# Semiclassical WKB Problem for the non-self-adjoint Dirac operator with an
analytic rapidly oscillating potential

Setsuro Fujiié∗ ∗ Department of Mathematical Sciences, Ritsumeikan University,
Japan<EMAIL_ADDRESS>https://research-
db.ritsumei.ac.jp/rithp/k03/resid/S000793?lang=en ,  Nicholas Hatzizisis† †
Department of Mathematics & Applied Mathematics, University of Crete, and
Institute of Applied and Computational Mathematics, FORTH, Greece
<EMAIL_ADDRESS>http://www.nikoshatzizisis.wordpress.com/home/ and
Spyridon Kamvissis‡ ‡ Department of Mathematics & Applied Mathematics,
University of Crete, and Institute of Applied and Computational Mathematics,
FORTH, Greece<EMAIL_ADDRESS>http://www.tem.uoc.gr/~spyros/

###### Abstract.

In this paper we examine the semiclassical behavior of the scattering data of
a non-self-adjoint Dirac operator with a rapidly oscillating potential that is
complex analytic in some neighborhood of the real line. Some of our results
are rigorous and quite general. On the other hand, complete and concrete
understanding requires the investigation of the WKB geometry of specific
examples. For such detailed computations we use a particular example that has
been investigated numerically more than 20 years ago by Bronski and Miller and
rely heavily on their numerical computations. Mostly employing the exact WKB
method, we provide the complete rigorous uniform semiclassical analysis of the
Bohr-Sommerfeld condition for the location of the eigenvalues across unions of
analytic arcs as well as the associated norming constants. For the reflection
coefficient as well as the eigenvalues near 0 in the spectral plane, we employ
instead an older theory that has been developed in great detail by Olver. Our
analysis is motivated by the need to understand the semiclassical behaviour of
the focusing cubic NLS equation with initial data $A\exp\\{iS/\epsilon\\}$, in
view of the well-known fact discovered by Zakharov and Shabat that the
spectral analysis of the Dirac operator enables the solution of the NLS
equation via inverse scattering theory.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Organization of the paper
    3. 1.3 Notation
  2. 2 Eigenvalues Away From The Real Axis
    1. 2.1 General strategy
    2. 2.2 The exact WKB method
    3. 2.3 Connection around a simple turning point
    4. 2.4 Eigenvalues near a generic point on the asymptotic spectral arc
    5. 2.5 What happens near the bifurcation point
    6. 2.6 The Stokes geometry and quantization in a particular case
  3. 3 Eigenvalues That Lie Near Zero
    1. 3.1 Preparations
    2. 3.2 Solutions in a neighborhood of only one simple turning point
    3. 3.3 Asymptotic estimates for the error terms
    4. 3.4 The connection theorem
    5. 3.5 The asymptotic form of the connection theorem
    6. 3.6 Application of the connection theorem
  4. 4 Norming Constants
  5. 5 Reflection Coefficient
    1. 5.1 Reflection away from zero.
    2. 5.2 Reflection near zero.
  6. 6 Conclusion: Application to the semiclassical focusing NLS
  7. A Modified Bessel Functions
    1. A.1 Primary solutions
    2. A.2 Secondary solutions
    3. A.3 Auxiliary functions

##  1\. Introduction

###  1.1. Motivation

Consider the semiclassical limit ($\epsilon\downarrow 0$) of the solution to
the initial value problem of the one-dimensional nonlinear Schrödinger
equation with cubic nonlinearity

$\left\\{\begin{array}[]{l}i\epsilon\partial_{t}\psi+\frac{\epsilon^{2}}{2}\partial_{x}^{2}\psi+|\psi|^{2}\psi=0,\quad(x,t)\in{\mathbb{R}}\times{\mathbb{R}}\\\
\psi(x,0)=A(x)\exp\\{iS(x)/\epsilon\\},\quad
x\in{\mathbb{R}}\end{array}\right.$ (1.1)

where $A$ and $S$ are real valued integrable functions defined on the real
line.

According to the seminal discovery in [28], this initial value problem can be
studied via the so-called inverse scattering method. In order for one to do
so, the first ingredient is the $direct$ spectral and scattering analysis of
the associated Dirac (or Zakharov-Shabat) operator $\mathfrak{D}_{\epsilon}$
given by

$\mathfrak{D}_{\epsilon}=\begin{bmatrix}-\frac{\epsilon}{i}\frac{d}{dx}&-iA(x)\exp\\{iS(x)/\epsilon\\}\\\
-iA(x)\exp\\{-iS(x)/\epsilon\\}&\frac{\epsilon}{i}\frac{d}{dx}\end{bmatrix}.$

Our main goal in this paper is to provide a rigorous investigation of the
semiclassical behavior of the scattering data (reflection coefficient,
eigenvalues and their associated norming constants) for the operator
$\mathfrak{D}_{\epsilon}$ and apply our conclusions to the semiclassical
investigation of the solutions to the focusing NLS equation.

We are interested in $A(x)$ and $S(x)$ that are real analytic functions which
can be extended holomorphically to at least a region in the whole complex
plane 111For the inverse scattering method to be applicable we will also need
that $S^{\prime}$ is integrable for real $x$.. Although a good deal of our
analysis will be fairly general, we will eventually focus on a very particular
choice of $A(x)$ and $S(x)$ so that we have a concrete configuration of the
geometry of turning points and Stokes lines. Our model will be the
$A(x)=S(x)=\sech(2x)$ since this has been the first case considered in detail:
a very careful numerical analysis of the eigenvalue problem was done by
Bronski [1] in $1996$ and a first investigation of the formal WKB theory,
involving numerical computations of turning points and Stokes lines, was
conducted by Miller [22] in $2001$. Another model has been studied numerically
recently in [18]; there, the authors use $A(x)=30/(1+x^{4})$ and
$S(x)=15x^{2}$.

If the parameter $\epsilon$ is not too small, then of course the solution is
described by a nonlinear superposition of breathers (corresponding to purely
imaginary eigenvalues of $\mathfrak{D}_{\epsilon}$), traveling solitons
(corresponding to all other eigenvalues of $\mathfrak{D}_{\epsilon}$) and a
background radiation contribution (corresponding to the continuous spectrum of
$\mathfrak{D}_{\epsilon}$). This is in a sense the content of the inverse
scattering method.

On the other hand, as $\epsilon\downarrow 0$, the solution attains a special
form which in particular regions attains a highly oscillatory behavior. Often
in such problems one expects the validity of the so-called finite gap ansatz
for the semiclassical asymptotics.

######  Finite Gap Ansatz.

Let $(x_{0},t_{0})$ be a generic point with $x_{0}\in\mathbb{R}$ and
$t_{0}>0$. The solution $\psi$ of (1.1) is asymptotically ($\epsilon\downarrow
0$) described (locally) as a slowly modulated $G$ phase wavetrain. More
precisely, setting $x=x_{0}+\epsilon\hat{x}$ and $t=t_{0}+\epsilon\hat{t}$ (so
that $x_{0},t_{0}$ are “slow” variables while $\hat{x},\hat{t}$ are “fast”
variables), there exist parameters (all of which depend on the slow variables
$x_{0}$, $t_{0}$ but not on $\hat{x},\hat{t}$)

  * •

$a,U_{0},k_{0},w_{0}$ in ${\mathbb{C}}$ and

  * •

${\bf U}=[U_{1},\dots,U_{G}]^{T}$, ${\bf k}=[k_{1},\dots,k_{G}]^{T}$, ${\bf
w}=[w_{1},\dots,w_{G}]^{T}$, ${\bf Y}=[Y_{1},\dots,Y_{G}]^{T}$, ${\bf
Z}=[Z_{1},\dots,Z_{G}]^{T}$ in ${\mathbb{C}}^{G}$

such that as $\epsilon\downarrow 0$,

$\psi(x,t)=\psi(x_{0}+\epsilon\hat{x},t_{0}+\epsilon\hat{t}\hskip 2.0pt)$

has the following leading order asymptotics

$\psi(x,t)\sim
a(x_{0},t_{0})\exp\Big{\\{}i\Big{(}\tfrac{U_{0}(x_{0},t_{0})}{\epsilon}+k_{0}(x_{0},t_{0})\hat{x}-w_{0}(x_{0},t_{0})\hat{t}\Big{)}\Big{\\}}\\\
\cdot\frac{\Theta\bigg{(}{\bf Y}(x_{0},t_{0})+i\Big{(}\frac{{\bf
U}(x_{0},t_{0})}{\epsilon}+{\bf k}(x_{0},t_{0})\hat{x}-{\bf
w}(x_{0},t_{0})\hat{t}\Big{)}\bigg{)}}{\Theta\bigg{(}{\bf
Z}(x_{0},t_{0})+i\Big{(}\frac{{\bf U}(x_{0},t_{0})}{\epsilon}+{\bf
k}(x_{0},t_{0})\hat{x}-{\bf w}(x_{0},t_{0})\hat{t}\Big{)}\bigg{)}}.$ (1.2)

All parameters here are defined in terms of an underlying Riemann surface $X$
which depends solely on $x_{0},t_{0}$. The moduli of $X$ vary slowly with
$x,t$, i.e. they depend on $x_{0},t_{0}$ but not on
$\epsilon,\hat{x},\hat{t}$. $\Theta$ is the G-dimensional Jacobi theta
function associated with $X$. The genus of $X$ can vary with $x_{0},t_{0}$. In
fact, the $(x,t)$-plane is divided into open regions in each of which G is
constant.

On the boundaries of such regions (sometimes called “caustics”; they are
unions of analytic arcs), some degeneracies appear in the mathematical
analysis (we may have “pinching” of the surfaces $X$ for example).

The above formula (1.2) gives asymptotics which is pointwise in $x,t$. We
refer to [19] and [20], for the justification of the above “finite gap”
formula in the case of special “bell-like” initial data $A(x)$ [and $S(x)=0$],
as well as the exact formulae for the parameters and the definition of the
theta functions. The formula is actually uniformly valid in compact
$(x,t)$-sets not containing points on the caustics. The mathematical theory
leading to the asymptotic formula is the well-known non-linear steepest
descent method. Near the caustics, boundary layers separating regions of
different genus appear. For an analysis of the somewhat more delicate behavior
(especially for higher order terms in $\epsilon$) near the first caustic one
can consult [5].

The following two pictures (prepared for us by Nikos Efremidis) exemplify this
behavior in two separate cases. In both cases $A(x)=2\sech x$; but in the
first picture $S$ is identically 0 while in the second one $S(x)=\sech x$.
Here $\epsilon=0.1$ and the time is actually rescaled (divided by 0.1). We
only show plots for $|\psi|^{2}$.

One observes the following:

  * (i)

There is no qualitative difference between the two cases, at least for not
very large times.

  * (ii)

The behavior of the solution is quite different in three distinct regions in
the $(x,t)$-plane.

  * (iii)

In the first region (for smaller times t) things are fairly “smooth”. There
are no oscillations. In the intermediate region fast oscillations appear. The
curve separating these two regions appears to be fairly continuous. There is
also a third region where the nature of the fast oscillations changes. Again
the intermediate and third regions are separated by a seemingly continuous
curve.

Figure 1.  A solution of the initial value problem (1.1) for $\epsilon=0.1$
with initial data $\psi(x,0)=2\sech x$ [i.e. $A(x)=2\sech x$ and $S(x)=0$]. In
this figure $|\psi|^{2}(x,t)$ is being plotted. The horizontal axis represents
$x$ while $t$ runs on the vertical axis (as shown on the left). The bar on the
right of the plot shows a graduated scale for the values of $|\psi|^{2}$
ranging from deep black (where $|\psi|^{2}$ is small) to bright white (where
$|\psi|^{2}$ is big). Figure 2.  A solution of the initial value problem (1.1)
for $\epsilon=0.1$ with initial data $\psi(x,0)=2(\sech
x)\exp\big{\\{}\tfrac{\sech x}{0.1}\big{\\}}$ [i.e. $A(x)=2\sech x$ and
$S(x)=\sech x$]. The figure shows the plot of $|\psi|^{2}(x,t)$. On the
horizontal axis lies $x$ while $t$ runs on the vertical axis (as shown on the
left). Also on the right of the plot there is a bar showing a graduated scale
for the values of $|\psi|^{2}$ ranging from deep black (where $|\psi|^{2}$ is
small) to bright white (where $|\psi|^{2}$ is big).

In [19] we have been able to identify the first region in Figure 1 with the
genus 0 region. The intermediate region is the genus 2 region and then comes
the genus 4 region. The separating curves are the “caustics”. In principle,
there can be a very large number of such caustics. But it is also worth
pointing out that the $2\sech x$ data, being a nonlinear superposition of many
(of order $\epsilon^{-1}$) breathers, is actually periodic in time with period
$\mathcal{O}(\epsilon^{-1})$. Of course, this is not seen in our picture,
since what happens at such large times is not shown.

If we were to run our numerics in Figure 2 for larger times, we would also
observe many separated traveling solitons at different non-zero speeds, since
(as we shall see later) the initial data give rise to many eigenvalues with
non-zero real part.

The question of the semiclassical approximation of the scattering data has a
deeper significance in view of the instabilities of the problem which apper in
many levels. In fact even in the non-semiclassical regime, the focusing NLS is
the main model for the so-called “modulational instability” (see [3] and [4]),
although for positive fixed $\epsilon$ the initial value problem is well-
posed. Semiclassically the instabilities become more pronounced. One way to
see this, is related to the underlying ellipticity of the formal semiclassical
limit. To be more specific, consider the well-known Madelung transformation
(see [21])

$\begin{cases}\rho=|\psi|^{2}\\\ \mu=\epsilon\Im(\psi^{*}\psi_{x})\end{cases}$

where $\psi^{*}$ denotes the complex conjugate of $\psi$. Then the initial
value problem (1.1) becomes

$\begin{cases}\rho_{t}+\mu_{x}=0\\\
\mu_{t}+\Big{(}\frac{\mu^{2}}{\rho}+\frac{\rho^{2}}{2}\Big{)}_{x}=\frac{\epsilon^{2}}{4}\partial_{x}[\rho(\log\rho)_{xx}]\\\
\rho(x,0)=A^{2}(x)\\\ \mu(x,0)=A^{2}(x)S^{\prime}(x).\end{cases}$

The formal limit, as $\epsilon\downarrow 0$, is

$\begin{cases}\rho_{t}+\mu_{x}=0\\\
\mu_{t}+\Big{(}\frac{\mu^{2}}{\rho}+\frac{\rho^{2}}{2}\Big{)}_{x}=0\\\
\rho(x,0)=A^{2}(x)\\\ \mu(x,0)=A^{2}(x)S^{\prime}(x).\end{cases}$

This is an initial value problem for an elliptic system of equations; and so
one expects that small perturbations of the initial data (independent of
$\epsilon$) can lead to large changes in the solution at any given time.

Instabilities also appear, independently, at the spectral analysis of the
related non-self-adjoint Dirac operator: indeed, small changes of the
potential are expected to result in relatively large changes in the discrete
spectrum; this is not true when the phase $S=0$ but it is true otherwise. We
refer to [2] for a numerical investigation of this fact in the special case
$A(x)=S(x)=sech(2x)$ and to [8] for the discussion of the more general
phenomenon in the theory of pseudo-differential operators. Instabilities are
evident also at the related equilibrium measure problem (it is a “max-min”
problem; see [20]), the related Whitham equations (they are also elliptic) and
even in the numerical studies of the problem. What was initially called the
modulational instability is just the tip of the iceberg!

One might even question whether the whole project is worthwhile studying in
detail, at least as a valid physical model, in view of all these
instabilities. To this, a first response is that the semiclassical analysis
turns out in practice to be relevant even for not so small values of the semi-
classical parameter $\epsilon$ (for example in applications to nonlinear
optics). From a mathematical point of view, it is a fascinating instance of a
difficult unstable problem that can be approximated by solvable stable
problems; this approximation is very singular and non-trivial and its study
involves important connections to different areas of mathematics (namely PDE
theory, spectral theory of non-self-adjoint operators, WKB analysis, potential
theory (cf. [20]), Riemann surface theory, hydrodynamic instability theory).

Going back to the pictures presented above, in the first (genus 0) region
there are no fast oscillations and actually strong semiclassical limits exist
for both $\rho$ and $\mu$. They actually satisfy the formally limiting PDE. In
the other (higher genus) regions, violent oscillations of frequency order
$\epsilon^{-1}$ appear and a strong pointwise limit does not exist. But to the
extent that we can perform a full asymptotic analysis, both at the direct
scatterring and the inverse scattering stage, we can show that there is at
least a weak limit and we are able to provide complete asymptotic formulae (as
above).

The semiclassical analysis of the NLS solution $\psi$ in [19] undertook the
asymptotic ($\epsilon\downarrow 0$) analysis of the inverse scattering
problem, to which Zakharov and Shabat have reduced the solution of the
equation; in particular, we worked on the formulation of that problem as a
Riemann-Hilbert factorisation problem and we applied and extended ideas and
calculations going back to the seminal work of Deift, Venakides and Zhou ([6],
[7]). But no careful semiclassical analysis of the $direct$ scattering problem
had been undertaken until recently. Instead, an ad hoc approximation of the
eigenvalues by their Bohr-Sommerfeld approximants was used as a starting point
(and the reflection coefficient was set identically to 0, in the same formal
spirit).

In [14] the rigorous semiclassical analysis of the scattering data of the
related Dirac (or Zakharov-Shabat) operator was completed in the case where
$A$ is real analytic, integrable, positive, symmetric, with only one local
maximum (and where for simplicity the second derivative of $A$ is non-zero).
We applied the so-called exact WKB theory, which will also be applied in the
present work.

In [15] the analyticity assumption was replaced by a mild smoothness
assumption and a different method was employed, going back to Langer and Olver
[23]. In a sequel [16], the general case with several local maxima and minima
was also completed, under the assumption that potentials are smooth and
positive. In all the above cases the initial phase $S(x)$ is identically zero.
Of course, even with an initial phase zero, the Madelung system above shows
that a non-trivial phase will appear immediately for any small time $t$. But
it is also reasonable to postulate a non-zero initial phase especially in
cases where the discrete spectrum is no more imaginary and the instability due
to the non-self-adjointness of the Dirac operator is instrumental.

The exact WKB method was first developed for the Schrödinger operator, but
here we apply it to the Dirac operator that is associated to the focusing NLS
equation. The method goes back to works of Ecalle [9] and Voros [27] but here
we argue along the lines of the papers of Gérard-Grigis [13] and Fujiié-
Lasser-Nédélec [10]. Rather than relying on the usual formal WKB method which
relies on asymptotic series that are in general divergent, we use a
“resummation” of the series and in fact construct “exact solutions” in terms
of convergent series, thus resolving a problem of “asymptotics beyond all
orders”.

For the study of the eigenvalues with small imaginary part, the exact WKB
method seems to break down and Olver’s [23] is not directly applicable since
we don’t know a priori that the eigenvalues are purely imaginary (as in [15]).
Fortunately the somewhat forgotten paper [24] which generalizes his earlier
work is proved to be useful here.

###  1.2. Organization of the paper

The plan of this paper is the following. In the next section we investigate
the eigenvalues of the Dirac operator that in the semiclassical limit lie away
from the real axis. In §2.1, we present the main ideas that will be used in
the following sections, including the definition of turning points,
progressive paths, Stokes lines, admissible contours and asymptotic spectral
arcs. In §2.2, we introduce the theory of the exact (resummed and converging)
WKB solutions and state a basic theorem about convergence and semiclassical
asymptotics. Then in §2.3 we prove a theorem that shows how different
solutions are connected in a neighborhood of a simple turning point. We use it
in §2.4, to give a rigorous justification of the Bohr-Sommerfeld asymptotic
conditions for the location of the eigenvalues that lie away from the
bifurcation point (see Figure 10). In §2.5 we refine the exact WKB analysis in
a small neighborhood of a bifurcation point. In the last section of this
paragraph, namely §2.6, we apply all the results above to the particular case
$A(x)=S(x)=\sech(2x)$, where we make heavy use of the numerical results and
figures of [22].

We are not able at this point to extend the exact WKB method for the
eigenvalues near $0$. In paragraph 3, we use instead Olver’s theory as it was
extended in [24]. We start in §3.1 with some preparatory material and move to
§3.2 where we encounter the behavior of solutions near a simple turning point
(compare this with §2.3). Next, comes §3.3 where we present asymptotics for
the error-terms in our solutions. The connection of the above solutions -in
the way of Olver- is achieved in §3.4. Using the asymptotics for the error-
terms, we arrive at the asymptotic form of these connection formulas in §3.5.
We then assemble all the previous results, and in §3.6 we provide a Bohr-
Sommerfeld quantization condition for the location of the eigenvalues that lie
near zero.

In paragraph 4 we are interested in the behavior of the corresponding norming
constants to the eigenvalues discussed previously. Then in paragraph 5, we use
Olver’s method as exploited in [15], to study the behavior of the reflection
coefficient; we prove that it is small away from the point 0 (in the spectral
real line) in §5.1. Also, in §5.2, we give asymptotics for the reflection
coefficient nearer to zero.

Finally, in paragraph 6, we present the application of the WKB results to the
focusing NLS problem and explain how the analysis of [19], [20] extends to our
present case in view of these results.

Since the approximation technique we are using for the near-zero eigenvalues
relies on modified Bessel functions, we have included in the appendix a
paragraph that contains all the material that we shall need.

###  1.3. Notation

Before we start our main exposition, we specify some notation used throughout
our work.

  * •

The symbol ${\mathbb{N}}_{0}$ is used for the set ${\mathbb{N}}\cup\\{0\\}$.

  * •

The Riemann sphere is denoted by
$\overline{{\mathbb{C}}}={\mathbb{C}}\cup\\{\infty\\}$.

  * •

The (topological) closure of a set $S\subseteq\overline{C}$ shall be denoted
by $\textsf{clos}(S)\subseteq\overline{C}$.

  * •

We use the notation ${\mathbb{R}}^{\pm}=\\{\,x\in{\mathbb{R}}\mid\pm x>0\,\\}$
for the sets of positive/negative numbers.

  * •

Complex conjugation is denoted with a star superscript, $*$; i.e. $z^{*}$ is
the complex conjugate of $z$.

  * •

The transpose of a matrix $M$ is denoted by $M^{T}$.

  * •

For the upper half-plane we write $\mathbb{H}^{+}=\\{\,z\in{\mathbb{C}}\mid\Im
z>0\,\\}$.

  * •

For a set $\Sigma\subseteq{\mathbb{R}}$, when we write $i\Sigma$ we mean the
set $\\{i\kappa\mid\kappa\in\Sigma\\}\subset{\mathbb{C}}$. Also, for
$z_{1},z_{2}\in{\mathbb{C}}$ the set $[z_{1},z_{2}]\subset{\mathbb{C}}$
denotes the closed line segment starting at $z_{1}$ and ending at $z_{2}$.

  * •

The set of holomorphic functions from a domain $A$ to a domain $B$ shall be
denoted by $\mathscr{H}(A;B)$. For the holomorphic functions from $A$ to
${\mathbb{C}}$, we simply write $\mathscr{H}(A)$.

  * •

The symbol $\sigma(a\rightsquigarrow b)$ is used to denote a curve in
${\mathbb{C}}$, parametrized by $\sigma$, starting at $a$ and ending at $b$.

  * •

Let ${\mathcal{W}}[\textbf{f},\textbf{g}]$ be defined as the determinant of
the $2\times 2$ matrix $[\textbf{f}\hskip 3.0pt\textbf{g}]$ consisting of the
two column vectors f, $\textbf{g}\in{\mathbb{C}}^{2}$.

  * •

The wronskian of the pair $\\{f,g\\}$ of functions is represented by $W[f,g]$.

  * •

For a continuous function $f$ in ${\mathbb{R}}$, the notation $\int f(t)dt$
denotes any of its primitives.

  * •

If $z\in{\mathbb{C}}$, we denote by $\text{ph}z\in(-\pi,\pi]$ its phase.

  * •

The abbreviations LHS and RHS stand for left-hand side and right-hand side
respectively.

  * •

EV and EF stand for eigenvalues and eigenfunctions respectively.

  * •

The acronym MBF means modified Bessel function.

##  2\. Eigenvalues Away From The Real Axis

###  2.1. General strategy

This paragraph presents a general scheme (see [22], and also [11] for the case
of Schrödinger operators) for finding the semiclassical EVs of the Dirac (or
Zakharov-Shabat) operator

$\mathfrak{D}_{\epsilon}=\begin{bmatrix}-\frac{\epsilon}{i}\frac{d}{dx}&\omega(x,\epsilon)\\\
-\omega^{*}(x,\epsilon)&\frac{\epsilon}{i}\frac{d}{dx}\end{bmatrix}$ (2.1)

where $\omega$ is the complex potential function

$\omega(x,\epsilon)=-iA(x)\exp\Big{\\{}i\frac{S(x)}{\epsilon}\Big{\\}}$

for some real-valued, analytic functions $A(x)$ and $S(x)$ defined on the real
line [$\omega^{*}(x,\epsilon)$ represents the complex conjugate of
$\omega(x,\epsilon)$] such that $A$, $S$ and $S^{\prime}$ are integrable. As
previously stated in the introduction, we shall be interested in the spectral
(generalized eigenvalue) problem

$\mathfrak{D}_{\epsilon}\textbf{u}=\lambda\textbf{u}$ (2.2)

where $\textbf{u}=[u_{1}\hskip 3.0ptu_{2}]^{T}$ is a function from
${\mathbb{R}}$ to ${\mathbb{C}}^{2}$ and $\lambda\in\mathbb{C}$ plays the role
of the spectral parameter.

The operator $\mathfrak{D}_{\epsilon}$ is non-self-adjoint and the eigenvalues
are distributed in the complex plane. But in the semiclassical limit, their
location is restricted close to the numerical range $\mathcal{R}$ of the
semiclassical symbol.

We shall see [see the reasoning that follows, eventually leading to (2.7)]
that the eigenvalues of $\mathfrak{D}_{\epsilon}$ coincide with those of the
operator

$\begin{bmatrix}-\frac{\epsilon}{i}\frac{d}{dx}-\frac{1}{2}S^{\prime}(x)&-iA(x)\\\
-iA(x)&\frac{\epsilon}{i}\frac{d}{dx}-\frac{1}{2}S^{\prime}(x)\end{bmatrix}.$
(2.3)

We have the following definition.

######  Definition 2.1.

The numerical range $\mathcal{R}$ of the symbol

$\begin{bmatrix}-\xi-\frac{1}{2}S^{\prime}(x)&-iA(x)\\\
-iA(x)&\xi-\frac{1}{2}S^{\prime}(x)\end{bmatrix}$

of the matrix/operator in (2.3) is defined to be the union of the set of
values of the eigenvalues $-\frac{1}{2}S^{\prime}(x)\pm\sqrt{\xi-A(x)^{2}}$ of
this matrix when $(x,\xi)$ varies in the whole phase space ${\mathbb{R}}^{2}$.

We see that

$\displaystyle\mathcal{R}$
$\displaystyle={\mathbb{R}}\cup{\mathcal{R}}_{0}\quad\text{where}\quad$
$\displaystyle{\mathcal{R}}_{0}$
$\displaystyle=-\frac{1}{2}\Big{[}\inf_{x\in{\mathbb{R}}}S^{\prime}(x),\sup_{x\in{\mathbb{R}}}S^{\prime}(x)\Big{]}\times\Big{[}-\sup_{x\in{\mathbb{R}}}|A(x)|,\sup_{x\in{\mathbb{R}}}|A(x)|\Big{]}.$
(2.4)

The following result concerning the semiclassical eigenvalues is well-known;
see e.g. [8].

######  Proposition 2.2.

For any compact set $K\subset{\mathbb{R}}^{2}$ with
$K\cap{\mathcal{R}}_{0}=\emptyset$, there exists $\epsilon_{0}>0$ such that
there is no eigenvalue in $K$ for $0<\epsilon<\epsilon_{0}$.

It is clear that (2.2) can be written equivalently as a first order system of
differential equations, namely

$\frac{\epsilon}{i}\frac{d}{dx}\textbf{u}(x,\lambda,\epsilon)=K(x,\lambda,\epsilon)\textbf{u}(x,\lambda,\epsilon)$
(2.5)

where

$K(x,\lambda,\epsilon)=\begin{bmatrix}-\lambda&\omega(x,\epsilon)\\\
\omega^{*}(x,\epsilon)&\lambda\end{bmatrix}.$ (2.6)

Applying first the transformation

$\textbf{u}(x,\lambda,\epsilon)=\begin{bmatrix}\exp\big{\\{}i\frac{S(x)}{2\epsilon}\big{\\}}&0\\\
0&\exp\big{\\{}-i\frac{S(x)}{2\epsilon}\big{\\}}\end{bmatrix}\tilde{\textbf{v}}(x,\lambda,\epsilon)$

where $\tilde{\textbf{v}}=[\tilde{v}_{1}\hskip 3.0pt\tilde{v}_{2}]^{T}$, takes
the system in (2.5) to a new form

$\frac{\epsilon}{i}\frac{d}{dx}\tilde{\textbf{v}}(x,\lambda,\epsilon)=\begin{bmatrix}-\lambda-\frac{1}{2}S^{\prime}(x)&-iA(x)\\\
iA(x)&\lambda+\frac{1}{2}S^{\prime}(x)\end{bmatrix}\tilde{\textbf{v}}(x,\lambda,\epsilon)$
(2.7)

where prime denotes differentiation with respect to $x$. Next, we apply the
mapping

$\tilde{\textbf{v}}(x,\lambda,\epsilon)=\begin{bmatrix}1&1\\\
-1&1\end{bmatrix}\textbf{v}(x,\lambda,\epsilon)$

where $\textbf{v}=[v_{1}\hskip 3.0ptv_{2}]^{T}$, to finally express the
initial system as

$\frac{\epsilon}{i}\frac{d}{dx}\textbf{v}(x,\lambda,\epsilon)=M(x,\lambda)\textbf{v}(x,\lambda,\epsilon)$
(2.8)

where

$M(x,\lambda)=\begin{bmatrix}0&g_{+}(x,\lambda)\\\
-g_{-}(x,\lambda)&0\end{bmatrix}$ (2.9) $\text{and}\quad
g_{\pm}(x,\lambda)=\mp[\lambda+\frac{1}{2}S^{\prime}(x)\pm iA(x)].$ (2.10)

The important function to study is

$\displaystyle V_{0}(x,\lambda)$ $\displaystyle=\det M(x,\lambda)$
$\displaystyle=g_{-}(x,\lambda)g_{+}(x,\lambda)$
$\displaystyle=-[\lambda+\tfrac{1}{2}S^{\prime}(x)]^{2}-A^{2}(x).$ (2.11)

The zeros of this function play an important role. So we state the following
definition.

######  Definition 2.3.

For a fixed value of $\lambda\in\mathbb{C}$, the zeros of
$V_{0}(\cdot,\lambda)$ in $\mathbb{C}$ are called turning points of (2.8).

In our application, the potential function $V_{0}(\cdot,\lambda)$ in (2.1),
i.e.

$V_{0}(x,\lambda)=-[\lambda+\tfrac{1}{2}S^{\prime}(x)]^{2}-A^{2}(x)$

is complex-valued for $\lambda\in{\mathbb{C}}\setminus{\mathbb{R}}$ and the
turning points are in general complex. Assuming that the functions $A(x)$ and
$S(x)$ extend analytically to some fixed complex neighborhood $\Omega_{0}$ of
the real axis, we may consider the asymptotics of solutions of (2.5)
[eventually of (2.8)] on some contour (in the complex $x$-plane), other than
the real $x$-axis, connecting $-\infty$ to $+\infty$ and forcing that contour
to pass through at least one pair of complex turning points.

Let us define

$z(x,\lambda,\alpha)=i\int_{\gamma(\alpha\rightsquigarrow
x)}\sqrt{-V_{0}(t,\lambda)}dt$

for a fixed point $\alpha\in\Omega_{0}$.

######  Definition 2.4.

A path $\gamma$ in $\Omega_{0}$ along which $\Re z$ is strictly monotone will
be called a progressive path.

######  Remark 2.5.

This notion does not depend on the branch of the square root nor the base
point $\alpha$. This is equivalent to $\gamma$ been transversal to the Stokes
curves defined later in Definition 2.6. When the branch of the square root and
the orientation of the path are specified, we also say a path is
$+$-progressive (resp. $-$-progressive) if $\Re z$ is strictly increasing
(resp. decreasing).

It is important to notice that, in our setting where $S^{\prime}$ and $A$ tend
to 0 at infinity, $\sqrt{-V_{0}(t,\lambda)}$ tends to $\pm\lambda$. Hence if
$\Im\lambda$ is non-zero, then the real axis is progressive near infinity.

The level curves of $\Re[z(x,\lambda,\alpha)]$ in the complex $x$-plane are
very important for what follows. Indeed, $(\pm)$-progressive curves are always
transversal to those. For that, they deserve a definition.

######  Definition 2.6.

The level curves of $\Re[z(x,\lambda,\alpha)]$ in the complex $x$-plane are
called the Stokes lines (or Stokes curves) of the system (2.8).

The geometric configuration of the Stokes curves is crucial in the
investigation of the domain of validity of the asymptotic expansion of the WKB
solutions. Notice that from a simple turning point [i.e. a simple zero of the
function $V_{0}(\cdot,\lambda)$ in (2.1)], exactly three Stokes lines emanate.
At such a point, the angles between two Stokes curves are all $2\pi/3$.

Fix $\lambda\in{\mathbb{C}}\setminus{\mathbb{R}}$. With the above intuition in
mind, we have the following definition.

######  Definition 2.7.

Let $\\{x_{-}(\lambda)$, $x_{+}(\lambda)\\}$ be a pair of simple complex roots
of $V_{0}(x,\lambda)=0$. A contour $C=C^{-}\cup C^{0}\cup C^{+}$ in
$\Omega_{0}$ shall be called admissible if the following four conditions hold
true:

  * (i)

$V_{0}$ is holomorphic in $x$, in the region of the complex $x$-plane enclosed
by $C$ and the real $x$-axis (this is needed to ensure that the approximate
eigenfunctions can be continued back to the real $x$-axis).

  * (ii)

The contour $C^{-}$ is a progressive path from $x_{-}(\lambda)$ to $-\infty$
which coincides with the real axis for $-\Re x\gg 1$.

  * (iii)

The contour $C^{+}$ is a progressive path from $x_{+}(\lambda)$ to $+\infty$
which coincides with the real axis for $\Re x\gg 1$.

  * (iv)

The contour $C^{0}$ is a path from $x_{-}(\lambda)$ to $x_{+}(\lambda)$ along
which $\Re z(x,\lambda,x_{-}(\lambda))$ vanishes identically; in other words,
$C^{0}$ is a Stokes line.

######  Remark 2.8.

The term admissible also appears in [22] but here, we impose a stronger
requirement on $C^{-}$ and $C^{+}$. This stronger definition is directly
applicable to the exact WKB method of the next section, and was first used for
the study of quantum resonances of the Schrödinger operator in [11].

Item (iv) in the list above can be reinterpreted as a differential equation
for the path $C^{0}$ in the complex $x$-plane. Indeed, if $x=u+iv$, where
$u,v\in\mathbb{R}$, then a field of curves is defined by the differential
relation

$\Re\bigg{\\{}i\sqrt{-V_{0}(u+iv,\lambda)}(du+idv)\bigg{\\}}=0.$ (2.12)

The idea guiding the definition above is that the set of $\lambda$ for which
such admissible contours exist is (hopefully) the locus of points in the
complex plane that attracts the actual eigenvalues as $\epsilon\to 0$. This
cannot be true in general, as we shall see soon, but an appropriately extented
generalisation of this statement probably is. See the last remark 2.25 of this
section.

Suppose now that for a pair of complex turning points
$\\{x_{-}(\lambda),x_{+}(\lambda)\\}$ we have the condition

$\Re\bigg{\\{}i\int_{C^{0}(x_{-}(\lambda)\rightsquigarrow
x_{+}(\lambda))}\sqrt{-V_{0}(t,\lambda)}dt\bigg{\\}}=0$ (2.13)

and let $\lambda$ vary. Given a pair of turning points depending on $\lambda$
that are distinct throughout a domain in the complex $\lambda$-plane, relation
(2.13) itself determines a curve in the complex $\lambda$-plane. If $\lambda$
is on one of these curves, the standard WKB procedure can be expected to apply
to determine whether $\lambda$ is in fact an $o(1)$ distance away from an
eigenvalue (of course, this is subject to the curve avoiding any singularities
and the existence of appropriate progressive paths $C^{-}$, $C^{+}$).

Therefore, as $\epsilon\downarrow 0$, we expect that the discrete eigenvalues
of (2.1) will accumulate on the union of curves in the complex $\lambda$-plane
described by formula (2.13), with the union being taken over pairs of complex
turning points. Hence we are led to the following.

######  Definition 2.9.

The curves in the complex $\lambda$-plane consisting of $\lambda$-points that
give rise to admissible contours $C$ (on the $x$-plane) will be called
asymptotic spectral arcs.

######  Definition 2.10.

The set that is defined as the union of all the asymptotic spectral arcs in
the complex $\lambda$-plane will be called the asymptotic spectrum of our
Dirac operator.

######  Remark 2.11.

It follows from the arguments above that this asymptotic spectrum is a union
of analytic arcs; if for each $\lambda$ there is only a finite number of
turning points, then the asymptotic spectrum is a finite union of asymptotic
spectral arcs.

Applying the exact WKB theory stated below in a neighborhood of the contour
$C$ (admissible contour), we shall rigorously obtain the eigenvalue
approximation by $\lambda$’s satisfying

$\frac{1}{\pi\epsilon}\Im\bigg{\\{}i\int_{C^{0}(x_{-}(\lambda)\rightsquigarrow
x_{+}(\lambda))}\sqrt{-V_{0}(t,\lambda)}dt\bigg{\\}}-\frac{1}{2}\in\mathbb{Z}$
(2.14)

which can be interpreted as a Bohr-Sommerfeld quantization rule. Such
$\lambda$’s are on the asymptotic spectral arc corresponding to the condition
(2.13) for the turning points $x_{-}(\lambda)$ and $x_{+}(\lambda)$.

We remark that, in general, there may well be eigenvalues which are not
associated with such an asymptotic spectral arc as defined so far. This is
illustrated by the example below.

######  Example 2.12.

Consider the simplest case where $S(x)$ is linear, i.e.
$S^{\prime}(x)=c\in{\mathbb{R}}$ is constant. In this case, the rectangular
region ${\mathcal{R}}_{0}$ corresponds to the segment

${\mathcal{R}}_{0}=\Big{\\{}-\frac{c}{2}\Big{\\}}\times\Big{[}-\sup_{x\in{\mathbb{R}}}|A|,\sup_{x\in{\mathbb{R}}}|A|\Big{]}.$

Additionally, if $A(x)$ is “bell-shaped”, i.e. if it decays at $\pm\infty$ and
$xA^{\prime}(x)<0$ in ${\mathbb{R}}^{\pm}$, then for any $\lambda$ lying in
either of the segments $[-\frac{c}{2},-\frac{c}{2}+i\sup A]$ or
$[-\frac{c}{2},-\frac{c}{2}-i\sup A]$, there exist exactly two real, simple
turning points $x_{-}(\lambda)$, $x_{+}(\lambda)$ connected by a Stokes line
$C^{0}=[x_{-}(\lambda),x_{+}(\lambda)]\subset{\mathbb{R}}$, and
$C_{-}=(-\infty,x_{-}(\lambda))$, $C_{+}=(x_{+}(\lambda),+\infty)$ are
progressive. Hence $C={\mathbb{R}}$ is an admissible contour and the vertical
segments $[-\frac{c}{2},-\frac{c}{2}+i\sup A]$ and
$[-\frac{c}{2},-\frac{c}{2}-i\sup A]$ are asymptotic spectral arcs.

Now, if we keep the linearity of $S$ but consider $A(x)$ to be double humped,
say $A(x)=[(x^{2}-1)^{2}+1]^{-1}$, the complex intervals
$[-\frac{c}{2},\frac{-c+i}{2}]$, $[-\frac{c}{2},\frac{-c-i}{2}]$ are
asymptotic spectral arcs just as above. In particular, when
$\lambda=\frac{-c\pm i}{2}$, there appears a double turning point at the
origin $x=0$ on the Stokes line $C^{0}=[-\sqrt{2},\sqrt{2}]$. However, for
$\lambda\neq\frac{-c\pm i}{2}$ in $[\frac{-c+i}{2},-\frac{c}{2}+i]$ or
$[\frac{-c-i}{2},-\frac{c}{2}-i]$, $C^{0}$ splits into two Stokes lines, and
such $\lambda$ does not belong to the asymptotic spectral arcs according to
the above definition (although there do exist eigenvalues near such $\lambda$,
see [17]).

But there is no reason why we should not allow two or more Stokes lines
apearing in an admissible path. We thus propose the following general
definition. We note however that for the specific example we eventually focus
on, $A(x)=S(x)=\sech(2x)$, there is at most one Stokes line appearing.

######  Definition 2.13.

Consider $N\in{\mathbb{N}}$. In general, we call a contour $C$ admissible if
the following conditions hold.

  * •

$V_{0}$ is holomorphic in $x$, in the region of the complex $x$-plane enclosed
by $C$ and the real $x$-axis.

  * •

$C$ coincides with ${\mathbb{R}}$ outside a compact set.

  * •

$C$ contains a finite number of pairs of simple turning points
$x_{-}^{i}(\lambda)$, $x_{+}^{i}(\lambda)$, where $i=1,...,N$, with

$V_{0}(x^{i}_{\pm}(\lambda),\lambda)=0,$

  * •

$C$ consists of a progressive curve $C^{-}$ connecting $-\infty$ and
$x_{-}^{1}(\lambda)$, a progressive curve $C^{+}$ connecting
$x_{+}^{N}(\lambda)$ and $+\infty$, progressive curves connecting
$x^{i}_{+}(\lambda)$ and $x^{i+1}_{-}(\lambda)$ ($i=1,...,N-1$) and Stokes
lines connecting $x^{i}_{-}(\lambda)$ and $x^{i}_{+}(\lambda)$, $i=1,...,N$.

Even this more general definition cannot be completely comprehensive as a
characterization of eigenvalues, since it does not take into account double
truning points, which can exist, at least non-generically, as we will
eventually see.

###  2.2. The exact WKB method

In this and next subsection 2.3, we review the local theory of the exact WKB
method applied to our Zakharov-Shabat operator in a fixed open set in
${\mathbb{C}}$.

Let us fix $\lambda$, let $\Omega$ be a simply connected subdomain of the
$x$-complex plane free from turning points, and take a fixed point
$\alpha\in\Omega$. We define a phase map

$\displaystyle z(\cdot,\lambda,\alpha):\Omega$
$\displaystyle\rightarrow\mathbb{C}\quad\text{with}$ $\displaystyle
z(x,\lambda,\alpha)=i\int_{\gamma(\alpha\rightsquigarrow x)}$
$\displaystyle\sqrt{-V_{0}(t,\lambda)}dt$ (2.15)

where of course the path of integration is irrelevant and we choose the branch
of $\sqrt{-V_{0}(x,\lambda)}$ which makes it positive for large $x>0$. Observe
that $z$ is uniquely defined in $\Omega$ and it maps $\Omega$ conformally to
$z(\Omega,\lambda,\alpha)$.

We look for solutions of (2.8) having the form

$\textbf{v}(x,\lambda,\epsilon,\alpha)=\exp\bigg{\\{}\pm\frac{z(x,\lambda,\alpha)}{\epsilon}\bigg{\\}}\tilde{\textbf{w}}^{\pm}[z(x,\lambda,\alpha)]$
(2.16)

for $\tilde{\textbf{w}}^{\pm}=[\tilde{w}^{\pm}_{1}\hskip
3.0pt\tilde{w}^{\pm}_{2}]^{T}$. Substituting this in (2.8), we get the system

$\frac{\epsilon}{i}\frac{d}{dz}\tilde{\textbf{w}}^{\pm}(z)=\begin{bmatrix}\pm
i&H(z)^{-2}\\\ -H(z)^{2}&\pm i\end{bmatrix}\tilde{\textbf{w}}^{\pm}(z)$ (2.17)

where [upon recalling (2.10)] the function $H$ is given by

$H(z)=\bigg{[}\frac{g_{-}(x,\lambda)}{g_{+}(x,\lambda)}\bigg{]}^{1/4}.$

Finally, we set

$P_{\pm}(z)=\begin{bmatrix}H(z)^{-1}&H(z)^{-1}\\\
iH(z)&-iH(z)\end{bmatrix}\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}^{\frac{1\pm
1}{2}}$

and apply the transformation

$\tilde{\textbf{w}}^{\pm}(z)=P_{\pm}(z)\textbf{w}^{\pm}(z)$

for $\textbf{w}^{\pm}=[w^{\pm}_{\rm even}\hskip 3.0ptw^{\pm}_{\rm odd}]^{T}$.
Then the system in (2.17) is written as

$\frac{\epsilon}{i}\frac{d}{dz}\textbf{w}^{\pm}(z)=\begin{bmatrix}0&\mathcal{H}(z)\\\
\mathcal{H}(z)&\mp\frac{2}{\epsilon}\end{bmatrix}\textbf{w}^{\pm}(z)$ (2.18)

in which we consider $\mathcal{H}$ to be defined by

$\mathcal{H}(z)=\frac{\dot{H}(z)}{H(z)}$ (2.19)

and where the dot denotes differentiation with respect to $z$. Easily, one can
arrive at

$\mathcal{H}(z)=\frac{d}{dz}\log[H(z)]=\frac{1}{4}\frac{A(x)S^{\prime\prime}(x)-2\lambda
A^{\prime}(x)-A^{\prime}(x)S^{\prime}(x)}{\big{\\{}[\lambda+\tfrac{1}{2}S^{\prime}(x)]^{2}+A^{2}(x)\big{\\}}^{3/2}}.$

In the usual WKB theory, the vector valued symbols $\textbf{w}^{\pm}$ are
constructed as a power series in the parameter $\epsilon$, which is in general
divergent. Here we would like to introduce the reader to the so-called exact
WKB method (along the lines of Gérard-Grigis [13] and Fujiié-Lasser-Nédélec
[10]). This method consists in the resummation of this divergent series in the
following way. By taking a point $x_{0}\in\Omega$, setting

$z_{0}\equiv z_{0}(\lambda)=z(x_{0},\lambda,\alpha)$

and postulating the series ansatz

$\textbf{w}^{\pm}=\sum_{n=0}^{\infty}\textbf{w}^{\pm}_{n}=\sum_{n=0}^{\infty}\begin{bmatrix}w^{\pm}_{2n}\\\
w^{\pm}_{2n-1}\end{bmatrix}\equiv\begin{bmatrix}w^{\pm}_{\rm even}\\\
w^{\pm}_{\rm odd}\end{bmatrix}$ (2.20)

we see from (2.18) that the functions $w^{\pm}_{n},~{}-1\leq n\in\mathbb{Z},$
can be defined inductively by

$w_{-1}^{\pm}\equiv 0,\quad w_{0}^{\pm}\equiv 1$

and for $1\leq n\in\mathbb{Z}$

$\left\\{\begin{array}[]{rl}\frac{d}{dz}w_{2n}^{\pm}(z)&=\mathcal{H}(z)w_{2n-1}^{\pm}(z)\\\
(\frac{d}{dz}\pm\frac{2}{\epsilon})w_{2n-1}^{\pm}(z)&=\mathcal{H}(z)w_{2n-2}^{\pm}(z)\\\
w_{n}^{\pm}(z_{0})&=0.\end{array}\right.$ (2.21)

These recurrence equations uniquely determine (at least in a neighborhood of
$x_{0}$) the sequence of scalar functions
$\\{w_{n}^{\pm}(z,\epsilon,z_{0})\\}_{n=-1}^{\infty}$ and hence the sequence
of vector-valued functions
$\\{\textbf{w}_{n}^{\pm}(z,\epsilon,z_{0})\\}_{n=0}^{\infty}$. We can write
the relations (2.21) in an integral form if we introduce the following
integral operators $\mathcal{I}_{\pm}$ and $\mathcal{J}$ taking functions from
$\mathscr{H}(z(\Omega,\lambda,\alpha))$ to functions in
$\mathscr{H}(z(\Omega,\lambda,\alpha))$ by

$\mathcal{I}_{\pm}[f](z)=\int_{\Gamma}e^{\pm
2(\zeta-z)/\epsilon}\mathcal{H}(\zeta)f(\zeta)d\zeta$ (2.22)
$\mathcal{J}[f](z)=\int_{\Gamma}\mathcal{H}(\zeta)f(\zeta)d\zeta$ (2.23)

where $\Gamma\equiv\Gamma(z_{0}\rightsquigarrow z)$ is the image by $z$ of a
path $\gamma\equiv\gamma(x_{0}\rightsquigarrow x)$ in $\Omega$ connecting
$x_{0}$, $x$; in other words $\Gamma=z(\gamma)$ (for notation cf. §1.3). Then
(2.21) for $n\geq 1$ becomes

$\left\\{\begin{array}[]{rl}w_{2n}^{\pm}&=\mathcal{J}[w_{2n-1}^{\pm}]\\\
w_{2n-1}^{\pm}&=\mathcal{I}_{\pm}[w_{2n-2}^{\pm}].\end{array}\right.$ (2.24)

Going back to the initial system (2.5), we have constructed (formal) WKB
solutions of the form

$\textbf{u}^{\pm}(x,\lambda,\epsilon,\alpha,x_{0})=\\\
\exp\Big{\\{}\pm\frac{z(x,\lambda,\alpha)}{\epsilon}\Big{\\}}Q(x,\lambda,\epsilon)\begin{bmatrix}0&1\\\
1&0\end{bmatrix}^{\frac{1\pm
1}{2}}\textbf{w}^{\pm}[z(x,\lambda,\alpha),\epsilon,z_{0}]$ (2.25)

where $Q(\cdot,\lambda,\epsilon)$ is a matrix-valued function with value

$Q(x,\lambda,\epsilon)=\begin{bmatrix}\exp\\{-iS(x)/(2\epsilon)\\}&0\\\
0&\exp\big{\\{}iS(x)/(2\epsilon)\\}\end{bmatrix}\begin{bmatrix}1&1\\\
-1&1\end{bmatrix}\begin{bmatrix}H(z)^{-1}&H(z)^{-1}\\\
iH(z)&-iH(z)\end{bmatrix}.$

Here, we added the superscript $\pm$ to the solution $\textbf{u}=[u_{1}\hskip
3.0ptu_{2}]^{T}$ -consequently arriving at the notation
$\textbf{u}^{\pm}=[u^{\pm}_{1}\hskip 3.0ptu^{\pm}_{2}]$\- to distinguish
between the two different cases in the previous discussion; also we keep in
mind that the WKB solutions in (2.25) depend on a base point $\alpha$ for the
phase map (2.15) and a base point $x_{0}$ for the symbols $\textbf{w}^{\pm}$.

Recall Definition 2.4 and write

$\Omega_{\pm}=\\{\,x\in\Omega\mid\exists\hskip 3.0pt(\pm)\text{-progressive
path}\hskip 3.0pt\gamma(x_{0}\rightsquigarrow x)\subset\Omega\,\\}.$

We have the following basic theorem; for the proof, we refer to [10].

######  Theorem 2.14.

The exact WKB solutions in (2.25) satisfy the following properties.

  1. (i)

The formal series in (2.20) is absolutely convergent in a neighborhood of
$x_{0}$.

  2. (ii)

For each $N\in\mathbb{N}$, we have

$\displaystyle\mathbf{w}^{\pm}-\sum_{n=0}^{N-1}\mathbf{w}^{\pm}_{n}$
$\displaystyle=\mathcal{O}(\epsilon^{N})\quad\text{as}\quad\epsilon\downarrow
0$ $\displaystyle w^{\pm}_{\rm even}-\sum_{n=0}^{N-1}w_{2n}^{\pm}$
$\displaystyle=\mathcal{O}(\epsilon^{N})\quad\text{as}\quad\epsilon\downarrow
0$ $\displaystyle w^{\pm}_{\rm odd}-\sum_{n=0}^{N-1}w_{2n-1}^{\pm}$
$\displaystyle=\mathcal{O}(\epsilon^{N})\quad\text{as}\quad\epsilon\downarrow
0$

uniformly in any compact subset of $\Omega_{\pm}$. In particular, there we
have

$\displaystyle w^{\pm}_{\rm even}$
$\displaystyle=1+\mathcal{O}(\epsilon)\quad\text{as}\quad\epsilon\downarrow 0$
$\displaystyle\quad w^{\pm}_{\rm odd}$
$\displaystyle=\mathcal{O}(\epsilon)\quad\text{as}\quad\epsilon\downarrow 0.$

  3. (iii)

Any two exact WKB solutions with different base points for the symbol satisfy

$\displaystyle{\mathcal{W}}[{\bf u}^{+}(x,\lambda,\epsilon,\alpha,x_{0}),{\bf
u}^{-}(x,\lambda,\epsilon,\alpha,x_{1})]$ $\displaystyle=4iw_{\rm
even}^{+}(z_{1},\epsilon,z_{0})$ (2.26) $\displaystyle{\mathcal{W}}[{\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{0}),{\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{1})]$
$\displaystyle=-4ie^{2z_{1}/\epsilon}w_{\rm odd}^{+}(z_{1},\epsilon,z_{0})$
(2.27)

where $x_{0}$, $x_{1}$ are two different points in $\Omega$ and
$z_{j}=z(x_{j},\lambda,\alpha)$ for $j=0,1$.

###  2.3. Connection around a simple turning point

We continue in this section by presenting a local connection formula for exact
WKB solutions near a simple turning point. Let $\alpha$ be a simple turning
point, i.e.

$\displaystyle\det M(\alpha)$ $\displaystyle=g_{+}(\alpha)g_{-}(\alpha)=0$
$\displaystyle(\det M)^{\prime}(\alpha)$
$\displaystyle=g_{+}(\alpha)g^{\prime}_{-}(\alpha)+g^{\prime}_{+}(\alpha)g_{-}(\alpha)\neq
0$

which means that $\alpha$ is a simple zero of one and only one of either
$g_{+}(x)$ or $g_{-}(x)$. Three Stokes lines $l_{0}$, $l_{1}$ and $l_{2}$,
numbered in the anti-clockwise sense, emanate from $\alpha$ and devide a
neighborhood $\omega$ of $\alpha$ into three regions $\omega_{0}$ bounded by
$l_{1}$ and $l_{2}$, $\omega_{1}$ bounded by $l_{2}$ and $l_{0}$ and
$\omega_{2}$ bounded by $l_{0}$ and $l_{1}$. In each of these regions, we are
going to define exact WKB solutions ${\bf u}_{0}$, ${\bf u}_{1}$ and ${\bf
u}_{2}$ respectively in such a way that its asymptotic behavior is known near
$\alpha$ by Theorem 2.14.

We have first to specify the branch of the multi-valued function
$z(x,\lambda,\alpha)$ defined by (2.15). Let us take a base point $x_{j}$ in
$\omega_{j}$ $(j=0,1,2)$. We put a branch cut on the Stokes line $l_{1}$, and
take the branch of the function $\sqrt{\det M}$ so that
$\Re\,z(x,\lambda,\alpha)$ increases from $x_{0}$ towards the turning point
$\alpha$. Then consequently $\Re\,z(x,\lambda,\alpha)$ decreases from $x_{1}$
towards $\alpha$ and increases from $x_{2}$ towards $\alpha$.

We define three exact WKB solutions:

$\displaystyle{\bf u}_{0}(x,\epsilon)={\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{0}),$ (2.28) $\displaystyle{\bf
u}_{1}(x,\epsilon)={\bf u}^{-}(x,\lambda,\epsilon,\alpha,x_{1}),$ (2.29)
$\displaystyle{\bf u}_{2}(x,\epsilon)={\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{2}).$ (2.30)

These solutions are constructed near $x_{0}$, $x_{1}$, $x_{2}$ respectively
but can be extended analytically to $\omega$. They cannot be linearly
independent since the vector space of the solutions is of dimension two. More
precisely, we have the following linear relation.

######  Proposition 2.15.

Let ${\bf u}_{0}$, ${\bf u}_{1}$, ${\bf u}_{2}$ be three solutions to the
differential equation (2.2) in a region $\omega$. Then the following identity
holds:

${\mathcal{W}}[{\bf u}_{1},{\bf u}_{2}]{\bf u}_{0}+{\mathcal{W}}[{\bf
u}_{2},{\bf u}_{0}]{\bf u}_{1}+{\mathcal{W}}[{\bf u}_{0},{\bf u}_{1}]{\bf
u}_{2}=0.$ (2.31)

###### Proof.

If the three determinants ${\mathcal{W}}[{\bf u}_{1},{\bf u}_{2}]$,
${\mathcal{W}}[{\bf u}_{2},{\bf u}_{0}]$ and ${\mathcal{W}}[{\bf u}_{0},{\bf
u}_{1}]$ are all 0, the identity holds obviously. Suppose that at least one of
those does not vanish, say ${\mathcal{W}}[{\bf u}_{1},{\bf u}_{2}]\neq 0$.
Then by taking the determinant of the LHS of (2.31) with ${\bf u}_{1}$ we find

${\mathcal{W}}[{\bf u}_{1},{\bf u}_{2}]{\mathcal{W}}[{\bf u}_{0},{\bf
u}_{1}]+{\mathcal{W}}[{\bf u}_{0},{\bf u}_{1}]{\mathcal{W}}[{\bf u}_{2},{\bf
u}_{1}]$

which is obviously zero. In the same way, we obtain that the determinant of
the LHS of (2.31) with ${\bf u}_{2}$ is zero. Since the pair of solutions
$\\{{\bf u}_{1},{\bf u}_{2}\\}$ makes a basis of the solution space, this
means that the LHS of (2.31) should be zero. ∎

Let us come back to our exact WKB solutions ${\bf u}_{0}$, ${\bf u}_{1}$ and
${\bf u}_{2}$ defined above and compute the asymptotic behavior of the
determinants appearing in this proposition.

######  Proposition 2.16.

As $\epsilon\downarrow 0$, the following asymptotics hold true

$\displaystyle{\mathcal{W}}[{\bf u}_{0},{\bf
u}_{1}]=2i(1+{\mathcal{O}}(\epsilon)),$ (2.32)
$\displaystyle{\mathcal{W}}[{\bf u}_{1},{\bf
u}_{2}]=-2i(1+{\mathcal{O}}(\epsilon)),$ (2.33)
$\displaystyle{\mathcal{W}}[{\bf u}_{2},{\bf u}_{0}]=\mp
2(1+{\mathcal{O}}(\epsilon)),$ (2.34)

where in (2.34) the upper sign (minus) is to be chosen in the case of $\alpha$
being a zero of $g_{+}$ and the lower sign (plus) in the case that $\alpha$ is
a zero of $g_{-}$. In particular, any two of the solutions ${\bf u}_{0}$,
${\bf u}_{1}$ and ${\bf u}_{2}$ are linearly independent for sufficiently
small $\epsilon$.

###### Proof.

Using Theorem 2.14 (iii), we find

${\mathcal{W}}[{\bf u}_{1},{\bf u}_{2}]=-2iw_{\rm
even}^{+}(z_{1},\epsilon,z_{2})$

where $z_{j}=z(x_{j},\lambda,\alpha)$. Since there exists a $(+)$-progressive
path from $x_{2}$ to $x_{1}$, one has $w_{\rm
even}^{+}(z_{1},\epsilon,z_{2})=1+{\mathcal{O}}(\epsilon)$ by Theorem 2.14
(ii), and hence we obtain (2.33). Similarly we have

${\mathcal{W}}[{\bf u}_{0},{\bf u}_{1}]=2iw_{\rm
even}^{+}(z_{1},\epsilon,z_{0})=2i(1+{\mathcal{O}}(\epsilon)).$

For the computation of the determinant ${\mathcal{W}}[{\bf u}_{2},{\bf
u}_{0}]$, we have to be careful since there is a branch cut between $x_{2}$
and $x_{0}$. Before applying the wronskian formula, we rewrite ${\bf u}_{2}$
on the Riemann sheet continued from $x_{0}$ passing across the branch. Let $x$
be a point near $x_{2}$ and $\hat{x}$ the same point as $x$ but on that
Riemann sheet. Then we have

$\displaystyle z(x,\lambda,\alpha)=-z(\hat{x},\lambda,\alpha),$ $\displaystyle
H(x)=\mp iH(\hat{x}),\,\text{ if }\alpha\text{ is a zero of
}g_{\pm}\quad\text{and}$ $\displaystyle
w^{\pm}(z,\epsilon,z_{2})=w^{\mp}(z,\epsilon,z_{2}).$

Observe that $\arg(x-\alpha)=\arg(\hat{x}-\alpha)+2\pi$ and hence the first
identity is obvious since the integrand of $z$ contains a multi-valued
function of type $\sqrt{x-\alpha}$. The second assertion holds due to the fact
that $H$ is the product of $g_{+}^{-1/4}$ and $g_{-}^{1/4}$. To verify the
third identity, let $w_{n}^{\pm}(z)$, $n\geq 1$, be a family of solutions to
the transport equations (2.21) satisfying the initial condition
$w_{n}^{\pm}(z_{2})=0$ at $z_{2}=z(x_{2})$, and set
$f_{n}^{\pm}(\hat{z})=w_{n}^{\pm}(z)$, $n\geq 1$. Inspecting (2.21) and using
$d/dz=-d/d\hat{z}$, it is easy to check that

$\displaystyle\left(\frac{d}{d\hat{z}}\mp\frac{2}{\epsilon}\right)f_{2n+1}^{\pm}(\hat{z})=\frac{dH(\hat{z})/d\hat{z}}{H(\hat{z})}f_{2n}^{\pm}(\hat{z}),$
$\displaystyle\frac{d}{d\hat{z}}f_{2n+2}^{\pm}(\hat{z})=\frac{dH(\hat{z})/d\hat{z}}{H(\hat{z})}f_{2n+1}^{\pm}(\hat{z})$

with $f_{n}^{\pm}(\hat{z}_{2})=0$ at $\hat{z}_{2}=z(\hat{x}_{2})$. By
uniqueness of solutions, it follows that $w_{n}^{\pm}(z)=w_{n}^{\mp}(\hat{z})$
with $w_{n}^{\mp}(\hat{z}_{2})=0$, which, in view of (2.20), implies that

${\bf
w}^{\pm}(z,\epsilon,z_{2})=\sum_{n=0}^{\infty}\left(\begin{array}[]{c}w_{2n}^{\pm}(z,z_{2})\\\
w_{2n+1}^{\pm}(z,z_{2})\end{array}\right)=\sum_{n=0}^{\infty}\left(\begin{array}[]{c}w_{2n}^{\mp}(\hat{z},\hat{z}_{2})\\\
w_{2n+1}^{\mp}(\hat{z},\hat{z}_{2})\end{array}\right)={\bf
w}^{\mp}(\hat{z},\epsilon,\hat{z}_{2}).$

Thus, with this point $\hat{x}_{2}$ as base point of the symbol, the solution
${\bf u}_{2}(x,\epsilon)$ is written as

${\bf u}_{2}(x,\epsilon)=\mp i{\bf
u}^{-}(x,\lambda,\epsilon,\alpha,\hat{x}_{2}),\,\text{ if }\alpha\text{ is a
zero of }g_{\pm}.$

We now apply Theorem 2.14 (iii) to compute the desired determinant, using the
above expression. Since there exists a progressive curve from $x_{0}$ to
$\hat{x}_{2}$, we have

${\mathcal{W}}[{\bf u}_{0},{\bf u}_{2}]=\pm i(\det Q)w_{\rm
even}^{+}(\hat{z}_{2},\epsilon,z_{0})=\pm 2(1+{\mathcal{O}}(\epsilon))$

when $\alpha$ is a zero of $g_{\pm}$. ∎

###  2.4. Eigenvalues near a generic point on the asymptotic spectral arc

In this section, we return to the eigenvalue problem (2.2) for the Zakharov-
Shabat operator $\mathfrak{D}_{\epsilon}$. We derive the quantization
condition for the eigenvalues in a small neighborhood of a fixed
$\lambda_{0}$, independent of $\epsilon$, belonging to an asymptotic spectral
arc under the following generic condition.

(H1):

$C^{0}$ contains no other turning point than $x_{-}(\lambda)$ and
$x_{+}(\lambda)$.

Let $C(\lambda_{0})$ be the admissible contour
$C(\lambda_{0})=C^{-}(\lambda_{0})\cup C^{0}(\lambda_{0})\cup
C^{+}(\lambda_{0})$ corresponding to $\lambda_{0}$ and denote by
$\alpha_{0}=\alpha(\lambda_{0})=x_{-}(\lambda_{0})$,
$\beta_{0}=\beta(\lambda_{0})=x_{+}(\lambda_{0})$ the simple turning points at
the extremities of $C^{0}(\lambda_{0})$. For $\lambda$ in a small enough
neighborhood of $\lambda_{0}$, the simple turning points
$\alpha=\alpha(\lambda)=x_{-}(\lambda)$ and
$\beta=\beta(\lambda)=x_{+}(\lambda)$ are still defined and analytic.

From point $\alpha$ there are three Stokes lines emanating, namely
$l_{0}^{\alpha}$, $l_{1}^{\alpha}$, $l_{2}^{\alpha}$ (in anti-clockwise
order); similarly from $\beta$ there emerge three Stokes lines which we denote
by $l_{0}^{\beta}$, $l_{1}^{\beta}$, $l_{2}^{\beta}$ (in clockwise order). For
the configuration we refer to Figure 3. Suppose $\lambda=\lambda_{0}$ and that
$l_{0}^{\alpha}$ and $l_{0}^{\beta}$ coincide with $C^{0}(\lambda)$. The five
Stokes lines $C^{0}(\lambda)$, $l_{1}^{\alpha}$, $l_{2}^{\alpha}$,
$l_{1}^{\beta}$ and $l_{2}^{\beta}$ divide a neighborhood $\omega$ of the
admissible contour into four regions which we denote as follows

  * •

$\omega_{\rm left}^{\alpha}$ is bounded by $l_{1}^{\alpha}$, $l_{2}^{\alpha}$

  * •

$\omega_{\rm right}^{\beta}$ is bounded by $l_{1}^{\beta}$, $l_{2}^{\beta}$

  * •

$\omega_{\rm up}^{\alpha,\beta}$ is bounded by $C^{0}(\lambda)$,
$l_{1}^{\alpha}$, $l_{1}^{\beta}$ and finally

  * •

$\omega_{\rm down}^{\alpha,\beta}$ is bounded by $C^{0}(\lambda)$,
$l_{2}^{\alpha}$, $l_{2}^{\beta}$.

Let us take a point in each of these regions: $x_{0}^{\alpha}$ in the region
$\omega_{\rm left}^{\alpha}$, $x_{0}^{\beta}$ in the region $\omega_{\rm
right}^{\beta}$, $x_{1}$ in the region $\omega_{\rm down}^{\alpha,\beta}$ and
$x_{2}$ in the region $\omega_{\rm up}^{\alpha,\beta}$.

Figure 3. Stokes lines near a simple turning point.

As in §2.3, we have to consider branch cuts; we choose one on the Stokes line
$l_{1}^{\alpha}$ and one on $l_{2}^{\beta}$. Assuming that $\Re z(x)$
increases towards $\alpha$ in $\omega_{\rm left}^{\alpha}$, we define exact
WKB solutions as follows.

$\displaystyle{\bf u}_{0}^{\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{0}^{\alpha}),\quad{\bf
u}_{0}^{\beta}(x,\epsilon)={\bf
u}^{-}(x,\lambda,\epsilon,\beta,x_{0}^{\beta})$ $\displaystyle{\bf
u}_{1}^{\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{-}(x,\lambda,\epsilon,\alpha,x_{1}),\quad{\bf
u}_{1}^{\beta}(x,\epsilon)={\bf u}^{-}(x,\lambda,\epsilon,\beta,x_{1})$
$\displaystyle{\bf u}_{2}^{\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{2}),\quad{\bf
u}_{2}^{\beta}(x,\epsilon)={\bf u}^{+}(x,\lambda,\epsilon,\beta,x_{2})$

Remark that all these exact WKB solutions are chosen so that their asymptotic
expansions in Theorem 2.14 (ii) are valid near the designated turning points.
In particular, ${\bf u}_{0}^{\alpha}(x,\epsilon)$ is a decaying solution along
$C^{-}(\lambda_{0})$ and ${\bf u}_{0}^{\beta}(x,\epsilon)$ is a decaying
solution along $C^{+}(\lambda_{0})$. This implies the following lemma.

######  Lemma 2.17.

For any $\lambda_{0}$ on the asymptotic spectral arcs, there exists a complex
neighbourhood $U=U(\lambda_{0})$ of $\lambda_{0}$ such that a $\lambda\in U$
is an eigenvalue of $\mathfrak{D}_{\epsilon}$ if and only if ${\bf
u}_{0}^{\alpha}(x,\epsilon)$ and ${\bf u}_{0}^{\beta}(x,\epsilon)$ are
linearly dependent.

Applying Proposition 2.15 near $\alpha$ and $\beta$, we have

$\displaystyle{\mathcal{W}}[{\bf u}_{1}^{\alpha},{\bf u}_{2}^{\alpha}]{\bf
u}_{0}^{\alpha}+{\mathcal{W}}[{\bf u}_{2}^{\alpha},{\bf u}_{0}^{\alpha}]{\bf
u}_{1}^{\alpha}+{\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf u}_{1}^{\alpha}]{\bf
u}_{2}^{\alpha}=0,$ (2.35) $\displaystyle{\mathcal{W}}[{\bf
u}_{1}^{\beta},{\bf u}_{2}^{\beta}]{\bf u}_{0}^{\beta}+{\mathcal{W}}[{\bf
u}_{2}^{\beta},{\bf u}_{0}^{\beta}]{\bf u}_{1}^{\beta}+{\mathcal{W}}[{\bf
u}_{0}^{\beta},{\bf u}_{1}^{\beta}]{\bf u}_{2}^{\beta}=0.$ (2.36)

On the other hand, there is an obvious relation between ${\bf u}_{j}^{\alpha}$
and ${\bf u}_{j}^{\beta}$ for $j=1,2$:

${\bf u}_{1}^{\beta}=e^{z(\beta,\lambda,\alpha)/\epsilon}{\bf
u}_{1}^{\alpha},\quad{\bf
u}_{2}^{\beta}=e^{-z(\beta,\lambda,\alpha)/\epsilon}{\bf u}_{2}^{\alpha}$
(2.37)

where $z(\beta,\lambda,\alpha)$ is the action integral between $\alpha$ and
$\beta$

$z(\beta,\lambda,\alpha)=i\int_{\gamma(\alpha\rightsquigarrow\beta)}\sqrt{-V_{0}(t,\lambda)}dt.$
(2.38)

It follows that ${\bf u}_{0}^{\alpha}(x,\epsilon)$ and ${\bf
u}_{0}^{\beta}(x,\epsilon)$ are linearly dependent if and only if

${\mathcal{W}}[{\bf u}_{2}^{\alpha},u_{0}^{\alpha}]{\mathcal{W}}[{\bf
u}_{0}^{\beta},{\bf u}_{1}^{\beta}]={\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf
u}_{1}^{\alpha}]{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]e^{2z(\beta,\lambda,\alpha)/\epsilon}.$

Recall the functions
$g_{\pm}(x,\lambda)=\mp[\lambda+\frac{1}{2}S^{\prime}(x)\pm iA(x)]$. For a
pair of simple turning points $\alpha$, $\beta$, we define an index
$\delta(\alpha,\beta)$ by

$\delta(\alpha,\beta)=\left\\{\begin{array}[]{cll}-1&\text{if}\quad
g_{+}(\alpha,\lambda)=g_{+}(\beta,\lambda)=0&\text{or}\quad
g_{-}(\alpha,\lambda)=g_{-}(\beta,\lambda)=0,\\\ 1&\text{if}\quad
g_{+}(\alpha,\lambda)=g_{-}(\beta,\lambda)=0&\text{or}\quad
g_{-}(\alpha,\lambda)=g_{+}(\beta,\lambda)=0.\end{array}\right.$ (2.39)

Hence we arrive at the following result.

######  Theorem 2.18.

For any $\lambda_{0}$ on the asymptotic spectral arcs satisfying condition
(H1), there exists a complex neighborhood $U$ of $\lambda_{0}$ such that
$\lambda\in U$ is an eigenvalue of $\mathfrak{D}_{\epsilon}$ if and only if

$m(\epsilon)e^{2z(\beta,\lambda,\alpha)/\epsilon}=1,\quad\text{with
}m(\epsilon)=\frac{{\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf
u}_{1}^{\alpha}]{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]}{{\mathcal{W}}[{\bf u}_{2}^{\alpha},{\bf
u}_{0}^{\alpha}]{\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf u}_{1}^{\beta}]}.$

As $\epsilon\downarrow 0$, $m(\epsilon)$ has the asymptotic behavior

$m(\epsilon)=\delta(\alpha_{0},\beta_{0})+{\mathcal{O}}(\epsilon).$

###### Proof.

It remains to check the asymptotic part of the statement. We already know from
Proposition 2.16 that as $\epsilon\downarrow 0$

${\mathcal{W}}[{\bf u}_{2}^{\alpha},{\bf u}_{0}^{\alpha}]=\mp
2+{\mathcal{O}}(\epsilon),\quad{\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf
u}_{1}^{\alpha}]=2i+{\mathcal{O}}(\epsilon)$ (2.40)

when $\alpha$ is a zero of $g_{\pm}$. Similarly, as $\epsilon\downarrow 0$ we
have

${\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]=2i+{\mathcal{O}}(\epsilon),\quad{\mathcal{W}}[{\bf
u}_{0}^{\beta},{\bf u}_{1}^{\beta}]=\mp 2+{\mathcal{O}}(\epsilon)$ (2.41)

when $\beta$ is a zero of $g_{\pm}$. These asymptotic formulas immediately
give the assertion. ∎

######  Remark 2.19.

(The situation at the endpoints $\lambda_{D}^{(j)},j=1,2$ of the branches) It
is clear from the proof that the statement of Theorem 2.18 is still valid for
$\lambda_{0}$ at a limit point of the asymptotic spectral arcs where
$x_{+}(\lambda_{0})$ and $x_{-}(\lambda_{0})$ coalesce to a double turning
point.

###  2.5. What happens near the bifurcation point

Now let us suppose that $\lambda_{0}$ lies on the asymptotic spectral arcs
with admissible contour $C(\lambda_{0})=C^{-}(\lambda_{0})\cup
C^{0}(\lambda_{0})\cup C^{+}(\lambda_{0})$ and that the Stokes line
$C^{0}(\lambda_{0})$ connecting the two simple turning points
$\alpha_{0}=\alpha(\lambda_{0})$ and $\beta_{0}=\beta(\lambda_{0})$ now also
contains a third simple turning point between of them, namely
$\gamma_{0}=\gamma(\lambda_{0})$; of course, $\alpha,\beta$ and $\gamma$ are
analytic functions defined in a neighborhood of $\lambda_{0}$.

Hence, $C^{0}(\lambda)$ consists of a Stokes line $l_{0}^{\alpha}$ connecting
$\alpha$ and $\gamma$ and a Stokes line $l_{0}^{\beta}$ connecting $\beta$ and
$\gamma$: $C^{0}(\lambda)=l_{0}^{\alpha}\cup l_{0}^{\beta}$. We suppose that
the third Stokes line from $\gamma$ divides the region $\omega_{\rm
down}^{\alpha,\beta}$ (defined as in section 2.4) into two subregions
$\omega_{\rm down}^{\alpha,\gamma}$ and $\omega_{\rm down}^{\beta,\gamma}$,
and take a point $x_{1}^{\alpha}$ in $\omega_{\rm down}^{\alpha,\gamma}$ and
$x_{1}^{\beta}$ in $\omega_{\rm down}^{\beta,\gamma}$ (see Figure 4).

Figure 4. Stokes lines near a bifurcation point.

Now we let $\lambda$ vary in a small neighborhood of $\lambda_{0}$. Then the
Stokes lines in $\omega$ change the geometric configuration. We denote by
$l_{j}^{\alpha}(\lambda)$, $l_{j}^{\beta}(\lambda)$ for $j=0,1,2$ and
$l^{\gamma}(\lambda)$, $l_{\alpha}^{\gamma}(\lambda)$,
$l_{\beta}^{\gamma}(\lambda)$ the three Stokes lines emanating from
$\alpha(\lambda)$, $\beta(\lambda)$ and $\gamma(\lambda)$ respectively such
that $l_{0}^{\alpha}(\lambda_{0})=l_{\alpha}^{\gamma}(\lambda_{0})$,
$l_{0}^{\beta}(\lambda_{0})=l_{\beta}^{\gamma}(\lambda_{0})$. The variation of
Stokes geometry as $\lambda$ turns around $\lambda_{0}$ is illustrated in
Figure 5. Remark that the Stokes lines stay close to those for
$\lambda=\lambda_{0}$ by continuity.

Figure 5. Variation of the Stokes geometry around a bifurcation point.

We put branch cuts along the Stokes lines $l_{1}^{\alpha}(\lambda)$,
$l_{1}^{\beta}(\lambda)$ and $l^{\gamma}(\lambda)$ (we emphasize at this point
that our selection for the branch cut placed on a Stokes line emerging from
$\beta$ in this case is different from the previous case where we dealt with
only two turning points), and suppose that $\Re z(x)$ increases towards
$\alpha$ in $\omega_{\rm left}^{\alpha}$. We define as before

$\displaystyle{\bf u}_{0}^{\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{0}^{\alpha}),\quad{\bf
u}_{0}^{\beta}(x,\epsilon)={\bf
u}^{+}(x,\lambda,\epsilon,\beta,x_{0}^{\beta})$ $\displaystyle{\bf
u}_{1}^{\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{-}(x,\lambda,\epsilon,\alpha,x_{1}^{\alpha}),\quad{\bf
u}_{1}^{\beta}(x,\epsilon)={\bf
u}^{-}(x,\lambda,\epsilon,\beta,x_{1}^{\beta})$ $\displaystyle{\bf
u}_{2}^{\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{+}(x,\lambda,\epsilon,\alpha,x_{2}),\quad{\bf
u}_{2}^{\beta}(x,\epsilon)={\bf u}^{+}(x,\lambda,\epsilon,\beta,x_{2})$
$\displaystyle{\bf u}_{1}^{\gamma,\alpha}(x,\epsilon)$ $\displaystyle={\bf
u}^{-}(x,\lambda,\epsilon,\gamma,x_{1}^{\alpha})$ $\displaystyle{\bf
u}_{1}^{\gamma,\beta}(x,\epsilon)$ $\displaystyle={\bf
u}^{-}(x,\lambda,\epsilon,\gamma,x_{1}^{\beta})$ $\displaystyle{\bf
u}_{2}^{\gamma}(x,\epsilon)$ $\displaystyle={\bf
u}^{+}(x,\lambda,\epsilon,\gamma,x_{2})$

For each one of the triples $\\{{\bf u}_{0}^{\alpha},{\bf u}_{1}^{\alpha},{\bf
u}_{2}^{\alpha}\\}$, $\\{{\bf u}_{2}^{\gamma},{\bf u}_{1}^{\gamma,\alpha},{\bf
u}_{1}^{\gamma,\beta}\\}$ and $\\{{\bf u}_{0}^{\beta},{\bf u}_{1}^{\beta},{\bf
u}_{2}^{\beta}\\}$, we have the following relations

$\displaystyle{\mathcal{W}}[{\bf u}_{1}^{\alpha},{\bf u}_{2}^{\alpha}]{\bf
u}_{0}^{\alpha}+{\mathcal{W}}[{\bf u}_{2}^{\alpha},{\bf u}_{0}^{\alpha}]{\bf
u}_{1}^{\alpha}+{\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf u}_{1}^{\alpha}]{\bf
u}_{2}^{\alpha}=0,$ (2.42) $\displaystyle{\mathcal{W}}[{\bf
u}_{1}^{\gamma,\alpha},{\bf u}_{1}^{\gamma,\beta}]{\bf
u}_{2}^{\gamma}+{\mathcal{W}}[{\bf u}_{1}^{\gamma,\beta},{\bf
u}_{2}^{\gamma}]{\bf u}_{1}^{\gamma,\alpha}+{\mathcal{W}}[{\bf
u}_{2}^{\gamma},{\bf u}_{1}^{\gamma,\alpha}]{\bf u}_{1}^{\gamma,\beta}=0,$
(2.43) $\displaystyle{\mathcal{W}}[{\bf u}_{1}^{\beta},{\bf
u}_{2}^{\beta}]{\bf u}_{0}^{\beta}+{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]{\bf u}_{1}^{\beta}+{\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf
u}_{1}^{\beta}]{\bf u}_{2}^{\beta}=0.$ (2.44)

Using the obvious relations

${\bf u}_{1}^{\beta}=e^{z(\beta,\lambda,\gamma)/\epsilon}{\bf
u}_{1}^{\gamma,\beta},\quad{\bf
u}_{1}^{\gamma,\alpha}=e^{z(\gamma,\lambda,\alpha)/\epsilon}{\bf
u}_{1}^{\alpha}$ ${\bf
u}_{2}^{\beta}=e^{-z(\beta,\lambda,\gamma)/\epsilon}{\bf
u}_{2}^{\gamma},\quad{\bf
u}_{2}^{\gamma}=e^{-z(\gamma,\lambda,\alpha)/\epsilon}{\bf u}_{2}^{\alpha}$

and

$z(\beta,\lambda,\gamma)+z(\gamma,\lambda,\alpha)=z(\beta,\lambda,\alpha)$

we deduce that the sum

${\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf u}_{0}^{\beta}]{\bf
u}_{1}^{\beta}+{\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf u}_{1}^{\beta}]{\bf
u}_{2}^{\beta}$

of the second and the third terms of (2.44) is

${\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]e^{z(\beta,\lambda,\gamma)/\epsilon}{\bf
u}_{1}^{\gamma,\beta}+{\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf
u}_{1}^{\beta}]e^{-z(\beta,\lambda,\gamma)/\epsilon}{\bf u}_{2}^{\gamma}$

and this is equal to

${\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf
u}_{1}^{\beta}]e^{-z(\beta,\lambda,\alpha)/\epsilon}{\bf
u}_{2}^{\alpha}-\frac{{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]}{{\mathcal{W}}[{\bf u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\alpha}]}e^{z(\beta,\lambda,\gamma)/\epsilon}\left({\mathcal{W}}[{\bf
u}_{1}^{\gamma,\alpha},{\bf u}_{1}^{\gamma,\beta}]{\bf
u}_{2}^{\gamma}+{\mathcal{W}}[{\bf u}_{1}^{\gamma,\beta},{\bf
u}_{2}^{\gamma}]{\bf u}_{1}^{\gamma,\alpha}\right)$

or

${\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf
u}_{1}^{\beta}]e^{-z(\beta,\lambda,\alpha)/\epsilon}{\bf u}_{2}^{\alpha}\\\
\qquad-\frac{{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]}{{\mathcal{W}}[{\bf u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\alpha}]}e^{z(\beta,\lambda,\gamma)/\epsilon}\left({\mathcal{W}}[{\bf
u}_{1}^{\gamma,\alpha},{\bf
u}_{1}^{\gamma,\beta}]e^{-z(\gamma,\lambda,\alpha)/\epsilon}{\bf
u}_{2}^{\alpha}+{\mathcal{W}}[{\bf u}_{1}^{\gamma,\beta},{\bf
u}_{2}^{\gamma}]e^{z(\gamma,\lambda,\alpha)/\epsilon}{\bf
u}_{1}^{\alpha}\right).$

After some calculations, the coefficient of ${\bf u}_{2}^{\alpha}$ in this
last expression is

${\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf
u}_{1}^{\beta}]e^{-z(\beta,\lambda,\alpha)/\epsilon}-\frac{{\mathcal{W}}[{\bf
u}_{2}^{\beta},{\bf u}_{0}^{\beta}]{\mathcal{W}}[{\bf
u}_{1}^{\gamma,\alpha},{\bf u}_{1}^{\gamma,\beta}]}{{\mathcal{W}}[{\bf
u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\alpha}]}e^{(z(\beta,\lambda,\gamma)-z(\gamma,\lambda,\alpha))/\epsilon}$

while the coefficient of ${\bf u}_{1}^{\alpha}$ is

$-\frac{{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]{\mathcal{W}}[{\bf u}_{1}^{\gamma,\beta},{\bf
u}_{2}^{\gamma}]}{{\mathcal{W}}[{\bf u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\alpha}]}e^{z(\beta,\lambda,\alpha)/\epsilon}.$

Comparing this with (2.42), we obtain the following theorem.

######  Theorem 2.20.

Let $\lambda_{0}$ be a point on the asymptotic spectral arcs such that
$C^{0}(\lambda_{0})$ consists of two Stokes lines; one from $\alpha_{0}$ to
$\gamma_{0}$ and another one from $\beta_{0}$ to $\gamma_{0}$ (for a simple
turning point $\gamma_{0}$; cf. Figure 4). There exists a neighborhood $U$ of
$\lambda_{0}$ such that $\lambda\in U$ is an eigenvalue of
$\mathfrak{D}_{\epsilon}$ if and only if the following condition holds

$m_{\alpha}(\epsilon)e^{2z(\gamma,\lambda,\alpha)/\epsilon}+m_{\beta}(\epsilon)e^{2z(\gamma,\lambda,\beta)/\epsilon}=1.$
(2.45)

Here the functions $m_{\alpha}(\epsilon)$ and $m_{\beta}(\epsilon)$ are given
by

$m_{\alpha}(\epsilon)=\frac{{\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf
u}_{1}^{\alpha}]{\mathcal{W}}[{\bf u}_{1}^{\gamma,\beta},{\bf
u}_{2}^{\gamma}]}{{\mathcal{W}}[{\bf u}_{2}^{\alpha},{\bf
u}_{0}^{\alpha}]{\mathcal{W}}[{\bf u}_{1}^{\gamma,\alpha},{\bf
u}_{1}^{\gamma,\beta}]},\quad m_{\beta}(\epsilon)=\frac{{\mathcal{W}}[{\bf
u}_{0}^{\beta},{\bf u}_{1}^{\beta}]{\mathcal{W}}[{\bf u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\alpha}]}{{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]{\mathcal{W}}[{\bf u}_{1}^{\gamma,\alpha},{\bf
u}_{1}^{\gamma,\beta}]}$

and as $\epsilon\downarrow 0$, they behave like

$m_{\alpha}(\epsilon)=\mp 1+{\mathcal{O}}(\epsilon),\quad
m_{\beta}(\epsilon)=\mp 1+{\mathcal{O}}(\epsilon)$

where the minus/plus sign for $m_{\alpha}(\epsilon)$ [resp.
$m_{\beta}(\epsilon)$] corresponds to the case where $\alpha$ (resp. $\beta$)
and $\gamma$ are zeros of the same/different $g_{+}$, $g_{-}$.

###### Proof.

For the second part, we have in fact modulo ${\mathcal{O}}(\epsilon)$,

${\mathcal{W}}[{\bf u}_{0}^{\alpha},{\bf
u}_{1}^{\alpha}]=2i,\quad{\mathcal{W}}[{\bf u}_{0}^{\beta},{\bf
u}_{1}^{\beta}]=2i,\quad{\mathcal{W}}[{\bf u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\alpha}]=2i,\quad{\mathcal{W}}[{\bf u}_{2}^{\gamma},{\bf
u}_{1}^{\gamma,\beta}]=2i,$ ${\mathcal{W}}[{\bf u}_{2}^{\alpha},{\bf
u}_{0}^{\alpha}]=\mp 2,\quad{\mathcal{W}}[{\bf u}_{2}^{\beta},{\bf
u}_{0}^{\beta}]=\pm 2,\quad{\mathcal{W}}[{\bf u}_{1}^{\gamma,\alpha},{\bf
u}_{1}^{\gamma,\beta}]=\pm 2,$

where the upper (resp. lower) sign corresponds to the case where the indicated
turning point is a zero of $g_{+}$ (resp. $g_{-}$). ∎

Let us investigate the quantization condition (2.45) for $\lambda$ close to
(but different from) $\lambda_{0}$. First, consider the actions

$\displaystyle z_{\alpha}(\lambda)$
$\displaystyle=z(\gamma(\lambda),\lambda,\alpha(\lambda))$ $\displaystyle
z_{\beta}(\lambda)$ $\displaystyle=z(\gamma(\lambda),\lambda,\beta(\lambda)).$

The configurations from I to VI of the Stokes lines in Figure 5 correspond to
$\lambda$ satisfying the following conditions on the actions $z_{\alpha}$,
$z_{\beta}$:

I:

$\Re z_{\alpha}(\lambda)=\Re z_{\beta}(\lambda)<0$,

II:

$\Re z_{\beta}(\lambda)<0=\Re z_{\alpha}(\lambda)$,

III:

$\Re z_{\beta}(\lambda)=0<\Re z_{\alpha}(\lambda)$,

IV:

$\Re z_{\alpha}(\lambda)=\Re z_{\beta}(\lambda)>0$,

V:

$\Re z_{\alpha}(\lambda)=0<\Re z_{\beta}(\lambda)$,

VI:

$\Re z_{\alpha}(\lambda)<0=\Re z_{\beta}(\lambda)$,

Each condition defines a curve in the $\lambda$-plane starting from
$\lambda_{0}$ (see Figure 6). For $\lambda$ not on any these curves, there is
no connection between any two turning points, and such $\lambda$ cannot be
o(1)-close to an eigenvalue for small enough $\epsilon$.

Figure 6. Real part of the actions $z_{\alpha}=z(\gamma,\lambda,\alpha)$ and
$z_{\beta}=z(\gamma,\lambda,\beta)$

Consider first conditions I, III or V. In fact, the right hand side of the
quantization condition (2.45) is exponentially small in case I and
exponentially large in cases III and V, and the quantization condition cannot
hold for such $\lambda$’s.

Finally, in the cases II, IV and VI, the quantization condition (2.45) reduces
to the usual one of Bohr-Sommerfeld type. We have

$\displaystyle\tilde{m}_{\alpha}(\epsilon)e^{2z(\gamma,\lambda,\alpha)/\epsilon}=1\quad\text{in
case II},$ (2.46)
$\displaystyle\tilde{m}_{\beta}(\epsilon)e^{2z(\gamma,\lambda,\beta)/\epsilon}=1\quad\text{in
case VI},$ (2.47)
$\displaystyle\tilde{m}_{\gamma}(\epsilon)e^{2z(\beta,\lambda,\alpha)/\epsilon}=1\quad\text{in
case IV},$ (2.48)

where, as $\epsilon\downarrow 0$,

$\displaystyle\tilde{m}_{\alpha}(\epsilon)=\frac{m_{\alpha}(\epsilon)}{1-m_{\beta}(\epsilon)e^{2z(\gamma,\lambda,\beta)/\epsilon}}=\delta(\alpha,\gamma)+{\mathcal{O}}(\epsilon),$
(2.49)
$\displaystyle\tilde{m}_{\beta}(\epsilon)=\frac{m_{\beta}(\epsilon)}{1-m_{\alpha}(\epsilon)e^{2z(\gamma,\lambda,\alpha)/\epsilon}}=\delta(\beta,\gamma)+{\mathcal{O}}(\epsilon),$
(2.50)
$\displaystyle\tilde{m}_{\gamma}(\epsilon)=-\frac{m_{\alpha}(\epsilon)}{m_{\beta}(\epsilon)-e^{2z(\gamma,\lambda,\beta)/\epsilon}}=\delta(\alpha,\beta)+{\mathcal{O}}(\epsilon).$
(2.51)

Notice that the above asymptotics holds a long as the absolute values of the
actions $z(\gamma,\lambda,\beta)$ in (2.49), $z(\gamma,\lambda,\alpha)$ in
(2.50), $z(\gamma,\lambda,\beta)$ in (2.51) are larger than
$c\epsilon\log(1/\epsilon)$ for some positive constant $c$, which means
$|\lambda-\lambda_{0}|\geq c^{\prime}\epsilon\log(1/\epsilon)$.

###  2.6. The Stokes geometry and quantization in a particular case

In this paragraph, we present the geometric configuration of the very
particular case of [22]

$A(x)=S(x)=\sech(2x),\quad x\in\mathbb{R}.$ (2.52)

Actually, there is nothing really special about this example; it just happens
that it was the first case studied numerically by Bronski in [1] and
subsequently by Miller in [22].

We begin with the turning points (see Definition 2.3) of our problem. For
fixed $\lambda\in\mathbb{C}$, these are the zeros (in the complex $x$-plane)
of [cf. (2.1)]

$-V_{0}(x,\lambda)=[\lambda-\tanh(2x)\sech(2x)]^{2}+\sech^{2}(2x).$

In the present case the potential $V_{0}(\cdot,\lambda)$ is periodic with
fundamental period $i\pi$. For this reason, we define the fundamental strip

$\mathcal{S}=\\{x\in\mathbb{C}|-\tfrac{\pi}{2}<\Im x\leq\tfrac{\pi}{2}\\}$
(2.53)

of the complex $x$-plane and restrict our attention only to what happens in
$\mathcal{S}$. For all $\lambda\in\mathbb{C}$, the potential
$V_{0}(\cdot,\lambda)$ has in $\mathcal{S}$ two fourth order $x$-poles located
at $x=\pm i\pi/4$. On the other hand, the zeros of $V_{0}(\cdot,\lambda)$ are
$\lambda$-dependent; there are precisely eight in $\mathcal{S}$ (counting
multiplicity).

Figure 7. A (qualitative) picture showing the turning point position with blue
& green circles (in the fundamental strip $\mathcal{S}$ on the $x$-complex
plane) for a purely imaginary $\lambda$ ($=0.2i$). The red diamonds denote the
poles of the potential.

When $\lambda$ is purely imaginary we have eight distinct (and therefore
simple) zeros in $\mathcal{S}$. There are four $x$-zeros that lie on the
imaginary axis, two between the poles and two outside. The remaining four
$x$-zeros (in the fundamental strip) make up the vertices of a rectangle. We
label the turning points in $\mathcal{S}$ as follows (the whole configuration
is depicted in Figure 7).

  1. (i)

The vertices of the rectangle

     * •

$x_{1}(\lambda)$ is the turning point with negative real part and the greatest
imaginary part.

     * •

$x_{2}(\lambda)=-x_{1}^{*}(\lambda)$.

     * •

$x_{3}(\lambda)$ is the turning point with the same real part to
$x_{1}(\lambda)$ and the smallest imaginary part.

     * •

$x_{4}(\lambda)=-x_{3}^{*}(\lambda)$.

  2. (ii)

The four purely imaginary turning points are labeled in order of increasing
imaginary part. More precisely

     * •

The turning point between $-i\pi/2$ and $-i\pi/4$ is $x_{5}(\lambda)$.

     * •

Between $-i\pi/4$ and $0$ lies $x_{6}(\lambda)$.

     * •

$x_{7}(\lambda)$ is to be found between $0$ and $i\frac{\pi}{4}$.

     * •

Finally, the turning point between $i\pi/4$ and $i\pi/2$ is $x_{8}(\lambda)$.

In Figure 7 we show the general qualitative picture which appears in the
actual numerics. The “indexing” process above provides us with an unambiguous
way for the labeling of the eight turning points $x_{j}(0.2i)$, $j=1,\dots 8$
in $\mathcal{S}$.

######  Remark 2.21.

Using this indexing, one can show that $x_{1}$, $x_{2}$, $x_{6}$ and $x_{8}$
are zeros of $g_{-}$ [see (2.10)] while $x_{3}$, $x_{4}$, $x_{5}$ and $x_{7}$
are zeros of $g_{+}$.

For $\lambda$ not purely imaginary, the nice symmetry of the complex turning
points breaks down. The turning points for non-imaginary values of $\lambda$
can be uniquely defined by analytically continuing the turning points
$x_{j}(0.2i)$, $j=1,\dots 8$ along the L-shaped path (see Figure 8) from
$0.2i$ to $\Re\lambda+0.2i$ (horizontally) and from $\Re\lambda+0.2i$ to
$\lambda$ (vertically).

Figure 8. The L-shaped analytic continuation in the $\lambda$-complex plane of
the turning points $x_{j}(0.2i)$, $j=1,\dots 8$ for a non-purely imaginary
$\lambda$.

For some $\lambda\in{\mathbb{C}}$, $V_{0}(\cdot,\lambda)$ has double zeros
which can be found by simply solving (with respect to $x$) simultaneously the
equations $V_{0}(x,\lambda)=\tfrac{d}{dx}V_{0}(x,\lambda)=0$. It is useful to
compute these double zeros since it turns out that they are related to the
end-points of the (semiclassical) asymptotic spectrum. One easily sees that
these double zeros are present only for four values of $\lambda$, namely

$\lambda_{D}=i\sigma\sqrt{\frac{1}{2}+\frac{1+i\tau\sqrt{7}}{8}}\Bigg{(}1-\frac{1+i\tau\sqrt{7}}{4}\Bigg{)}\quad\text{where}\quad\sigma,\tau=\pm
1$ (2.54)

and are given by the solutions of the transcendental equations 222Each one of
these four equations provides us with a double turning point in $\mathcal{S}$.

$\tanh(2x)=\frac{1+i\tau\sqrt{7}}{4i\sigma}\quad\text{where}\quad\sigma,\tau=\pm
1$

We adopt the notation $\lambda_{D}^{(j)}$, $j=1,\dots,4$, by requiring that
the point $\lambda_{D}^{(j)}$ is located in the $j^{\rm th}$ quadrant in the
complex $\lambda$-plane. Observe that the set of the four $\lambda_{D}$’s in
(2.54) can be written as $(\lambda,-\lambda,\lambda^{*},-\lambda^{*})$. These
four $\lambda_{D}$’s turn out to be -as we shall eventually discover
numerically- the endpoints of the four branches of a
${{\mathrel{\begin{subarray}{c}\Ydown\\\\[-2.87996pt]
\Yup\end{subarray}}}}$-shaped asymptotic spectrum in the complex
$\lambda$-plane, symmetric to both of the coordinate axes. The Y-shaped set
consisting of the points in the upper half-plane $\mathbb{H}^{+}$ is the union
of three arcs which intersect at a point $i\mu_{\otimes}$ (the numerical
experiments in [22] indicate that $\mu_{\otimes}\approx 0.28$).

We now proceed to investigate for which $\lambda\in{\mathbb{C}}$ there exists
an admissible contour (as defined in paragraph §2). For these values of
$\lambda$ the WKB analysis for locating the eigenvalues will be possible. For
each $j,k=1,\dots,8$ with $j\neq k$, we now define the complex-valued action
integrals

$I_{jk}(\lambda)=z(x_{k}(\lambda),\lambda,x_{j}(\lambda))=i\int_{C^{0}(x_{j}(\lambda)\rightsquigarrow
x_{k}(\lambda))}\sqrt{-V_{0}(t,\lambda)}dt.$ (2.55)

We construct well-defined branches for these functions by defining signs of
the square root for $\lambda=0.2i$ and applying analytic continuation along
the same L-shaped path as was used to define the turning points
$x_{j}(\lambda)$, $j=1,\dots,8$ (again refer to Figure 8). For reasons that
will be explained later, we restrict our attention only to $I_{12}$, $I_{16}$
and $I_{26}$.

Figure 9.  Candidates for the asymptotic spectral arcs on the $\lambda$-upper-
half-plane. The dashed curve in this figure (which is Fig. 3 in [22]) is the
so called real-turning-point curve (it is the set
$\\{\,\lambda\in\mathbb{C}\mid\exists\hskip 2.0ptr\in\mathbb{R}\hskip
2.0pt:\hskip 2.0ptV_{0}(r,\lambda)=0\,\\}$); it plays no role in our analysis.

We confine ourselves to the upper $\lambda$-half-plane, since the spectrum of
the Dirac operator in (2.1) is symmetric with respect to the real axis. The
three conditions $\Re[I_{12}(\lambda)]=0$, $\Re[I_{16}(\lambda)]=0$ and
$\Re[I_{26}(\lambda)]=0$ [recall (2.13)] yield three curves in the upper
$\lambda$-half-plane (see Figure 9). We label them as follows

$\displaystyle\Lambda_{12}$
$\displaystyle=\\{\,\lambda\in\mathbb{H}^{+}\mid\Re[I_{12}(\lambda)]=0\,\\},$
$\displaystyle\Lambda_{16}$
$\displaystyle=\\{\,\lambda\in\mathbb{H}^{+}\mid\Re[I_{16}(\lambda)]=0\,\\},$
$\displaystyle\Lambda_{26}$
$\displaystyle=\\{\,\lambda\in\mathbb{H}^{+}\mid\Re[I_{26}(\lambda)]=0\,\\}.$

Actually, the curve $\Lambda_{12}$ coincides with the upper imaginary semi-
axis while the curves $\Lambda_{16}$ and $\Lambda_{26}$ have negative and
positive slopes respectively. There is only one point of intersection in the
upper $\lambda$-half-plane, which lies on the imaginary axis and is denoted by
$\lambda_{\otimes}=i\mu_{\otimes}$. It is easily observed that
$\lambda_{D}^{(1)}$ belongs to the curve $\Lambda_{26}$ while
$\lambda_{D}^{(2)}=-\lambda_{D}^{(1)\hskip 2.0pt*}$ belongs to the curve
$\Lambda_{16}$.

Figure 10. The asymptotic spectral arcs on the $\lambda$-plane.

Our understanding is that [22] only considers $I_{12}$, $I_{16}$ and $I_{26}$
because the curves described by the rest of the equations
$\Re[I_{jk}(\lambda)]=0$ (as well as curves in the $\lambda$-plane arising
from equations corresponding to pairs of turning points in different period
strips) play no new role in the WKB analysis of the particular potentials
(2.52). More precisely, even though Stokes curves connecting the respective
turning points exist, there are no progressive paths connecting to
$\pm\infty$. For similar reasons one eventually eliminates portions of the
curves $\Lambda_{12}$, $\Lambda_{16}$ and $\Lambda_{26}$. Specifically, we
only consider (see Figure 10)

  * •

$\tilde{\Lambda}_{12}$, the part of ${\Lambda}_{12}$ that lies between $0$ and
$\lambda_{\otimes}$, endpoints included

  * •

$\tilde{\Lambda}_{26}$, which lies between $\lambda_{\otimes}$ and
$\lambda_{D}^{(1)}$, endpoints included

  * •

$\tilde{\Lambda}_{16}$, which lies between $\lambda_{\otimes}$ and
$\lambda_{D}^{(2)}$, endpoints included.

These are the asymptotic spectral arcs defined in Definition 2.9, in our
specific case (2.52). The union of these three arcs defines a Y-shaped set in
the upper $\lambda$-half-plane which is the part of the asymptotic spectrum
that lies on the upper-half-plane. It has been numerically observed in [22]
that the semiclassical spectrum observed by Bronski in [1] coincides with this
asymptotic spectrum computed here!

We must point out however, that we have a different argument why there are no
eigenvalues away from this union of asymptotic spectral arcs. The numerics of
[22] show that for $\lambda$ away from this union of asymptotic spectral arcs
there is always a progressive path from $-\infty$ to $+\infty$ which coincides
with the real axis for large $|x|$. From standard WKB theory these $\lambda$’s
cannot be eigenvalues 333See also Remark 2.24 at end of this section..

Figure 11. The behavior of Stokes lines near a pole in $x$-plane. This is Fig.
5 in [22].

We now examine carefully the geometry of the solutions of the differential
equation (2.12) in the complex $x$-plane. From each simple turning point
$x_{j}(\lambda)$ three orbits of (2.12) are emerging at angles of $2\pi/3$ (on
the other hand there is an infinite number of orbits meeting at each forth-
order pole; see Figure 11). These are Stokes curves of (2.12).

Figure 12. On the left, the $\lambda$-plane with a small square indicating
point $\lambda=0.2i\in\tilde{\Lambda}_{12}$, below the bifurcation point. On
the right, for that particular of $\lambda$, the fundamental strip
$\mathcal{S}$ in the $x$-plane, showing clearly the Stokes lines that emanate
from all turning points. This is Fig. 4 in [22].

For $\lambda=0.2i\in\tilde{\Lambda}_{12}$ (located below $\lambda_{\otimes}$
on the imaginary axis), the corresponding situation of the $x$-plane is shown
in Figure 12. We observe that there is a Stokes line $C^{0}$ connecting
$x_{1}(0.2i)$ and $x_{2}(0.2i)$. Furthermore, there exist progressive paths
$C^{-}$, $C^{+}$ joining $x_{1}(0.2i)$ to $-\infty$ and $x_{2}(0.2i)$ to
$+\infty$ respectively 444 Note here that while the Stokes line is uniquely
defined, there is some freedom in the choice of the two progressive paths..

Figure 13. On the left, the $\lambda$-plane, with a small square indicating a
point just below close the bifurcation point $\lambda_{\otimes}\approx 0.28i$.
On the right, for that particular value of $\lambda$, the strip $\mathcal{S}$
of the $x$-plane and the Stokes lines in $\mathcal{S}$ that emanate from all
turning points. Compare with case IV of Figure 5. This is Fig. 6 in [22].

For a value of $\lambda$ on the imaginary axis coming very close to the triple
intersection point $\lambda_{\otimes}$ (but still imaginary and just below
$\lambda_{\otimes}$), the situation on the $x$-plane still looks similar to
that we already examined (cf. Figure 13). The complex turning point
$x_{6}(\lambda)$ is moving up very close to the Stokes line that continues to
connect $x_{1}(\lambda)$ and $x_{2}(\lambda)$. This proximity becomes more
evident as $\lambda$ approaches the triple intersection point
$\lambda_{\otimes}$. The progressive paths emerging from $x_{1}(\lambda)$,
$x_{2}(\lambda)$ continue to exist.

Figure 14. On the left, the $\lambda$-plane in the extreme case when $\lambda$
is right at the bifurcation point $\lambda_{\otimes}$. On the right, the
$x$-plane with the Stokes lines in $\mathcal{S}$. Compare with Figure 4 or the
central picture of Figure 5. This is Fig. 7 in [22].

When $\lambda=\lambda_{\otimes}$ (see Figure 14) there is no Stokes line
connecting $x_{1}(\lambda_{\otimes})$ and $x_{2}(\lambda_{\otimes})$ without
passing through another turning point. Parts of the two paths previously
connecting $x_{6}(\lambda_{\otimes})$ to $x_{5}(\lambda_{\otimes})$ have now
partly coalesced with the path connecting $x_{1}(\lambda_{\otimes})$ and
$x_{2}(\lambda_{\otimes})$. Hence $x_{1}(\lambda_{\otimes})$ and
$x_{2}(\lambda_{\otimes})$ are now connected to $x_{6}(\lambda_{\otimes})$ via
Stokes lines. But the remaining Stokes line from $x_{6}(\lambda_{\otimes})$
continues to pass through the pole at $-i\pi/4$. The three conditions
$\Re[I_{12}(\lambda_{\otimes})]=0$, $\Re[I_{26}(\lambda_{\otimes})]=0$,
$\Re[I_{16}(\lambda_{\otimes})]=0$ are satisfied simultaneously and the
corresponding Stokes lines meet at $x_{6}(\lambda_{\otimes})$ and at a
$2\pi/3$ angle from each other. For this special value of $\lambda$, there is
a connected sequence of two Stokes lines and the progressive paths continue to
exist.

Figure 15. Progressive paths (dashed curves) in the $x$-plane when $\lambda$
is slightly off the asymptotic spectrum; cf. Figure 5, cases I, III, V. Figure
16. The $\lambda$-plane on the left, for a specific $\lambda\in\Lambda_{12}$
just above the bifurcation point $\lambda_{\otimes}$. On the right, the
$x$-plane with the Stokes lines in $\mathcal{S}$. Compare with case I of
Figure 5. Note that one can draw a progressive path from $-\infty$ to
$+\infty$ which passes above $x_{1}$, just below $x_{6}$ and above $x_{2}$
(see Figure 15, top). Hence there is no eigenvalue semiclassically. This is
Fig. 8 in [22].

Having completed the description of what happens when
$\lambda\in\tilde{\Lambda}_{12}$, there are five more remaining directions in
the complex $\lambda$-plane all meeting at the triple intersection point
$\lambda_{\otimes}$ (cf. Figure 10).

Figure 17. The $\lambda$-plane on the left, for a particular
$\lambda\in\Lambda_{26}$ just below $\lambda_{\otimes}\approx 0.28i$. On the
right, the $x$-plane with the Stokes lines in $\mathcal{S}$. Compare with case
III of Figure 5. Note that one can draw a progressive path from $-\infty$ to
$+\infty$ which passes below $x_{1}$, just above $x_{6}$ and above $x_{2}$
(see Figure 15, bottom left). Hence there is no eigenvalue semiclassically.
This is Fig. 9 in [22].

Now suppose we continue along the imaginary axis above $\lambda_{\otimes}$
(see Figure 16). There is no more admissible connection between the turning
points $x_{1}(\lambda)$, $x_{2}(\lambda)$. There are paths emanating from
$x_{1}(\lambda)$, $x_{2}(\lambda)$ (see Definition 2.7) meeting at the double
pole of $\sqrt{-V_{0}(\cdot,\lambda)}$ so the holomorphic deformation is
impossible, requiring us to restrict the WKB analysis described so far only to
the section $\tilde{\Lambda}_{12}$. Note the existence of a progressive path
from $-\infty$ to $+\infty$ which passes above $x_{1}$, just below $x_{6}$ and
above $x_{2}$ (see dashed curve at the top of Figure 15). Hence there is no
eigenvalue semiclassically.

Figure 18. The $\lambda$-plane on the left, for a particular
$\lambda\in\tilde{\Lambda}_{26}$ just above $\lambda_{\otimes}\approx 0.28i$.
On the right, the $x$-plane with the Stokes lines in $\mathcal{S}$. Compare
with case VI of Figure 5. This is Fig. 10 in [22].

Same observations for different pairs of turning points show that the WKB
analysis only works for $I_{12}$, $I_{16}$ and $I_{26}$ (and indeed only for
the sections $\tilde{\Lambda}_{12}$, $\tilde{\Lambda}_{26}$ and
$\tilde{\Lambda}_{16}$). One can examine the rest of the cases by checking
Figures 17 through 21.

Figure 19. The $\lambda$-plane on the left, for a particular
$\lambda\in\tilde{\Lambda}_{26}$ with $\Im\lambda$ greater than that shown in
Figure 18 but less than $\Im\lambda_{D}^{(1)}$. On the right, the $x$-plane
with the Stokes lines in $\mathcal{S}$. This is Fig. 11 in [22].

Finally we give the quantization condition of eigenvalues along
$\tilde{\Lambda}_{12}$, $\tilde{\Lambda}_{26}$ and $\tilde{\Lambda}_{16}$ away
from the bifurcation point $\lambda_{\otimes}$, and in a neighborhood of
$\lambda_{\otimes}$ using the results of the previous subsections.

Figure 20. On the left, the $\lambda$-plane for a specific
$\lambda\in\tilde{\Lambda}_{26}$ just below $\lambda_{D}^{(1)}$. On the right,
the $x$-plane with the Stokes lines in $\mathcal{S}$. This is Fig. 12 in [22].

######  Theorem 2.22.

Let $(j,k)$ be $(1,2),(2,6)$ or $(1,6)$. For any $\epsilon$-independent
$\delta>0$ and $\lambda_{0}\neq 0$ on $\tilde{\Lambda}_{jk}$, fixed and
independent of $\epsilon$, satisfying
$|\lambda_{0}-\lambda_{\otimes}|>\delta$, there exist a complex neighborhood
$U$ of $\lambda_{0}$ and a function $m_{jk}(\epsilon)$ with a uniform
asymptotic behavior $m_{jk}(\epsilon)=-1+{\mathcal{O}}(\epsilon)$ in $U$, such
that $\lambda\in U$ is an eigenvalue of $\mathfrak{D}_{\epsilon}$ if and only
if

$m_{jk}(\epsilon)e^{2z(\beta,\lambda,\alpha)/\epsilon}=1.$

Figure 21. On the left, the $\lambda$-plane for a specific
$\lambda\in\Lambda_{26}$ just above $\lambda_{D}^{(1)}$. On the right, the
$x$-plane with the Stokes lines in $\mathcal{S}$. Note that one can draw a
progressive path that is a deformation of the real line in such a way that it
crosses the Stokes line from $x_{2}$ to $x_{6}$ transversally. Hence there is
no eigenvalue semiclassically. This is Fig. 13 in [22].

######  Theorem 2.23.

There exists a neighborhood $U$ of $\lambda_{\otimes}$ and functions
$m_{\alpha}(\epsilon)$, $m_{\beta}(\epsilon)$ with uniform asymptotic
behaviors $m_{\alpha}(\epsilon)=-1+{\mathcal{O}}(\epsilon)$,
$m_{\beta}(\epsilon)=-1+{\mathcal{O}}(\epsilon)$ in $U$, such that $\lambda\in
U$ is an eigenvalue of $\mathfrak{D}_{\epsilon}$ if and only if

$m_{\alpha}(\epsilon)e^{2z(\gamma,\lambda,\alpha)/\epsilon}+m_{\beta}(\epsilon)e^{2z(\gamma,\lambda,\beta)/\epsilon}=1.$
(2.56)

######  Remark 2.24.

We claim that no eigenvalue can appear away from the asymptotic spectral arcs.
Although we do not present a rigorous proof of this fact, it follows from the
numerical investigation of the geometry of turning points and Stokes lines for
different values of $\lambda$. For example, Figure 16, Figure 17 and Figure 21
show that for $\lambda$ slightly off the asymptotic spectral arcs, a
progressive path from $x=-\infty$ to $x=+\infty$ does exist, which is in fact
real near $x=\pm\infty$. It follows easily from standard WKB theory that such
$\lambda$ cannot be eigenvalues. Indeed, in view of part (ii) of Theorem 2.14,
and the fact that $S$ is real near $x=\pm\infty$, the important factor in
formula (2.25) for ${\bf u}^{+}$ that determines the behavior at
$x=\pm\infty$, is the exponential $\exp(z/\epsilon)$. If $\Re z$ is monotonic
along a path from $-\infty$ to $+\infty$ then it is impossible that the
solution decays at both ends.

######  Remark 2.25.

The question arises whether there is a possibility of a general proof of the
fact that no eigenvalue can appear away from the asymptotic spectral arcs. A
consequence of this would be a general proof that the limiting spectrum (the
minimal set such that every actual eigenvalue lies $\epsilon$-close to this
set) would be what we have called the asymptotic spectrum (which we defined in
terms of the existence of an admissible path). In turn that would prove that
the limiting spectrum is a union of analytic arcs.

A possible strategy is suggested by an analogous problem appearing in the
semiclassical analysis of the inverse scattering problem in [19] where a
crucial ingredient of the proof is the existence of appropriate steepest
descent paths. The existence proof in [20] requires the study of an associated
max-min variational problem.

In our own problem here, we can define a generalized admissible contour as a
contour that is either admissible in the sense of Definition 2.13 (extended to
allow for double turning points as degenerate cases) or a progressive path
from $x=-\infty$ to $x=+\infty$. The inequalities defining progressiveness and
the equations defining Stokes lines can be interpreted as variational Euler-
Lagrange conditions for a max-min problem. We expect that a proof along the
lines of [20] will show the existence of the desired contour which is either a
progressive path from $x=-\infty$ to $x=+\infty$ (hence no eigenvalue) or an
admissible contour in the sense of Definition 2.13 (hence an asymptotic
spectral arc).

##  3\. Eigenvalues That Lie Near Zero

Note: In this section $A(x)=S(x)=\sech(2x)$ for simplicity. But the theory
applies very generally.

###  3.1. Preparations

We now turn our attention to the semiclassical behavior of eigenvalues (and
their corresponding norming constants) of the Zakharov-Shabat operator that
lie near zero. We will focus on a neighborhood $\mathscr{D}$ of the real axis
(to be appropriately specified later on) and start with the problem

$\mathfrak{D}_{\epsilon}\mathbf{u}(x)=\lambda\mathbf{u}(x),\quad
x\in\mathscr{D}$

where

  * •

the Dirac (or Zakharov-Shabat) operator is defined by

$\mathfrak{D}_{\epsilon}=\begin{bmatrix}-\frac{\epsilon}{i}\frac{d}{dx}&-iA(x)\exp\\{iA(x)/\epsilon\\}\\\
-iA(x)\exp\\{-iA(x)/\epsilon\\}&\frac{\epsilon}{i}\frac{d}{dx}\end{bmatrix}$

  * •

$\epsilon>0$ is the semiclassical parameter,

  * •

$\mathbf{u}=[u_{1}\hskip 3.0ptu_{2}]^{T}$ is a function from $\mathscr{D}$ to
$\mathbb{C}^{2}$ and

  * •

$\lambda\in{\mathbb{C}}$ represents the spectral parameter.

First, we apply the change of variables [cf. equation (4) in [22]]

$w_{\pm}=\frac{u_{2}\exp\\{iA/(2\epsilon)\\}\pm
u_{1}\exp\\{-iA/(2\epsilon)\\}}{\sqrt{A\pm i(\lambda+A^{\prime}/2)}}.$ (3.1)

Then for the lower sign (minus), the equation reads

$w_{-}^{\prime\prime}=\Bigg{[}-\epsilon^{-2}\Big{\\{}A(x)^{2}+[\lambda+\tfrac{A^{\prime}(x)}{2}]^{2}\Big{\\}}\\\
+\frac{3}{4}\bigg{\\{}\frac{A^{\prime}(x)-i\frac{A^{\prime\prime}(x)}{2}}{A(x)-i[\lambda+\tfrac{A^{\prime}(x)}{2}]}\bigg{\\}}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)-i\frac{A^{\prime\prime\prime}(x)}{2}}{A(x)-i[\lambda+\tfrac{A^{\prime}(x)}{2}]}\Bigg{]}w_{-}$
(3.2)

where we dropped the dependence of $w_{-}$ on $(x,\lambda,\epsilon)$ for
notational simplicity and let prime denote differentiation with respect to
$x$. Hence, dropping the subscript, and considering only the upper half
$\lambda$-plane due to symmetry, we are led to the following proposition.

######  Proposition 3.1.

Finding the discrete spectrum of $\mathfrak{D}_{\epsilon}$ is equivalent to
finding the values of $\lambda\in\mathbb{H}^{+}$ for which equation

$\frac{d^{2}w}{dx^{2}}=[\epsilon^{-2}V_{0}(x,\lambda)+g(x,\lambda)]w$ (3.3)

has an $L^{2}(\mathscr{D})$ solution. In this equation, the leading order
potential $V_{0}(x,\lambda)$ satisfies

$V_{0}(x,\lambda)=-\Big{\\{}A(x)^{2}+[\lambda+\tfrac{A^{\prime}(x)}{2}]^{2}\Big{\\}}$
(3.4)

while the correction $g(x,\lambda)$ is given by

$g(x,\lambda)=\frac{3}{4}\bigg{\\{}\frac{A^{\prime}(x)-i\frac{A^{\prime\prime}(x)}{2}}{A(x)-i[\lambda+\tfrac{A^{\prime}(x)}{2}]}\bigg{\\}}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)-i\frac{A^{\prime\prime\prime}(x)}{2}}{A(x)-i[\lambda+\tfrac{A^{\prime}(x)}{2}]}.$
(3.5)

In this section, we are only interested in the behavior of the eigenvalues
that lie near zero. So, let us first fix some set $\mathscr{O}$ on the
spectral plane that satisfies the following condition.

(H2):

The set $\mathscr{O}$ is an open and simply connected set, proper subset of
${\mathcal{R}}_{0}\cap\mathbb{H}^{+}$ [recall the formula for the mumerical
range (2.4) and of course here $S(x)=A(x)=\sech(2x)$] and located in such a
way that its intersection with $\tilde{\Lambda}_{12}$ (see Figure 10) is a
single interval and has no common points with either $\tilde{\Lambda}_{16}$ or
$\tilde{\Lambda}_{26}$.

From now on, we shall only consider eigenvalues $\lambda$ that lie in a set
$\mathscr{O}$ satisfying (H2).

###  3.2. Solutions in a neighborhood of only one simple turning point

In this subsection we shall confine the study of the differential equation
(3.3) to some domain in which it has only one simple turning point. Let us be
more precise.

######  Assumption 3.2.

For $\epsilon>0$, consider the differential equation

$\frac{d^{2}w}{dx^{2}}=[\epsilon^{-2}V_{0}(x,\lambda)+g(x,\lambda)]w,\quad(x,\lambda)\in\mathscr{D}\times\mathscr{O}$

where $V_{0}(x,\lambda)$ is given by (3.4) and $g(x,\lambda)$ by (3.5) for
$A(x)=\sech(2x)$. We assume that the following hold true.

  * •

We denote by $c=c(\lambda)$, one of the following (simple) turning points:
$x_{1}$, $x_{2}$, $x_{6}$ or $x_{8}$ (cf. §2.6) which are (simple) zeros of
$g_{-}$ [see (2.10), recalling that $V_{0}=g_{-}g_{+}$].

  * •

The set $\mathscr{D}\subset{\mathbb{C}}$ (in the $x$-plane) is an open and
simply connected neighborhood of the real axis so that $c$ is an interior
point and the only transition point (of the equation) in it.

  * •

The set $\mathscr{O}$ satisfies condition $\mathbf{(H2})$.

######  Remark 3.3.

Clearly, we can always choose $\mathscr{D}$ and $\mathscr{O}$ so that the
assumption above holds.

######  Remark 3.4.

From the assumptions above, it follows that

  * •

the function $(x-c)^{-1}V_{0}(x,\lambda)$ is holomorphic and non-vanishing
throughout $\mathscr{D}$ (including $c$).

  * •

the function $g(x,\lambda)$ is holomorphic in $\mathscr{D}$.

Having placed the assumptions on our equation, we continue as follows. We
introduce a new variable
$\zeta=\zeta(\cdot,\lambda):\mathscr{D}\rightarrow{\mathbb{C}}$ defined by

$\zeta=\bigg{(}\int_{\gamma(c\rightsquigarrow
x)}V_{0}^{1/2}(t,\lambda)dt\bigg{)}^{2/3}.$ (3.6)

In order to choose the branch, we proceed as follows. We expand
$(t-c)^{-1}V_{0}(t,\lambda)$ in a Taylor-series in the neighborhood of $c$,
namely

$(t-c)^{-1}V_{0}(t,\lambda)=f_{0}+f_{1}(t-c)+f_{2}(t-c)^{2}+\dots$

for a sequence $\\{f_{n}\\}_{n\in{\mathbb{N}}_{0}}$ of complex numbers
(depending on $\lambda$) where $f_{0}\neq 0$. Substituting for $V_{0}$ in
(3.6) and integrating term by term, we find that $\zeta$ has a Taylor-series
expansion (in the neighborhood of $c$) that begins with

$\zeta=\Big{(}\frac{2}{3}\Big{)}^{2/3}f_{0}^{1/3}(x-c)\Big{[}1+\frac{1}{5}\frac{f_{1}}{f_{0}}(x-c)+\dots\Big{]}$
(3.7)

We select any branch of the coefficient $f_{0}^{1/3}$ that is convenient to
us, and this fixes the relation between $\zeta$ and $x$ in the neighborhood of
$c$; elsewhere, $\zeta$ is defined by continuity.

Next, we define $\Delta=\zeta(\mathscr{D},\lambda)$. We suppose that
$\mathscr{D}$ is restricted in such a way that the mapping
$\zeta(\cdot,\lambda):\mathscr{D}\rightarrow\Delta$ is one-to-one [it is
already onto; of course this implies that $c$ is the only point in
$\mathscr{D}$ at which $\zeta(\cdot,\lambda)$ vanishes]. Hence, it follows
that $\zeta(\cdot,\lambda)\in\mathscr{H}(\mathscr{D})$ and therefore the
$\zeta(\cdot,\lambda)$-mapping from $\mathscr{D}$ on to $\Delta$ is conformal.
Also, we define the following.

######  Definition 3.5.

Let $x(\cdot,\lambda)$ be the inverse of $\zeta(\cdot,\lambda)$ and $S_{j}$,
$j=0,1,2$ be the sectors defined by

$(2j-1)\frac{\pi}{3}\leq\mathrm{ph}\zeta\leq(2j+1)\frac{\pi}{3}.$

The sets

$\mathscr{D}_{j}=x(\Delta\cap S_{j},\lambda),\quad\text{for}\quad j=0,1,2$

shall be called the principal regions associated with the transition point
$c$.

Furthermore, we define

$\hat{f}(x,\lambda)\equiv\frac{4}{9}\frac{V_{0}(x,\lambda)}{\zeta(x,\lambda)}=\Big{[}\frac{d\zeta}{dx}(x,\lambda)\Big{]}^{2}$
(3.8)

where it is clear that $\hat{f}(\cdot,\lambda)$ is holomorphic and non-
vanishing in $\mathscr{D}$. Subsequently, we introduce the notion of a
balancing function.

######  Definition 3.6.

Consider a function $\Omega(t)$ which is any conveniently chosen positive
function of the complex variable $t$ that is continuous and satisfies the
asymptotics

$\Omega(t)=\mathcal{O}(t^{1/2})\quad\text{uniformly in the neighborhood
of}\quad t=\infty.$

Such a function shall be called a balancing function, and for $j,k=0,1,2$ with
$j\neq k$ we set

$\rho_{j,k}=\sup_{t\in S_{j}\cup S_{k}}\\{\Omega(t)\mathsf{M}_{j,k}^{2}(t)\\}$
(3.9)

where the $\mathsf{M}_{j,k}$ are the auxiliary functions found in appendix A
(they are related to the modified Bessel functions).

######  Remark 3.7.

Using the bounds we have for $\mathsf{M}_{j,k}(t)$, the fact that
$\mathsf{M}_{j,k}(t)$ is continuous in $S_{j}\cup S_{k}$ and the asymptotics
for the balancing function, it is easy to realize that the quantities
$\rho_{j,k}$ are always finite.

We fix a way to measure the error of the approximation by setting an error-
control function.

######  Definition 3.8.

We define an error-control function $H(x,\lambda,\epsilon)$ of equation (3.3)
to be any primitive of

$\frac{1}{\Omega(\epsilon^{-2/3}\zeta)}\bigg{\\{}\frac{1}{\hat{f}^{1/4}(x,\lambda)}\frac{d^{2}}{dx^{2}}\Big{[}\frac{1}{\hat{f}^{1/4}(x,\lambda)}\Big{]}-\frac{g(x,\lambda)}{\hat{f}^{1/2}(x,\lambda)}\bigg{\\}}$

where for $\hat{f}^{1/4}$ and $\hat{f}^{1/2}$ we choose any continuous
branches in $\mathscr{D}$ such that $\hat{f}^{1/2}=(\hat{f}^{1/4})^{2}$. Using
(3.8) and $V_{0}$ instead of $\hat{f}$ the above becomes

$\frac{3}{2}\frac{\zeta^{1/2}}{\Omega(\epsilon^{-2/3}\zeta)}\bigg{\\{}\frac{1}{V_{0}^{1/4}(x,\lambda)}\frac{d^{2}}{dx^{2}}\Big{[}\frac{1}{V_{0}^{1/4}(x,\lambda)}\Big{]}-\frac{g(x,\lambda)}{V_{0}^{1/2}(x,\lambda)}\bigg{\\}}-\frac{5}{16}\frac{1}{\zeta^{2}\Omega(\epsilon^{-2/3}\zeta)}.$

Finally, before stating the theorem about the solutions of (3.3), let us fix
some complementary notation.

######  Definition 3.9.

Take two points $x_{j}\in\textsf{clos}(\mathscr{D}_{j})$ and
$y_{k}\in\textsf{clos}(\mathscr{D}_{k})$ where $j,k\in\\{0,1,2\\}$. A path
$\mathscr{P}:\gamma(x_{j}\rightsquigarrow y_{k})$ [meaning a Jordan arc
comprising of a finite chain of $R_{2}$ arcs; where an $R_{2}$ arc is an arc
$\sigma=\sigma(s)$ ($s$ being the arc parameter) such that
$\sigma^{\prime\prime}(s)$ is continuous and $\sigma^{\prime}(s)$ does not
vanish], shall be called a progressive path joining $x_{j},y_{k}$ if it
satisfies the following “monotonicity condition”: as $x$ travels along
$\mathscr{P}$ from $x_{j}$ to $y_{k}$, the function

$\Re\int_{\gamma(x_{j}\rightsquigarrow x)}V_{0}^{1/2}(t,\lambda)dt$ (3.10)

is non-decreasing 555This definition is somewhat different from that of the
previous section where we impose that the same expression is strictly
increasing for a path to be progressive.. We choose the branch of the integral
in (3.10) so that the whole function is continuous, staying non-positive in
$\mathscr{D}_{j}$ while non-negative in $\mathscr{D}_{k}$. Furthermore, we
define

$\mathsf{H}_{k}(x_{j})=\\{\,x\in\mathscr{D}_{j}\cup\mathscr{D}_{k}\mid\exists\hskip
3.0pt\text{a progressive path}\hskip 3.0pt\gamma(x_{j}\rightsquigarrow
x)\subset\mathscr{D}_{j}\cup\mathscr{D}_{k}\,\\}.$

######  Definition 3.10.

Let $\mathscr{P}:\gamma(a\rightsquigarrow b)$ be a progressive path between
the points $a$ and $b$. We define the variation of the error-control function
$H$ along $\mathscr{P}$ to be

$\mathcal{V}_{\mathscr{P}}[H](\lambda,\epsilon)\equiv\mathcal{V}_{a,b}[H](\lambda,\epsilon)\\\
=\int_{\gamma(a\rightsquigarrow
b)}\bigg{|}\frac{1}{\Omega(\epsilon^{-2/3}\zeta)}\bigg{\\{}\frac{1}{\hat{f}^{1/4}(t,\lambda)}\frac{d^{2}}{dt^{2}}\Big{[}\frac{1}{\hat{f}^{1/4}(t,\lambda)}\Big{]}-\frac{g(t,\lambda)}{\hat{f}^{1/2}(t,\lambda)}\bigg{\\}}\bigg{|}dt.$

All these, lead us to the next important theorem about the solutions to
equation (3.3).

######  Theorem 3.11.

Consider Assumption 3.2, let $a,b\in{\mathbb{C}}$ be arbitrary and choose a
reference point $a_{j}\in\textsf{clos}(\mathscr{D}_{j})$, $j=0,1,2$ (the
reference point could be $\infty$). Also, for $k\in\\{0,1,2\\}$ such that
$k\neq j$, consider the set $\mathsf{H}_{k}(a_{j})$ of points in
$\mathscr{D}_{j}\cup\mathscr{D}_{k}$ that can be joined to $a_{j}$ with a
progressive path $\mathscr{P}$ in $\mathscr{D}_{j}\cup\mathscr{D}_{k}$ [with
the understanding that if $\zeta(a_{j})$ is at infinity, then we suppose that
it is the point at infinity on a path $P$ in $S_{j}$ and require
$\zeta(\mathscr{P})$ to coincide with $P$ in a neighborhood of
$\zeta(a_{j})$]. Then for every $\epsilon>0$, equation (3.3), namely

$\frac{d^{2}w}{dx^{2}}=[\epsilon^{-2}V_{0}(x,\lambda)+g(x,\lambda)]w$

with $V_{0},g$ given by (3.4) and (3.5) respectively, has a solution
$w(x,\lambda,\epsilon)$ (depending of course on $a,b$ and $a_{j}$) that is
holomorphic in $\mathscr{D}\setminus\\{c\\}$ and continuous at $c$.
Furthermore, when $x\in\mathsf{H}_{k}(a_{j})$ we have

$w(x,\lambda,\epsilon)=\hat{f}^{-1/4}(x,\lambda)\big{[}aU_{j}(\epsilon^{-2/3}\zeta)+bU_{k}(\epsilon^{-2/3}\zeta)+R(\zeta,\lambda,\epsilon)\big{]}$

where $U_{j}$, $U_{k}$ are the MBF defined in appendix A. The remainder $R$
satisfies the following precise estimates

$\frac{|R(\zeta,\lambda,\epsilon)|}{\mathsf{M}_{j,k}(\epsilon^{-2/3}\zeta)},\frac{\Big{|}\frac{\partial
R}{\partial\zeta}(\zeta,\lambda,\epsilon)\Big{|}}{\epsilon^{-2/3}\mathsf{N}_{j,k}(\epsilon^{-2/3}\zeta)},\frac{\Big{|}\frac{\partial[\zeta^{1/4}R(\zeta,\lambda,\epsilon)]}{\partial\zeta}\Big{|}}{\epsilon^{-2/3}|\zeta|^{1/4}\hat{\mathsf{N}}_{j,k}(\epsilon^{-2/3}\zeta)}\\\
\leq\frac{\sigma_{j,k}}{\rho_{j,k}}\mathsf{E}_{j,k}(\epsilon^{-2/3}\zeta)\bigg{(}\exp\Big{\\{}\frac{\rho_{j,k}}{3|\lambda_{j,k}|}\epsilon^{2/3}\mathcal{V}_{a_{j},x}[H](x,\lambda,\epsilon)\Big{\\}}-1\bigg{)}$
(3.11)

where the variation of $H$ is being evaluated along $\mathscr{P}$ (see
Definition 3.10), the $\rho_{j,k}$ are given by (3.9) and the quantities
$\sigma_{j,k},\lambda_{j,k}$ satisfy

$\sigma_{j,k}=\sup_{p\in\zeta(\mathscr{P})}\bigg{\\{}\frac{\Omega(\epsilon^{-2/3}p)\mathsf{M}_{j,k}(\epsilon^{-2/3}p)}{\mathsf{E}_{j,k}(\epsilon^{-2/3}p)}\big{|}aU_{j}(\epsilon^{-2/3}p)+bU_{k}(\epsilon^{-2/3}p)\big{|}\bigg{\\}}$
(3.12)

and

$\lambda_{j,k}=\frac{\sin\Big{[}(k-j)\frac{\pi}{3}\Big{]}}{\sin\frac{\pi}{3}}$
(3.13)

respectively. When $x\in\mathscr{D}_{k}$, the bounds (3.11) for the error,
apply only to the branch of $w$ obtained by analytic continuation from a
neighborhood of $c$ in $\mathscr{D}_{j}$ by rotation through an angle
$2(k-j)\frac{\pi}{3}$.

###### Proof.

For the proof see §§3.3-3.6 in Olver’s [24] with $m=3$. The statement of the
theorem can be found in §3.3 as Theorem 3.1. ∎

######  Remark 3.12.

For the theorem to be meaningful, it is necessary that the right-hand side of
(3.11) is finite. Recall that $\rho_{j,k}$ are finite (cf. Remark 3.7) and
that in our case $1/|\lambda_{j,k}|$ are finite as well [see (3.13) and recall
that $(k-j)\neq 0\hskip 2.0pt{\rm mod}3$]. But $\mathcal{V}_{\mathscr{P}}[H]$
has to converge (the variation of $H$ is calculated along $\mathscr{P}$) and
also the $\sigma_{j,k}$ have to be finite. It is seen that the quantity in
brackets in (3.12) is finite at all finite points $p\in S_{j}\cup S_{k}$ and
bounded as $p\rightarrow\infty$ in $S_{k}$. As $p\rightarrow\infty$ in an
internal part of $S_{j}$ (see appendix A) however, the quantity in brackets is
unbounded, unless $b=0$. Accordingly, if $\zeta(a_{j})=\infty$ in an internal
part of $S_{j}$, then the theorem may be applied only in the case $b=0$.

######  Remark 3.13.

In order to deal with the behavior near the remaining (simple) turning points,
namely $x_{3}$, $x_{4}$, $x_{5}$ and $x_{7}$, instead of the differential
equation for $w_{-}$, we consider the differential equation for $w_{+}$ [see
(3.1)] and apply exactly the same strategy like the one just presented.

###  3.3. Asymptotic estimates for the error terms

Let us now consider a circumstance in which the error-term $R$ is sufficiently
small for the theorem to supply meaningful approximations when $\epsilon$ is
small. But let us first define the function

$F(x,\lambda)=\int\bigg{\\{}\frac{1}{V_{0}^{1/4}(x,\lambda)}\frac{d^{2}}{dx^{2}}\Big{[}\frac{1}{V_{0}^{1/4}(x,\lambda)}\Big{]}-\frac{g(x,\lambda)}{V_{0}^{1/2}(x,\lambda)}\bigg{\\}}dx$
(3.14)

where for this integral we shall not use paths that intersect $c$, while for
$V_{0}^{1/4},V_{0}^{1/2}$ we adopt any branches, provided they are continuous
and the latter is the square of the former. With this in mind, we have the
following result.

######  Theorem 3.14.

Let Assumption 3.2 be satisfied (observe that $V_{0},g$ are independent of
$\epsilon$), consider $a_{j},a_{k}$ to be arbitrary points in
$\textsf{clos}(\mathscr{D}_{j})$ and $\textsf{clos}(\mathscr{D}_{k})$
respectively (including the point at infinity) and let $\mathscr{P}$ be a
progressive path joining $a_{j}$ and $a_{k}$. Then as $\epsilon\downarrow 0$,
we have

$\mathcal{V}_{\mathscr{P}}[H](\lambda,\epsilon)=\mathcal{O}(\epsilon^{1/3}).$

###### Proof.

Referring to Theorem 4.1(ii) when $m=3$ in [24], one only needs to check that

  * •

the function $1/\zeta(x)$ [refer to (3.6)] and

  * •

the function $F(x,\lambda)$ in (3.14)

are of bounded variation as $x\rightarrow a_{j}$ or $a_{k}$ along
$\mathscr{P}$. ∎

###  3.4. The connection theorem

In this section, we will state a theorem that concerns the connection of
solutions of our equation between two principal regions near a turning point.
But first, we reformulate the geometric concepts introduced in §3.2 using a
new variable $\xi_{c}$, in order to avoid referring explicitly to the variable
$\zeta$. We begin with a definition (see Definition 2.6).

######  Definition 3.15.

The curves in the complex $x$-plane having the equation

$\Re\int_{\gamma(c\rightsquigarrow x)}V_{0}^{1/2}(t,\lambda)dt=0$

are called the Stokes curves associated with the turning point $c$ 666 Olver
actually calls them anti-Stokes curves or principal curves.. Either branch may
be used for the square root of $V_{0}$, provided continuity is maintained.

Since $c$ is a simple turning point of equation (3.3), there are three
distinct Stokes curves intersecting at $c$ at an angle of $2\pi/3$. All of
them occupy the same Riemann sheet. Using the conformality of the mapping one
can prove that: (i) a Stokes curve can either terminate at $c$ or at a
boundary point of $\mathscr{D}$, and (ii) that no Stokes curve intersects
itself or any other Stokes curve on the same Riemann sheet (except c).

Evidently, the Stokes curves are boundaries of principal regions (each
principal region includes its bounding Stokes curves). From here on, we
suppose that in each principal region there is a one-to-one relation between
$x$ and any continuous branch of the integral $\int V_{0}^{1/2}(t,\lambda)dt$
(which is equivalent to the fact that the mapping from $\mathscr{D}$ to
$\Delta$ is one-to-one).

We start by fixing an arbitrarily chosen principal region and name it
$\mathscr{D}_{0}$. We then define $\mathscr{D}_{1}$ to be the successive
principal domain encountered as we pass around $c$ in a counterclockwise
direction and similarly set $\mathscr{D}_{2}$ be the successive principal
region (to $\mathscr{D}_{0}$) encountered as we pass around $c$ in a clockwise
direction. Whatever choice we make for $\mathscr{D}_{0}$, the labeling can be
made consistent with that chosen before, by appropriately choosing the branch
of $f_{0}^{1/3}$ in (3.7).

######  Definition 3.16.

Take $j\in\\{0,1,2\\}$. We shall refer to
$\mathscr{D}_{j}\cap\mathscr{D}_{(j+1)\hskip 2.0pt{\rm mod}3}$ as the left
boundary of $\mathscr{D}_{j}$. Also we will call
$\mathscr{D}_{j}\cap\mathscr{D}_{(j-1)\hskip 2.0pt{\rm mod}3}$, the right
boundary of $\mathscr{D}_{j}$.

Next, in each principal region $\mathscr{D}_{j}$, $j\in\\{0,1,2\\}$ associated
with $c$ we define

$\xi_{c}(x,\lambda)=\int_{\gamma(c\rightsquigarrow
x)}V_{0}^{1/2}(t,\lambda)dt$ (3.15)

taking the branch that is continuous with $\Re\xi_{c}(x,\lambda)\geq 0$. This
determines a function of $x$ that is continuous in $\mathscr{D}$ except on the
Stokes curves. Fix a $j\in\\{0,1,2\\}$. Clearly, for $x$ on the left boundary
of $\mathscr{D}_{j}$ we have $\xi_{c}(x,\lambda)\in i{\mathbb{R}}^{+}$, while
on the right boundary of $\mathscr{D}_{j}$ we have $\xi_{c}(x,\lambda)\in
i{\mathbb{R}}^{-}$. Since the left boundary of $\mathscr{D}_{j}$ is also the
right boundary of $\mathscr{D}_{(j+1)\hskip 2.0pt{\rm mod}3}$ and the right
boundary of $\mathscr{D}_{j}$ is also the left boundary of
$\mathscr{D}_{(j-1)\hskip 2.0pt{\rm mod}3}$, we realize that
$\xi_{c}(x,\lambda)$ changes sign when it crosses a principal curve. Finally
we redefine the notion of the progressive path (cf. Definition 3.9).

######  Definition 3.17.

Any Jordan arc $\mathscr{P}$ comprising of a finite chain of $R_{2}$ arcs and
having the property that $\Re\xi_{c}(\cdot,\lambda)$ is monotonic on the
intersection of $\mathscr{P}$ with any principal region, shall (also) be
called a progressive path associated with $c$.

Before stating the main theorem of this section, let us formulate an important
lemma first.

######  Lemma 3.18.

Consider the differential equation (3.3), taking into account Assumption 3.2.
Also, let $j\in\\{0,1,2\\}$, pick a boundary point or point at infinity
$a_{j}\in\textsf{clos}(\mathscr{D}_{j})$ and consider a progressive path
$\mathscr{P}$ in $\mathscr{D}_{j}$ passing from $a_{j}$. Then equation (3.3)
has a unique solution $w(x,\lambda,\epsilon)\in\mathscr{H}(\mathscr{D})$ and
satisfies

$\displaystyle V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon)$ $\displaystyle\sim
e^{-\xi_{c}(x,\lambda)/\epsilon}\quad\text{as}\quad x\rightarrow
a_{j}\quad\text{along}\quad\mathscr{P}$
$\displaystyle\frac{\partial[V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon)]}{\partial
x}$
$\displaystyle\sim-\frac{1}{\epsilon}V_{0}^{1/2}(x,\lambda)e^{-\xi_{c}(x,\lambda)/\epsilon}\quad\text{as}\quad
x\rightarrow a_{j}\quad\text{along}\quad\mathscr{P}.$

###### Proof.

Using Theorem 11.1 in chapter 6 of Olver’s book [25] one needs to realize that

  * •

$\Re\xi_{c}(x,\lambda)\rightarrow+\infty$ as $x\rightarrow a_{j}$ along
$\mathscr{P}$ and

  * •

$\mathcal{V}_{a_{j},x}[F]$ converges as $x\rightarrow a_{j}$ along
$\mathscr{P}$.

It follows that such a solution exists, is holomorphic and additionally
satisfies the desired asymptotics. Since
$\Re\xi_{c}(x,\lambda)\rightarrow+\infty$ as $x\rightarrow a_{j}$ along
$\mathscr{P}$, this solution is recessive and hence unique. ∎

We are now ready for current section’s main theorem.

######  Theorem 3.19.

Consider the differential equation (3.3), taking into account Assumption 3.2.
Also, let $j,k\in\\{0,1,2\\}$ with $k\neq j$, pick two boundary points or
points at infinity, namely $a_{j}\in\textsf{clos}(\mathscr{D}_{j})$,
$a_{k}\in\textsf{clos}(\mathscr{D}_{k})$ (different from $c$) and consider a
progressive path $\mathscr{P}$ in $\mathscr{D}_{j}\cup\mathscr{D}_{k}$ joining
them. Then the unique solution $w(x,\lambda,\epsilon)$ of equation (3.3)
provided by the previous lemma, satifies the following asymptotics as
$x\rightarrow a_{k}$ along $\mathscr{P}$

$\displaystyle V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon)$ $\displaystyle\sim
i^{k-j-1}(\lambda_{j,k}+\kappa_{0})e^{\xi_{c}(x,\lambda)/\epsilon}$
$\displaystyle\frac{\partial[V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon)]}{\partial
x}$
$\displaystyle\sim\frac{i^{k-j-1}}{\epsilon}(\lambda_{j,k}+\kappa_{0})V_{0}^{1/2}(x,\lambda)e^{\xi_{c}(x,\lambda)/\epsilon}$

where $\kappa_{0}$ is independent of $x$ and subject to the bound

$|\kappa_{0}|\leq(1+\lambda_{j,k}^{2})^{1/2}\bigg{(}\exp\Big{\\{}\frac{\rho_{j,k}}{3|\lambda_{j,k}|}\epsilon^{2/3}\mathcal{V}_{\mathscr{P}}[H]\Big{\\}}-1\bigg{)}.$

###### Proof.

The proof of this theorem follows from Theorem 5.1 in [24] merely by checking
that

  * •

$\Re\xi_{c}(x,\lambda)\rightarrow+\infty$ as $x\rightarrow a_{j}$ along
$\mathscr{P}$,

  * •

$\mathcal{V}_{a_{j},x}[F]$ converges as $x\rightarrow a_{j}$ along
$\mathscr{P}$ and

  * •

$\Re\xi_{c}(x,\lambda)\rightarrow+\infty$ as $x\rightarrow a_{k}$ along
$\mathscr{P}$.

∎

###  3.5. The asymptotic form of the connection theorem

In this section, we construct asymptotic estimates for the error-terms in
Theorem 3.19 for small $\epsilon>0$ by applying the results from §3.3. Let us
first fix some notation that will be used in this and the following sections.
We have

  * •

We use the notation “$\doteq$” to state that besides the validity of the
equation shown, the corresponding equation obtained by formally
differentiating with respect to $x$, ignoring the differentiation of the
$\mathcal{O}$-terms present, also stands true.

  * •

When an $\mathcal{O}$-term appears in an equation it is understood to hold
uniformly for all the values of $x$ associated with that equation.

  * •

With $\chi(\epsilon)$, we denote any positive function of $\epsilon$ such that
as $\epsilon\downarrow 0$ it satisfies

$\chi(\epsilon)\rightarrow 0\quad\text{and}\quad
1/\chi(\epsilon)=\mathcal{O}(\epsilon^{-1}).$ (3.16)

  * •

Lastly, let $a,b$ be two points on a path $\mathscr{P}$. We write
$[a,b]_{\mathscr{P}}$, $(a,b)_{\mathscr{P}}$, $[a,b)_{\mathscr{P}}$ and
$(a,b]_{\mathscr{P}}$ to denote the part of $\mathscr{P}$ that lies between
$a$ and $b$, with endpoints included or not, accordingly.

With these in mind, we formulate the following theorem.

######  Theorem 3.20.

For $\epsilon>0$, consider the differential equation

$\frac{d^{2}w}{dx^{2}}=[\epsilon^{-2}V_{0}(x,\lambda)+g(x,\lambda)]w,\quad(x,\lambda)\in\mathscr{D}\times\mathscr{O}$
(3.17)

and assume Assumption 3.2. Also, let $j,k\in\\{0,1,2\\}$ with $k\neq j$ and
consider a progressive path $\mathscr{P}$ in
$\mathscr{D}_{j}\cup\mathscr{D}_{k}$. Pick points $a_{j},b_{j},b_{k},a_{k}$,
in that order on $\mathscr{P}$, neither of which depends on $\epsilon$ nor
coincides with $c$, and such that $a_{j}\in\textsf{clos}(\mathscr{D}_{j})$,
$a_{k}\in\textsf{clos}(\mathscr{D}_{k})$, $b_{j}\in\mathscr{D}_{j}$ and
$b_{k}\in\mathscr{D}_{k}$ [thus $a_{j},a_{k}$ (but not $b_{j},b_{k}$) may be
boundary points of $\mathscr{D}$, including the point at infinity]. Finally,
let the function $w(x,\lambda,\epsilon)$ denote the solution of the
differential equation (3.17) (as provided by Lemma 3.18).Then, on
$[b_{k},a_{k})_{\mathscr{P}}$, the analytic continuation of
$w(x,\lambda,\epsilon)$ [obtained by passing from $\mathscr{D}_{j}$ to
$\mathscr{D}_{k}$ in the same sense as the sign of $(k-j)$], is given by

$\displaystyle\text{case A}:\quad w(x,\lambda,\epsilon)$ $\displaystyle=w_{\rm
I}(x,\lambda,\epsilon)\quad\text{or}$ $\displaystyle\text{case B}:\quad
w(x,\lambda,\epsilon)$ $\displaystyle=w_{\rm I,1}(x,\lambda,\epsilon)+w_{\rm
L}(x,\lambda,\epsilon)\quad\text{or}$ $\displaystyle\text{case C}:\quad
w(x,\lambda,\epsilon)$ $\displaystyle=w_{\rm I,-1}(x,\lambda,\epsilon)+w_{\rm
R}(x,\lambda,\epsilon)$

depending on whether $b_{k}$ is an interior point of $\mathscr{D}_{k}$ (case
A), or $[b_{k},a_{k})_{\mathscr{P}}$ lies on the left boundary of
$\mathscr{D}_{k}$ (case B), or $[b_{k},a_{k})_{\mathscr{P}}$ lies on the right
boundary of $\mathscr{D}_{k}$ (case C). Here, $w_{\rm I}(x,\lambda,\epsilon)$,
$w_{\rm I,\pm 1}(x,\lambda,\epsilon)$, $w_{\rm L}(x,\lambda,\epsilon)$ and
$w_{\rm R}(x,\lambda,\epsilon)$ are solutions of the differential equation
(3.17) so that on $[b_{k},a_{k})_{\mathscr{P}}$ they have the following
asymptotic forms as $\epsilon\downarrow 0$

$\begin{rcases}V_{0}^{1/4}(x,\lambda)w_{\rm I}(x,\lambda,\epsilon)\\\
V_{0}^{1/4}(x,\lambda)w_{\rm I,1}(x,\lambda,\epsilon)\\\
V_{0}^{1/4}(x,\lambda)w_{\rm I,-1}(x,\lambda,\epsilon)\end{rcases}\doteq
i^{k-j-1}[\lambda_{j,k}+\mathcal{O}(\hat{\chi})]e^{\xi_{c}(x,\lambda)/\epsilon}$
$V_{0}^{1/4}(x,\lambda)w_{\rm L}(x,\lambda,\epsilon)\doteq
i^{k-j}[\lambda_{j,k+1}+\mathcal{O}(\hat{\chi})]e^{-\xi_{c}(x,\lambda)/\epsilon}$
$V_{0}^{1/4}(x,\lambda)w_{\rm
R}(x,\lambda,\epsilon)\doteq-i^{k-j}[\lambda_{j,k-1}+\mathcal{O}(\hat{\chi})]e^{-\xi_{c}(x,\lambda)/\epsilon}.$

In these relations, $\lambda_{j,k}$ are given by (3.13),
$V_{0}^{1/4}(x,\lambda)$ denotes the branch obtained from that used in Lemma
3.18 [equivalently see (3.19) below] by analytic continuation in the same
manner as for $w(x,\lambda,\epsilon)$ and the function
$\hat{\chi}=\hat{\chi}(\epsilon)$ is defined by

$\hat{\chi}(\epsilon)=\max\\{\epsilon,\chi(\epsilon)\\}.$ (3.18)

###### Proof.

For the proof, the reader can refer to Theorem 6.1 in [24] with $m=3$ and the
function $g$ analytic at $c$. It suffices to see that the following are true.

  * (i)

The function $1/\xi_{c}(x,\lambda)$ [see (3.15)] is of bounded variation on
$(a_{j},b_{j}]_{\mathscr{P}}$ and $[b_{k},a_{k})_{\mathscr{P}}$

  * (ii)

The function $F(x,\lambda)$ given by (3.14) is of bounded variation on
$(a_{j},b_{j}]_{\mathscr{P}}$ and $[b_{k},a_{k})_{\mathscr{P}}$ and

  * (iii)

The function $w(x,\lambda,\epsilon)$ satisfies the following.

    * –

For $x\in(a_{j},b_{j}]_{\mathscr{P}}$

$V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon)\doteq[1+\mathcal{O}(\chi)]e^{-\xi_{c}(x,\lambda)/\epsilon}\quad\text{as}\quad\epsilon\downarrow
0$ (3.19)

    * –

As $x\rightarrow a_{j}$ along $\mathscr{P}$, the functions

$\displaystyle e^{\xi_{c}(x,\lambda)/\epsilon}$ $\displaystyle
V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon),$ $\displaystyle
V_{0}^{-1/2}(x,\lambda)e^{\xi_{c}(x,\lambda)/\epsilon}$
$\displaystyle\frac{\partial[V_{0}^{1/4}(x,\lambda)w(x,\lambda,\epsilon)]}{\partial
x}$

tend to non-zero finite limits.

∎

######  Remark 3.21.

Even though $w_{\rm I}(x,\lambda,\epsilon)$ and $w_{\rm I,\pm
1}(x,\lambda,\epsilon)$ have the same asymptotics as $\epsilon\downarrow 0$ on
$[b_{k},a_{k})_{\mathscr{P}}$, it should be emphasized that they do represent
distinct solutions of equation (3.17).

###  3.6. Application of the connection theorem

In this section, we apply Theorem 3.20 to prove a Bohr-Sommerfeld type
quantization condition for the eigenvalues that lie near zero. Once again, we
begin by some geometric formulations that will pave the way to the final
result.

Let now $\mathscr{D}$ denote a simply connected open neighborhood of the real
axis that contains only two simple turning points of our differential
equation. Let us be more precise and make the following assumption.

######  Assumption 3.22.

Consider the differential equation

$\frac{d^{2}w}{dx^{2}}=[\epsilon^{-2}V_{0}(x,\lambda)+g(x,\lambda)]w,\quad(x,\lambda)\in\mathscr{D}\times\mathscr{O}$

where

  * •

The functions $V_{0}(x,\lambda)$ and $g(x,\lambda)$ are given by

$V_{0}(x,\lambda)=-\Big{\\{}A(x)^{2}+[\lambda+\tfrac{A^{\prime}(x)}{2}]^{2}\Big{\\}}$
$g(x,\lambda)=\frac{3}{4}\bigg{\\{}\frac{A^{\prime}(x)-i\frac{A^{\prime\prime}(x)}{2}}{A(x)-i[\lambda+\tfrac{A^{\prime}(x)}{2}]}\bigg{\\}}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)-i\frac{A^{\prime\prime\prime}(x)}{2}}{A(x)-i[\lambda+\tfrac{A^{\prime}(x)}{2}]}$

for $A(x)=\sech(2x)$.

  * •

The set $\mathscr{O}$ is an open and simply connected set (of the
$\lambda$-plane), proper subset of ${\mathcal{R}}_{0}\cap\mathbb{H}^{+}$ [see
(2.4) with $S(x)=A(x)=\sech(2x)$] located in such a way that its intersection
with $\tilde{\Lambda}_{12}$ is a single interval and moreover it does not have
common points with either $\tilde{\Lambda}_{16}$ or $\tilde{\Lambda}_{26}$.
Also, let $c\equiv x_{1}$ and $\hat{c}\equiv x_{2}$ be the two distinct simple
zeros of $g_{-}$ as introduced in §2.6 [see Remark 2.21; recall that
$V_{0}=g_{-}g_{+}$ where $g_{\pm}$ are given by (2.10)].

  * •

The set $\mathscr{D}\subset{\mathbb{C}}$ is an open and simply connected
domain (of the $x$-plane) so that it contains the real axis and that the only
transition points of the differential equation in it, are $c$ and $\hat{c}$.

######  Remark 3.23.

Observe that the assumption above implies that

  * (i)

the function $(x-c)^{-1}(x-\hat{c})^{-1}V_{0}(x,\lambda)$ is holomorphic and
non-vanishing throughout $\mathscr{D}$ (including $c,\hat{c}$) and

  * (ii)

the function $g(x,\lambda)$ is holomorphic in $\mathscr{D}$.

Let the WKB approximation of a solution of our differential equation be given
at a point in $\textsf{clos}(\mathscr{D})$ other than one of the turning
points. Then the WKB approximation of the same solution at any other point
(except at a turning point) can be found by at most two applications of
Theorem 3.20.

As was done previously, we associate with the two transition points the
functions

$\displaystyle\xi_{c}(x,\lambda)$
$\displaystyle=\int_{\gamma(c\rightsquigarrow x)}V_{0}^{1/2}(t,\lambda)dt$
$\displaystyle\xi_{\hat{c}}(x,\lambda)$
$\displaystyle=\int_{\sigma(\hat{c}\rightsquigarrow
x)}V_{0}^{1/2}(t,\lambda)dt.$

For the former, namely $\xi_{c}(x,\lambda)$, we take a branch of the integral
so that $\Re\xi_{c}(x,\lambda)\geq 0$ and so that it makes
$\xi_{c}(x,\lambda)$ continuous except at $\Re\xi_{c}(x,\lambda)=0$. The last
relation defines a set of three Stokes curves emanating from $c$. We refer to
any Stokes curve emanating from the turning point $c$ as a $c$-Stokes curve
777 Olver calls them $c$-principal curves.. We argue similarly for
$\xi_{\hat{c}}(x,\lambda)$.

On the $\xi$-plane, where

$\xi(x,\lambda)=\int V_{0}^{1/2}(x,\lambda)dx$

in which the integration constant is arbitrary and the branch is chosen in a
continuous way, we have the following. The $c$-Stokes curves and the
$\hat{c}$-Stokes curves are mapped to straight lines parallel to the imaginary
axis. Conformal mapping theory shows that the $c$-Stokes curves intersect on
the same Riemann sheet only at $c$ and a similar assertion holds for
$\hat{c}$. Also, $c$-Stokes curves and $\hat{c}$-Stokes curves do not
intersect on the same sheet. However, in our case $c$ and $\hat{c}$ are linked
together by a common Stokes curve: the interval $[\xi(c),\xi(\hat{c})]$ (on
the $\xi$-plane) is parallel to the imaginary axis.

Figure 22. The configuration on the $x$-plane for the case of near-zero
eigenvalues.