# Semiclassical resonance asymptotics for systems with degenerate crossings of
classical trajectories

Marouane Assal  Marouane Assal, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR
5251, F-33400 Talence, France. e-mail<EMAIL_ADDRESS>,
Setsuro Fujiie  Setsuro Fujiie, Department of Mathematical Sciences,
Ritsumeikan University, 111 Noji-Higashi, Kusatsu, 525-8577, Japan. e-mail:
<EMAIL_ADDRESS>and  Kenta Higuchi  Kenta Higuchi, Graduate School
of Science and Engineering, Ehime University/ Bunkyocho 3, Matsuyama, Ehime,
790-8577, Japan. e-mail<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with the asymptotics of resonances in the
semiclassical limit $h\to 0^{+}$ for $2\times 2$ matrix Schrödinger operators
in one dimension. We study the case where the two underlying classical
Hamiltonian trajectories cross tangentially in the phase space. In the setting
that one of the classical trajectories is a simple closed curve whereas the
other one is non-trapping, we show that the imaginary part of the resonances
is of order $h^{(m_{0}+3)/(m_{0}+1)}$, where $m_{0}$ is the maximal contact
order of the crossings. This principal order comes from the subprincipal terms
of the transfer matrix at crossing points which describe the propagation of
microlocal solutions from one trajectory to the other. In addition, we compute
explicitly the leading coefficient of the resonance widths in terms of the
probability amplitudes associated with all the generalized classical
trajectories escaping to infinity from the closed trajectory.

##  1\. Introduction

In this work, we are interested in the asymptotic distribution of quantum
resonances in the semiclassical limit for matrix Schrödinger operators,
especially in the effect of crossings of the underlying classical
trajectories. We study the one-dimensional model of Schrödinger systems
arising in the context of the Born-Oppenheimer approximation of diatomic
molecular Hamiltonians:

(1.1) $P=P(h):=\begin{pmatrix}P_{1}&hU\\\ hU^{*}&P_{2}\end{pmatrix}.$

The diagonal part consists of scalar Schrödinger operators

$P_{j}=P_{j}(h):=(hD_{x})^{2}+V_{j}(x)\quad(j=1,2),\;\;\;\;D_{x}:=-i\frac{d}{dx},$

with smooth real-valued potentials $V_{1}$ and $V_{2}$ on $\mathbb{R}$, and
the anti-diagonal part describing the interaction between $P_{1}$ and $P_{2}$
consists of ($h$ times) a first order semiclassical differential operator $U$
and its adjoint $U^{*}$ with smooth real-valued coefficients on $\mathbb{R}$.
We consider the situation where, at an energy-level $E_{0}$, the classical
trajectories of $p_{1}$ are trapping while the ones of $p_{2}$ are non-
trapping. Here $p_{j}(x,\xi):=\xi^{2}+V_{j}(x)$ is the classical Hamiltonian
associated with $P_{j}$ $(j=1,2)$, and its classical trajectories are the
integral curves of the corresponding Hamiltonian vector field
$H_{p_{j}}=\partial_{\xi}p_{j}\partial_{x}-\partial_{x}p_{j}\partial_{\xi}$.
In our one-dimensional framework, the classical trajectories of $p_{j}$
coincide with the characteristic set $\Gamma_{j}(E_{0}):=p_{j}^{-1}(E_{0})$.
The interaction between $P_{1}$ and $P_{2}$ may shift the real eigenvalues
near $E_{0}$ of $P_{1}$ to the lower half complex plane as resonances of $P$
(Fermi’s golden rule). The semiclassical asymptotics of the imaginary parts
(widths) of the resonances has been studied in this setting in [17, 18, 3, 1]
and [8, 9, 10]. Under a simple well condition on $P_{1}$, the first-mentioned
series of works studied the exponential decay rate in crossing free cases
while the second one gave polynomial asymptotics in the presence of crossings.

The aim of this paper is to generalize [10]. In that paper, a simplest model
where the classical trajectories of $p_{1}$ and $p_{2}$ cross transversally in
the phase space $T^{*}\mathbb{R}=\mathbb{R}_{x}\times\mathbb{R}_{\xi}$ is
considered. The asymptotic of the resonance widths is explicitly computed at
the principal level. In particular, its polynomial power in $h$ is 2, which is
determined only by the transversality of the crossings. In the present paper,
we relax the transversality condition on crossings, and give a precise
asymptotics of the resonance widths for degenerate crossings. In this case, we
prove that the polynomial power of the principal asymptotics of the resonance
width is $(m+3)/(m+1)$ (Theorem 1), where $m$ stands for the contact order
(2.4) of the crossings (including the transversal case $m=1$). The number of
crossing points is allowed to be finitely many. In that case, $m$ should be
the maximum contact order among them.

The power $(m+3)/(m+1)$ is determined by the microlocal transfer matrix at
each crossing point. The transfer matrix is a $2\times 2$ matrix describing
the microlocal outgoing data in terms of the incoming data, and we show in
Theorem 3 that it is approximated by $I_{2}+h^{1/(m+1)}T_{\operatorname{sub}}$
with an anti-diagonal matrix $T_{\operatorname{sub}}$. In the transversal
case, this asymptotic formula was computed in [10] for matrix Schrödinger
operators and generalized in [2] for general pseudodifferential systems by
applying the normal form reduction due to Helffer-Sjöstrand [11] (see also
[20, 5]). In our degenerate case, such a reduction is not valid and we develop
a new approach which brings the study of microlocal solutions to the system
near a given crossing point
$\rho=(\rho_{x},\rho_{\xi})\in\Gamma_{1}\cap\Gamma_{2}$ to that of exact local
solutions near $\rho_{x}$ constructed by the method established in [8].
Microlocally near each classical trajectory, incoming to or outgoing from a
crossing point, we compute the asymptotic behavior of these solutions. The
principal term of the anti-diagonal entries is expressed as the inner product
of the WKB solutions to $P_{1}$ and $P_{2}$, and its asymptotic expansion is
computed by the stationary phase method. The critical points of this integral
are the crossing points, and the degeneracy of the critical point coincides
with the contact order of the crossing point. From this, we obtain the
exponent $1/(m+1)$ of the anti-diagonal entries. Here we only consider the
generic case, where the energy level is different from the crossing level.
Otherwise, a crossing point is also a turning point, and the WKB approximation
is not valid there. In such a case the analysis would be different.

A Bohr-Sommerfeld type “microlocal quantization condition” for resonances is
given using the above microlocal transfer matrices. Resonances will be
approximated by energies for which the “monodromy matrix” has one as
eigenvalue (we call them pseudo-resonances). This is an extension of the
simple well problem to our system, where a non-trapping trajectory
$\Gamma_{2}$ exists in addition to the closed curve $\Gamma_{1}$. The usual
Bohr-Sommerfeld quantization condition for the scalar operator $P_{1}$ is the
condition that the multiplication factor $-\exp(2i{\mathcal{A}}(E)/h)$ after a
continuation along $\Gamma_{1}$ of a microlocal solution equals one, where
$2{\mathcal{A}}(E)$ is the action of $\Gamma_{1}$. We justify this
approximation by pseudo-resonances in the following way. The non-existence of
resonance outside an ${\mathcal{O}}(h^{\infty})$-neighborhood of the pseudo-
resonances is proved by applying the argument in [4]. The one-to-one
correspondence between the pseudo-resonances and the resonances is shown using
the Kato-Rellich theorem. This microlocal method enables us to consider not
only a simple model but also general cases with finitely many crossings.

To obtain the principal term of the widths of pseudo-resonances, one needs to
compute the asymptotic expansion of the microlocal transfer matrix up to
${\mathcal{O}}(h^{2/(m+1)})$. Instead, we employ the following additional
argument. The width of a resonance can be expressed as $h$ times the square of
the amplitude of the corresponding suitably normalized resonant state near
infinity. The asymptotic behavior of the resonant state coincide with that of
the microlocal WKB asymptotics on the outgoing trajectories tending to
infinity. We will see in Lemma 4.12 that, for each outgoing trajectory tending
to infinity, this microlocal behavior is given as the sum of “probability
amplitude” over all “paths” from $\Gamma_{1}$ to the trajectory. Here, paths
are defined by crossing points and classical trajectories as in the graph
theory (see Section 4). The probability amplitude associated with a path is
defined in Definition 4.1 in terms of the actions of the classical
trajectories and entries of the microlocal transfer matrices at the crossing
points.

The paper is organized as follows. The next section is devoted to the precise
assumptions and the statement of our main result. In section 3, we establish a
microlocal connection formula at a crossing point in the phase space (Theorems
2 and 3). In the last section, we define the monodromy matrix and pseudo-
resonances by applying the transfer matrix given in the previous section, and
show the asymptotic distribution of resonances (Theorem 1) in relation with
that of pseudo-resonances.

##  2\. Assumptions and main result

###  2.1. Precise assumptions

Let $p_{j}(x,\xi):=\xi^{2}+V_{j}(x)$, $(x,\xi)\in T^{*}\mathbb{R}$, be the
classical Hamiltonian associated with $P_{j}$, and let
$H_{p_{j}}:=2\xi\partial_{x}-V_{j}^{\prime}(x)\partial_{\xi}$ be the
corresponding Hamiltonian vector-field, $j=1,2$. For an energy
$E\in\mathbb{R}$, denote by $\Gamma_{j}(E)$ the characteristic set associated
with $P_{j}$ given by

$\Gamma_{j}(E):=p_{j}^{-1}(E)=\\{(x,\xi)\in
T^{*}\mathbb{R};\,p_{j}(x,\xi)=E\\}\quad(j=1,2),$

and set

$\Gamma(E):=\Gamma_{1}(E)\cup\Gamma_{2}(E).$

Introduce the interaction operator

$U=U(x,hD_{x}):=r_{0}(x)+ir_{1}(x)hD_{x}.$

Throughout the paper, we assume the following condition.

######  Assumption 1.

$V_{1},V_{2},r_{0}$ and $r_{1}$ are smooth real-valued functions on
$\mathbb{R}$. They can be extended to bounded holomorphic functions on an
angular complex domain near infinity of the form

$\Sigma=\Sigma(\theta_{0},R_{0}):=\\{x\in\mathbb{C};\,\left|{\rm Im}\hskip
1.0ptx\right|<(\tan\theta_{0})\left|{\rm Re}\hskip 1.0ptx\right|,\ \left|{\rm
Re}\hskip 1.0ptx\right|>R_{0}\\}$

for some constants $0<\theta_{0}<\pi/2$ and $R_{0}\geq 0$. Moreover, for each
$j=1,2$, $V_{j}$ admits limits $V_{j}^{\pm}$ as ${\rm Re}\hskip
1.0ptx\to\pm\infty$ in $\Sigma$.

In the sequel we fix a reference energy-level $E_{0}\in\mathbb{R}$ such that
$E_{0}\neq V_{j}^{\pm}$, $j=1,2$. For $\delta_{1},\delta_{2}>0$ possibly
$h$-dependent, one introduces the complex box

(2.1)
$\mathcal{R}=\mathcal{R}_{E_{0}}(\delta_{1},\delta_{2}):=[E_{0}-\delta_{1},E_{0}+\delta_{1}]+i[-\delta_{2},\delta_{2}].$

Under Assumption 1, the operator $P$ is self-adjoint in
$L^{2}(\mathbb{R};\mathbb{C}^{2})$ with domain
$H^{2}(\mathbb{R};\mathbb{C}^{2})$. One can define the resonances of $P$ in
$\mathcal{R}$, for sufficiently small $\delta_{1}$ and $\delta_{2}$, e.g., as
the values $E\in\mathbb{C}$ such that the equation $Pw=Ew$ has a non-trivial
outgoing solution $w$, that is, a non-identically vanishing solution such that
for some $\theta>0$ sufficiently small, the function $x\mapsto
w(\zeta_{\theta}(x))$ is in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ (see e.g.
[15]), where $\zeta_{\theta}\in C^{\infty}(\mathbb{R};\mathbb{C})$ is such
that $\zeta_{\theta}(x)=x$ for $\left|x\right|\leq R_{0}$ and
$\zeta_{\theta}(x)=xe^{i\theta}$ for $|x|\geq 2R_{0}$. Equivalently, the
resonances can be defined as the eigenvalues of a non-self adjoint operator
$P_{\theta}$ obtained from $P$ by the method of analytic distortion (see
Section 4). In particular, the imaginary part of each resonance is negative.
We denote $\operatorname{Res}(P)$ the set of resonances of $P$.

We now make some assumptions on the geometry of the characteristic set
$\Gamma(E_{0})$ and on the set of crossing points

$\mathcal{V}=\mathcal{V}(E_{0}):=\Gamma_{1}(E_{0})\cap\Gamma_{2}(E_{0}).$

######  Assumption 2.

  * (i)

There exist real numbers $a_{0}<b_{0}$ such that

$\frac{V_{1}(x)-E_{0}}{(x-a_{0})(x-b_{0})}>0\quad\forall x\in\mathbb{R}.$

  * (ii)

The classically allowed region $\\{x\in\mathbb{R};\,V_{2}(x)\leq E_{0}\\}$
consists of unbounded intervals, and $V_{2}^{\prime}(x)\neq 0$ for all
$x\in\\{x\in\mathbb{R};V_{2}(x)=E_{0}\\}$.

Assumption 2 (i) means that $V_{1}$ admits a simple well at $E_{0}$ and hence
$\Gamma_{1}(E_{0})$ is a simple closed curve symmetric with respect to the
$x$-axis, while (ii) implies that $E_{0}$ is a non-trapping energy for
$p_{2}$.

For $E$ near $E_{0}$, we define the action integral

${\mathcal{A}}(E):=\int_{\Gamma_{1}(E)}\xi dx.$

This is a smooth function near $E_{0}$. Under Assumption 2 (i), it is well
known that the spectrum of the scalar Schrödinger operator $P_{1}$ in a small
neighborhood of $E_{0}$ consists of eigenvalues approximated by the roots of
the Bohr-Sommerfeld quantization rule (see e.g., [16, 22])

(2.2) $\cos\left(\frac{\mathcal{A}(E)}{2h}\right)=0.$

For $\delta_{1},h>0$ small enough, we define an $h$-dependent discrete set
$\mathfrak{B}_{h}\subset[E_{0}-\delta_{1},E_{0}+\delta_{1}]$ by

(2.3)
$\mathfrak{B}_{h}=\mathfrak{B}_{h}(\delta_{1}):=\left\\{E\in[E_{0}-\delta_{1},E_{0}+\delta_{1}];\,E\
\text{satisfies}\,\eqref{BSR}\right\\}.$

The distance of two successive points in $\mathfrak{B}_{h}$ is of order
$\mathcal{O}(h)$, more precisely, if $\delta_{1}=\mathcal{O}(h)$, then for any
$E\in\mathfrak{B}_{h}$ one has

$\operatorname{dist}(\mathfrak{B}_{h}\setminus\\{E\\},E)=\frac{2\pi
h}{{\mathcal{A}}^{\prime}(E_{0})}+{\mathcal{O}}(h^{2}).$

######  Assumption 3.

The set of crossing points $\mathcal{V}$ is non-empty, and
$\mathcal{V}\cap\\{\xi=0\\}=\emptyset$. Moreover, the contact order of
$\Gamma_{1}(E_{0})$ and $\Gamma_{2}(E_{0})$ at each crossing point
$\rho\in\mathcal{V}$ defined by

(2.4) $m_{\rho}:=\min\left\\{m\in\mathbb{N};\,H_{p_{1}}^{m}p_{2}(\rho)\neq
0\right\\}=\min\left\\{m\in\mathbb{N};\,H_{p_{2}}^{m}p_{1}(\rho)\neq
0\right\\}$

is finite.

Notice that since $\Gamma_{1}(E_{0})$ is compact, the number of crossing
points is finite under the above assumption. The condition
$\mathcal{V}\cap\\{\xi=0\\}=\emptyset$ means the non-existence of a crossing-
turning point. It is equivalent to $V_{2}(a_{0})\neq E_{0}$ and
$V_{2}(b_{0})\neq E_{0}$. Since $\rho=(\rho_{x},\rho_{\xi})\in\mathcal{V}$
implies $\breve{\rho}:=(\rho_{x},-\rho_{\xi})\in\mathcal{V}$, the crossing
points appear in pairs. The contact order $m_{\rho}$ at a crossing point
$\rho=(\rho_{x},\rho_{\xi})\in\mathcal{V}$ coincides with the contact order of
the potentials $V_{1}$ and $V_{2}$ at $\rho_{x}$, more precisely one has

(2.5) $m_{\rho}=\min\\{k\in\mathbb{N};V_{1}^{(k)}(\rho_{x})\neq
V_{2}^{(k)}(\rho_{x})\\}.$

In fact, for any $m\in\mathbb{N}$ and $(x,\xi)\in\mathbb{R}^{2}$, one has

$H_{p_{1}}^{m}p_{2}(x,\xi)=2^{m}(V_{2}^{(m)}(x)-V_{1}^{(m)}(x))\xi^{m}+\sum_{k=0}^{m-1}\alpha_{k,m}(x)\xi^{k},$

with $\alpha_{k,m}$ of the form
$\alpha_{k,m}(x)=\sum_{j=0}^{m-1}(V_{2}^{(j)}(x)-V_{1}^{(j)}(x))\beta_{j,k,m}(x)$,
where $\beta_{j,k,m}$ are smooth functions on $\mathbb{R}$.

###  2.2. Main result

In the following, we use the same letter $U$ for the principal symbol of the
operator $U$, that is, $U(x,\xi):=r_{0}(x)+ir_{1}(x)\xi$, $(x,\xi)\in
T^{*}\mathbb{R}$. Set

$m_{0}:=\max\left\\{m_{\rho};\rho\in\mathcal{V}\right\\}.$

Our main result is the following.

######  Theorem 1.

Let Assumptions 1, 2 and 3 hold, and let $\mathfrak{B}_{h}$ and $\mathcal{R}$
be given by (2.3) and (2.1) with $\delta_{1}=\delta_{2}=Lh$ for an arbitrarily
fixed $L>0$. Then there exist a bijective map

$z_{h}:\mathfrak{B}_{h}\to{\rm Res}(P)\cap{\mathcal{R}}$

and a non-negative smooth function $\mathcal{D}(E)=\mathcal{D}(E;h)$ of $E$
near $E_{0}$ uniformly bounded with respect to $h>0$ small enough (given
explicitly by (4.29)) such that for any $E\in\mathfrak{B}_{h}$ one has

(2.6) $|z_{h}(E)-E|=\mathcal{O}(h^{\frac{m_{0}+3}{m_{0}+1}}),$ (2.7) ${\rm
Im}\,z_{h}(E)=-\mathcal{D}(E)h^{\frac{m_{0}+3}{m_{0}+1}}+{\mathcal{O}}(h^{\frac{m_{0}+3}{m_{0}+1}+\kappa}),$

uniformly as $h\to 0^{+}$, where
$\kappa=\min\left\\{1/3,\,1/(m_{0}+1)\right\\}$.

The leading coefficient $\mathcal{D}(E)$ in the asymptotics (2.7) may vanish
at some energies $E\in\mathfrak{B}_{h}$ for each small $h$. This means that,
in such a case, there exist resonances closer to the real axis than those with
imaginary part of order $h^{(m_{0}+3)/(m_{0}+1)}$. This phenomenon of double
lines of resonances occurs due to the existence of a directed cycle of the
characteristic set $\Gamma(E_{0})$ created by crossings. We illustrate this in
the next paragraph on a simple model. We conclude this section with the
definition of directed cycle.

######  Definition 2.1.

We call a curve $\gamma:[0,T]\to\Gamma$ a generalized classical trajectory if
it is parametrized by the time variable of Hamiltonian flow on each
$\Gamma_{j}$ $(j=1,2)$. In particular, we call it directed cycle if it is a
simple closed curve.

In other words, a generalized classical trajectory is a usual classical
trajectory of $p_{1}$ on $\Gamma_{1}\setminus\Gamma_{2}$ and of $p_{2}$ on
$\Gamma_{2}\setminus\Gamma_{1}$, but it is allowed to change from one to the
other at a crossing point. One may find an analogous terminology in [21]. In
[13], one of the authors showed that a directed cycle made by two non-trapping
classical trajectories creates resonances with width of order $h\log(1/h)$.

###  2.3. Example

We restate the above result with the explicit value of ${\mathcal{D}}(E)$ for
a simplest model including the one studied in [10] which corresponds to the
case $m_{0}=1$.

Suppose that the set $\\{V_{1}=V_{2}\leq E_{0}\\}$ consists of only one point.
Without loss of generality we assume that

$E_{0}>0,\quad\quad\\{x\in\mathbb{R};\,V_{1}(x)=V_{2}(x)\leq
E_{0}\\}=\\{0\\}\quad\text{and}\quad V_{1}(0)=V_{2}(0)=0.$

Suppose moreover that $V_{1}$ satisfies the simple well condition given by
Assumption 2 (i) with $a_{0}<0<b_{0}$, and there exists $c_{0}\in\mathbb{R}$
such that (see Fig. 1)

$\frac{V_{2}(x)-E_{0}}{x-c_{0}}>0,\quad\forall x\in\mathbb{R}.$

Under these conditions, the set of crossing points $\mathcal{V}$ in the phase
space consists of two points $\rho_{\pm}=(0,\pm\sqrt{E_{0}})$, and there
exists a unique directed cycle $\gamma\subset\Gamma(E_{0})$ other than
$\Gamma_{1}(E_{0})$. The associated action integral defined by

$S_{\gamma}(E):=\int_{\gamma}\xi dx,$

can be explicitly written as

(2.8)
$S_{\gamma}(E)=2\int_{a(E)}^{0}\sqrt{E-V_{1}(x)}\,dx+2\int_{0}^{c(E)}\sqrt{E-V_{2}(x)}\,dx,$

where $a(E)$ denotes the unique solution to $V_{1}(x)=E$ near $a_{0}$ and
$c(E)$ denotes the unique solution to $V_{2}(x)=E$ near $c_{0}$ (in
particular, $a(E_{0})=a_{0}$ and $c(E_{0})=c_{0}$). Finally assume that

$m_{0}:=\min\\{k\in\mathbb{N};V_{1}^{(k)}(0)\neq V_{2}^{(k)}(0)\\}<\infty,$

and set $v_{0}:=|V_{2}^{(m_{0})}(0)-V_{1}^{(m_{0})}(0)|$.

Figure 1. Typical model

Then, with the notations of Theorem 1, the leading coefficient of the width of
the resonances of $P$ is given by

$\mathcal{D}(E)=\frac{2E^{-\frac{m_{0}}{m_{0}+1}}}{{\mathcal{A}}^{\prime}(E)}\left[{\rm
Im}\hskip
1.0pt\left(\eta_{m_{0}}\overline{U(\rho_{+})}e^{iS_{\gamma}(E)/2h}\right)\right]^{2},$

where $\eta_{m_{0}}\neq 0$ is given by

(2.9) $\displaystyle\eta_{m_{0}}=\left\\{\begin{aligned}
&\exp\left(i\frac{\pi}{2(m_{0}+1)}\right)\bm{\Gamma}\left(\frac{m_{0}+2}{m_{0}+1}\right)\left(\frac{2(m_{0}+1)!}{v_{0}}\right)^{\frac{1}{m_{0}+1}}&&\text{if
$m_{0}$ is odd},\\\
&\cos\left(\frac{\pi}{2(m_{0}+1)}\right)\bm{\Gamma}\left(\frac{m_{0}+2}{m_{0}+1}\right)\left(\frac{2(m_{0}+1)!}{v_{0}}\right)^{\frac{1}{m_{0}+1}}&&\text{if
$m_{0}$ is even}.\end{aligned}\right.$

Here $\bm{\Gamma}$ is the usual gamma function defined by

(2.10) $\bm{\Gamma}(z):=\int_{0}^{+\infty}t^{z-1}e^{-t}dt,\quad{\rm Re}\,z>0.$

Notice that even when $U(\rho_{+})\neq 0$, the coefficient $\mathcal{D}(E)$
vanishes on a discrete subset determined by the action $S_{\gamma}(E)$
associated to the directed cycle $\gamma$. This set is given by the energies
$E$ such that

$\left\\{\begin{aligned}
&h^{-1}S_{\gamma}(E)-\pi+2\tan^{-1}\left(\frac{r_{0}(0)}{r_{1}(0)\sqrt{E_{0}}}\right)+\frac{\pi}{(m_{0}+1)}&&\text{if
$m_{0}$ is odd},\\\
&h^{-1}S_{\gamma}(E)-\pi+2\tan^{-1}\left(\frac{r_{0}(0)}{r_{1}(0)\sqrt{E_{0}}}\right)&&\text{if
$m_{0}$ is even},\end{aligned}\right.$

belongs to $2\pi\mathbb{Z}$. We here set
$\tan^{-1}\left(\frac{r_{0}(0)}{r_{1}(0)\sqrt{E_{0}}}\right)=\frac{\pi}{2}$
when $r_{1}(0)=0$.

##  3\. Microlocal connection at a crossing point

###  3.1. Definition and asymptotics for the microlocal transfer matrix

We recall briefly some basic notions of semiclassical and microlocal analysis,
referring to the textbooks [6, 17, 23] for more details. For $N\in\mathbb{N}$
and $m\in\mathbb{R}$, we introduce the class $S_{N}^{m}$ of matrix-valued
symbols

$S_{N}^{m}:=\left\\{a\in C^{\infty}(\mathbb{R}^{2};\mathbb{C}^{N\times
N})\,;\,\|\partial_{x}^{k}\partial_{\xi}^{l}a(x,\xi)\|_{N\times N}\leq
C_{k,l}(1+\xi^{2})^{m/2},\ \forall k,l\in\mathbb{N}\right\\}.$

For a symbol $a\in S_{N}^{m}$, the corresponding $h$-pseudodifferential
operator, denoted $a^{w}(x,hD_{x})$, is defined as an unbounded operator in
$L^{2}(\mathbb{R};\mathbb{C}^{N})$ via the $h$-Weyl quantization

(3.1) $\displaystyle a^{w}(x,hD_{x})u(x):=\frac{1}{2\pi
h}\int_{\mathbb{R}^{2}}e^{i(x-y)\xi/h}a\left(\frac{x+y}{2},\xi\right)u(y)dyd\xi.$

This is a bounded operator in $L^{2}(\mathbb{R};\mathbb{C}^{N})$ if $m\leq 0$
according to the Calderón-Vaillancourt Theorem, see e.g. [6, Theorem 7.11].

Let $(x_{0},\xi_{0})\in\mathbb{R}^{2}$ and $f=f(x;h)\in
L^{2}(\mathbb{R};\mathbb{C}^{N})$ with $\|f\|_{L^{2}}\leq 1$. We say that $f$
is microlocally $0$ near $(x_{0},\xi_{0})$ and we write

$f(x;h)\equiv 0\quad\text{near }\ (x_{0},\xi_{0}),$

if there exists a symbol $\chi=\chi(x,\xi)\in S_{N}^{0}$ with
$\det\,\chi(x_{0},\xi_{0})\neq 0$ such that

$\|\chi^{w}(x,hD_{x})f\|_{L^{2}(\mathbb{R})}={\mathcal{O}}(h^{\infty}).$

We also say that $f$ is microlocally $0$ near a set
$\Omega\subset\mathbb{R}^{2}$ if it is microlocally $0$ near each point of
$\Omega$. We define $\operatorname{WF}_{h}(f)$, the semiclassical wave front
set of $f$, as the set of all points of $\mathbb{R}^{2}$ where $f$ is not
microlocally $0$. The following results for scalar pseudo-differential
operators are well-known (see e.g. [23, Theorem 12.5]). Let $a\in S^{0}_{1}$.
If $f$ satisfies $a^{w}(x,hD)f\equiv 0$ microlocally in $U$, we have
$\operatorname{WF}_{h}(f)\cap U\subset\\{a=0\\}$. Moreover, if $\partial a\neq
0$ on $\\{a=0\\}$, the semiclassical wave front set $\operatorname{WF}_{h}(f)$
is invariant under the Hamiltonian flow of $a$.

Throughout this section, we fix $E_{0}\in\mathbb{R}$ as in the previous
section and consider $E$ belonging to our reference box $\mathcal{R}$ defined
by (2.1) with $\delta_{1}=\delta_{2}=Lh$ for an arbitrarily fixed $L>0$. We
set $\Gamma=\Gamma(E_{0}):=\Gamma_{1}(E_{0})\cup\Gamma_{2}(E_{0})$.

The following proposition is a simple consequence of a general fact that the
equation $a^{w}(x,hD_{x})u=0$ for a matrix-valued symbol $a(x,\xi)\in
S_{N}^{m}$ reduces microlocally near $(x_{0},\xi_{0})\in\mathbb{R}^{2}$ to a
scalar equation if $0$ is a simple eigenvalue of $a(x_{0},\xi_{0})$.

######  Proposition 3.1.

The space of microlocal solutions to the system $(P-E)w=0$ near each connected
subset of $\Gamma\setminus\mathcal{V}$ is one dimensional.

Microlocally near each point of $\Gamma\setminus\mathcal{V}$, except at the
turning points $\Gamma\cap\\{\xi=0\\}$, we have the following WKB microlocal
solution to the system $(P-E)w=0$ constructed in [10, Proposition 5.4]. Here
and in what follows, $\hat{j}$ stands for the complement of $j$ in the set
$\\{1,2\\}$.

######  Proposition 3.2.

[10, Proposition 5.4] Let $j\in\\{1,2\\}$. For any
$\rho_{j}=(x_{j},\xi_{j})\in\Gamma_{j}(E_{0})\setminus\mathcal{V}$,
$\xi_{j}\neq 0$, there exists a microlocal solution $f_{\rho_{j}}$ to the
equation

(3.2) $(P-E)w\equiv 0\quad\text{ near }\ \rho_{j},$

of the form

(3.3)
$f_{\rho_{j}}(x,h;x^{*})=e^{(\operatorname{sgn}\xi_{j})i\phi_{j}(x;x^{*})/h}\begin{pmatrix}\sigma_{j,1}(x,h)\\\
\sigma_{j,2}(x,h)\end{pmatrix},$

where the phase function $\phi_{j}$ normalized at some point $x^{*}$ in the
same connected component of the classically allowed region of $P_{j}-E$ as
$x_{j}$ is given by

(3.4) $\phi_{j}(x;x^{*}):=\int_{x^{*}}^{x}\sqrt{E_{0}-V_{j}(y)}\,dy,$

and the symbols $\sigma_{j,k}(x,h)$ are smooth functions of $x$ with
asymptotic expansion in powers of $h$ of the form
$\sigma_{j,k}(x,h)\sim\sum_{l\geq 0}h^{l}\sigma_{j,k,l}(x)$ such that

$\sigma_{j,j,0}(x)=(E_{0}-V_{j}(x))^{-1/4},\quad\sigma_{j,\hat{j},0}(x)=0\quad\text{and}\quad\sigma_{j,j,l}(x_{j})=0\quad\text{for
all }l\geq 1.$

Now let $\rho=(\rho_{x},\rho_{\xi})\in\mathcal{V}$ be a crossing point. Recall
from Assumption 3 that $\rho_{\xi}\neq 0$. We study the behavior of microlocal
solutions to $(P-E)w\equiv 0$ near an $h$-independent small neighborhood of
$\rho$. In this neighborhood, each $\Gamma_{j}(E_{0})$, $j=1,2$, has an
incoming classical trajectory ${\bm{e}}_{j}^{\flat}$ and outgoing one
${\bm{e}}_{j}^{\sharp}$ consisting of points $\exp tH_{p_{j}}(\rho)$ with
negative and positive small $t$. We take a point $\rho_{j}^{\flat}$ on
${\bm{e}}_{j}^{\flat}$ and a point $\rho_{j}^{\sharp}$ on
${\bm{e}}_{j}^{\sharp}$.

Let $f_{j}^{\bullet}:=f_{\rho_{j}^{\bullet}}(x,h;\rho_{x})$ be the WKB
microlocal solution to the system near $\rho_{j}^{\bullet}$ given by the
previous Proposition, normalized at $x^{*}=\rho_{x}$. If $w\in
L^{2}(\mathbb{R};\mathbb{C}^{2})$ satisfies

(3.5) $(P-E)w\equiv 0\quad\text{ near }\ \rho,$

then according to Proposition 3.1, there exist constants
$\alpha_{j}^{\bullet}=\alpha_{j}^{\bullet}(w,E,h)$ $(j=1,2$,
${\bullet}={\sharp},{\flat})$ such that

(3.6) $w\equiv\alpha_{j}^{\bullet}f_{j}^{\bullet}\quad\text{ near }\
\rho_{j}^{\bullet}.$

Let $m=m_{\rho}$ be the contact order at $\rho$ of $\Gamma_{1}(E_{0})$ and
$\Gamma_{2}(E_{0})$ defined by (2.4), which is finite according to Assumption
3.

The following two theorems are the main results of this section. The first one
ensures the well-definedness of the transfer matrix at the crossing point
$\rho$, and the second one gives the semiclassical asymptotics of this matrix.

######  Theorem 2.

The space of microlocal solutions to the equation $(P-E)w\equiv 0$ near $\rho$
is two dimensional. More precisely, there exists a matrix
$T_{\rho}=T_{\rho}(E,h;\rho_{1}^{\flat},\rho_{2}^{\flat},\rho_{1}^{\sharp},\rho_{2}^{\sharp})$
such that for any microlocal solution $w\in L^{2}(\mathbb{R};\mathbb{C}^{2})$
near $\rho$ with $\|w\|_{L^{2}}\leq 1$, one has

(3.7) $\biggl{(}\begin{matrix}\alpha_{1}^{\sharp}\\\
\alpha_{2}^{\sharp}\end{matrix}\biggr{)}=T_{\rho}(E,h)\biggl{(}\begin{matrix}\alpha_{1}^{\flat}\\\
\alpha_{2}^{\flat}\end{matrix}\biggr{)}.$

######  Theorem 3.

Let $L>0$ be an $h$-independent constant. For $E$ in
$\mathcal{R}_{E_{0}}(Lh,Lh)$, one has

(3.8) $T_{\rho}(E,h)={\rm Id}_{2}+h^{\frac{1}{m+1}}T_{\rm
sub}+{\mathcal{O}}(h^{\frac{2}{m+1}}),$

uniformly with respect to $h>0$ small enough, where ${\rm Id}_{2}$ is the
$2\times 2$ identity matrix and

$T_{\rm sub}=-i\begin{pmatrix}0&\overline{\omega_{\rho}}\\\
\omega_{\rho}&0\end{pmatrix},$

where $\omega_{\rho}\in\mathbb{C}$ is a constant independent of
$\rho_{j}^{\bullet}$ $(j=1,2,{\bullet}=\flat,\sharp)$ explicitly given by
(3.40) (see also Remark 3.12).

The rest of this section is devoted to the proofs of these results.

###  3.2. Outline of the proof of the theorems

We first remark that if $\rho=(\rho_{x},\rho_{\xi})$ is a crossing point with
$\rho_{\xi}\neq 0$, then $\breve{\rho}=(\rho_{x},-\rho_{\xi})$ is also a
crossing point.

Theorem 2 follows from the two propositions below. The first one brings the
study of microlocal solutions near $\rho=(\rho_{x},\rho_{\xi})\in\mathcal{V}$
to that of local exact solutions to the system $(P-E)w=0$, and the second one
shows that the space of exact solutions supported microlocally near the
crossing point is two dimensional.

######  Proposition 3.3.

Let $I$ be a real interval containing $\rho_{x}$ in its interior. For any
microlocal solution $w\in L^{2}(I;\mathbb{C}^{2})$ to the system $(P-E)w=0$
near $\rho$ with $\|w\|_{L^{2}(I)}\leq 1$, there exists an exact solution
$\widetilde{w}$ such that

(3.9) $w\equiv\widetilde{w}\quad\text{ near }\ \rho.$

######  Proposition 3.4.

There exists a basis $(w_{1}^{+},w_{2}^{+},w_{1}^{-},w_{2}^{-})$ of exact
solutions on $I$ to $(P-E)w=0$ such that

(3.10) $\|w_{j}^{\pm}\|_{L^{2}(I)}=1\quad\text{and }\ w_{j}^{-}\equiv
0\quad\text{ near }\ \rho,\quad j=1,2.$

We prove these propositions in Section 3.4.

We briefly explain the scheme of the construction of these solutions due to
[8]. In the following, we use the notations $U_{1}=U$ and $U_{2}=U^{*}$.

Fix a compact interval $I$ containing $\rho_{x}$. For each $j=1,2$, let
$(u_{j},\breve{u}_{j})$ be a pair of two linearly independent solutions to the
scalar homogeneous equation $(P_{j}-E)u=0$ on $I$. We define an integral
operator $K_{j}=K(\breve{u}_{j},u_{j},s_{j},t_{j})$ on $C(I)$ depending on
$s_{j},t_{j}\in I$ by

(3.11)
$K_{j}[f](x):=\frac{1}{h^{2}{\mathcal{W}}(\breve{u}_{j},u_{j})}\left(\breve{u}_{j}(x)\int_{s_{j}}^{x}u_{j}(y)f(y)dy-
u_{j}(x)\int_{t_{j}}^{x}\breve{u}_{j}(y)f(y)dy\right).$

Here ${\mathcal{W}}(\breve{u}_{j},u_{j})=\det(\breve{u}_{j},u_{j})$ stands for
the Wronskian of $\breve{u}_{j}$ and $u_{j}$.

If at least one of the infinite sums $J_{1}=J_{1}(K_{1},K_{2})$ or
$J_{2}=J_{2}(K_{1},K_{2})$ defined by

(3.12) $J_{1}(K_{1},K_{2}):=\sum_{k\geq
0}(\widetilde{K}_{1}\widetilde{K}_{2})^{k},\quad
J_{2}(K_{1},K_{2}):=\sum_{k\geq
0}(\widetilde{K}_{2}\widetilde{K}_{1})^{k}\quad\text{with }\
\widetilde{K}_{j}:=-hK_{j}U_{j}$

converges in some suitable space, then for any solution $v_{1}$ to
$(P_{1}-E)v=0$ or for any solution $v_{2}$ to $(P_{2}-E)v=0$, the vector-
valued function $w_{1}$ or $w_{2}$ defined by

(3.13) $w_{1}(K_{1},K_{2},v_{1}):=\begin{pmatrix}J_{1}v_{1}\\\
\widetilde{K}_{2}J_{1}v_{1}\end{pmatrix},\quad
w_{2}(K_{1},K_{2},v_{2}):=\begin{pmatrix}\widetilde{K}_{1}J_{2}v_{2}\\\
J_{2}v_{2}\end{pmatrix},$

is an exact solution to the system $(P-E)w=0$.

Suppose $\rho_{\xi}>0$. This means that the incoming trajectories are on the
left of the crossing point $\rho$ and the outgoing ones on the right. We take
$u_{j}$, $\breve{u}_{j}$, $j=1,2$ as in Proposition 3.6 below, i.e. $u_{j}$
and $\breve{u}_{j}$ are WKB solutions to $(P_{j}-E)u=0$ near $\rho$ and
$\breve{\rho}$ respectively. If we take $s_{j}$, $t_{j}$ both on the left of
$\rho_{x}$, the exact solutions $w_{1}^{\flat}=w_{1}(K_{1},K_{2},u_{1})$,
$w_{2}^{\flat}=w_{2}(K_{1},K_{2},u_{2})$ defined by (3.13) satisfies

(3.14) $w_{1}^{\flat}\equiv\left\\{\begin{aligned}
&f_{1}^{\flat}+\mathcal{O}(h)&&\text{ near}\;\rho_{1}^{\flat},\\\
&\mathcal{O}(h)&&\text{ near}\;\rho_{2}^{\flat},\end{aligned}\right.\qquad
w_{2}^{\flat}\equiv\left\\{\begin{aligned} &\mathcal{O}(h)&&\text{
near}\;\rho_{1}^{\flat},\\\ &f_{2}^{\flat}+\mathcal{O}(h)&&\text{
near}\;\rho_{2}^{\flat}.\end{aligned}\right.$

The main work consists in the computation of the microlocal asymptotic
behavior of these solutions on the outgoing trajectories:

(3.15) $w_{1}^{\flat}\equiv\left\\{\begin{aligned}
&t_{1,1}^{\sharp}f_{1}^{\sharp}+\mathcal{O}(h)&&\text{
near}\;\rho_{1}^{\sharp},\\\
&t_{2,1}^{\sharp}f_{2}^{\sharp}+\mathcal{O}(h)&&\text{
near}\;\rho_{2}^{\sharp},\end{aligned}\right.\qquad
w_{2}^{\flat}\equiv\left\\{\begin{aligned}
&t_{1,2}^{\sharp}f_{1}^{\sharp}+\mathcal{O}(h)&&\text{
near}\;\rho_{1}^{\sharp},\\\
&t_{2,2}^{\sharp}f_{2}^{\sharp}+\mathcal{O}(h)&&\text{
near}\;\rho_{2}^{\sharp}.\end{aligned}\right.$

Theorem 2 implies that the coefficients $t_{j,k}^{\sharp}$ give the transfer
matrix modulo ${\mathcal{O}}(h)$:

(3.16) $T_{\rho}=T^{\sharp}+{\mathcal{O}}(h),\quad
T^{\sharp}=(t_{j,k}^{\sharp})_{1\leq j,k\leq 2}.$

For the computation of (3.15) (as well as for the convergence of the infinite
series $J_{1}$ and $J_{2}$), we apply the following degenerate stationary
phase estimate in one dimension (see for instance [14]) for the oscillatory
integral

(3.17)
$\mathcal{I}(h):=\int_{I}\sigma(x)\exp\left(\frac{i}{h}\phi(x)\right)dx$

with a smooth real valued phase $\phi(x)$ and a smooth symbol $\sigma(x,h)$
uniformly bounded in $h$ with its derivative. In the application, $\phi$ will
be in the form

(3.18) $\phi=\pm\int^{x}\left(\sqrt{E-V_{1}(y)}\pm\sqrt{E-V_{2}(y)}\right)dy.$

The critical points of the phase function $\phi$ exist only when the sign
between the two square roots is $-$, and, in that case, they are the crossing
points of the potentials $V_{1}$ and $V_{2}$.

######  Lemma 3.5.

Let $I\subset\mathbb{R}$ be an interval, and suppose that $\phi(x)$ and
$\sigma(x,h)$ are smooth functions in a neighborhood of $\bar{I}$. Suppose
moreover that $\phi$ is real-valued, and $\sigma,$ $\partial_{x}\sigma$ are
bounded as $h\to 0^{+}$.

  1. (1)

If $\phi^{\prime}(x)$ never vanishes in $\bar{I}$, then we have
$\mathcal{I}(h)={\mathcal{O}}(h)$.

  2. (2)

If $x_{0}\in\abxring{I}$ is the unique stationary point of $\phi$ in
$\bar{I}$, and if $\phi^{\prime}(x_{0})=\cdots=\phi^{(m)}(x_{0})=0$,
$\phi^{(m+1)}(x_{0})\neq 0$, then we have

$\mathcal{I}(h)=\widetilde{\omega}e^{\frac{i}{h}\phi(x_{0})}h^{\frac{1}{m+1}}+{\mathcal{O}}(h^{\frac{2}{m+1}}),$

where

(3.19)
$\widetilde{\omega}=\mu_{m}\sigma(x_{0})\left(\frac{(m+1)!}{\left|\phi^{(m+1)}(x_{0})\right|}\right)^{\frac{1}{m+1}}\bm{\Gamma}\left(\frac{m+2}{m+1}\right),$
$\mu_{m}=\left\\{\begin{aligned}
&e^{\frac{i\pi}{2(m+1)}\operatorname{sgn}\phi^{(m+1)}(x_{0})}&&\quad\text{when
$m$ is odd, }\ \\\ &\cos\left(\frac{\pi}{2(m+1)}\right)&&\quad\text{when $m$
is even. }\ \end{aligned}\right.$

Here $\bm{\Gamma}$ is the usual Gamma function defined by (2.10).

###  3.3. Asymptotics of the transfer matrix at a crossing point (Proof of
Theorem 3)

In this paragraph, we suppose that Theorem 2 holds, that is, the transfer
matrix $T_{\rho}$ is well defined, and we prove Theorem 3. The proof of
Theorem 2 will be given in the next paragraph.

We assume without loss of generality that $\rho_{x}=0$ and
$V_{1}(0)=V_{2}(0)=0$. Assumption 3 then implies $E_{0}>0$. Let $I$ be an
$h$-independent small interval containing $0$ inside such that $0$ is the only
zero of $V_{1}$ and $V_{2}$ in $I$. We restrict our study to the region
$I\times\mathbb{R}_{\xi}$ and set
$\Gamma^{I}=\Gamma^{I}(E_{0}):=\Gamma(E_{0})\cap(I\times\mathbb{R}_{\xi})$,
$\Gamma_{j}^{I}(E_{0}):=\Gamma_{j}(E_{0})\cap(I\times\mathbb{R}_{\xi})$.

The two crossing points $\rho=(0,\sqrt{E_{0}})$ and
$\breve{\rho}=(0,-\sqrt{E_{0}})$ divide the characteristic set $\Gamma^{I}$
into eight curves (see Fig. 2): For $j=1,2$,

$\displaystyle{\bm{e}}_{j}^{\sharp}:=\Gamma_{j}^{I}\cap\\{x>0,\xi>0\\},\quad\breve{\bm{e}}_{j}^{\sharp}:=\Gamma_{j}^{I}\cap\\{x<0,\xi<0\\},$
$\displaystyle{\bm{e}}_{j}^{\flat}:=\Gamma_{j}^{I}\cap\\{x<0,\xi>0\\},\quad\breve{\bm{e}}_{j}^{\flat}:=\Gamma_{j}^{I}\cap\\{x>0,\xi<0\\}.$

Figure 2. The characteristic set near a pair of crossing points

We use bases of solutions to the scalar equation $(P_{j}-E)u=0$ $(j=1,2)$ to
construct solutions to the system $(P-E)w=0$. The scalar equations can be
solved locally by the method of successive approximation (see [19]) since
$E_{0}-V_{j}$ is $h$-independently positive on $I$.

######  Proposition 3.6.

For $j=1,2$, there exists a pair of solutions
$(\breve{u}_{j},u_{j})=(\breve{u}_{j}(x,h,E),u_{j}(x,h,E))$ to the scalar
equation $(P_{j}-E)u=0$ satisfying the following properties:

  1. (1)

$\breve{u}_{j}$ and $u_{j}$ admit the following asymptotic behaviors

(3.20)
$\breve{u}_{j}=\frac{1+{\mathcal{O}}(h)}{\sqrt[4]{E-V_{j}(x)}}e^{-i\int_{0}^{x}\sqrt{E-V_{j}(t)}\,dt/h},\quad
u_{j}=\frac{1+{\mathcal{O}}(h)}{\sqrt[4]{E-V_{j}(x)}}e^{i\int_{0}^{x}\sqrt{E-V_{j}(t)}\,dt/h},$

as $h\to 0^{+}$ uniformly on $I$.

  2. (2)

For real $E$ near $E_{0}$, one has $\breve{u}_{j}=\overline{u_{j}}$, and

(3.21)
$\operatorname{WF}_{h}(\breve{u}_{j})\subset\Gamma_{j}^{I}(E_{0})\cap\\{\xi<0\\},\quad\operatorname{WF}_{h}(u_{j})\subset\Gamma_{j}^{I}(E_{0})\cap\\{\xi>0\\}.$

  3. (3)

They are normalized to satisfy

(3.22)
${\mathcal{W}}(\breve{u}_{j},u_{j})=\breve{u}_{j}(u_{j})^{\prime}-(\breve{u}_{j})^{\prime}u_{j}=2ih^{-1}.$

######  Remark 3.7.

Only the first property (3.20) of the three previous properties is essential
for the existence. Once one obtains a pair of solutions which satisfies
(3.20), the pair $(\breve{u}_{j},u_{j})$ satisfying the other two properties
is obtained by a suitable linear combination.

We denote by $C(I)$ the Banach space of continuous functions on $I$ endowed
with the norm $\left\|f\right\|_{C(I)}=\sup_{x\in I}\left|f(x)\right|$. For
$j=1,2$ we take $s_{j}$, $t_{j}\in I$ and define operators by

(3.23) $K_{j}:=K(\breve{u}_{j},u_{j},s_{j},t_{j}):C(I)\to C(I)$

######  Proposition 3.8.

The operator $\tilde{K}_{j}=-hK_{j}U_{j}$ is uniformly bounded for
$E\in\mathcal{R}$ and $h$ small enough. Moreover, we have the following
uniform estimates

(3.24)
$\left\|\widetilde{K}_{1}\widetilde{K}_{2}\right\|_{\mathcal{B}(C(I))}={\mathcal{O}}(h^{1/(m+1)}),\quad\left\|\widetilde{K}_{2}\widetilde{K}_{1}\right\|_{\mathcal{B}(C(I))}={\mathcal{O}}(h^{1/(m+1)}).$

###### Proof.

We follow the scheme of [10, Proposition 3.1], where the authors estimated the
norm of similar operators for $m=1$. Remark that the estimate $h^{1/2}$ here
in (3.24) is better than that of $h^{1/3}$ in [10, Proposition 3.1]. This is
because the latter is a global estimate on $(-\infty,0)$ which contains
turning points.

For simplicity, we assume that the interaction operator is a multiplication
operator $r_{0}(x)$. The general case of first differential operator can be
treated similarly after an integration by parts as in [10].

The operators $\widetilde{K}_{j}$ are obviously bounded in $C(I)$ since

$\displaystyle\widetilde{K}_{j}f(x)$
$\displaystyle=-hK_{j}U_{j}f(x)=\frac{i}{2}\left(\breve{u}_{j}(x)\int_{s_{j}}^{x}r_{0}(y)u_{j}(y)f(y)dy-
u_{j}(x)\int_{t_{j}}^{x}r_{0}(y)\breve{u}_{j}(y)f(y)dy\right)$

and $r_{0}$, $u_{j}$ and $\breve{u}_{j}$ are of ${\mathcal{O}}(1)$ on $I$.

As for the estimate for the composed operator
$\widetilde{K}_{1}\widetilde{K}_{2}$, we study the integral of the form

(3.25)
$\int_{s}^{x}r_{0}(y)v_{1}(y)v_{2}(y)\left(\int_{s^{\prime}}^{y}r_{0}(t)v_{3}(t)f(t)dt\right)dy$

for $s,s^{\prime}\in I$ and
$(v_{1},v_{2},v_{3})=(u_{1},\breve{u}_{2},u_{2}),(u_{1},u_{2},\breve{u}_{2}),(\breve{u}_{1},\breve{u}_{2},u_{2})$
and $(\breve{u}_{1},u_{2},\breve{u}_{2})$. If we write (3.20) as
$\breve{u}_{j}=\breve{a}_{j}(x,h)e^{-i\phi_{j}(x)/h},\quad{u}_{j}=a_{j}(x,h)e^{i\phi_{j}(x)/h}$,
$\phi_{j}(x)=\int_{0}^{x}\sqrt{E-V_{j}(t)}\,dt$, then (3.25) can be written in
the form

(3.26) $\int_{s}^{x}\sigma(y,h)e^{i\phi(y)/h}dy,$

where

$\sigma(y,h)=r_{0}(y)b_{1}(y,h)b_{2}(y,h)\int_{s^{\prime}}^{y}r_{0}(t)v_{3}(t)f(t)dt\,\,\text{
with }b_{j}=a_{j}\text{ or }\breve{a}_{j},$
$\phi(y)=\int_{0}^{y}\left(\pm\sqrt{E-V_{1}(t)}\pm\sqrt{E-V_{2}(t)}\right)dt.$

Notice that the symbol $\sigma(y)$ and its derivative $D_{y}\sigma(y)$ are
both uniformly bounded with respect to $h$. Lemma 3.5 then implies that this
integral is estimated by ${\mathcal{O}}(h^{1/(m+1)})$. ∎

The following proposition suggests an appropriate choice of the endpoints
$s_{j}$ and $t_{j}$ for the definition of the $K_{j}$’s.

######  Proposition 3.9.

Let $K_{1}$ and $K_{2}$ be given by (3.23). If $s_{1},$ $s_{2}$, $t_{1}$,
$t_{2}$ are negative, then for $j=1,2$, we have

(3.27) $w_{j}(K_{1},K_{2},u_{j})\equiv\left\\{\begin{aligned}
&f_{j}^{\flat}+{\mathcal{O}}(h)\quad\text{ near }\ {\bm{e}}_{j}^{\flat},\\\
&{\mathcal{O}}(h)\quad\text{ near }\
{\bm{e}}_{\hat{j}}^{\flat}\cup\breve{\bm{e}}_{1}^{\sharp}\cup\breve{\bm{e}}_{2}^{\sharp},\end{aligned}\right.$
(3.28) $w_{j}(K_{1},K_{2},\breve{u}_{j})\equiv\left\\{\begin{aligned}
&\breve{f}_{j}^{\sharp}+{\mathcal{O}}(h)\quad\text{ near }\
\breve{\bm{e}}_{j}^{\sharp},\\\ &{\mathcal{O}}(h)\quad\text{ near }\
\breve{\bm{e}}_{\hat{j}}^{\sharp}\cup{\bm{e}}_{1}^{\flat}\cup{\bm{e}}_{2}^{\flat}.\end{aligned}\right.$

Similarly, if $s_{1},$ $s_{2}$, $t_{1}$, $t_{2}$ are positive, then we have

(3.29) $w_{j}(K_{1},K_{2},u_{j})\equiv\left\\{\begin{aligned}
&f_{1}^{\sharp}+{\mathcal{O}}(h)\quad\text{ near }\ {\bm{e}}_{j}^{\sharp},\\\
&{\mathcal{O}}(h)\quad\text{ near }\
{\bm{e}}_{\hat{j}}^{\sharp}\cup\breve{\bm{e}}_{1}^{\flat}\cup\breve{\bm{e}}_{2}^{\flat},\end{aligned}\right.$
(3.30) $w_{j}(K_{1},K_{2},\breve{u}_{j})\equiv\left\\{\begin{aligned}
&\breve{f}_{j}^{\flat}+{\mathcal{O}}(h)\quad\text{ near }\
\breve{\bm{e}}_{j}^{\flat},\\\ &{\mathcal{O}}(h)\quad\text{ near }\
\breve{\bm{e}}_{\hat{j}}^{\flat}\cup{\bm{e}}_{1}^{\sharp}\cup{\bm{e}}_{2}^{\sharp}.\end{aligned}\right.$

###### Proof.

We only show (3.27) for $j=1$ supposing $s_{2},t_{2}<0$. For this, it is
enough to prove a local estimate

(3.31) $w_{1}(K_{1},K_{2},u_{1})=\begin{pmatrix}u_{1}\\\
0\end{pmatrix}+{\mathcal{O}}(h).$

on an interval $[-\delta,-\delta/2]$ with a small $\delta>0$. In fact, we see
from Proposition 3.2 and (3.20) that the vector-valued function
${}^{t}(u_{1},0)$ has the same microlocal asymptotic behavior as the right
hand side of (3.27).

To prove (3.31), it is enough to show the estimate

(3.32) $\widetilde{K}_{2}(u_{1})(x)={\mathcal{O}}(h)\quad\text{uniformly for
}\ x\in[-\delta,-\delta/2].$

If $U_{2}=r_{0}(x)$, for simplicity as in the proof of the previous lemma,

$\widetilde{K}_{2}(u_{1})(x)=-hK_{2}U_{2}u_{1}(x)=\frac{i}{2}\left(\breve{u}_{j}(x)\int_{s_{2}}^{x}r_{0}(y)u_{j}(y)u_{1}(y)dy-
u_{j}(x)\int_{t_{2}}^{x}r_{0}(y)\breve{u}_{j}(y)u_{1}(y)dy\right)$

is again an oscillatory integral of type (3.17) with a phase function (3.18).
The critical point is the crossing point $x=0$, and it is not contained in the
integration intervals $[s_{2},x],[t_{2},x]$ since $s_{2},t_{2},x$ are all
negative. Thus the first assertion of Lemma 3.5 applies and (3.32) is
obtained. ∎

Define the operators $K_{j,L}$ and $K_{j,R}$ acting on $C(I)$ using (3.11) by

(3.33) $K_{j,L}:=K(\breve{u}_{j},u_{j},-\delta,-\delta),\quad
K_{j,R}:=K(\breve{u}_{j},u_{j},\delta,\delta),$

and solutions $w_{j}^{\sharp}$, $w_{j}^{\flat}$ $(j=1,2)$ to the equation
$(P-E)w=0$ using (3.13) by

$\displaystyle w_{j}^{\sharp}:=w_{j}(K_{1,R},K_{2,R},u_{j}),\quad$
$\displaystyle w_{j}^{\flat}:=w_{j}(K_{1,L},K_{2,L},u_{j}),$
$\displaystyle\breve{w}_{j}^{\sharp}:=w_{j}(K_{1,L},K_{2,L},\breve{u}_{j}),\quad$
$\displaystyle\breve{w}_{j}^{\flat}:=w_{j}(K_{1,R},K_{2,R},\breve{u}_{j}).$

These solutions are well-defined according to Proposition 3.8. By Proposition
3.9, we know the microlocal asymptotic behavior of $w_{j}^{\flat}$ and
$\breve{w}_{j}^{\sharp}$ on the left of $\rho_{x}(=0)$, and that of
$w_{j}^{\sharp}$ and $\breve{w}_{j}^{\flat}$ on the right of $\rho_{x}$. In
particular, $w_{1}^{\flat}$ and $w_{2}^{\flat}$ satisfy (3.14). We compute the
coefficients $t_{j,k}^{\sharp}$’s in (3.15) by representing $w_{1}^{\flat}$
and $w_{2}^{\flat}$ as linear combinations of the elements of the basis
$(\breve{w}_{1}^{\flat},w_{1}^{\sharp},\breve{w}_{2}^{\flat},w_{2}^{\sharp})$.

######  Proposition 3.10.

We have the following linear relationship between the two bases

(3.34)
$(\breve{w}_{1}^{\sharp},w_{1}^{\flat},\breve{w}_{2}^{\sharp},w_{2}^{\flat})=(\breve{w}_{1}^{\flat},w_{1}^{\sharp},\breve{w}_{2}^{\flat},w_{2}^{\sharp})A(E,h).$

Here, the $4\times 4$-matrix $A$ is given by
$A=(I_{4}+A_{R})^{-1}(I_{4}+A_{L})$ with

$\displaystyle
A_{S}=\begin{pmatrix}\breve{\beta}_{1,S}\breve{u}_{1}&\breve{\beta}_{1,S}u_{1}&\breve{\alpha}_{2,S}\breve{u}_{2}&\breve{\alpha}_{2,S}u_{2}\\\
\beta_{1,S}\breve{u}_{1}&\beta_{1,S}u_{1}&\alpha_{2,S}\breve{u}_{2}&\alpha_{2,S}u_{2}\\\
\breve{\alpha}_{1,S}\breve{u}_{1}&\breve{\alpha}_{1,S}u_{1}&\breve{\beta}_{2,S}\breve{u}_{2}&\breve{\beta}_{2,S}u_{2}\\\
\alpha_{1,S}\breve{u}_{1}&\alpha_{1,S}u_{1}&\beta_{2,S}\breve{u}_{2}&\beta_{2,S}u_{2}\end{pmatrix},\quad
S=L,R,$

where $\alpha_{j,S}$, $\breve{\alpha}_{j,S}$, $\beta_{j,S}$,
$\breve{\beta}_{j,S}$ $(j=1,2,$ $S=L,R)$ are linear functionals defined by

$\displaystyle\alpha_{j,L}f=-{\frac{i}{2}}\int_{-\delta}^{0}\breve{u}_{\hat{j}}(y)U_{\hat{j}}J_{j,L}f(y)dy,\quad$
$\displaystyle\breve{\alpha}_{j,L}f={\frac{i}{2}}\int_{-\delta}^{0}u_{\hat{j}}(y)U_{\hat{j}}J_{j,L}f(y)dy,$
$\displaystyle\beta_{j,L}f=-{\frac{i}{2}}\int_{-\delta}^{0}\breve{u}_{j}(y)U_{j}\widetilde{K}_{\hat{j},L}J_{j,L}f(y)dy,\quad$
$\displaystyle\breve{\beta}_{j,L}f={\frac{i}{2}}\int_{-\delta}^{0}u_{j}(y)U_{j}\widetilde{K}_{\hat{j},L}J_{j,L}f(y)dy,$
$\displaystyle\alpha_{j,R}f={\frac{i}{2}}\int_{0}^{\delta}\breve{u}_{\hat{j}}(y)U_{\hat{j}}J_{j,R}f(y)dy,\quad$
$\displaystyle\breve{\alpha}_{j,R}f=-{\frac{i}{2}}\int_{0}^{\delta}\breve{u}_{\hat{j}}(y)U_{\hat{j}}J_{j,R}f(y)dy,$
$\displaystyle\beta_{j,R}f={\frac{i}{2}}\int_{0}^{\delta}\breve{u}_{j}(y)U_{j}\widetilde{K}_{\hat{j},R}J_{j,R}f(y)dy,\quad$
$\displaystyle\breve{\beta}_{j,R}f=-{\frac{i}{2}}\int_{0}^{\delta}u_{j}(y)U_{j}\widetilde{K}_{\hat{j},R}J_{j,R}f(y)dy,$

where $J_{j,S}=J_{j}(K_{1,S},K_{2,S})$.

###### Proof.

Let $\bm{u}_{j}$ and $\breve{\bm{u}}_{j}$ be the vectors given by

$\bm{u}_{1}:=\begin{pmatrix}u_{1}(0)\\\ 0\\\ u_{1}^{\prime}(0)\\\
0\end{pmatrix},\quad\breve{\bm{u}}_{1}:=\begin{pmatrix}\breve{u}_{1}(0)\\\
0\\\ \breve{u}_{1}^{\prime}(0)\\\
0\end{pmatrix},\quad\bm{u}_{2}:=\begin{pmatrix}0\\\ u_{2}(0)\\\ 0\\\
u_{2}^{\prime}(0)\end{pmatrix},\quad\breve{\bm{u}}_{2}:=\begin{pmatrix}0\\\
\breve{u}_{2}(0)\\\ 0\\\ \breve{u}_{2}^{\prime}(0)\end{pmatrix}.$

Then by the same arguments as in the proof of Proposition 5.2 of [8], we have

$\displaystyle\begin{pmatrix}\breve{w}_{1}^{\sharp}&w_{1}^{\flat}&\breve{w}_{2}^{\sharp}&w_{2}^{\flat}\\\
(\breve{w}_{1}^{\sharp})^{\prime}&(w_{1}^{\flat})^{\prime}&(\breve{w}_{2}^{\sharp})^{\prime}&(w_{2}^{\flat})^{\prime}\end{pmatrix}\big{|}_{x=0}=\begin{pmatrix}\breve{\bm{u}}_{1}&\bm{u}_{1}&\breve{\bm{u}}_{2}&\bm{u}_{2}\end{pmatrix}(I+A_{L}),$
$\displaystyle\begin{pmatrix}\breve{w}_{1}^{\flat}&w_{1}^{\sharp}&\breve{w}_{2}^{\flat}&w_{2}^{\sharp}\\\
(\breve{w}_{1}^{\flat})^{\prime}&(w_{1}^{\sharp})^{\prime}&(\breve{w}_{2}^{\flat})^{\prime}&(w_{2}^{\sharp})^{\prime}\end{pmatrix}\big{|}_{x=0}=\begin{pmatrix}\breve{\bm{u}}_{1}&\bm{u}_{1}&\breve{\bm{u}}_{2}&\bm{u}_{2}\end{pmatrix}(I+A_{R}).$

This means

$\begin{pmatrix}\breve{\bm{u}}_{1}&\bm{u}_{1}&\breve{\bm{u}}_{2}&\bm{u}_{2}\end{pmatrix}(I+A_{L})=\begin{pmatrix}\breve{\bm{u}}_{1}&\bm{u}_{1}&\breve{\bm{u}}_{2}&\bm{u}_{2}\end{pmatrix}(I+A_{R})A,$

and the proposition follows since the matrix
$\begin{pmatrix}\breve{\bm{u}}_{1}&\bm{u}_{1}&\breve{\bm{u}}_{2}&\bm{u}_{2}\end{pmatrix}$
is invertible. ∎

Writing $A=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3},\bm{a}_{4})$,
$\bm{a}_{j}={}^{t}(a_{j1},a_{j2},a_{j3},a_{j4})$, we have obatined

(3.35)
$w_{1}^{\flat}=(\breve{w}_{1}^{\flat},w_{1}^{\sharp},\breve{w}_{2}^{\flat},w_{2}^{\sharp})\bm{a}_{2},\quad
w_{2}^{\flat}=(\breve{w}_{1}^{\flat},w_{1}^{\sharp},\breve{w}_{2}^{\flat},w_{2}^{\sharp})\bm{a}_{4}.$

This with Proposition 3.9 shows

(3.36) $T^{\sharp}=\begin{pmatrix}a_{22}&a_{24}\\\
a_{42}&a_{44}\end{pmatrix}+{\mathcal{O}}(h).$

The following lemma gives the asymptotic formula for these coefficients, and
thus ends the proof of Theorem 3.

######  Lemma 3.11.

For each $(j,S)\in\\{1,2\\}\times\\{L,R\\}$, we have

(3.37)
$\alpha_{j,S}u_{j},\quad\alpha_{j,S}\breve{u}_{j},\quad\breve{\alpha}_{j,S}u_{j},\quad\breve{\alpha}_{j,S}\breve{u}_{j}={\mathcal{O}}(h^{1/(m+1)}),$
(3.38)
$\beta_{j,S}u_{j},\quad\beta_{j,S}\breve{u}_{j},\quad\breve{\beta}_{j,S}u_{j},\quad\breve{\beta}_{j,S}\breve{u}_{j}={\mathcal{O}}(h^{2/(m+1)}).$

Moreover, with the notations $F_{j}:=\alpha_{j,L}-\alpha_{j,R}$,
$\breve{F}_{j}:=\breve{\alpha}_{j,L}-\breve{\alpha}_{j,R}$, we have

(3.39) $A=\begin{pmatrix}1&0&\breve{F}_{2}\breve{u}_{2}&\breve{F}_{2}u_{2}\\\
0&1&F_{2}\breve{u}_{2}&F_{2}u_{2}\\\
\breve{F}_{1}\breve{u}_{1}&\breve{F}_{1}u_{1}&1&0\\\
F_{1}\breve{u}_{1}&F_{1}u_{1}&0&1\end{pmatrix}+{\mathcal{O}}(h^{2/(m+1)}),$

and

$F_{1}u_{1}=-i\omega h^{1/(m+1)}+{\mathcal{O}}(h^{2/(m+2)}),\quad
F_{2}u_{2}=-i\overline{\omega}h^{1/(m+1)}+{\mathcal{O}}(h^{2/(m+2)}),$

where $\omega=\omega_{\rho}$ is given by

(3.40)
$\omega_{\rho}=\mu_{m}\left(\frac{2(m+1)!}{|v_{1}-v_{2}|}\right)^{\frac{1}{m+1}}E_{0}^{-\frac{m}{2(m+1)}}\bm{\Gamma}\left(\frac{m+2}{m+1}\right)\overline{U(\rho)},$
(3.41) $\mu_{m}=\left\\{\begin{aligned}
&e^{\frac{i\pi}{2(m+1)}\operatorname{sgn}(v_{2}-v_{1})}&&\quad\text{when $m$
is odd, }\ \\\ &\cos\left(\frac{\pi}{2(m+1)}\right)&&\quad\text{when $m$ is
even. }\ \end{aligned}\right.$

######  Remark 3.12.

In the general case, where the crossing point is
$\rho=(\rho_{x},\rho_{\xi})\in\mathcal{V}$, one has to replace $E_{0}$ by
$E_{0}-V_{j}(\rho_{x})$ and $v_{j}$ by $V_{j}^{\prime}(\rho_{x})$. In
addition, when $m$ is odd, $\mu_{m}$ is given by

$\mu_{m}=e^{\frac{i\pi}{2(m+1)}\operatorname{sgn}\rho_{\xi}(v_{2}-v_{1})}.$

###### Proof.

The estimates (3.37) and (3.38) are proved in the same manner as Proposition
3.8.

We compute the principal term of
$F_{1}u_{1}=(\alpha_{1,L}-\alpha_{1,R})u_{1}$. Since $J_{1,L}$ and $J_{1,R}$
are both identity at the principal level, we have

$F_{1}u_{1}=-{\frac{i}{2}}\int_{-\delta}^{\delta}\breve{u}_{2}(y)U_{2}u_{1}(y)dy+{\mathcal{O}}(h^{2/(m+1)}).$

This is again an oscillatory integral (3.17) with the phase function

$\phi(y)=\int_{0}^{y}\left(\sqrt{E-V_{1}(t)}-\sqrt{E-V_{2}(t)}\right)dt$

which behaves like

(3.42)
$\displaystyle\phi(y)=\int_{0}^{y}\frac{V_{2}(t)-V_{1}(t)}{\sqrt{E-V_{1}(t)}+\sqrt{E-V_{2}(t)}}dt=\frac{V_{2}^{(m)}(0)-V_{1}^{(m)}(0)}{2\sqrt{E}(m+1)!}\left(y^{m+1}+{\mathcal{O}}(y^{m+2})\right)$

near the stationary point $0\in I=[-\delta,\delta]$, and the amplitude
$\sigma(y,h)$ satisfying

$\sigma(y,h)=-i\frac{(r_{0}(y)-ir_{1}(y)\sqrt{E-V_{1}(y)})}{2(E-V_{1}(y))^{1/4}(E-V_{2}(y))^{1/4}}+{\mathcal{O}}(h).$

Then the asymptotic formula for $F_{1}u_{1}$ is computed using Lemma 3.5. ∎

###  3.4. The space of microlocal solutions near a crossing point (Proof of
Theorem 2)

Recall from Proposition 3.9 that

$w_{j}^{\flat}\equiv\left\\{\begin{aligned}
&f_{j}^{\flat}+{\mathcal{O}}(h)\quad\text{ near }\ {\bm{e}}_{j}^{\flat},\\\
&{\mathcal{O}}(h)\quad\text{ near }\
{\bm{e}}_{\hat{j}}^{\flat}\cup\breve{\bm{e}}_{1}^{\sharp}\cup\breve{\bm{e}}_{2}^{\sharp},\end{aligned}\right.\quad\breve{w}_{j}^{\sharp}\equiv\left\\{\begin{aligned}
&\breve{f}_{j}^{\sharp}+{\mathcal{O}}(h)\quad\text{ near }\
\breve{\bm{e}}_{j}^{\sharp},\\\ &{\mathcal{O}}(h)\quad\text{ near }\
\breve{\bm{e}}_{\hat{j}}^{\sharp}\cup{\bm{e}}_{1}^{\flat}\cup{\bm{e}}_{2}^{\flat}.\end{aligned}\right.$

We can arrange $w_{j}^{\flat}$ so that it is microlocally zero near
$\breve{\rho}$ instead of ${\mathcal{O}}(h)$, and similarly arrange
$\breve{w}_{j}^{\sharp}$ so that it is microlocally zero near $\rho$.

######  Lemma 3.13.

There exist a basis $(w_{1}^{+},w_{2}^{+},w_{1}^{-},w_{2}^{-})$ of exact
solutions to the equation $(P-E)w=0$ such that, for $j=1,2$,

(3.43) $w_{j}^{+}\equiv\left\\{\begin{aligned} &w_{j}^{\flat}&&\text{near
}\rho,\\\ &\,\,0&&\text{near }\breve{\rho},\end{aligned}\right.\quad
w_{j}^{-}\equiv\left\\{\begin{aligned} &\,\,0&&\text{near }\rho,\\\
&\breve{w}_{j}^{\sharp}&&\text{near }\breve{\rho}.\end{aligned}\right.$

Figure 3. Cutoff functions for the construction of a quasi-mode

This lemma immediately implies Proposition 3.4. We first prove Proposition 3.3
using this lemma.

###### Proof of Proposition 3.3.

Let $w\in L^{2}(I;\mathbb{C}^{2})$ be a microlocal solution near $\rho$ with
$\|w\|_{L^{2}(I)}\leq 1$. Lemma 3.13 with the microlocal behavior derived by
Proposition 3.9 of $w_{j}^{\flat}$ and $\breve{w}_{j}^{\sharp}$ guarantees
that there exist exact solutions $w^{\flat}$ and $w^{\sharp}$ such that

$w\equiv w^{\bullet}\quad\text{microlocally near }\
{\bm{e}}_{1}^{\bullet}\cup{\bm{e}}_{2}^{\bullet},\quad\operatorname{WF}_{h}(w^{\bullet})\subset\\{\xi>0\\}$

for each ${\bullet}={\sharp},{\flat}$. We define the function
$w_{\text{quasi}}$ by

(3.44)
$w_{\text{quasi}}=\chi(x)\varphi(hD_{x})w+\chi_{-}(x)w^{\flat}+\chi_{+}(x)w^{\sharp},$

where the functions $\varphi,\chi,\chi_{\pm},\in C^{\infty}(\mathbb{R};[0,1])$
satisfy $\chi+\chi_{-}+\chi_{+}=1$, $\varphi(\xi)=1$ near $\xi=\sqrt{E_{0}}$,
$\chi(x)=1$ near $x=0$, $\chi_{\pm}(x)=0$ for $\pm x<0$,

$\displaystyle(\operatorname{supp}\chi\times\operatorname{supp}\varphi)\cap\operatorname{WF}_{h}(w)\subset(\Gamma^{I}\cap\\{\xi>0\\})\subset\\{\operatorname{supp}\varphi(\xi)=1\\},$

and that the support of $\varphi$, $\chi$ and of $\chi_{I^{\prime}}$ are
compact (see Figure 3). More precisely, the operator $\varphi(hD_{x})$ is
defined by the following sense: Fix an interval
$I^{\prime}\subset\joinrel\subset I$, take $\chi_{I^{\prime}}\in
C_{0}^{\infty}(I;[0,1])$ such that $\chi_{I^{\prime}}=1$ on $I^{\prime}$, and

$\varphi(hD_{x})f(x):=\frac{1}{2\pi
h}\iint_{\mathbb{R}^{2}}e^{i(x-y)\xi/h}\varphi(\xi)\chi_{I^{\prime}}(y)f(y)dyd\xi.$

In what follows, we denote $I^{\prime}$ by $I$.

Let us show

(3.45) $w\equiv w_{\text{quasi}}\quad\text{ near }\
\rho,\quad\|(P(h)-E)w_{\text{quasi}}\|_{L^{2}(I)}={\mathcal{O}}(h^{\infty})$

for $w_{\text{quasi}}$ given by (3.44). By writing

$(P(h)-E)w_{\text{quasi}}=[P,\chi]\varphi(hD_{x})w+\chi[P,\varphi(hD_{x})]w+\chi\varphi(hD_{x})(P-E)w+[P,\chi_{-}]w^{\flat}+[P,\chi_{+}]w^{\sharp},$

we see that the terms $\chi[P,\varphi(hD_{x})]w$ and
$\chi\varphi(hD_{x})(P-E)w$ are ${\mathcal{O}}(h^{\infty})$ since the
essential support of the Weyl symbol of the pseudo-differential operator
$\chi[P,\varphi(hD_{x})]$ and of $\chi\varphi(hD_{x})$ is compact and does not
intersect with the semiclassical wavefront set of $w$ and of $(P-E)w$,
respectively. By the equality $[P,\chi]=-([P,\chi_{-}]+[P,\chi_{+}])$, the
other terms are sum of the following two terms:

$\displaystyle[P,\chi_{-}](w^{\flat}-\varphi(hD_{x})w)=[P,\chi_{-}](1-\varphi(hD_{x}))w^{\flat}+{\mathcal{O}}(h^{\infty}),$
$\displaystyle[P,\chi_{+}](w^{\sharp}-\varphi(hD_{x})w)=[P,\chi_{+}](1-\varphi(hD_{x}))w^{\sharp}+{\mathcal{O}}(h^{\infty}).$

Since the functions $w^{\flat}$ and $w^{\sharp}$ are exact solution to
$(P-E)w^{\bullet}=0$ and microlocally zero away from
$\Gamma^{I}\cap\\{\xi>0\\}$, the above two terms are also
${\mathcal{O}}(h^{\infty})$.

By applying Cramer’s rule, we have

(3.46)
$w_{\text{quasi}}(x)=\frac{1}{\omega}\sum_{k=1}^{4}\omega_{k}(x)w_{k},\quad\omega:=\det\begin{pmatrix}w_{1}&w_{2}&w_{3}&w_{4}\\\
w_{1}^{\prime}&w_{2}^{\prime}&w_{3}^{\prime}&w_{4}^{\prime}\end{pmatrix},$

where the function $\omega_{k}$ is given by the Wronskian of the tuple
$(w_{1},w_{2},w_{3},w_{4})$ replaced $w_{k}$ by $w_{\text{quasi}}$ for each
$k$, for example,

$\omega_{1}(x):=\det\begin{pmatrix}w_{\text{quasi}}(x)&w_{2}(x)&w_{3}(x)&w_{4}(x)\\\
w_{\text{quasi}}^{\prime}(x)&w_{2}^{\prime}(x)&w_{3}^{\prime}(x)&w_{4}^{\prime}(x)\end{pmatrix},$

Then $\omega_{k}^{\prime}={\mathcal{O}}(h^{\infty})$ uniformly on $I$, and

(3.47)
$w_{\text{quasi}}(x)=\widetilde{w}+{\mathcal{O}}(h^{\infty})\quad\text{uniformly
on }\ I\quad\text{with }\
\widetilde{w}=\frac{1}{\omega}\sum_{k=1}^{4}\omega_{k}(0)w_{k}(x)+{\mathcal{O}}(h^{\infty}).$

This with (3.45) proves Proposition 3.3. ∎

###### Proof of Lemma 3.13.

Let $w$ be a bounded exact solution to $(P-E)w=0$ on $I$. The asymptotic
behavior (3.14) of $w_{1}^{\flat}$ and $w_{2}^{\flat}$ implies that, for each
$j=1,2$, there exists a linear combination of them which coincides with
$f_{j}^{\flat}$ microlocally near ${\bm{e}}_{j}^{\flat}$ and with $0$ near
${\bm{e}}_{\hat{j}}^{\flat}$. Consequently, there exist constants $c_{1}$ and
$c_{2}$ such that

(3.48) $w_{0}:=w-(c_{1}w_{1}^{\flat}+c_{2}w_{2}^{\flat})\equiv 0\quad\text{
near }\ {\bm{e}}_{1}^{\flat}\cup{\bm{e}}_{2}^{\flat}.$

We claim that for each $n\in\mathbb{N}$, there exists a basis
$(w_{1,n}^{-},\,w_{1,n}^{+},\,w_{2,n}^{-},\,w_{2,n}^{+})$ of exact solutions
such that

$w_{j,n}^{-}\equiv\left\\{\begin{aligned}
&(1+{\mathcal{O}}(h^{2/(m+1)}))\breve{f}_{j}^{\bullet}&&\quad\text{ near }\
\breve{\bm{e}}_{j}^{\bullet},\\\
&{\mathcal{O}}(h^{1/(m+1)})\breve{f}_{\hat{j}}^{\bullet}&&\quad\text{ near }\
\breve{\bm{e}}_{\hat{j}}^{\bullet},\\\ &{\mathcal{O}}(h^{n})&&\quad\text{ near
}\ \rho,\end{aligned}\right.$

and

$w_{j,n}^{+}\equiv\left\\{\begin{aligned}
&(1+{\mathcal{O}}(h^{2/(m+1)}))f_{j}^{\bullet}&&\quad\text{ near }\
{\bm{e}}_{j}^{\bullet},\\\
&{\mathcal{O}}(h^{1/(m+1)})f_{\hat{j}}^{\bullet}&&\quad\text{ near }\
{\bm{e}}_{\hat{j}}^{\bullet},\\\ &{\mathcal{O}}(h^{n})&&\quad\text{ near }\
\breve{\rho},\end{aligned}\right.$

for each $(j,\hat{j})=(1,2),(2,1)$ and ${\bullet}={\flat},{\sharp}$. We write
$w_{0}$ as a linear combination of this basis:

$w_{0}=c_{1,n}w_{1,n}^{+}+c_{2,n}w_{2,n}^{+}+c_{3,n}w_{1,n}^{-}+c_{4,n}w_{2,n}^{-}.$

The boundedness of $\|w_{0}\|_{L^{2}(I)}$ implies that the coefficients
$c_{k,n}$ $(k=1,2,3,4)$ are bounded as $h\to 0^{+}$, and in particular, we
have

(3.49) $\displaystyle
w_{0}-(c_{1,n}w_{1,n}^{+}+c_{2,n}w_{2,n}^{+})=c_{3,n}w_{1,n}^{-}+c_{4,n}w_{2,n}^{-}\equiv{\mathcal{O}}(h^{n})\text{
near }\rho.$

The above formula (3.49) with (3.48) implies that $c_{1,n}$ and $c_{2,n}$ are
${\mathcal{O}}(h^{n})$ as $h\to 0^{+}$, and

(3.50) $w_{0}\equiv{\mathcal{O}}(h^{n})\quad\text{ near }\ \rho.$

Since $n\in\mathbb{N}$ is arbitrarily chosen, we conclude here that

(3.51) $w\equiv c_{1}w_{1}^{\flat}+c_{2}w_{2}^{\flat}\quad\text{ near }\
\rho.$

In the above argument, we also proved for any exact solution $w$ the existence
of an exact solution $w_{0}$ such that

(3.52) $w_{0}\equiv\left\\{\begin{aligned} &0&&\text{near }\rho,\\\
&w+{\mathcal{O}}(h)&&\text{near }\breve{\rho}.\end{aligned}\right.$

We see that the required solutions $w_{1}^{-}$ and $w_{2}^{-}$ are given by
linear combinations of $w_{0}$ corresponding to $\breve{w}_{1}^{\sharp}$ and
to $\breve{w}_{2}^{\sharp}$. The solutions $w_{1}^{+},w_{2}^{+}$ are given in
the symmetric way.

We then prove the existence of such a basis
$(w_{1,n}^{-},w_{1,n}^{+},w_{2,n}^{-},w_{2,n}^{+})$. We here only discuss on
$w_{1,n}^{+}$. Others are also shown in a symmetric way. It suffices to
construct an approximate solution $g_{k}$ such that

(3.53) $(P(h)-E)g_{k}=\begin{pmatrix}hU_{1}r_{k}\\\ 0\end{pmatrix}$

with a function $r_{k}$ which is of order $h^{(2k-1)/(m+1)}$ uniformly on $I$.
In particular, $r_{k}={\mathcal{O}}(h^{n})$ for a large enough
$k\geq((m+1)n+1)/2$. Then the required solution $w_{1,n}^{+}$ is given by

(3.54) $w_{1,n}^{+}=g_{k}+\begin{pmatrix}J_{1,L}r_{k}\\\
\widetilde{K}_{2,L}J_{1,L}r_{k}\end{pmatrix}.$

Such an approximate solution $g_{k}$ is constructed inductively. The facts

$(P_{2}-E)\widetilde{K}_{2,L}u_{1}=-hU_{2}u_{1},\quad\operatorname{WF}_{h}(-hU_{2}u_{1})\subset\operatorname{WF}_{h}(u_{1})\subset\Gamma_{1}\cap\\{\xi>0\\}$

imply that $\operatorname{WF}_{h}(\widetilde{K}_{2,L}u_{1})$ is contained in
$(\Gamma_{1}\cap\\{\xi>0\\})\cup\Gamma_{2}$. The usual propagation of
singularities theorem with the fact
$\operatorname{WF}_{h}(-hU_{2}u_{1})\cap(\Gamma_{2}\cap\\{\xi<0\\})=\emptyset$
shows that there exists a constant $c_{2,1}\in\mathbb{C}$ such that

$\widetilde{K}_{2,L}u_{1}\equiv c_{2,1}\breve{u}_{2}\quad\text{microlocally
near }\ \Gamma_{2}\cap\\{\xi<0\\}.$

According to the estimate (3.32), this constant admits
$c_{2,1}={\mathcal{O}}(h)$ as $h\to 0^{+}$. By putting $g_{1,1}:=u_{1}$ and
$g_{2,1}:=\widetilde{K}_{2,L}u_{1}-c_{2,1}\breve{u}_{2}$, we have

(3.55)
$\operatorname{WF}_{h}(g_{1,1})\subset\Gamma\cap\\{\xi>0\\},\quad\operatorname{WF}_{h}(g_{2,1})\subset\Gamma\cap\\{\xi>0\\}.$

We then apply the above argument for $g_{1,1}=u_{1}$ to $g_{2,1}$. Then there
exists $c_{1,2}={\mathcal{O}}(h^{2})$ such that one has

(3.56)
$\operatorname{WF}_{h}(g_{1,2})\subset\Gamma\cap\\{\xi>0\\}\quad\text{for }\
g_{1,2}:=g_{1,1}+\widetilde{K}_{1,L}g_{2,1}-c_{1,2}\breve{u}_{1}.$

By repeating this process, we obtain such functions $g_{1,k}$, $g_{2,k}$ and
constants $c_{j,k}={\mathcal{O}}(h^{2k+j-3})$ $(j=1,2)$ for each
$k\in\mathbb{N}$ that we have

(3.57)
$\displaystyle\operatorname{WF}_{h}(g_{1,k})\subset\Gamma\cap\\{\xi>0\\}\quad\text{for
}\ g_{1,k}:=g_{1,k-1}+\widetilde{K}_{1,L}g_{2,k-1}-c_{1,k}\breve{u}_{1},$
$\displaystyle\operatorname{WF}_{h}(g_{2,k})\subset\Gamma\cap\\{\xi>0\\}\quad\text{for
}\ g_{2,k}:=g_{2,k-1}+\widetilde{K}_{2,L}g_{1,k}-c_{2,k}\breve{u}_{2}.$

Then these functions satisfy

$(P(h)-E)g_{k}=\begin{pmatrix}hU_{1}r_{k}\\\ 0\end{pmatrix}\quad\text{with }\
g_{k}=\begin{pmatrix}g_{1,k}\\\ g_{2,k}\end{pmatrix},\quad
r_{k}=\widetilde{K}_{2,L}g_{1,k}-c_{2,k}\breve{u}_{2}.$

The estimates on the resolvent operators given by Proposition 3.8 implies
$r_{k}={\mathcal{O}}(h^{(2k-1)/(m+1)})$. ∎

##  4\. Resonance asymptotics

This section is devoted to the proof of Theorem 1.

###  4.1. Probability amplitude

We call a complex number $E\in\mathcal{R}$ a pseudo-resonance of $P$ if there
exists an outgoing microlocal solution to the system $(P-E)w=0$ near an enough
large bounded subset of $\Gamma$. We make this definition precise by
introducing the notions of probability amplitude and monodromy matrix. We will
see that the pseudo-resonances approximate the resonances in the semiclassical
limit.

On the characteristic set $\Gamma$, we define what we call probability
amplitude. The probability amplitude is a complex number associated with each
generalized classical trajectory (see Definition 2.1). We recall that
$S_{\gamma}=\int_{\gamma}\xi dx$ is the action along the trajectory $\gamma$.

Recall first from the Maslov theory that the microlocal WKB solutions change
their asymptotic behavior when continued beyond a turning point (see [10] in
our system case). More precisely, let $\gamma_{\rm{cl}}$ be a classical
trajectory for $p_{j}$ starting from
$\rho_{-}=(x_{-},\xi_{-})\in\Gamma_{j}\setminus(\mathcal{V}\cup\\{\xi=0\\})$
and ending at
$\rho_{+}=(x_{+},\xi_{+})\in\Gamma_{j}\setminus(\mathcal{V}\cup\\{\xi=0\\})$
without passing through any crossing point:
$\gamma_{\rm{cl}}\cap\mathcal{V}=\emptyset$. Then there exists
$\nu_{\gamma_{\rm{cl}}}=\nu_{\gamma_{\rm{cl}}}(h)$ which coincides with the
number of turning points in $\gamma_{\rm{cl}}$ modulo ${\mathcal{O}}(h)$ such
that the two microlocal solutions $f_{\rho_{-}}(x,h;x_{-})$ and
$f_{\rho_{+}}(x,h;x_{+})$ to the system near $\rho_{-}$ and $\rho_{+}$,
respectively (see Proposition 3.2), are related with the formula:

(4.1)
$f_{\rho_{-}}(x,h;x_{-})=\exp\left(\frac{i}{h}S_{\gamma_{\rm{cl}}}-\frac{i\pi}{2}\nu_{\gamma_{\rm{cl}}}\right)f_{\rho_{+}}(x,h;x_{+}).$

Let $\gamma:[0,T]\to\Gamma$ be a generalized classical trajectory whose
extremities are supposed not to be a crossing point nor a turning point.
Thanks to Assumption 3 which guarantees that a turning point cannot be a
crossing point, one can take a finite partition of the time interval

(4.2) $[0,T]=\bigsqcup_{l\in L}I_{l}\sqcup\bigsqcup_{n\in N}I_{n}$

such that, for each $l\in L$, $\gamma_{l}:=\gamma(I_{l})$ passes through a
single turning point and no crossing point, and for $n\in N$,
$\gamma_{n}:=\gamma(I_{n})$ passes through a single crossing point
$\rho_{n}\in\mathcal{V}$ and no turning point. Let
$\nu_{\gamma_{l}}=1+\mathcal{O}(h)$, $l\in L$, be defined by (4.1). For each
$n\in N$, let $T_{\rho_{n}}$ be the transfer matrix at $\rho_{n}$ defined with
the microlocal WKB solutions at the extremities of $\gamma_{n}$, given by
Theorem 2. We denote $\tau_{\gamma_{n}}$ the entry of $T_{\rho_{n}}$
corresponding to the trajectory $\gamma_{n}$, that is, if the incoming part of
$\gamma_{n}$ belongs to $\Gamma_{j}$ and the outgoing part belongs to
$\Gamma_{k}$, then $\tau_{\gamma_{n}}=(T_{\rho_{n}})_{k,j}$ is the
$(k,j)$-entry of $T_{\rho_{n}}$. The following definition of the probability
amplitude does not depend of the choice of the partition of the time interval
(4.2).

######  Definition 4.1.

For a generalized classical trajectory $\gamma$ whose extremities are neither
a turning point nor a crossing point, we define the probability amplitude
$\mathscr{P}(\gamma)$ by

(4.3)
$\mathscr{P}(\gamma):=\exp\left(\frac{i}{h}S_{\gamma}-\frac{i\pi}{2}\sum_{l\in
L}\nu_{\gamma_{l}}\right)\prod_{n\in N}\tau_{\gamma_{n}}.$

######  Remark 4.2.

The probability amplitude maps the addition of generalized classical
trajectories to the multiplication. More precisely, if
$\gamma_{1}:[0,T_{1}]\to\Gamma$ and $\gamma_{2}:[0,T_{2}]\to\Gamma$ are two
consecutive generalized classical trajectories, that is, the terminal point
$\gamma_{1}(T_{1})$ coincides with the initial point $\gamma_{2}(0)$, then one
has

(4.4)
$\mathscr{P}(\gamma_{1}\cup\gamma_{2})=\mathscr{P}(\gamma_{1})\mathscr{P}(\gamma_{2}).$

Here $\gamma_{1}\cup\gamma_{2}:[0,T_{1}+T_{2}]\to\Gamma$ stands for the usual
composition:

$(\gamma_{1}\cup\gamma_{2})(t)=\left\\{\begin{aligned} &\gamma_{1}(t)&&0\leq
t<T_{1},\\\ &\gamma_{2}(t-T_{1})&&T_{1}\leq t\leq
T_{1}+T_{2}.\end{aligned}\right.$

Using the terminology of the graph theory, we call vertex a crossing point and
edge a classical trajectory joining two adjacent vertices (i.e. an edge does
not contain any vertex except at the extremities), and write $\mathcal{E}$ the
set of all edges (we already use the notation $\mathcal{V}$ for the set of
vertices). For an edge ${\bm{e}}$, we denote ${\bm{e}}^{-}$ its starting point
and ${\bm{e}}^{+}$ its endpoint.

###  4.2. Monodromy matrix

We take a base point $\rho_{\bm{e}}$ on each edge ${\bm{e}}\in\mathcal{E}$,
and define a matrix $M(E,h)$ the size of which is the total number of edges:

(4.5)
$M(E,h)=(m_{{\bm{e}},{\bm{e}}^{\prime}}(E,h))_{{\bm{e}},{\bm{e}}^{\prime}\in\mathcal{E}}.$

The $({\bm{e}},{\bm{e}}^{\prime})$-entry is 0 if
${\bm{e}}^{-}\neq({\bm{e}}^{\prime})^{+}$ and otherwise it is given by the
probability amplitude $\mathscr{P}({\gamma})$ of the generalized classical
trajectory $\gamma$ from $\rho_{{\bm{e}}^{\prime}}$ to $\rho_{\bm{e}}$ passing
through exactly one crossing point ${\bm{e}}^{-}=({\bm{e}}^{\prime})^{+}$.

######  Definition 4.3.

We call zeros in $\mathcal{R}$ of $\det(I-M(E,h))$ pseudo-resonances of $P$,
and denote the set of pseudo-resonances by $\operatorname{Res}_{0}(P)$.

######  Remark 4.4.

The pseudo-resonances are independent of the choice of the base points
$\rho_{\bm{e}}$.

######  Remark 4.5.

If $E\in\mathcal{R}$ and if $w$ is an outgoing microlocal solution to the
equation $(P-E)w=0$ near some neighborhood of $\mathcal{E}$, then $E$ is a
pseudo-resonance, and there exists an eigenvector
$\alpha=(\alpha_{\bm{e}})_{{\bm{e}}\in\mathcal{E}}$ of the eigenvalue 1 of
$M(E,h)$ such that

(4.6) $w\equiv\alpha_{\bm{e}}f_{\rho_{\bm{e}}}\quad\text{ near }\ {\bm{e}}$

for any ${\bm{e}}\in\mathcal{E}$. Here $f_{\rho_{\bm{e}}}$ is the WKB
microlocal solution to the system $(P-E)w=0$ near $\rho_{\bm{e}}$ given by
Proposition 3.2.

We will see in the next subsection that the pseudo-resonances are a good
approximation of the resonances in $\mathcal{R}$. Here we compute the
asymptotic behavior of the determinant $\det(I-M(E,h))$ and observe that the
condition of the pseudo-resonances coincide with the Bohr-Sommerfeld
quantization condition at principal level.

######  Proposition 4.6.

As $h\to 0^{+}$, we have

(4.7)
$\displaystyle\det(I-M(E,h))=1+e^{i\mathcal{A}(E)/h}+{\mathcal{O}}(h^{\frac{2}{m_{0}+1}}),$
$\displaystyle
h\partial_{E}\det(I-M(E,h))=i{\mathcal{A}}^{\prime}(E)e^{i{\mathcal{A}}(E)/h}+{\mathcal{O}}(h^{\frac{2}{m_{0}+1}}),$

uniformly for $E\in\mathcal{R}$.

As a consequence of the above asymptotics one immediately gets the following
result:

######  Corollary 4.7.

The pseudo-resonances are simple zeros of $\det(I-M(E,h))$ and there exists a
bijection $\widetilde{z}_{h}:\mathfrak{B}_{h}\to\operatorname{Res}_{0}(P)$
such that as $h\to 0^{+}$ one has,

(4.8) $\widetilde{z}_{h}(E)=E+{\mathcal{O}}(h^{\frac{m_{0}+3}{m_{0}+1}}),$

uniformly for any $E\in\mathfrak{B}_{h}$.

As in the standard graph theory, we call a consecutive sequence of edges a
path, and a path whose initial and terminal point coincide a cycle (we
identify two cycles by ignoring their initial and terminal point). A cycle is
said to be primitive if it does not contain any sub-cycles. Let $\mathcal{C}$
be the set of primitive cycles.

###### Proof of Proposition 4.6.

By definition, each non-zero entry of $M(E,h)$ stands for the probability
amplitude. According to (4.4), the multiplication of probability amplitudes
corresponds to the composition of consecutive generalized classical
trajectories. Consequently, the determinant $\det(I-M)$ can be written in
terms of the probability amplitudes associated with the cycles as follows:

(4.9) $\det(I-M)=1+\sum_{K}\prod_{{\gamma}\in K}(-\mathscr{P}({\gamma})),$

where the summation is taken over the nonempty subsets $K$ of $\mathcal{C}$
such that ${\gamma}\cap{\gamma}^{\prime}=\emptyset$ holds for any ${\gamma},$
${\gamma}^{\prime}\in K$ with ${\gamma}\neq{\gamma}^{\prime}$. According to
Theorem 3, for each crossing point $\rho\in\mathcal{V}$, the $(j,k)$-entry of
$T_{\rho}$ is $1+{\mathcal{O}}(h^{2/(m_{\rho}+1)})$ if $j=k$ and is
${\mathcal{O}}(h^{1/(m_{\rho}+1)})$ if $j\neq k$. Then by the definition of
the probability amplitude,

(4.10) $\mathscr{P}({\gamma})={\mathcal{O}}(h^{2/(m_{0}+1)})$

holds for any primitive directed cycle ${\gamma}$ containing some edges of
$\mathcal{E}_{2}$. We also have

(4.11)
$\mathscr{P}(\Gamma_{1})=(-1+{\mathcal{O}}(h^{2/(m_{0}+1)}))e^{i{\mathcal{A}}(E)/h}.$

The representation (4.9) of $\det(I-M)$ with Formulae (4.10) and (4.11) imply
the asymptotic formula for $\det(I-M)$. For the derivative, we apply the fact
that multiplying by $h$ the derivative with respect to $E$ of each entry of
the transfer matrix $T_{\rho}$ keeps the same order as itself. ∎

######  Example.

Let us consider the model in Subsection 2.3. We label the three edges as in
Figure 4, and for each $j\in\\{1,2,3\\}$ we take the base point
$\rho_{{\bm{e}}_{j}}={\bm{e}}_{j}^{-}$. Then the monodromy matrix $M$ is given
by $M=T\Phi$, where $T$ and $\Phi$ are the $3\times 3$ matrices given by

(4.12)
$\Phi=\exp\left(\frac{i}{h}\operatorname{diag}\left({\mathcal{A}}_{1,L}-\frac{h\pi}{2},{\mathcal{A}}_{1,R}-\frac{h\pi}{2},{\mathcal{A}}_{2,R}-\frac{h\pi}{2}\right)\right),$

with

${\mathcal{A}}_{1,L}:=2\int_{a(E)}^{0}\sqrt{E-V_{1}(x)}\,dx,\quad{\mathcal{A}}_{1,R}:=2\int_{0}^{b(E)}\sqrt{E-V_{1}(x)}\,dx,$
${\mathcal{A}}_{2,R}:=2\int_{0}^{c(E)}\sqrt{E-V_{2}(x)}\,dx,$

and

(4.13) $T=\begin{pmatrix}0&(T_{\rho_{-}})_{1,1}&(T_{\rho_{-}})_{1,2}\\\
(T_{\rho_{+}})_{1,1}&0&0\\\ (T_{\rho_{+}})_{2,1}&0&0\end{pmatrix}.$

We have ${\mathcal{A}}_{1,L}+{\mathcal{A}}_{1,R}={\mathcal{A}}$, and
${\mathcal{A}}_{1,L}+{\mathcal{A}}_{2,R}=S_{\gamma}$ where $S_{\gamma}$ is the
action along the directed cycle $\gamma$ defined for this example by (2.8),
and obtain

$\displaystyle\det(I-M)=1+e^{i{\mathcal{A}}/h}(T_{\rho_{+}})_{1,1}(T_{\rho_{-}})_{1,1}+e^{iS_{\gamma}/h}(T_{\rho_{+}})_{2,1}(T_{\rho_{-}})_{1,2}.$

Figure 4. Labeling the edges in typical examples

###  4.3. Bijection between resonances and pseudo-resonances

In this subsection, we prove a one to one correspondence between the set of
resonances and the set of pseudo-resonances in the domain $\mathcal{R}$ and
estimate the distance between them.

######  Proposition 4.8.

For any $h$-independent $C>0$ and for $h$ small enough, there is no resonance
outside a $Ch^{2}$-neighborhood of pseudo-resonances.

######  Remark 4.9.

In fact, for any real number $n$, there is no resonance outside a
$Ch^{n}$-neighborhood of pseudo-resonances for $h$ small enough.

###### Proof.

We proceed by a contradiction argument. Suppose that the proposition is not
true. Then there would exist a subset $\mathfrak{h}\subset(0,1]$ and maps $E$
and $w$ defined on $\mathfrak{h}$ with values in
$\\{z\in\mathcal{R};\,\operatorname{dist}(z,\operatorname{Res}_{0}(P(h)))>Ch^{2}\\}$
and $L^{2}(\mathbb{R};\mathbb{C}^{2})$ respectively, such that
$0\in\bar{\mathfrak{h}}$, $\|w(h)\|_{L^{2}}=1$ and

(4.14) $(P_{\theta}(h)-E(h))w(h)=0\quad\text{for }\ h\in\mathfrak{h}.$

Then the equality (4.6) for $w$ holds with a vector
$\alpha=(\alpha_{\bm{e}})_{{\bm{e}}\in\mathcal{E}}$ such that
$(I-M(E,h))\alpha={\mathcal{O}}(h^{\infty})$. According to Proposition 4.6,
the inverse $(I-M(E,h))^{-1}$ exists and is polynomially bounded with respect
to $h$ for any $E\in\mathcal{R}$ with
$\operatorname{dist}(E,\operatorname{Res}_{0}(P(h)))>Ch^{2}$, and consequently
$w\equiv 0$ near $\Gamma(E_{0})$. This with Formula (4.14) implies that
$\|w\|_{L^{2}}={\mathcal{O}}(h^{\infty})$ which contradicts with
$\|w\|_{L^{2}}=1$. ∎

Now, using Proposition 4.8 and the Kato-Rellich theorem we prove the unique
existence of a resonance in ${\mathcal{O}}(h^{2})$-neighborhood of each
pseudo-resonance:

######  Proposition 4.10.

For each $E\in\mathfrak{B}_{h}$, there uniquely exists a resonance $z_{h}(E)$
verifying

(4.15) $\left|z_{h}(E)-\widetilde{z}_{h}(E)\right|\leq h^{2},$

for small $h>0$.

To prove this proposition, we introduce the family $P_{\theta}(s,h)$,
$s\in\mathbb{C},\,\left|s\right|<2$ of operators defined by

(4.16) $\displaystyle
P_{\theta}(s,h):=P_{\theta}^{\operatorname{diag}}(h)+shQ_{\theta},\quad
P_{\theta}^{\operatorname{diag}}(h):=\mathcal{U}_{\theta}P^{\operatorname{diag}}(h)\mathcal{U}_{\theta}^{-1},\quad
Q_{\theta}:=\mathcal{U}_{\theta}Q\mathcal{U}_{\theta}^{-1},$ $\displaystyle
P^{\operatorname{diag}}(h):=\begin{pmatrix}P_{1}(h)&0\\\
0&P_{2}(h)\end{pmatrix},\quad Q:=\begin{pmatrix}0&U_{1}\\\
U_{2}&0\end{pmatrix},\quad\mathcal{U}_{\theta}w:=\left|\zeta_{\theta}^{\prime}(x)\right|^{1/2}w\circ\zeta_{\theta},$

where $\zeta_{\theta}$ is defined in Section 2. This is first defined for
analytic functions in $\Sigma$, and then extended to an operator on
$L^{2}(\mathbb{R};\mathbb{C}^{2})$ by using the analyticity of the
coefficients (Assumption 1). The operator $P_{\theta}$ is non-self-adjoint in
$L^{2}(\mathbb{R};\mathbb{C}^{2})$, and its spectrum is discrete in
$\mathcal{R}$ due to the assumption $E_{0}\neq V_{j}^{\pm}$, $j=1,2$ (see [12,
Appendix B]), and is independent of $\theta$. The resonances in $\mathcal{R}$
of $P$ are equivalently defined as the eigenvalues of $P_{\theta}$.

For each fixed small $h>0$, this is an analytic family in the sense of Kato,
that is, $P_{\theta}(s)=P_{\theta}(s,h)$ satisfies the following conditions:

  1. (1)

$P_{\theta}(s)$ is closed and the resolvent set of $P_{\theta}(s)$ is not
empty for each $\left|s\right|<2$.

  2. (2)

For any $\left|s_{0}\right|<2$, there exists an element $\lambda_{0}$ of the
resolvent set of $P_{\theta}(s_{0})$ such that
$(P_{\theta}(s)-\lambda_{0})^{-1}$ is an analytic operator-valued function of
$s$ near $s_{0}$.

In fact, for fixed small $h$, we can take $C_{0}>0$ large enough such that

(4.17)
$\left\|\left(P_{\theta}^{\operatorname{diag}}-(E_{0}-iC_{0}h)\right)^{-1}\right\|_{L^{2}\to
H^{2}}\leq(h\|Q\|_{H^{1}\to L^{2}})^{-1},$

since we know that the spectrum near $E_{0}$ of $P_{\theta}(0)$ consists only
of the eigenvalues of $P_{1}$ on the real line, the essential spectrum and the
resonances of $P_{2}$ are far at least of order $h\log(1/h)$. Then the
resolvent of $P_{\theta}(s)$ is given by the Neumann series which depends
analytically on $s$:

$(P_{\theta}(s)-\lambda_{0})^{-1}=\left(I+sh(P_{\theta}^{\operatorname{diag}}-\lambda_{0})^{-1}Q_{\theta}\right)^{-1}(P_{\theta}^{\operatorname{diag}}-\lambda_{0})^{-1},\quad(\lambda_{0}=E_{0}-iC_{0}h).$

Our previous arguments for $P(h)$ and $P_{\theta}(h)$ remain true for

$P(s,h):=P^{\operatorname{diag}}+shQ=\begin{pmatrix}P_{1}&shU_{1}\\\
shU_{2}&P_{2}\end{pmatrix}\quad(s\in[0,1])$

and $P_{\theta}(s,h)$ $(s\in[0,1])$ (we may replace $U_{1},$ $r_{k}$ by
$sU_{1}$, $sr_{k}$ $(k=0,1)$). Note that our considering operator and the
reference diagonal operator are $P(h)=P(1,h)$ and
$P^{\operatorname{diag}}=P(0)$, respectively. Then we conclude by applying the
Kato-Rellich theorem that for each $E=E(h)\in\mathfrak{B}_{h}$, there uniquely
exists an analytic function $z_{h}(s,E(h))$ of $s$ such that
$z_{h}(0,E(h))=E(h)+{\mathcal{O}}(h^{2})$ is an eigenvalue of $P_{1}$ and
$z_{h}(s,E(h))$ is an eigenvalue of $P_{\theta}(s,h)$, that is, a resonance of
$P(s,h)$.

###### Proof of Proposition 4.10.

Proposition 4.6 shows that each pseudo-resonance is away of order $h$ from the
others. Then the complement in $\mathcal{R}$ of the resonance free region
(shown by Proposition 4.8 with $C=1$)

$\mathcal{R}\setminus\\{\operatorname{dist}(z,\operatorname{Res}_{0}(P(h)))<h^{2}\\}$

is written in the disjoint sum

$\\{\operatorname{dist}(z,\operatorname{Res}_{0}(P(h)))<h^{2}\\}\bigsqcup_{E\in\mathfrak{B}_{h}}\\{\left|z-\widetilde{z}_{h}(s,E)\right|<h^{2}\\}.$

In particular for $s=0$, each eigenvalue
$z_{h}(0,E(h))=E(h)+{\mathcal{O}}(h^{2})$ of $P_{1}$ has to satisfy
$\left|z_{h}(0,E(h))-\widetilde{z}_{h}(0,E)\right|<h^{2}$. We also know that
there is no other eigenvalue of the diagonal operator $P_{\theta}(0,h)$ in
$\mathcal{R}$. Then by the analyticity of $z_{h}(s,E)$, it is the unique
resonance with $\left|z_{h}(s,E)-\widetilde{z}_{h}(s,E)\right|<h^{2}$ for any
$s\in[0,1]$. ∎

###  4.4. Precise width of resonances

Proposition 4.10 with Corollary 4.7 immediately implies that each resonance
$z_{h}(E)$ corresponding to $E\in\mathfrak{B}_{h}$ admits the asymptotic
formula

(4.18) $z_{h}(E)=E+{\mathcal{O}}(h^{\frac{m_{0}+3}{m_{0}+1}}).$

The imaginary part of the subprincipal term of
${\mathcal{O}}(h^{(m_{0}+3)/(m_{0}+1)})$ of this formula would give the width
of the resonances. To obtain this subprincipal term, we need to compute the
subpricipal term of ${\mathcal{O}}(h^{2/(m_{0}+1)})$ of the diagonal entries
of the transfer matrices.

However we here employ an alternative method used in [10]. It consists in the
expression of the width of resonances in terms of the resonant state $w$ near
infinity. Due to the outgoing property of the resonant state, it is enough to
compute the microlocal asymptotics of the resonant state on the outgoing tails
of $\Gamma$. We restate this fact in the following lemma. Let
$\rho_{1}=(x_{1},\xi_{1})$ [resp. $\rho_{2}=(x_{2},\xi_{2})$] be a point on
the outgoing tail tending to $-\infty$ [resp. $+\infty$], if it exists, such
that $x_{1}<a_{0}<b_{0}<x_{2}$.

######  Lemma 4.11.

Let $f_{k}$ be the microlocal WKB solution defined in Proposition 3.2 near
$\rho_{k}$, whose phase is based on $\rho_{k}$, and $\alpha_{k}$ defined by

(4.19) $w\equiv\alpha_{k}f_{k}\text{ near }\rho_{k},\quad k=1,2.$

Then we have

(4.20) ${\rm Im}\hskip
1.0ptz=-\frac{h(1+{\mathcal{O}}(h))}{\|w\|^{2}_{L^{2}(x_{1},x_{2})}}(|\alpha_{1}|^{2}+|\alpha_{2}|^{2}),$

with the convention that $\alpha_{k}=0$ when
$\Gamma_{2}\cap\\{x=x_{k}\\}=\emptyset$.

###### Proof.

We have, by integration by parts from ${\rm Im}\hskip
1.0pt\langle(P(h)-z)w,w\rangle_{L^{2}(x_{1},x_{2})}=0$,

(4.21) $({\rm Im}\hskip 1.0ptz)\|w\|^{2}_{L^{2}(x_{1},x_{2})}=h^{2}{\rm
Im}\hskip
1.0pt\left[-v_{1}^{\prime}\bar{v}_{1}-v_{2}^{\prime}\bar{v}_{2}+r_{1}v_{2}\bar{v}_{1}\right]^{x_{2}}_{x_{1}},\quad
w=\begin{pmatrix}v_{1}\\\ v_{2}\end{pmatrix},$

where $[g]_{x_{1}}^{x_{2}}=g(x_{2})-g(x_{1})$.

For each $k=1,2$, $w={\mathcal{O}}(h^{\infty})$ near $x_{k}$ if
$\Gamma_{2}\cap\\{x=x_{k}\\}=\emptyset$. Otherwise, the line $\\{x=x_{k}\\}$
intersects with $\Gamma_{2}$ at two points (since $\Gamma_{2}$ is symmetric
with respect to $\\{\xi=0\\}$), where one is on an outgoing trajectory
corresponding to $p_{2}$ while the other is on an incoming trajectory. We
denote by $\rho_{k}=(x_{k},\xi_{k})$, $k=1,2$, the one on the outgoing
trajectory.

Remark that in the subset $\omega_{k}\times\mathbb{R}_{\xi}$ of the phase
space, where $\omega$ is a small neighborhood of $x_{k}$, $w$ has its
semiclassical wave front set only on the outgoing tail. This fact together
with the microlocal expression of $w$ on the outgoing tail implies that $w$
admits the local expression in $\omega_{k}$

(4.22) $w=\alpha_{k}\chi^{w}f_{k}+{\mathcal{O}}(h^{\infty})$

where $\chi\in C_{0}^{\infty}(T^{*}\mathbb{R};\mathbb{R})$ is identically 1
and supported only near $\rho_{k}$. Since $v_{1}$ is ${\mathcal{O}}(h)$ on
$\Gamma_{2}$, we have

$-v_{1}^{\prime}\bar{v}_{1}+r_{1}v_{2}\bar{v}_{1}\Big{|}_{x=x_{k}}={\mathcal{O}}(h),\quad
v_{2}^{\prime}\bar{v}_{2}\Big{|}_{x=x_{k}}=(-1)^{k}ih^{-1}\left|\alpha_{k}\right|^{2}(1+{\mathcal{O}}(h)).$

Thus we obtain the lemma. ∎

Thanks to this lemma, it is enough to compute the asymptotics of
$\|w\|_{L^{2}(x_{1},x_{2})}$ and $\alpha_{k}$.

Let ${\bm{e}}_{0}$ be the edge on $\Gamma_{1}$ such that its endpoint
${\bm{e}}_{0}^{+}$ coincides with the starting point of the outgoing tail
containing $\rho_{1}$ on it. We normalize $w$ in such a way that

(4.23) $w\equiv f_{\rho_{0}}\,\,\,\text{ near }\rho_{0},$

where $\rho_{0}$ is the base point of ${\bm{e}}_{0}$.

The following Proposition asserts that the microlocal behavior (4.23) of $w$
on ${\bm{e}}_{0}$ uniquely determines that on all other edges, if we “cut” the
periodic trajectory $\Gamma_{1}$ at the point $\rho_{0}$: Let
$\operatorname{Path}(\rho_{1},\rho_{2})$ be the set of all generalized
classical trajectories starting from $\rho_{1}$ and ending at $\rho_{2}$
without passing through $\rho_{0}$. This is an infinite set in general.

######  Lemma 4.12.

Let $w$ be a resonant state satisfying (4.23). Then at any point $\rho$ on
$\Gamma\setminus\mathcal{V}$, different from $\rho_{0}$, we have the following
microlocal expression of $w$:

(4.24) $w\equiv\alpha_{\rho}f_{\rho}\quad\text{ near }\ \rho\quad\text{with }\
\alpha_{\rho}=\sum_{{\gamma}\in\operatorname{Path}(\rho_{0},\rho)}\mathscr{P}({\gamma}).$

The sum on the right hand side of (4.24) absolutely converges for $h$ small
enough.

###### Proof.

Put $\widetilde{M}:=(I-E_{{\bm{e}}_{0},{\bm{e}}_{0}})M$ where
$E_{{\bm{e}}_{0},{\bm{e}}_{0}}$ stands for the matrix unit. Then for $k\geq
1$, each $({\bm{e}},{\bm{e}}^{\prime})$-entry of $\widetilde{M}^{k}$ stands
for the sum of the probability amplitude associated with generalized classical
trajectories from $\rho_{{\bm{e}}^{\prime}}$ to $\rho_{{\bm{e}}}$ passing
through $k$-crossing points without coming back to $\rho_{0}$:

(4.25)
$(\widetilde{M}^{k})_{{\bm{e}},{\bm{e}}^{\prime}}=\sum_{\begin{subarray}{c}\gamma\in\operatorname{Path}(\rho_{{\bm{e}}^{\prime}},\rho_{\bm{e}}),\\\
\\#\\{t;\,\gamma(t)\in\mathcal{V}\\}=k\end{subarray}}\mathscr{P}(\gamma).$

For $k$ large enough, such a generalized classical trajectory always
intersects with both $\Gamma_{1}$ and $\Gamma_{2}$ since
$\rho_{0}\in\Gamma_{1}$. This implies that the absolute value of each
eigenvalue of $\widetilde{M}$ is smaller than one, and
$(I-\widetilde{M})^{-1}=\sum_{k\geq 0}\widetilde{M}^{k}$. Then for any
${\bm{e}}$,

(4.26) $w\equiv\alpha_{\bm{e}}f_{\rho_{\bm{e}}}\quad\text{ near }\
\rho_{\bm{e}}$

holds with $\alpha=(\alpha_{\bm{e}})_{{\bm{e}}\in\mathcal{E}}$ given by

$\alpha:=(I-\widetilde{M})^{-1}\delta_{{\bm{e}}_{0}},\quad\delta_{{\bm{e}}_{0}}=(\delta_{{\bm{e}}_{0}}({\bm{e}}))_{{\bm{e}}\in\mathcal{E}}\quad\text{(Kronecker's
delta). }\ $

According to (4.25), we have

$\alpha_{\bm{e}}=\sum_{\gamma\in\operatorname{Path}(\rho_{0},\rho_{\bm{e}})}\mathscr{P}(\gamma),$

for any ${\bm{e}}\in\mathcal{E}$. This with (4.26) shows the required formula
(4.24) when $\rho$ is a point on an edge since the base point can be chosen
arbitrarily. Apply (4.4) for otherwise. ∎

In the same way as in [10], for a resonant state $w$ satisfying (4.23), we
have

(4.27)
$\left\|w\right\|_{L^{2}(x_{1},x_{2})}^{2}=2{\mathcal{A}}^{\prime}(E)+{\mathcal{O}}(h^{1/3}+h^{1/(m_{0}+1)}),$

where the error ${\mathcal{O}}(h^{1/3})$ comes from the turning point while
${\mathcal{O}}(h^{1/(m_{0}+1)})$ comes from the crossing points (in [10], the
later is ${\mathcal{O}}(h)$ $(m_{0}=1)$ and contained in the former). We
finally obtain the formula

(4.28) ${\rm Im}\hskip
1.0ptz_{h}(E)=-\frac{h}{2\left|{\mathcal{A}}^{\prime}(E_{0})\right|}\sum_{k=1,2}\Bigl{|}\sum_{\gamma\in\operatorname{Path}(\rho_{0},\rho_{k})}\mathscr{P}(\gamma)\Bigr{|}^{2}+{\mathcal{O}}(h^{\frac{m_{0}+3}{m_{0}+1}+\kappa}),$

where $\kappa=\min\\{1/3,1/(m_{0}+1)\\}$. This is of
${\mathcal{O}}(h^{\frac{m_{0}+3}{m_{0}+1}})$ since each generalized classical
trajectory changes between $\Gamma_{1}$ and $\Gamma_{2}$ at least one time.

The principal terms in the sum of the right hand side of (4.28) come from the
generalized classical trajectories which change trajectory once from
$\Gamma_{1}$ to $\Gamma_{2}$. Therefore we have the following proposition.

######  Proposition 4.13.

The coefficient $\mathcal{D}$ of the principal term of the resonance width
admits the formula

(4.29)
$\mathcal{D}(E,h)=\frac{h^{-2/(m_{0}+1)}}{2\left|{\mathcal{A}}^{\prime}(E_{0})\right|}\sum_{k=1,2}\left|\sum_{\gamma\in\operatorname{Path}^{1}(\rho_{0},\rho_{k})}\mathscr{P}(\gamma)\right|^{2},$

where $\operatorname{Path}^{1}(\rho_{0},\rho_{k})$ is the set of
$\gamma\in\operatorname{Path}(\rho_{0},\rho_{k})$ (see before Lemma 4.12 for
the definition) which changes trajectory from $\Gamma_{1}$ to $\Gamma_{2}$
exactly once. Here the sum over $k$ is for such a $k$ that the corresponding
outgoing tail exists.

######  Remark 4.14.

The constant $\mathcal{D}(E,h)$ defined by (4.29) is obviously independent of
the choice of $\rho_{0}$, $\rho_{1}$ and $\rho_{2}$ because of the absolute
value taken for the sum over the general classical trajectories with the same
starting and ending points. This absolute value is of order $h^{1/(m_{0}+1)}$
because they change characteristic set once, and hence $\mathcal{D}(E,h)$ is
bounded as $h\to 0^{+}$.

## Acknowledgement

A part of this work was done during the stay of the second and the third
authors in VIASM (Vietnam Institute of Advanced Study in Mathematics). They
thank VIASM for the financial support and the hospitality. The second author
was also supported by the JSPS KAKENHI Grant Number 18K03384. The third author
was supported by the Grant-in-Aid for JSPS Fellows Grant Number 22J00430. The
first author acknowledges the financial support of the CONICYT FONDECYT Grant
Number 3180390.

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