Eigenvalue Splitting of Polynomial Order for a System of Schrödinger Operators with Energy-Level Crossing

Abstract

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrödinger operators each of which has a simple well potential. Focusing on the cases where the two underlying classical periodic trajectories cross to each other, we give Bohr–Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal crossing cases. Our main results consist in the eigenvalue splitting which occurs when the two action integrals along the closed trajectories coincide. The splitting is of polynomial order $h^{{\frac{4}{3}}}$ in the tangential case and of order $h^{{\frac{3}{2}}}$ in the transversal case, and the coefficients of the leading terms reflect the geometry of the crossing.

Appendices

Appendix A. Solutions to the Scalar Equations

In this section we recall the construction of solutions to the scalar equations $(P_j-E)u=0$, $j=1,2$, together with some asymptotic properties needed in our study. We refer to [Ol, Ya, FMW1] for the details and the proofs. Since both $P_1$ and $P_2$ have the same type simple well potential, we drop the subscripts 1 and 2.

For E small enough, we introduce the auxiliary function

$$\begin{aligned} \xi _\beta (x) := \left\{ \begin{array}{lll} - \left( {{\frac{3}{2}}} \int _{x}^{\beta (E)} \sqrt{E-V(t)} dt \right) ^{{{\frac{2}{3}}}} &{}\quad \mathrm{for} \;\; x\in (\alpha (E),\beta (E)), \\ \left( {{\frac{3}{2}}} \int _{\beta (E)}^x \sqrt{V(t)-E} dt \right) ^{{{\frac{2}{3}}}} &{}\quad \mathrm{for} \;\; x\in [\beta (E), +\infty ), \end{array}\right. \end{aligned}$$

where we recall that $\alpha (E)<\beta (E)$ are the roots of $V(x)-E$. Let $\mathrm{Ai}$ and $\mathrm{Bi}$ be solutions to the Airy equation $u''(t)=tu(t)$ defined by

$$\begin{aligned} \mathrm{Ai}(t):= & {} \frac{1}{\pi } \int _0^{+\infty } \cos \left( \frac{\eta ^3}{3} + \eta t\right) d\eta , \end{aligned}$$

(A.1)

$$\begin{aligned} \mathrm{Bi}(t):= & {} \frac{1}{\pi } \int _0^{+\infty } \exp \left( -\frac{\eta ^3}{3} + \eta t\right) d\eta + \frac{1}{\pi } \int _0^{+\infty } \sin \left( \frac{\eta ^3}{3} + \eta t\right) d\eta . \end{aligned}$$

(A.2)

Proposition A.1

Under the assumptions (A1) and (A2), for E small enough, the equation $(P - E)u=0$ admits two real-valued solutions $u_{R}^{\pm }$ on ${\mathbb {R}}$ such that,

$$\begin{aligned} u_{R}^-(x)= & {} 2 (\xi _{\beta }'(x))^{-\frac{1}{2}} \mathrm{Ai}\,(h^{{-\frac{2}{3}}}\xi _{\beta }(x))+{\mathcal {O}}(h), \end{aligned}$$

(A.3)

$$\begin{aligned} u_{R}^+(x)= & {} (\xi _{\beta }'(x))^{-\frac{1}{2}} \mathrm{Bi}\,(h^{{-\frac{2}{3}}}\xi _{\beta }(x))+\mathcal {O}(h), \end{aligned}$$

(A.4)

as $h\rightarrow 0^+$ uniformly on any h-independent compact set in $(\alpha (E), +\infty )$. Moreover, we have

$$\begin{aligned} u_{R}^{\pm } (x) \sim (1+ \mathcal {O}(h)) \frac{h^\frac{1}{6}}{\sqrt{\pi }} (V(x)-E)^{-1/4} e^{\pm \frac{1}{h} \int _{\beta (E)}^x \sqrt{V(t)-E} dt} \;\; \text {as} \;\; x\rightarrow +\infty . \end{aligned}$$

(A.5)

Remark A.2

The error estimate in (A.3) and (A.4) can in fact be improved to $\mathcal {O}(h^{\frac{7}{6}})$ as long as x is away from $\alpha (E)$ with distance $h^{\frac{2}{3}}$.

On the other hand, similar constructions on $(-\infty , \beta (E))$ starting from the introduction of an auxiliary function

$$\begin{aligned} \xi _\alpha (x) := \left\{ \begin{array}{lll} \left( {{\frac{3}{2}}} \int ^{x}_{\alpha (E)} \sqrt{E-V(t)} dt \right) ^{{{\frac{2}{3}}}} &{}\quad \mathrm{for} \;\; x\in (\alpha (E),\beta (E)), \\ -\left( {{\frac{3}{2}}} \int ^{\alpha (E)}_x \sqrt{V(t)-E} dt \right) ^{{{\frac{2}{3}}}} &{}\quad \mathrm{for} \;\; x\in (-\infty , \alpha (E)], \end{array}\right. \end{aligned}$$

lead to solutions $u_{L}^\pm$ to the equation $(P-E)u=0$, satisfying

$$\begin{aligned} u_{L}^-(x)= & {} 2 (\xi _{\alpha }'(x))^{-\frac{1}{2}} {\check{\mathrm{Ai}}}\,(h^{{{-\frac{2}{3}}}}\xi _{\alpha }(x))+\mathcal {O}(h), \end{aligned}$$

(A.6)

$$\begin{aligned} u_{L}^+(x)= & {} (\xi _{\alpha }'(x))^{-\frac{1}{2}} {\check{\mathrm{Bi}}}\,(h^{{-\frac{2}{3}}}\xi _{\alpha }(x))+\mathcal {O}(h), \end{aligned}$$

(A.7)

as $h\rightarrow 0^+$ uniformly on any h-independent compact set in $(-\infty , \beta (E))$, and

$$\begin{aligned} u_{L}^{\pm } (x) \sim (1+ \mathcal {O}(h)) \frac{h^\frac{1}{6}}{\sqrt{\pi }} (V(x)-E)^{-1/4} e^{\pm \frac{1}{h} \int _x^{\alpha _1(E)} \sqrt{V(t)-E} dt} \;\; \mathrm{as} \;\;\; x \rightarrow -\infty .\nonumber \\ \end{aligned}$$

(A.8)

Here ${\check{f}}(x):=f(-x)$. These solutions $u_{L}^{\pm }$ are connected with $u_{R}^{\pm }$ by the following relations, which can be obtained by computing the Wronskian of these solutions at a point in the well $(\alpha (E), \beta (E))$.

Proposition A.3

We have

$$\begin{aligned} u_{L}^{\pm }(x) = \gamma _{}^\pm u_{R}^-(x) + \delta _{}^\pm u_{R}^+(x), \end{aligned}$$

(A.9)

where the coefficients $\gamma ^\pm$ and $\delta ^\pm$ behave as $h\rightarrow 0^+$ like

$$\begin{aligned} \gamma _{}^- = \sin \left( \frac{\mathcal {A}(E)}{h}\right) + \mathcal {O}(h) , \;\;\; \delta _{}^- = 2 \cos \left( \frac{\mathcal {A}(E)}{h}\right) + \mathcal {O}(h) ,\\ \gamma _{}^+ = \frac{1}{2}\cos \left( \frac{\mathcal {A}(E)}{h} \right) + \mathcal {O}(h) , \;\;\; \delta _{}^+ = - \sin \left( \frac{\mathcal {A}(E)}{h}\right) + \mathcal {O}(h). \end{aligned}$$

Finally, we recall the following formulae for the Wronskians. We have

$$\begin{aligned} \mathcal {W}(u_{L}^+,u_{L}^-)= & {} \frac{2}{\pi } h^{{-\frac{2}{3}}} + \mathcal {O}(h^{{\frac{1}{3}}}),\nonumber \\ \mathcal {W}(u_{R}^-,u_{R}^+)= & {} \frac{2}{\pi } h^{{-\frac{2}{3}}} + \mathcal {O}(h^{{\frac{1}{3}}}). \end{aligned}$$

(A.10)

$$\begin{aligned} \mathcal {W}(u_{R}^-,u_{L}^-)= & {} \frac{4}{\pi } h^{{-\frac{2}{3}}} \cos \left( \frac{\mathcal {A}(E)}{h}\right) + \mathcal {O}(h^{{\frac{1}{3}}}), \nonumber \\ \mathcal {W}(u_{L}^\pm ,u_{R}^\mp )= & {} \frac{2}{\pi } h^{{-\frac{2}{3}}} \sin \left( \frac{\mathcal {A}(E)}{h}\right) + \mathcal {O}(h^{{\frac{1}{3}}}). \end{aligned}$$

(A.11)

Appendix B. Microlocal Study of Solutions at a Crossing Point

1.1 B.1. Semiclassical and microlocal terminologies

We recall briefly some basic notions of semiclassical and microlocal analysis, referring to the textbooks [DS, Ma, Zw] for more details. We introduce the class of matrix-valued symbols

$$\begin{aligned} S_N^0:= \left\{ q=q(\cdot ,\cdot ;h)\in C^{\infty }({\mathbb {R}}^2; {\mathbb {C}}^{N\times N}); \, \big \Vert \partial _x^{\alpha }\partial _{\xi }^{\beta } q(x,\xi ;h) \big \Vert _{N\times N} = \mathcal {O}_{\alpha ,\beta }(1), \; \forall \alpha ,\beta \in {\mathbb {N}} \right\} . \end{aligned}$$

For a symbol $q\in S_N^0$, the corresponding h-pseudodifferential operator denoted $\mathcal {Q}(h) = \mathrm{Op}_h^w(q)$ can be defined using the h-Weyl quantization by

$$\begin{aligned} \mathcal {Q}(h) u(x) := \frac{1}{2\pi h} \int _{{\mathbb {R}}^2} e^{i(x-y)\xi /h} q\left( \frac{x+y}{2},\xi ;h\right) u(y) dy d\xi , \;\;\; u\in C_0^{\infty }({\mathbb {R}}). \end{aligned}$$

For $u\in \mathcal {S}'({\mathbb {R}})$, we denote $\mathcal {T}u$ the so-called semiclassical FBI-transform of u given by

$$\begin{aligned} \mathcal {T}u(x,\xi ;h) := 2^{-1/2} (\pi h)^{-3/4} \int _{{\mathbb {R}}} e^{i(x-y)\xi /h - (x-y)^2/2h} u(y) dy. \end{aligned}$$

The function $\mathcal {T}u$ is a $C^{\infty }$ function on $\mathbb {R}^2$. Let $\rho _0$ be a point of the phase space ${\mathbb {R}}^2$ and $u=u(x;h)\in L^2({\mathbb {R}})$ with $\Vert u\Vert _{L^2}\le 1$. The following notion of Frequency set is analogous to the notion of microsupport in the analytic framework (see [Ma, HeSj3]). We use the standard asymptotic notation $f_h=\mathcal {O}(h^{\infty })$ which means that $f_h=\mathcal {O}(h^k)$ for all $k\in {\mathbb {N}}$ and $h>0$ small enough.

Definition B.1

• (i) We say that u is microlocally 0 at $\rho _0$ and we write $u\sim 0$ microlocally at $\rho _0$, if there exists a neighborhood $\Omega$ of $\rho _0$ in ${\mathbb {R}}^2$ such that $\Vert \mathcal {T} u\Vert _{L^2(\Omega )} = \mathcal {O}(h^{\infty })$. The closed set of points where u is not microlocally 0 is called the frequency set of u and denoted $\mathrm{FS}\,(u)$.

• (ii) Let $\Omega \subset {\mathbb {R}}^2$ be an open set and $\mathcal {Q}(h)$ an h-pseudodifferential operator. We say that u is a microlocal solution to the equation $\mathcal {Q}(h)u=0$ in $\Omega$ and we write $\mathcal {Q}(h) u \sim 0$ microlocally in $\Omega$, if $\Omega \cap \mathrm{FS}\,(\mathcal {Q}(h)u)=\emptyset$.

Let $\mathcal {Q}(h)$ be an h-pseudodifferential operator with symbol $q(x,\xi ;h) \sim \sum _{k\ge 0} h^k q_k(x,\xi )$ in $S_N^0$. We say that $\mathcal {Q}(h)$ is microlocally elliptic at $\rho _0\in {\mathbb {R}}^2$ if $\mathrm{det}\,q_0(\rho _0)\ne 0$. The following is a basic property of the microsupport of microlocal solutions (see for instance [Ma]).

Proposition B.2

Suppose that $u=u(x;h)\in L^2({\mathbb {R}})$ with $\Vert u\Vert _{L^2}\le 1$ is a microlcal solution to $\mathcal {Q}(h)u=0$ at $\rho _0$. If $\mathcal {Q}(h)$ is microlocally elliptic at $\rho _0$ then $u\sim 0$ at $\rho _0$.

1.2 B.2. Microlocal WKB solutions

In this paragraph, we prove Proposition 5.1 where a basis of microlocal solutions to the system (5.5) on each curve $\Gamma _{q_j}^{\pm }$, $j=1,2$, is given. On $\Gamma _{q_1}^{\pm }$, the operator $\mathcal {Q}_1$ is of real principal type while $\mathcal {Q}_2$ is elliptic, and the same is true on $\Gamma _{q_2}^{\pm }$ by interchanging $\mathcal {Q}_1$ and $\mathcal {Q}_2$. Hence microlocally on each of the four curves $(\Gamma _{q_j}^{\pm })_{j=1,2}$, the system (5.5) is reduced to a scalar one-dimensional equation. Thus, the space of microlocal solutions on each of these curves is one-dimensional. Our construction is based on formal computations using standard pseudodifferential calculus.

Let M be an h-pseudodifferential operator with symbol $m=m(x,\xi )\in S_1^0$ and $\phi \in C^\infty ({\mathbb {R}})$. Let us start by computing $e^{-i \phi /h} M(e^{i\phi /h} a)$ for $a\in C_0^\infty ({\mathbb {R}})$. We have

$$\begin{aligned} e^{-i \phi (x)/h} M(e^{i\phi /h} a)(x) = \frac{1}{2\pi h} \int _{{\mathbb {R}}^2} e^{i(x-y)\xi /h} e^{-i(\phi (x)-\phi (y))/h} m\left( \frac{x+y}{2},\xi \right) a(y)dy d\xi . \end{aligned}$$

Writing $\phi (x)-\phi (y)=(x-y)\psi (x,y)$ with $\psi (x,y)=\int _0^1 \phi '(sx + (1-s)y) ds$, we have, after a change of variable $(y,\xi -\psi (x,y)) \mapsto (y,\xi )$,

$$\begin{aligned} e^{-i \phi (x)/h} M(e^{i\phi /h} a)(x)&= \frac{1}{2\pi h} \int _{{\mathbb {R}}^2} e^{i(x-y)\xi /h} m\left( \frac{x+y}{2},\xi + \psi (x,y)\right) a(y)dy d\xi \\&= \frac{1}{2\pi h} \int _{{\mathbb {R}}^2} e^{-iz\xi /h} m\left( x+\frac{z}{2},\xi + \psi (x,x+z)\right) a(x+z)dz d\xi . \end{aligned}$$

The right hand side of the above identity is an oscillatory integral with quadratic phase $(z,\xi )\mapsto -z\xi$. Hence the stationary phase theorem (see e.g. [Ma, Corollary 2.6.3]) gives the asymptotic expansion

$$\begin{aligned} e^{-i \phi (x)/h} M(e^{i\phi /h} a)(x) \sim \sum _{k=0}^{+\infty } \frac{h^k}{i^k k!} (\partial _z \partial _{\xi })^k c(x,0,0), \end{aligned}$$

where $c(x,z,\xi ):= m\left( x+\frac{z}{2},\xi + \psi (x,x+z)\right) a(x+z)$. In particular, using that $\psi (x,x)=\phi '(x)$ and $\partial _y\psi (x,x)=\frac{1}{2} \phi ''(x)$, we obtain

$$\begin{aligned} e^{-i\phi (x)/h} M(e^{i\phi /h}a)(x) = m(x,\phi '(x)) a - ih S_{m}(x,\partial _x)a+ \mathcal {O}(h^2), \end{aligned}$$

(B.1)

where $S_{m}(x,\partial _x)$ is a first order differential operator given by

$$\begin{aligned} S_{m}(x,\partial _x) = \partial _{\xi }m(x,\phi '(x))\partial _x + \frac{1}{2} \left( \partial _x\partial _{\xi }m(x,\phi '(x)) + \phi ''(x) \partial _{\xi }^2m(x,\phi '(x)) \right) . \end{aligned}$$

(B.2)

Now, we substitute (5.6) into (5.5). Then, for $f_{q_1}^{\pm }(x;h) \sim \begin{pmatrix} a_{q_1}(x;h) \\ b_{q_1}(x;h) \end{pmatrix} e^{i\phi _{q_1}(x)/h}$ with $a_{q_1}(x;h)\sim \sum _{k\ge 0} h^k a_{q_1,k}(x)$ and $b_{q_1}(x;h)\sim \sum _{k\ge 0} h^k b_{q_1,k}(x)$, using the above computation, we obtain,

$$\begin{aligned} e^{-i\phi _{q_1}/h} \mathcal {Q} f_{q_1}^{\pm } \sim \begin{pmatrix} q_1(x,\phi _{q_1}'(x)) a_{q_1} - ih S_{q_1} a_{q_1} + h r_1(x,\phi _{q_1}'(x)) b_{q_1} + \mathcal {O}(h^2)\\ h r_2(x,\phi _{q_1}'(x)) a_{q_1} + q_2(x,\phi _{q_1}'(x)) b_{q_1} - ih S_{q_2} b_{q_1}+ \mathcal {O}(h^2) \end{pmatrix}. \end{aligned}$$

The RHS is a power series of h, and in order that $f_{q_1}^{\pm }$ is a microlocal solution, each coefficient should vanish.

The coefficient of the 0th order is ${}^t\big ( q_1(x,\phi _{q_1}'(x))a_{q_1,0}, q_2(x,\phi _{q_1}'(x))b_{q_1,0}\big )$. We are looking for a microlocal WKB solution supported on $\Gamma _{q_1}$, and the phase function $\phi _{q_1}$ should satisfy the Eikonal equation (5.7). In particular, one has

$$\begin{aligned} \phi _{q_1}'(x) = - \frac{\partial _x q_1(0,0)}{\partial _{\xi } q_1(0,0)} x + \mathcal {O}(x^2) \;\; \mathrm{as}\; x\rightarrow 0. \end{aligned}$$

Then the first entry of the above 0th order coefficient is 0. In order that the second entry is 0, $b_{q_1,0}$ should vanish since

$$\begin{aligned} q_2(x,\phi _{q_1}'(x)) = \frac{ \{q_1,q_2\}(0,0) }{ \partial _{\xi }q_1(0,0) } x + \mathcal {O}(x^2) \end{aligned}$$

does not vanish near 0 except at 0.

The coefficient of the 1st order is ${}^t\big ( -i S_{q_1}a_{q_1,0}, r_2(x,\phi _{q_1}'(x)) a_{q_1,0} + q_2(x,\phi _{q_1}'(x)) b_{q_1,1} \big )$. Hence, $a_{q_1,0}$ and $b_{q_1,1}$ should satisfy the equations

$$\begin{aligned} S_{q_1} a_{q_1,0}= & {} 0, \end{aligned}$$

(B.3)

$$\begin{aligned} b_{q_1,1}= & {} - \frac{r_2(x,\phi _{q_1}'(x)) }{q_2(x, \phi _{q_1}'(x))} a_{q_1,0}. \end{aligned}$$

(B.4)

Notice that in (B.2), the coefficient $\partial _{\xi }q_1(x,\phi '(x))= \partial _{\xi }q_1(0,0)+ \mathcal {O}(x)$ does not vanish near $x=0$. Hence $a_{q_1,0}$ is uniquely determined by (B.3B.4) under the initial condition $a_{q_1,0}(0)=1$ and it is a non-zero analytic function near zero. More precisely, we have

$$\begin{aligned} a_{q_1,0}(x) = \exp \left( -\int _0^x \frac{ \partial _x\partial _{\xi }q_1(t,\phi _{q_1}'(t)) + \phi _{q_1}''(t)\partial _{\xi }^2 q_1(t,\phi _{q_1}'(t)) }{ 2\partial _{\xi } q_1(t,\phi _{q_1}'(t)) } dt\right) = 1+ \mathcal {O}(x) \end{aligned}$$

as $x\rightarrow 0$.

We also see from the condition (5.3) that $b_{q_1,1}(x)$ has a pole of order one at $x=0$. In particular,

$$\begin{aligned} b_{q_1,1}(x) = - \frac{\partial _{\xi }q_1(0,0)r_2(0,0)}{\{q_1,q_2\}(0,0)} \left( \frac{1+ \mathcal {O}(x)}{x}\right) \;\; \mathrm{as}\; x\rightarrow 0. \end{aligned}$$

In the same way, $a_{q_1,k}$, $k\ge 1$, is uniquely determined under $a_{q_1,k}(0)=0$ by an inhomogenious differential equation of the form $S_{q_1} a_{q_1,k}=F_k$ with $F_k$ depending on $a_{q_1,j}$, $j\le k-1$, and $b_{q_1,j}$, $j\le k$, and $b_{q_1,k}$ is determined algebraically from $a_{q_1,j}$, $j\le k-1$ and $b_{q_1,j}$, $j\le k-1$ and their derivatives.

1.3 B.3. Proof of Theorem 5.3

This paragraph is devoted to the proof of Theorem 5.3 which relies on many steps. The first step consists in the reduction of the system (5.5) to a scalar equation using the ellipticity condition (5.4) and then to solve this equation by means of a normal form in the spirit of [HeSj3, Sj, CdvPa].

1.3.1 B.3.1. Reduction to a scalar equation and normal form

Setting $u={}^t(u_1,u_2)$ and using the ellipticity of $\mathcal {R}_1$ at $\rho _0$ according to assumption (5.4), the system (5.5) is reduced microlocally near the origin to a scalar equation of $u_1$. More precisely, there exists a small neighborhood $\mathcal {V}\subset {\mathbb {R}}^2$ of $\rho _0$ such that microlocally in $\mathcal {V}$, the system $\mathcal {Q} u \sim 0$ is reduced to

$$\begin{aligned} \left\{ \begin{array}{lll} \mathcal {L}u_1 \sim 0,\\ u_2 \sim - h^{-1} {\mathcal {R}}_1^{-1} \mathcal {Q}_1 u_1, \end{array}\right. \end{aligned}$$

(B.5)

where ${\mathcal {R}}_1^{-1}$ denotes a parametrix of $\mathcal {R}_1$ in $\mathcal {V}$ and $\mathcal {L}$ is the h-pseudodifferential operator defined by

$$\begin{aligned} \mathcal {L} := \mathcal {R}_1 \mathcal {Q}_2 {\mathcal {R}}_1 ^{-1}\mathcal {Q}_1 - h^2 \mathcal {R}_1 \mathcal {R}_2, \end{aligned}$$

with semiclassical Weyl symbol $\ell (x,\xi ;h)= \sum _{j\ge 0} h^j \ell _j(x,\xi )$. In particular, by the pseudodifferential symbolic calculus, we have

$$\begin{aligned} \ell _0= q_1 q_2; \;\; \ell _1(\rho _0) = \frac{i}{2} \{q_1,q_2\}(\rho _0). \end{aligned}$$

The crossing point $\rho _0$ is a hyperbolic fixed point of the Hamiltonian vector field of $\ell$. We set

$$\begin{aligned} \alpha := \partial _{x} q_1(\rho _0), \; \beta := \partial _{\xi } q_1(\rho _0), \; \gamma := \partial _{x} q_2(\rho _0), \; \delta :=\partial _{\xi } q_2(\rho _0), \; D:=\{q_1,q_2\}(\rho _0).\nonumber \\ \end{aligned}$$

(B.6)

Without loss of generality, we assume that $\beta \delta >0$ and $D>0$.

In our one-dimensional case, we have the following normal form for the quantization ${\mathcal {L}}$:

Lemma B.3

There exist a small neighborhood $\Omega \subset {\mathbb {R}}^2$ of (0, 0), a Fourier integral operator U with associated canonical transformation $\kappa$ sending $\mathcal {V}$ to $\Omega$ and $\kappa (\rho _0)=(0,0)$, and a classical symbol $F(t;h)\sim \sum _{k\ge 0} h^k F_k(t)\in C^{\infty }$ defined near $t=0$, with

$$\begin{aligned} F(0;h) = -\frac{i}{2} h + \mu h^2, \end{aligned}$$

(B.7)

where $\mu =\mu (h)\sim \sum _{k\ge 0} h^k \mu _{k}$ is a classical symbol of order 0, such that

$$\begin{aligned} U F(\mathcal {L};h) U^{-1} \sim \mathcal {G}:= \frac{1}{2} (yh D_y + h D_y\cdot y) \;\;\; \mathrm{microlocally\, in} \;\; \Omega . \end{aligned}$$

(B.8)

Proof

The normal form (B.8) is due to [Sj]. Notice that in this work, this result was proved for self-adjoint operators, but it still holds for our non-self-adjoint operator $\mathcal {L}$ which is self-adjoint at the principal level $\mathrm{Op}_h^w(\ell _0)$. In the following, we prove (B.7).

The FIO U is associated with the canonical transform $\kappa : (x,\xi )\mapsto (y,\eta )$ satisfying

$$\begin{aligned} F(\ell (x,\xi ;h)) = y \eta . \end{aligned}$$

In particular, we can choose

$$\begin{aligned} \kappa (x,\xi )=\kappa _0(x,\xi ) + \mathcal {O}((x,\xi )^2),\quad \kappa _0(x,\xi ) = \frac{1}{\sqrt{D}} \left( \gamma x + \delta \xi , \alpha x + \beta \xi \right) . \end{aligned}$$

After a normalization, we can write $U^{-1}$ in the form

$$\begin{aligned} U^{-1} v(x;h) = \int _{{\mathbb {R}}} e^{i\psi (x,y)/h} c(x,y;h) v(y) dy, \end{aligned}$$

where $c(x,y;h)\sim \sum _{k\ge 0} c_k(x,y)$ is a symbol with $c_0(0,0)=1$ and the phase function $\psi (x,y)$ is a generating function of $\kappa ^{-1}$, in the sense that $\kappa ^{-1}: (y,-\nabla _y \psi )\mapsto (x,\nabla _x \psi )$. In particular, near $(x,y)=\rho _0$, we have

$$\begin{aligned} \psi (x,y)=\frac{1}{2\delta }(-\gamma x^2+ 2 \sqrt{D}xy- \beta y^2) + \mathcal {O}((x,y)^3). \end{aligned}$$

(B.9)

At the levels of principal and sub-principal symbols, the relation (B.8) implies that

$$\begin{aligned}&F_0(\ell _0(\kappa ^{-1}(y,\eta ))) = y \eta ,\\&F_1(\ell _0(\kappa ^{-1}(y,\eta ))) + \ell _1(\kappa ^{-1}(y,\eta )) F_0'(\ell _0(\kappa ^{-1}(y,\eta ))) = 0. \end{aligned}$$

In particular, the first equation at $(y,\eta )=(0,0)$ implies that $F_0(0)=0$ and $F_0'(0)= \frac{1}{D}$, and the second one gives

$$\begin{aligned} F_1(0)=-\ell _1(\rho _0)F_0'(0)=-\frac{i}{2}, \end{aligned}$$

since $\ell _1(\rho _0)=\frac{iD}{2}$. Thus the symbol F(0; h) has the form (B.7). $\square$

1.3.2 B.3.2. Microlocal solutions near the crossing point

Setting $\tilde{u}_1:= U u_1$, the equation $\mathcal {L}u_1\sim 0$ microlocally in $\mathcal {V}$ is equivalent to

$$\begin{aligned} \mathcal {G} \tilde{u}_1 \sim F(0;h) \tilde{u}_{1}\;\;\; \mathrm{microlocally \; in}\;\; \Omega , \end{aligned}$$

(B.10)

which can be rewritten as

$$\begin{aligned} y \tilde{u}_1' \sim i \mu h \tilde{u}_1 \;\;\; \mathrm{microlocally \; in}\;\; \Omega . \end{aligned}$$

The space of microlocal solutions of this equation is two dimensional and a basis is given by the two functions

$$\begin{aligned} g_{\mu }^+(y) := H(y) y^{i\mu h}, \quad g_{\mu }^-(y) := H(-y) \vert y \vert ^{i\mu h}, \end{aligned}$$

(B.11)

where H denotes the Heaviside function, i.e., $H(y)=1$ for $y\ge 0$ and $H(y) = 0$ for $y<0$. In particular, we have

$$\begin{aligned} \mathrm{FS}( g_{\mu }^{\pm })=\{\pm y> 0, \eta =0\} \cup \{y=0\}. \end{aligned}$$

Thus, $u_1^{\pm } := U^{-1} g_{\mu }^{\pm }$ are solutions to the equation $\mathcal {L}u_1\sim 0$ microlocally in $\mathcal {V}$, and we have

$$\begin{aligned} \mathrm{FS}(u_1^{\pm }) \cap \mathcal {V} \subset \big (\Gamma _{q_1}^{\pm } \cup \Gamma _{q_2}\big ) \cap \mathcal {V}. \end{aligned}$$

More precisely, we have the following asymptotic formulae for $u_1^{\pm }$.

Proposition B.4

There exist symbols $\sigma ^{\pm }(x;h)\sim \sum _{k\ge 0}h^k \sigma _k^{\pm }(x), \;\;\; \eta ^{\pm }(x;h)\sim \sum _{k\ge 0}h^k \eta _k^{\pm }(x),$ with leading terms given by

$$\begin{aligned} \sigma _0^\pm (x) = \sqrt{\frac{\delta }{\beta }} e^{-i\frac{\pi }{4}} + \mathcal {O}(x) , \quad \eta _0^\pm (x) = \pm \frac{i \delta }{\sqrt{D}x}(1+\mathcal {O}(x)), \end{aligned}$$

such that, modulo $\mathcal {O}(h^{\infty })$ as $h\rightarrow 0^+$, we have

$$\begin{aligned} u_1^+(x;h) = \left\{ \begin{array}{lll} \sqrt{2\pi h}\, \sigma ^+(x;h) e^{i\phi _{q_1}(x)/h} + h^{1+ i\mu h} \eta _+(x;h) e^{i\phi _{q_2}(x)/h} \;\;\; &{} (x>0) \\ h^{1+ i\mu h} \, \eta _+(x;h) e^{i\phi _{q_2}(x)/h} \;\;\; &{} (x<0) \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} u_1^-(x;h) = \left\{ \begin{array}{lll} h^{1+ i\mu h} \,\eta _-(x;h) e^{i\phi _{q_2}(x)/h} \;\;\; &{} (x>0) \\ \sqrt{2\pi h} \, \sigma ^-(x;h) e^{i\phi _{q_1}(x)/h} + h^{1+ i\mu h} \eta _-(x;h) e^{i\phi _{q_2}(x)/h} \;\;\; &{} (x<0). \end{array}\right. \end{aligned}$$

Proof

We only prove the formula for $u_1^+$. By definition, we have

$$\begin{aligned} u_1^+(x;h) = U^{-1} g_{\mu }^+(x;h)= \int _0^{+\infty } e^{i\psi (x,y)/h} c(x,y;h) y^{i\mu h} dy, \end{aligned}$$

(B.12)

where the phase function $\psi$ satisfies (B.9). The right hand side of (B.12) is an oscillatory integral and up to $\mathcal {O}(h^{\infty })$, its asymptotic behavior as $h\rightarrow 0$ is governed by the contributions of the critical points of the phase function $y \mapsto \psi (x,y)$ and the end point $y=0$ of $y\mapsto y^{i\mu h}$.

For $x>0$, the function $y \mapsto \psi (x,y)$ has a positive non degenerate critical point $y_c(x)$ which behaves like $y_c(x) = \frac{\sqrt{D}}{\beta }x + \mathcal {O}(x^2)$ as $x\rightarrow 0$. The critical value $\psi (x,y_c(x))$ coincides with the generating function $\phi _{q_1}(x)$ of $\Gamma _{q_1}$, and $\psi (x,y_c(x))= -\frac{\alpha }{2\beta }x^2 + \mathcal {O}(x^3)$ as $x\rightarrow 0$. Moreover, we have $\partial _y^2\psi (x,y_c(x))= -\frac{\beta }{\delta }+ \mathcal {O}(x)<0$. Then, for a cutoff function $\chi (y)\in C_0^\infty (\mathbb {R})$ identically 1 near 0 and supported in a small neighborhood of $y=0$ so that $y_c(x)\notin \mathrm{supp}\, \chi$, we have, by the stationary phase theorem (see e.g. [Ma, Corollary 2.6.3]),

$$\begin{aligned} \int _0^{+\infty } e^{i\psi (x,y)/h} c(x,y;h) y^{i\mu h} (1-\chi (y))dy=\sqrt{2\pi h} \,\sigma ^+(x;h) e^{i\phi _{q_1}(x)/h}, \end{aligned}$$

(B.13)

where $\sigma ^{+}(x;h)\sim \sum _{k\ge 0} h^k \sigma ^+_{k}(x)$ is a symbol with leading term

$$\begin{aligned} \sigma _0^+(x) = e^{-i\frac{\pi }{4}} \vert \partial _y^2\psi (x,y_c(x)) \vert ^{-\frac{1}{2}} c_0(x,y_{c}(x)) = \sqrt{\frac{\delta }{\beta }} e^{-i\frac{\pi }{4}} + \mathcal {O}(x). \end{aligned}$$

(B.14)

On the other hand, we have

$$\begin{aligned} \int _0^{+\infty } e^{i\psi (x,y)/h} c(x,y;h) y^{i\mu h} \chi (y)dy=h^{1+i\mu h} \eta _+(x;h) e^{i\phi _{q_2}(x)/h}, \end{aligned}$$

where $\eta _+(x;h) \sim \sum _{k\ge 0} h^k \eta ^+_{k}(x)$ with

$$\begin{aligned} \eta _0^+(x) = \frac{i\delta }{\sqrt{D}x}c_0(x,0) = \frac{i\delta }{\sqrt{D}x} (1+ \mathcal {O}(x)). \end{aligned}$$

For the study of this contribution from the endpoint, we develop $\psi (x,y)$ in Taylor expansion at $y=0$: $\psi (x,y)=\phi _{q_2}(x)+\left( \frac{\sqrt{D}}{\delta }+{\mathcal O}(x)\right) y+{{\mathcal {O}}}(y^2).$ Then using the fact that $D\ne 0$, we reduce the integral to a Laplace integral after a change of variable which eliminates the term ${{\mathcal {O}}}(y^2)$. The above asymptotic formula results from the term by term integration which is known as Watson’s lemma.

For $x<0$, there is no positive critical points of $y\mapsto \psi (x,y)$, and hence on this side, the asymptotic expansion of $u_1^+$ comes only from the endpoint of the integral, which can be computed similarly as above. $\square$

Now, we construct another pair of solutions $v^{\pm }={}^t(v_1^{\pm },v_2^{\pm })$ to the system (5.5) that are microlocally zero on one of $\Gamma _{q_2}^+$ and $\Gamma _{q_2}^-$. To do this, we proceed in a similar way as above but now by reducing the system (5.5) to a scalar equation of $v_2$ instead of $v_1$. Setting $v={}^t(v_1,v_2)$ and using the ellipticity of $\mathcal {R}_2$ at $\rho _0$, the system $\mathcal {Q}v\sim 0$ is reduced microlocally near $\rho _0$ to

$$\begin{aligned} \left\{ \begin{array}{lll} \widehat{\mathcal {L}} v_2 \sim 0,\\ v_1 \sim - h^{-1} \mathcal {R}_2^{-1} \mathcal {Q}_2 v_2, \end{array}\right. \end{aligned}$$

(B.15)

where $\mathcal {R}_2^{-1}$ denotes a parametrix of $\mathcal {R}_2$ in a neighborhood of $\rho _0$ and $\widehat{\mathcal {L}}$ is the h-pseudodifferential operator defined by

$$\begin{aligned} \widehat{\mathcal {L}} := \mathcal {R}_2 \mathcal {Q}_1 \mathcal {R}_2^{-1} \mathcal {Q}_2 - h^2 \mathcal {R}_2 \mathcal {R}_1 . \end{aligned}$$

As before, we can construct two microlocal solutions $v_2^\pm$using a normal form reduction.

Proposition B.5

There exist microlocal solutions $v_2^\pm$ to $\widehat{\mathcal {L}} v_2\sim 0$ in a neighborhood of $\rho _0$ such that, modulo $\mathcal {O}(h^{\infty })$ as $h\rightarrow 0^+$, we have

$$\begin{aligned} v_2^+(x;h) = \left\{ \begin{array}{lll} h^{1+ i\widehat{\mu }h} \, \widehat{\eta }_+(x;h) e^{i\phi _{q_1}(x)/h} \;\;\; &{}\quad (x>0) \\ \sqrt{2\pi h}\, \widehat{\sigma }^+(x;h) e^{i\phi _{q_2}(x)/h} + h^{1+ i\widehat{\mu }h} \widehat{\eta }_+(x;h) e^{i\phi _{q_1}(x)/h} \;\;\; &{}\quad (x<0) \ \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} v_2^-(x;h) = \left\{ \begin{array}{lll} \sqrt{2\pi h} \, \widehat{\sigma }^-(x;h) e^{i\phi _{q_2}(x)/h} + h^{1+ i\widehat{\mu }h} \widehat{\eta }_-(x;h) e^{i\phi _{q_1}(x)/h} \;\;\; &{}\quad (x>0) \\ h^{1+ i\widehat{\mu } h} \, \widehat{\eta }_-(x;h) e^{i\phi _{q_1}(x)/h} \;\;\; &{}\quad (x<0). \end{array}\right. \end{aligned}$$

where $\widehat{\sigma }^{\pm }(x;h)\sim \sum _{k\ge 0}h^k \widehat{\sigma }_k^{\pm }(x), \;\;\; \widehat{\eta }^{\pm }(x;h)\sim \sum _{k\ge 0}h^k \widehat{\eta }_k^{\pm }(x)$, $\widehat{\mu }(h)\sim \sum _{k\ge 0} h^k \widehat{\mu }_{-,k}$ with

$$\begin{aligned} \widehat{\sigma }_0^\pm (x) = \sqrt{ \frac{\beta }{\delta } } e^{i\frac{\pi }{4}} + \mathcal {O}(x) , \quad \widehat{\eta }_0^\pm (x) = \pm \frac{i \beta }{\sqrt{D}x}(1+\mathcal {O}(x)). \end{aligned}$$

Summing up, we have then constructed 4 microlocal solutions to the system (5.5) microlocally in a small neighborhood $\mathcal {V}$ of $\rho _0$

$$\begin{aligned} u^{\pm }={}^t(u_1^{\pm },u_2^{\pm }), \quad v^{\pm }={}^t(v_1^{\pm },v_2^{\pm }), \end{aligned}$$

with

$$\begin{aligned} \mathrm{FS}(u^{\pm }) \cap \mathcal {V} \subset \big (\Gamma _{q_1}^{\pm } \cup \Gamma _{q_2}\big ) \cap \mathcal {V}, \quad \mathrm{FS}(v^{\pm }) \cap \mathcal {V} \subset \big ( \Gamma _{q_2}^{\pm } \cup \Gamma _{q_1} \big ) \cap \mathcal {V}, \end{aligned}$$

where $u_1^{\pm }$ and $v_2^{\pm }$ are defined above and

$$\begin{aligned} u_2^{\pm } \sim -h^{-1} \mathcal {R}_1^{-1} \mathcal {Q}_1 u_1^{\pm },\;\; v_1^{\pm } \sim -h^{-1} (\mathcal {R}_2)^{-1} \mathcal {Q}_2 v_2^{\pm }. \end{aligned}$$

1.3.3 B.3.3. Proof of Theorem 5.3

Now we connect our microlocal solutions $u^{\pm }$ and $v^{\pm }$ to the WKB solutions $f_{q_j}^{\pm }$, $j=1,2$, given by Proposition 5.1 and we deduce the transfer matrix at the crossing point. The following result is an immediate consequence of Propositions B.4 and B.5.

Proposition B.6

There exist symbols

$$\begin{aligned} A_{q_1}^{\pm }(h)&\sim \sum _{k\ge 0} h^kA_{q_1,k}^{\pm }, \; A_{q_2}^{\pm ,\pm }(h) \sim \sum _{k\ge 0} h^kA_{q_2,k}^{\pm ,\pm }, \; B_{q_1}^{\pm ,\pm }(h) \sim \sum _{k\ge 0} h^kB_{q_1,k}^{\pm ,\pm }, \; B_{q_2}^{\pm }(h) \\&\sim \sum _{k\ge 0} h^kB_{q_2,k}^{\pm }, \end{aligned}$$

with leading terms

$$\begin{aligned} A_{q_1,0}^+= & {} A_{q_1,0}^- = \sqrt{\frac{2\pi \delta }{\beta }} e^{-i\frac{\pi }{4}}, \;\; A_{q_2,0}^{+,+} = A_{q_2,0}^{+,-}= -A_{q_2,0}^{-,+} = - A_{q_2,0}^{-,-}= \frac{i\sqrt{D}}{r_1(\rho _0)}, \nonumber \\\end{aligned}$$

(B.16)

$$\begin{aligned} B_{q_2,0}^+= & {} B_{q_2,0}^- = \sqrt{\frac{2\pi \beta }{\delta }} e^{i\frac{\pi }{4}}, \;\; - B_{q_1,0}^{+,+} = - B_{q_1,0}^{+,-}= B_{q_1,0}^{-,+} = B_{q_1,0}^{-,-}= \frac{i\sqrt{D}}{r_2(\rho _0)},\nonumber \\ \end{aligned}$$

(B.17)

such that

$$\begin{aligned}&u^+ \sim \left\{ \begin{array}{lll} A_{q_1}^+ h^{\frac{1}{2}} f_{q_1}^+ \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^+ \cap \mathcal {V} \\ 0 \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^- \cap \mathcal {V} \\ A_{q_2}^{+,+} h^{i\mu h} f_{q_2}^{+} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^{+} \cap \mathcal {V} \\ A_{q_2}^{+,-} h^{i\mu h} f_{q_2}^{-} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^{-} \cap \mathcal {V} \end{array}\right. , \quad u^- \sim \left\{ \begin{array}{lll} 0 \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^+ \cap \mathcal {V} \\ A_{q_1}^- h^{\frac{1}{2}} f_{q_1}^+ \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^- \cap \mathcal {V} \\ A_{q_2}^{-,+} h^{i\mu h} f_{q_2}^{+} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^{+} \cap \mathcal {V} \\ A_{q_2}^{-,-} h^{i\mu h} f_{q_2}^{-} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^{-} \cap \mathcal {V} \end{array}\right. \\&v^+ \sim \left\{ \begin{array}{lll} B_{q_2}^+ h^{\frac{1}{2}} f_{q_2}^+ \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^+ \cap \mathcal {V} \\ 0 \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^- \cap \mathcal {V} \\ B_{q_1}^{+,+} h^{i\widehat{\mu } h} f_{q_1}^{+} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^{+} \cap \mathcal {V}\\ B_{q_1}^{+,-} h^{i\widehat{\mu } h} f_{q_1}^{-} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^{-} \cap \mathcal {V} \end{array}\right. , \quad v^- \sim \left\{ \begin{array}{lll} 0 \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^+ \cap \mathcal {V} \\ B_{q_2}^- h^{\frac{1}{2}} f_{q_2}^+ \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_2}^- \cap \mathcal {V} \\ B_{q_1}^{-,+} h^{i\widehat{\mu } h} f_{q_1}^{+} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^{+} \cap \mathcal {V} \\ B_{q_1}^{-,-} h^{i\widehat{\mu } h} f_{q_1}^{-} \;\;\; &{}\quad \mathrm{on} \;\; \Gamma _{q_1}^{-} \cap \mathcal {V}. \end{array}\right. \end{aligned}$$

We set

$$\begin{aligned} \begin{pmatrix} t_{1}^{+}(h) \\ t_{2}^{+}(h) \end{pmatrix} = \begin{pmatrix} s_{1,1}(h) &{}\quad s_{1,2}(h) \\ s_{2,1}(h) &{}\quad s_{2,2}(h) \end{pmatrix} \begin{pmatrix} t_{1}^{-}(h) \\ t_{2}^{-}(h) \end{pmatrix}. \end{aligned}$$

Observe that if $t_1^-(h) = 1$ and $t_2^-(h)=0$ then u should be equal to $(B_{q_1}^{+,-} h^{i\widehat{\mu } h})^{-1} v^+$ microlocally near $\rho _0$, and therefore we have

$$\begin{aligned} s_{2,1}(h)= & {} t_2^+(h) = h^{\frac{1}{2} - i \widehat{\mu } h} \kappa _{2,1}(h) \;\;\; \mathrm{with} \;\; \kappa _{2,1}(h) := \frac{B_{q_2}^+(h)}{B_{q_1}^{+,-}(h)},\\ s_{1,1}(h)= & {} t_1^+(h) = \frac{B_{q_1}^{+,+}(h)}{B_{q_1}^{+,-}(h)}. \end{aligned}$$

Analogously, if $t_1^-(h) = 0$ and $t_2^-(h)=1$ then u should be equal to $(A_{q_2}^{+,-} h^{i\mu h})^{-1} u^+$ microlocally near $\rho _0$, and therefore we have

$$\begin{aligned} s_{1,2}(h)= & {} t_1^+(h) = h^{\frac{1}{2} - i \mu h} \kappa _{1,2}(h) \;\;\; \mathrm{with} \;\; \kappa _{1,2}(h) := \frac{A_{q_1}^+(h)}{A_{q_2}^{+,-}(h)},\\ s_{2,2}(h)= & {} t_2^+(h) = \frac{A_{q_2}^{+,+}(h)}{A_{q_2}^{+,-}(h)}. \end{aligned}$$

This ends the proof of Theorem 5.3. $\square$