}\tau)\varphi^{\prime},$ $\displaystyle
n_{2}$ $\displaystyle=-\varphi^{\prime 2}+\kappa
c_{\mu}\partial_{\tau}+c_{\mu}\kappa(\chi_{1}(\xi)-\tau)b_{a}\tau^{2}+2\kappa\tau
c_{\mu}(\chi_{1}(\xi)-b_{a}\tau)^{2}$ $\displaystyle\quad+\kappa\tau
c_{\mu}^{\prime}\left(\tau\right),$ $\displaystyle\textrm{ Re}\,n_{3}$
$\displaystyle=0,$ $\displaystyle\tilde{r}_{\hbar}$
$\displaystyle={\mathcal{O}}(\tau^{4},\tau^{2}\partial_{\tau}),$

where $\mathcal{O}(\tau^{4},\tau^{2}\partial_{\tau})$ is understood in the
sense of (6.5).

Then one can construct a parametrix of $\textrm{ Op}^{\textrm{
W}}_{\hbar}(\mathcal{P}_{z})$ (see [7, Thm. 3.5] for details)

$\mathcal{L}_{z}^{[3]}=\left(\begin{array}[]{ll}q_{z}&q_{z}^{+}\\\
q_{z}^{-}&q_{z}^{\pm}\end{array}\right),\quad\textrm{ Op}^{\textrm{
W}}_{\hbar}(\mathcal{L}_{z}^{[3]})\,\textrm{ Op}^{\textrm{
W}}_{\hbar}(\mathcal{P}_{z})=\textrm{
Id}+\hbar^{2}\mathcal{O}(\langle\tau\rangle^{6}),$

where

$q_{z}^{\pm}=q_{0,z}^{\pm}+\hbar^{1/2}q_{1,z}^{\pm}+\hbar
q_{2,z}^{\pm}+\hbar^{3/2}q_{3,z}^{\pm}\,,$

with

$\displaystyle q_{0,z}^{\pm}$ $\displaystyle=z-\mu_{1}(\chi_{1}(\xi)),$ (6.11)
$\displaystyle q_{1,z}^{\pm}$
$\displaystyle=-i\varphi^{\prime}(\sigma)\partial_{\xi}\mu_{1}(\chi_{1}(\xi)),$
$\displaystyle q_{2,z}^{\pm}$
$\displaystyle=k(\sigma)C_{1}(\xi,\mu)+C_{2}(\xi,z)\varphi^{\prime}(\sigma)^{2},$
$\displaystyle\textrm{ Re}\,q_{3,z}^{\pm}$ $\displaystyle=0,$

and

$\displaystyle C_{1}(\xi,\mu)=$
$\displaystyle-\langle\big{(}c_{\mu}\partial_{\tau}+c_{\mu}(\chi_{1}(\xi)-\tau)b_{a}\tau^{2}+2\tau
c_{\mu}(\chi_{1}(\xi)-b_{a}\tau)^{2}\big{)}v_{\xi},v_{\xi}\rangle$ (6.12)
$\displaystyle\qquad-\langle\tau
c_{\mu}^{\prime}\partial_{\tau}\big{)}v_{\xi},v_{\xi}\rangle,$ $\displaystyle
C_{2}(\xi,\mu)=$ $\displaystyle
1-\langle(p_{0}-z)^{-1}\Pi^{\bot}(\chi_{1}(\xi)-\tau)v_{\xi},(\chi_{1}(\xi)-\tau)v_{\xi}\rangle.$

Here $\Pi=\Pi_{\xi}$ is the orthogonal projection on $v_{\xi}$ and
$\Pi^{\bot}=\textrm{ Id}-\Pi$. Note that, by (6.6), Remark 9.1 and (9.3),

$C_{1}(\hat{\zeta}_{a},0)=-M_{3}(a),\quad
C_{2}(\hat{\zeta}_{a},\beta_{a})=\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}.$

Now we argue like [7, Prop 4.4]. Recall that
$|z-\beta_{a}-M_{3}(a)k_{\max}\hbar|\leqslant c_{0}\hbar^{2}$ and that $\mu\to
0$ as $h\to 0$. Expanding $C_{1}(\xi,\mu)$ and $C_{2}(\xi,z)$ near
$\xi=\hat{\zeta}_{a}$, we get

$\displaystyle C_{1}(\xi,\mu)$
$\displaystyle=-M_{3}(a)k_{\max}\hbar+\mathcal{O}\big{(}\hbar\min(1,|\xi-\hat{\zeta}_{a}|)\big{)},$
$\displaystyle C_{2}(\xi,z)$
$\displaystyle=\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{2}+\mathcal{O}\big{(}\hbar\min(1,|\xi-\hat{\zeta}_{a}|)\big{)}.$

Furthermore, since $\mu^{\prime}_{a}(\zeta_{a})=0$ and
$\mu^{\prime\prime}_{a}(\zeta_{a})>0$, there exists a constant $c_{1}>0$ such
that

$\mu(\chi_{1}(\xi))-z\geqslant c_{1}\min(1,|\xi-\hat{\zeta}_{a}|^{2}).$

Now we have the following lower bound

$-\textrm{
Re}\,q_{z}^{\pm}\geqslant\hbar(\mathfrak{v}(\sigma)-C_{2}(\hat{\zeta}_{a},\beta_{a})\varphi^{\prime}(\sigma)^{2}\big{)}-C\hbar^{2},$

where $\mathfrak{v}(\sigma)$ is introduced in Assumption 6.1. We apply the
Fefferman-Phong inequality [12, Thm. 3.2] (see also [37, Thm. 4.3.2]) on the
symbol

$\mathfrak{a}(\hat{\sigma},\hat{\xi};\hbar):=\mathscr{A}(\hbar^{1/2}\hat{\sigma},\hbar^{1/2}\hat{\xi};\hbar),$

where

$\mathscr{A}(\sigma,\xi;\hbar):=-\textrm{
Re}\,q_{z}^{\pm}(\sigma,\xi)-\hbar(\mathfrak{v}(\sigma)-C_{2}(\hat{\zeta}_{a},\beta_{a})\varphi^{\prime}(\sigma)^{2}\big{)}-C\hbar^{2}.$

In that way, we have

$-\textrm{ Re}\langle\textrm{ Op}_{\hbar}^{\textrm{
W}}(q_{z}^{\pm})\psi,\psi\rangle\geqslant\hbar\int_{\mathbb{R}}\left(\mathfrak{v}(\sigma)-\frac{\mu^{\prime\prime}_{a}(\zeta_{a})}{2}|\varphi^{\prime}(\sigma)|^{2}{-C\hbar}\right)|\psi|^{2}d\sigma,$

from which we get the following estimate (see [7, Thm. 4.2])

$cR^{2}\hbar^{2}\|\psi\|\leqslant\|(\textrm{ Op}_{\hbar}^{\textrm{
W}}(p_{\hbar})-z)\psi\|+C_{R}\hbar^{2}\|\chi_{0}(\hbar^{-1/2}R^{-1}(\sigma-
s_{r}))\psi\|+\hbar^{2}\|\tau^{6}\psi\|\,,$

which is almost the inequality in Theorem 6.2, but with the operator $\textrm{
Op}_{\hbar}^{\textrm{ W}}(p_{\hbar})$ instead of the operator
$\mathcal{N}_{\hbar}^{\varphi}=\textrm{ Op}_{\hbar}^{\textrm{ W}}(n_{\hbar})$.
The only difference between the two operators is the frequency cut-off in the
symbol, which can be removed following the same argument as in [7, Thm. 5.1].

###  6.2. Applications

By appropriate choices of the function $\varphi$ in Theorem 6.2, we get
optimal tangential estimates for the bound states of the ‘single’ and ‘double’
well operators. For details, see [7, Corol. 5.7, Corol. 6.1 & Prop. 6.2].

######  Proposition 6.3 (Decay of bound states).

Let $\theta,\varepsilon\in(0,1)$ and $K>0$. There exist $C,\hbar_{0}>0$ such
that for all $\hbar\in(0,\hbar_{0})$, the following is true.

If $\lambda$ eigenvalue of the operator $\mathscr{N}_{\hbar}$ in (4.6),
$\left|\lambda-\left(\beta_{a}+M_{3}(a)k_{\max}\hbar\right)\right|\leqslant
K\hbar^{2}$ and $u\in\textsf{ Dom}(\mathscr{N}_{\hbar})$ is an eigenfunction
associated to $\lambda$, then

$\int_{[-L,L)\times\mathbb{R}}e^{2\varphi/\hbar^{\frac{1}{2}}}|u|^{2}dsd\tau\leqslant
Ce^{\varepsilon/\hbar^{\frac{1}{2}}}\|u\|^{2}_{L^{2}([-L,L)\times\mathbb{R})}\,.$

where

$\varphi=(1-\theta)^{1/2}\min(\tilde{\Phi}_{r},\tilde{\Phi}_{\ell})\,,$

and $\tilde{\Phi}_{r},\tilde{\Phi}_{\ell}$ are $2L$-periodic functions
satisfying, for $\eta$ sufficiently small,

$\displaystyle\tilde{\Phi}_{r}(\sigma)\,|_{-L\leqslant\sigma\leqslant
s_{\ell}-\eta}$
$\displaystyle=\Phi_{r}(\sigma):=\sqrt{\frac{-2M_{3}(a)}{\mu_{a}^{\prime\prime}(\zeta_{a})}}\int_{[s_{r},\sigma]}\sqrt{k_{\max}-k(s)}\,ds,$
$\displaystyle\tilde{\Phi}_{\ell}(\sigma)\,|_{-L\leqslant\sigma\leqslant
s_{r}-\eta}$
$\displaystyle=\Phi_{\ell}(\sigma):=\sqrt{\frac{-2M_{3}(a)}{\mu_{a}^{\prime\prime}(\zeta_{a})}}\int_{[s_{\ell},\sigma]}\sqrt{k_{\max}-k(s)}\,ds\,.$

######  Remark 6.4.

The estimate in Proposition 6.3 continues to hold if $\lambda$ is an
eigenvalue of the right or left well operator, $\mathcal{N}_{\hbar,r}$ or
$\mathcal{N}_{\hbar,\ell}$, and $u$ is a corresponding eigenfunction.

Returning to the ‘one well’ operators, $\mathcal{N}_{\hbar,r,\gamma_{0}}$ and
$\mathcal{N}_{\hbar,\ell,\gamma_{0}}$ introduced in (5.1) and (5.7)
respectively, we get from Proposition 6.3 and the min-max principle, the
following rough estimate, important for the analysis of tunneling later on,

$\mu_{1}^{\textrm{
sw}}(\hbar)-\tilde{\mathcal{O}}(e^{-\mathsf{S}^{a}/\sqrt{\hbar}})\leqslant\nu_{1,a}(\hbar)\leqslant\nu_{2,a}(\hbar)\leqslant\mu_{1}^{\textrm{
sw}}(\hbar)+\tilde{\mathcal{O}}(e^{-\mathsf{S}^{a}/\sqrt{\hbar}}),$ (6.13)

where $\mathsf{S}^{a}$ is introduced in (3.15),

$\mu_{1}^{\textrm{
sw}}(\hbar)=\inf\sigma(\mathcal{N}_{\hbar,r,\gamma_{0}})=\inf\sigma(\mathcal{N}_{\hbar,r,\gamma_{0}})\,,$

$(\nu_{j,a}(\hbar))_{j\geqslant 1}$ is the sequence of eigenvalues of the
operator $\mathcal{N}_{\hbar}$, and the notation
$\tilde{\mathcal{O}}(e^{-\mathsf{S}^{a}/\sqrt{\hbar}})$ means

${\mathcal{O}}(e^{(\epsilon-\mathsf{S}^{a})/\sqrt{\hbar}})\quad\textrm{
for\leavevmode\nobreak\ any\leavevmode\nobreak\ fixed\leavevmode\nobreak\
}\varepsilon>0.$ (6.14)

The analysis of the tunneling requires an explicit approximation of the ground
state of the single well operators. Recall the Agmon distance $\Phi_{r}$ and
the WKB solution $\Psi_{\hbar,r}$ introduced in (5.9) and Theorem 5.1
respectively. Consider the flux free ‘right well’ operator
$\mathcal{N}_{\hbar,r}:=\mathcal{N}_{\hbar,r,0}$. By (5.5), the low lying
eigenvalues of this operator are simple; we denote by $\Pi_{r}$ the orthogonal
projection on its first eigenspace. By [7, Prop. 6.3], it results from Theorem
6.2:

######  Proposition 6.5 (WKB approximation).

We have

$\|\psi_{\hbar,r}-\Pi_{r}\psi_{\hbar,r}\|_{L^{2}(\mathbb{R}^{2})}=\mathcal{O}(\hbar^{\infty})$

and

$\langle\tau\rangle\,e^{\Phi_{r}/\sqrt{\hbar}}(\Psi_{\hbar,r}-u_{\hbar,r})=\mathcal{O}(\hbar^{\infty})\quad\textrm{
in}\leavevmode\nobreak\ \mathscr{C}^{1}(K;L^{2}(\mathbb{R})),$ (6.15)

where $K\subset
I_{2\hat{\eta},r}:=(s_{\ell}-2L+\hat{\eta},s_{\ell}-\hat{\eta})$ is a compact
set,

$\psi_{\hbar,r}(\sigma,\tau):=\chi_{\hat{\eta},r}(\sigma)\Psi_{\hbar,\tau}(\sigma,\tau)\,,$

and $\chi_{\hat{\eta},r}$ is a cut-off function supported in
$I_{\hat{\eta},r}$ such that $\chi_{\hat{\eta},r}=1$ on $I_{2\hat{\eta},r}$.

##  7\. Interaction matrix and tunneling

We return to the operator $\mathcal{N}_{\hbar}$ introduced in (4.6). In order
to estimate the splitting between the first and second eigenvalues,
$\nu_{2}(\hbar)-\nu_{1}(\hbar)$, we will write the matrix of this operator in
a specific basis of

$E=\oplus_{i=1}^{2}\textrm{
Ker}\big{(}\mathcal{N}_{\hbar}-\nu_{i}(\hbar)\big{)}.$

Let $\Pi$ be the orthogonal projection on $E$. We introduce the two functions

$f_{\hbar,r}=\chi_{\hat{\eta},r}\phi_{\hbar,r},\qquad\qquad
f_{\hbar,\ell}=\chi_{\hat{\eta},\ell}\phi_{\hbar,\ell},$

where $\chi_{\hat{\eta},r}$ is the cut-off function introduced in Proposition
6.5, $\chi_{\hat{\eta},\ell}=U\chi_{\hat{\eta},r}$ is defined by the symmetry
operator (see Sec. 5.2), $\phi_{\hbar,r}$, $\phi_{\hbar,\ell}$ enjoy
periodicity properties and are defined by inspiration from the functions in
(5.6) and (5.8). In fact, $\phi_{\hbar,r}(\sigma,\tau)$ need to be defined in
the support of $\chi_{\hat{\eta},r}$. Starting on
$[-L,s_{\ell}-\frac{\hat{\eta}}{2})\times\mathbb{R}$, we take
$\phi_{\hbar,r}(\sigma,\tau)$ the same as the function in (5.6); on
$[s_{\ell}+\frac{\hat{\eta}}{2},L)\times\mathbb{R}$, we do a change of
variable, and modify the function in (5.6) so that its module satisfies a
periodic boundary condition on $\pm L$. More precisely, we have

$\phi_{\hbar,r}(\sigma,\tau)=\begin{cases}e^{-i\gamma_{0}\sigma/\hbar^{2}}u_{\hbar,r}(\sigma,\tau),&\mbox{if
}-L\leqslant\sigma\leqslant s_{\ell}-\frac{\hat{\eta}}{2},\\\
e^{-i\gamma_{0}(\sigma-2L)/\hbar^{2}}u_{\hbar,r}(\sigma-2L,\tau),&\mbox{if
}s_{\ell}+\frac{\hat{\eta}}{2}<\sigma<L,\end{cases}$ (7.1)

and so $f_{\hbar,r}$ is well defined on $[-L,L)$. In a similar fashion,

$\phi_{\hbar,\ell}(\sigma,\tau)=\begin{cases}e^{-i\gamma_{0}(\sigma+2L)/\hbar^{2}}u_{\hbar,\ell}(\sigma+2L,\tau),&\mbox{if
}-L\leqslant\sigma\leqslant s_{r}-\frac{\hat{\eta}}{2},\\\
e^{-i\gamma_{0}\sigma/\hbar^{2}}u_{\hbar,\ell}(\sigma,\tau),&\mbox{if
}s_{r}+\frac{\hat{\eta}}{2}<\sigma<L,\end{cases}$ (7.2)

and $f_{\hbar,\ell}$ is well defined on $[-L,L)$.

Also we introduce the following actual bound states by projecting on the
eigenspace $E$,

$g_{\hbar,r}=\Pi f_{\hbar,r},\qquad\qquad g_{\hbar,\ell}=\Pi
f_{\hbar,\ell}\,.$

We use the notation $\tilde{\mathcal{O}}$ in (6.14). By Proposition 6.3, we
have (see [7, Sec. 7.1] and [9, Sec. 3] for details)

$\displaystyle\|f_{\hbar,r}\|^{2}$
$\displaystyle=1+\tilde{\mathcal{O}}(e^{-2\mathsf{S}^{a}/\sqrt{\hbar}}),\quad\|f_{\hbar,\ell}\|^{2}=1+\tilde{\mathcal{O}}(e^{-2\mathsf{S}^{a}/\sqrt{\hbar}}),$
$\displaystyle\langle f_{\hbar,r},f_{\hbar,\ell}\rangle$
$\displaystyle=\tilde{\mathcal{O}}(e^{-\mathsf{S}^{a}/\sqrt{\hbar}}),$

and

$\|g_{\hbar,\alpha}-f_{\hbar,\alpha}\|+\|\partial_{\tau}(g_{\hbar,\alpha}-f_{\hbar,\alpha})\|=\tilde{\mathcal{O}}(e^{-\mathsf{S}^{a}/\sqrt{\hbar}})\qquad\alpha\in\\{r,\ell\\}.$

Now, we construct an orthonormal basis
$\mathcal{B}_{\hbar}:=\\{\tilde{g}_{\hbar,r},\tilde{g}_{\hbar,\ell}\\}$ of $E$
from $\\{g_{\hbar,r},g_{\hbar,\ell}\\}$ by the Gram-Schmidt process. Let
$\mathsf{M}$ be the basis of $\mathcal{N}_{\hbar}$ relative to the basis
$\mathcal{B}_{\hbar}$. Then

$\nu_{2}(\hbar)-\nu_{1}(\hbar)=2|w_{\ell,r}|+\tilde{\mathcal{O}}(e^{-2\mathsf{S}/\sqrt{\hbar}})\,,\quad
w_{\ell,r}=\langle r_{\hbar,\ell},f_{\hbar,r}\rangle,$ (7.3)

where

$r_{\hbar,\ell}=(\mathcal{N}_{\hbar,\ell}-\mu^{\textrm{
sw}}(\hbar))f_{\hbar,\ell}.$

All we have to do now is the computation of $w_{\ell,r}$ by the WKB
approximation in Proposition 6.5. By [7, Lem. 7.1](which is essentially an
integration by parts formula)

$w_{\ell,r}=i\hbar\int_{\mathbb{R}}a_{\hbar}^{-1}\left(\phi_{\hbar,\ell}\overline{\mathscr{D}_{\hbar}\phi_{r}}+[\mathscr{D}_{\hbar}\phi_{\hbar,\ell}]\overline{\phi_{\hbar,r}}\right)(0,\tau)\\\
-i\hbar\int_{\mathbb{R}}a_{\hbar}^{-1}\left(\phi_{\hbar,\ell}\overline{\mathscr{D}_{\hbar}\phi_{\hbar,r}}+[\mathscr{D}_{\hbar}\phi_{\ell}]\overline{\phi_{\hbar,r}}\right)(-L,\tau)d\tau,$

where

$\mathscr{D}_{\hbar}=\hbar
D_{\sigma}+\hbar^{-1}\gamma_{0}-b_{a}(\tau)\tau+\hbar
c_{\mu}\frac{k}{2}b_{a}(\tau)\tau^{2}\,.$

Writing $a_{\hbar}=1+o(1)$, $\hbar c_{\mu}\tau^{2}=o(\hbar^{-2\eta})$, and
approximating $\phi_{\hbar,r}$, $\phi_{\hbar,\ell}$ by using (7.1), (7.2) and
Proposition 6.5, we get (see [7, Sec. 7.2.2] for details)

$w_{r,\ell}=i\hbar(w_{r,\ell}^{\textsf{ u}}+w_{r,\ell}^{\textsf{ d}})$ (7.4)

where

$\displaystyle w_{\ell,r}^{\textsf{ u}}$
$\displaystyle=\int_{\mathbb{R}}a_{\hbar}^{-1}\Psi_{\hbar,\ell}\overline{\left(\hbar
D_{\sigma}-b_{a}\tau+\hbar
c_{\mu}\frac{\kappa}{2}b_{a}\tau^{2}\right)\Psi_{\hbar,r}}(0,\tau)d\tau$
$\displaystyle+\int_{\mathbb{R}}a_{\hbar}^{-1}\left[\left(\hbar
D_{\sigma}-b_{a}\tau+\hbar
c_{\mu}\frac{\kappa}{2}b_{a}\tau^{2}\right)\Psi_{\hbar,\ell}\right]\overline{\Psi_{\hbar,r}}(0,\tau)d\tau+\mathcal{O}(\hbar^{\infty})e^{-\mathsf{S}_{\mathsf{u}}^{a}/\hbar^{1/2}},$
and $\displaystyle w_{\ell,r}^{\textsf{ d}}$
$\displaystyle=\int_{\mathbb{R}}a_{\hbar}^{-1}\Psi_{\hbar,\ell}\overline{\left(\hbar
D_{\sigma}-b_{a}\tau+\hbar
c_{\mu}\frac{\kappa}{2}b_{a}\tau^{2}\right)\Psi_{\hbar,r}}(-L,\tau)d\tau$
$\displaystyle+\int_{\mathbb{R}}a_{\hbar}^{-1}\left[\left(\hbar
D_{\sigma}-b_{a}\tau+\hbar
c_{\mu}\frac{\kappa}{2}b_{a}\tau^{2}\right)\Psi_{\hbar,\ell}\right]\overline{\Psi_{\hbar,r}}(-L,\tau)d\tau+\mathcal{O}(\hbar^{\infty})e^{-\mathsf{S}_{\mathsf{d}}^{a}/\hbar^{1/2}}.$

Eventually, using (3.13) and (5.11), we get (see [7, Eq. (7.11)])

$\hbar^{\frac{1}{4}}e^{\mathsf{S}_{\mathsf{u}}^{a}/\hbar^{1/2}}w_{\ell,r}^{\textsf{
u}}=-i\hbar^{\frac{1}{2}}\mu_{a}^{\prime\prime}(\zeta_{a})\pi^{-\frac{1}{2}}g_{a}^{\frac{1}{2}}\sqrt{V_{a}(0)}\mathsf{A}_{\mathsf{u}}^{a}e^{-2i\alpha_{0}(0)}+\mathcal{O}(\hbar)$

and, in a similar fashion,

$\hbar^{\frac{1}{4}}e^{\mathsf{S}_{\mathsf{d}}^{a}/\hbar^{1/2}}w_{\ell,r}^{\textsf{
d}}\\\
=-i\hbar^{\frac{1}{2}}\mu_{a}^{\prime\prime}(\zeta_{a})\pi^{-\frac{1}{2}}g_{a}^{\frac{1}{2}}\sqrt{V_{a}(-L)}\textsf{A}_{\mathsf{d}}^{a}e^{-2i\alpha_{0}(-L)}e^{i(-2L\gamma_{0}/\hbar^{2}-2L\zeta_{a}/\hbar)}+\mathcal{O}(\hbar)\,,$

where $\alpha_{0}$ is the function introduced in Remark 5.14.

Collecting (7.4) and (7.3) and using that $\hbar=h^{1/2}$, we get

$\displaystyle\nu_{2}(\hbar)-\nu_{1}(\hbar)$
$\displaystyle=2|e^{iLf(h)}w_{\ell,r}|+\tilde{\mathcal{O}}(e^{-2\mathsf{S}^{a}/\sqrt{\hbar}})$
$\displaystyle=h^{-1}|\tilde{w}_{a}(h)|+\tilde{\mathcal{O}}(h^{-1}e^{-2\mathsf{S}^{a}/\sqrt{\hbar}}),$

where $f(h)$ and $\tilde{w}_{a}(h)$ are the expressions in Theorem 3.4. In
light of Proposition 4.1, this finishes the proof of Theorem 3.4.

##  8\. Conclusion and open problems

Until now, examples of magnetic tunneling effects are rare in the literature.
Very few articles have been dealing with the measure of the tunneling effect
due to the presence of the magnetic field. In the presence of an electric
potential with multiple wells, the article [27] was only considering a case
when the magnetic field was a perturbation and the tunneling was mainly
created by the electric potential. Other examples include the case of a pure
flux [31]. After the recent contributions of Bonnaillie-Hérau-Raymond [9] and
Fefferman-Shapiro-Weinstein [14] (see also references therein), we have
presented a new magnetic tunneling effect due to the curvature of the magnetic
edge.

Both for the Neumann problem occurring in surface superconductivity [22] or
for the problem considered here [2], it would be interesting to consider the
$(3D)$-case.

Excluded in this paper is the case $a=-1$, where localization near the
point(s) of maximum curvature no more occurs ($M_{3}(a)=0$ in the asymptotics
of Theorem 3.2). On the contrast, this case seems to feature an interesting
new phenomenon where localization near the whole edge $\Gamma$ occurs, which
also has a nice analogy to what was observed in the multiple wells situation
in [26]. We hope to come back to the treatment of this case rather soon.

Finally we mention that the standard purely magnetic double well problem seems
at the moment a difficult challenge. Here we consider a purely semi-classical
magnetic Laplacian (say in $\mathbb{R}^{2}$) where the magnetic field has two
symmetric non degenerate positive minima.

##  9\. Appendix: Regularized resolvent and moments

Let us return back to the flat edge model in Sec. 3.1 and recall some
necessary computational results.

We can invert the operator $\mathfrak{h}_{a}[\zeta_{a}]$ on the orthogonal
complement of the ground state $\phi_{a}$. Extending by linearity, we get the
regularized resolvent $\mathfrak{R}_{a}$ defined on $L^{2}(\mathbb{R})$ by

$\mathfrak{R}_{a}\,u=\begin{cases}0&\mathrm{if}\leavevmode\nobreak\
u\parallel\phi_{a}\\\
(\mathfrak{h}_{a}[\zeta_{a}]-\beta_{a})^{-1}u&\mathrm{if}\leavevmode\nobreak\
u\perp\phi_{a}\end{cases}\,.$ (9.1)

By [5], $(\zeta_{a}+b_{a}(\tau)\tau)\phi_{a}$ and $\phi_{a}$ are orthogonal in
$L^{2}(\mathbb{R})$:

$\int_{\mathbb{R}}(\zeta_{a}+b_{a}(\tau)\tau)|\phi_{a}(\tau)|^{2}\,d\tau=0\,.$
(9.2)

Later on we will encounter the following integral [3, Prop. 2.5]

$I_{2}(a):=\int_{\mathbb{R}}\phi_{a}(\tau)\mathfrak{R}_{a}[(\zeta_{a}+b_{a}(\tau)\tau)\phi_{a}]\,d\tau=\frac{1}{4}-\frac{\mu_{a}^{\prime\prime}(\zeta_{a})}{8}\,.$
(9.3)

We recall some identities from [5] involving for $n\in\mathbb{N}$ quantities
of the form

$M_{n}(a)=\int_{\mathbb{R}}\frac{1}{b_{a}(\tau)}(\zeta_{a}+b_{a}(\tau)\tau)^{n}|\phi_{a}(\tau)|^{2}\,d\tau\,.$
(9.4)

We have

$\displaystyle M_{1}(a)$ $\displaystyle=0\,,$ (9.5) $\displaystyle M_{2}(a)$
$\displaystyle=-\frac{1}{2}\beta_{a}\int_{\mathbb{R}}\frac{1}{b_{a}(\tau)}|\phi_{a}(\tau)|^{2}\,d\tau+\frac{1}{4}\Big{(}\frac{1}{a}-1\Big{)}\zeta_{a}\phi_{a}(0)\phi_{a}^{\prime}(0)\,,$
(9.6) $\displaystyle M_{3}(a)$
$\displaystyle=\frac{1}{3}\Big{(}\frac{1}{a}-1\Big{)}\zeta_{a}\phi_{a}(0)\phi_{a}^{\prime}(0)\,.$
(9.7)

The case $a=-1$ is special because

$M_{3}(-1)=0\quad\mathrm{and}\quad M_{3}(a)<0\quad\mathrm{for}\quad-1<a<0\,.$

######  Remark 9.1.

The next identities follow in a straightforward manner [3, Rem. 2.3],

$\displaystyle\int_{\mathbb{R}}\tau(\zeta_{a}+b_{a}(\tau)\tau)|\phi_{a}(\tau)|^{2}\,d\tau$
$\displaystyle=M_{2}(a),$
$\displaystyle\int_{\mathbb{R}}\tau(\zeta_{a}+b_{a}(\tau)\tau)^{2}|\phi_{a}(\tau)|^{2}\,d\tau$
$\displaystyle=M_{3}(a)-\zeta_{a}M_{2}(a),$
$\displaystyle\int_{\mathbb{R}}b_{a}(\tau)\tau^{2}(\zeta_{a}+b_{a}(\tau)\tau)|\phi_{a}(\tau)|^{2}\,d\tau$
$\displaystyle=M_{3}(a)-2\zeta_{a}M_{2}(a),$

and

$\displaystyle\int_{\mathbb{R}}\tau|\phi_{a}(\tau)|^{2}\,d\tau$
$\displaystyle=-\zeta_{a}\int_{\mathbb{R}}\frac{1}{b_{a}(\tau)}|\phi_{a}(\tau)|^{2}\,d\tau,$
$\displaystyle\int_{\mathbb{R}}\tau|\phi^{\prime}_{a}(\tau)|^{2}\,d\tau$
$\displaystyle=\beta_{a}\zeta_{a}\int_{\mathbb{R}}\frac{1}{b_{a}(\tau)}|\phi_{a}(\tau)|^{2}\,d\tau+2M_{3}(a)-2\zeta_{a}M_{2}(a).$

### Funding

Part of this work was done while SF and AK visited the Laboratoire Jean Leray
at the university of Nantes. They wish to thank _la fédération de recherche
mathématique des Pays de Loire_ and _Nantes Université_ for supporting the
visit.

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