# SEMICLASSICAL WKB PROBLEM FOR THE NON-SELF-ADJOINT DIRAC OPERATOR WITH A
DECAYING POTENTIAL

Nicholas Hatzizisis † † Mathematics Building, University of Crete
700 13 Voutes, Greece<EMAIL_ADDRESS>http://www.nikoshatzizisis.wordpress.com/home/ and  Spyridon Kamvissis ‡ ‡
Mathematics Building, University of Crete
700 13 Voutes, 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 fairly smooth -but not necessarily
analytic- potential decaying at infinity. In particular, using ideas and
methods going back to Langer and Olver, we provide a rigorous semiclassical
analysis of the scattering coefficients, the Bohr-Sommerfeld condition for the
location of the eigenvalues and their corresponding norming constants. Our
analysis is motivated by the potential applications to the focusing cubic NLS
equation, in view of the well-known fact discovered by Zakharov and Shabat
that the spectral analysis of the Dirac operator is the basis of the solution
of the NLS equation via inverse scattering theory. This paper complements and
extends a previous work of Fujiié and the second author, which considered a
more restricted problem for a strictly analytic potential.

##  1\. Introduction

Consider the initial value problem (IVP) of the one-dimensional focusing
nonlinear Schrödinger equation (focusing NLS) for the complex field $u(x,t)$,
i.e.

$\begin{cases}i\hbar\partial_{t}u+\frac{\hbar^{2}}{2}\partial_{x}^{2}u+|u|^{2}u=0,\quad(x,t)\in{\mathbb{R}}\times{\mathbb{R}}\\\
\ u(x,0)=A(x),\quad x\in{\mathbb{R}}\end{cases}$ (1.1)

for a real valued function $A$ and a fixed positive number $\hbar$.

Zakharov and Shabat in [22] have proved back in 1972 that the focusing NLS
equation is integrable via the Inverse Scattering Transform (IST). A crucial
step of the method is the analysis of the following Zakharov-Shabat (or Dirac)
eigenvalue problem

$\hbar\begin{bmatrix}v_{1}^{\prime}(x,\lambda,\hbar)\\\
v_{2}^{\prime}(x,\lambda,\hbar)\end{bmatrix}=\begin{bmatrix}-i\lambda&A(x)\\\
-A(x)&i\lambda\end{bmatrix}\begin{bmatrix}v_{1}(x,\lambda,\hbar)\\\
v_{2}(x,\lambda,\hbar)\end{bmatrix}$ (1.2)

where $\lambda\in{\mathbb{C}}$ is a “spectral parameter”; here prime denotes
differentiation with respect to $x$.

Now let us suppose that $\hbar$ is small compared to the $x,t$ we are
interested in. The question raised is then: what is the behavior of solutions
of the IVP (1.1) as $\hbar\downarrow 0$? The rigorous analysis of this problem
was initiated in [9]. Because of the work of Zakharov and Shabat, the first
step in the study of this IVP in the semiclassical limit $\hbar\downarrow 0$
has to be the asymptotic spectral analysis of the scattering problem (1.2) as
$\hbar\downarrow 0$, keeping the function $A$ fixed.

The eigenvalue (EV) problem (1.2) is not and cannot be written as an EV
problem for a self-adjoint operator. What we study here is a semiclassical WKB
problem (or LG problem) for the corresponding non-self-adjoint Dirac operator
with potential $A$.

The question of the semiclassical approximation of the scattering data has a
deep significance in view of the instability of the NLS problem which appears
in many levels. In fact even away from the semiclassical regime, the focusing
NLS is the main model for the so-called modulational instability (as in [4]),
although for positive fixed $\hbar$ 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

$\begin{cases}\boldsymbol{\rho}=|u|^{2}\\\
\boldsymbol{\mu}=\hbar\Im\,(u^{*}u_{x})\end{cases}$

where $u^{*}$ denotes the complex conjugate of $u$. Then the IVP (1.1) becomes

$\begin{cases}\boldsymbol{\rho}_{t}+\boldsymbol{\mu}_{x}=0\\\
\boldsymbol{\mu}_{t}+\Big{(}\frac{\boldsymbol{\mu}^{2}}{\boldsymbol{\rho}}+\frac{\boldsymbol{\rho}^{2}}{2}\Big{)}_{x}=\par\frac{\hbar^{2}}{4}\partial_{x}[\boldsymbol{\rho}(\log\boldsymbol{\rho})_{xx}]\end{cases}$

with initial data $\boldsymbol{\rho}(x,0)=|u|^{2}(x,0)=A^{2}(x)$ and
$\boldsymbol{\mu}(x,0)=0$.

The formal limit as $\hbar\downarrow 0$ is

$\begin{cases}\boldsymbol{\rho}_{t}+\boldsymbol{\mu}_{x}=0\\\
\boldsymbol{\mu}_{t}+\Big{(}\frac{\boldsymbol{\mu}^{2}}{\boldsymbol{\rho}}+\frac{\boldsymbol{\rho}^{2}}{2}\Big{)}_{x}=0\end{cases}$

with initial data $\boldsymbol{\rho}(x,0)=|u|^{2}(x,0)=A^{2}(x)$ and
$\boldsymbol{\mu}(x,0)=0$. This is an IVP for an elliptic system of equations
and so one expects that small perturbations of the initial data (independent
of $\hbar$) can lead to large changes in the solution, at any given time.

Instabilities appear also at different stages of the analysis: the spectral
analysis of the related non-self-adjoint Dirac operator, the related
equilibrium measure problem (see [10]), the related Whitham equations (cf.
[9]) which are also elliptic, the possibility of the appearance of rogue waves
(see [3]) and even the numerical studies of the problem (as in [15]).

The semiclassical approximation of the scattering data results in small
changes of the initial data; changes $that~{}depend~{}on~{}\hbar$. It is a
priori unclear whether these small changes can have a significant effect in
the semiclassical asymptotics of the solution of the IVP (1.1) as
$\hbar\downarrow 0$. Our ultimate aim is to provide a proof that they do not.

Our work complements the paper [6] of S. Fujiié and the second author where
the potential is considered to be a real analytic bell-shaped function and in
which the so-called exact WKB method (cf. [5], [7], [8] and [19]) is employed.
In this work, we instead suppose that the bell-shaped potential function $A$
has only some prescribed smoothness which we specify explicitly in §2. Our
methods are necessarily different since the exact WKB method requires
analyticity. Our ideas are rather influenced by the papers [20] and [21] of D.
R. Yafaev where an analogous problem is treated for the self-adjoint
Schrödinger operator, which in turn rely on works [16] and [17] of F. W. J.
Olver 111Olver’s work draws upon the studies of N. D. Kazarinoff, R. E. Langer
and R.W. McKelvey (see the references in [16]).. More precisely, Yafaev uses
results mainly from [17] while we rely heavily on [16] as well.

The present paper is arranged as follows. In section §2 we state all the
necessary assumptions on the potential function $A$ so that Olver’s work can
be applied in our case. In section §3 we introduce a simple transformation
that maps the Dirac problem to an equivalent Schrödinger problem. Sections §4
and §5 show how the Liouville transformation changes our Schrödinger equation
into one containing an error term which is a continuous function on the
Liouville plane. By controlling this error term in section §6 we obtain
approximate solutions expressed with the help of Parabolic Cylinder Functions
(PCFs) in a Liouville variable $\zeta\geq 0$.

In section §7 we illustrate the previously mentioned results for the special
case where the potential function is $x\mapsto\frac{1}{1+x^{2}}$. Then in §8
we find the asymptotic behavior of the approximants introduced in §6. In §9 we
present some connection formulas that relate the approximate solutions for
$\zeta\geq 0$ to the ones for $\zeta\leq 0$. The significance of this
connection becomes clear in §10 where we find Bohr-Sommerfeld quantization
conditions for the EVs of our problem, uniformly away from zero.

Next in §11 we study the EVs that lie closer to zero and we are able to arrive
at uniform bounds all the way to zero. We only do this here for two specific
(but quite inclusive) families of functions $A$. It may be that more general
conditions can alternatively be posed instead; conditions that would ensure
the same results for a wide class of data. But we are not able to do this at
this point. Our situation is somewhat comparable to [6] where extra (general
but complicated) conditions had to be added, in order to assure a good
behavior of the EVs near $0$. Here however, the formula (11.3) we have for the
function $\psi$, makes it very easy to check if indeed the behavior of the EVs
near zero, is good enough for any family of potentials $A$ defined by
explicitly prescribed asymptotics at infinity.

Section §12 is concerned with the amplitudes of the transmission and
reflection coefficients (both away and close to zero). Unlike [6], we do not
need any extra conditions on $A$ to control the reflection coefficient (cf.
§12) near $0$ (even though our estimates are somewhat weaker, they are still
good enough for applications). Finally, we make some concluding remarks in
section §13.

For the sake of the reader, as the approximate solutions to our problems
involve Airy and Parabolic Cylinder Functions, we present all the necessary
results concerning these functions in sections A and B of the appendix. There,
the reader can also find a section (section C) on a theorem concerning
integral equations which is the primary tool in the proof of the main Theorem
6.3.

Notationwise, a bar over a letter (or number) does not denote complex
conjugation. For complex conjugation we have reserved the superscript $*$;
i.e. $z^{*}$ (and not $\bar{z}$) is the complex conjugate of $z$. The letter
$C$ denotes generically a positive constant (appearing in estimates) and
$\mathbb{R}_{+}$, $\mathbb{R}_{-}$ represent the sets of positive and negative
numbers respectively. Also, we denote the Wronskian of two functions $f$, $g$
by $\mathcal{W}[f,g]$. Furthermore, we write $f\sim g$ when $\tfrac{f}{g}$
tends to $1$. Finally, the notation $f^{2}(x)$ denotes the square of the value
of the function $f$ at $x$. Hence, the symbols $f^{2}(x)$ and $f(x)^{2}$ are
used interchangeably and are not to be confused with the composition $f\circ
f$ of $f$ with itself.

##  2\. Assumptions on the Potential

In this section we state precise assumptions on the potential function $A$
which are sufficient to ensure that all the techniques and methods developed
in the following sections can go through easily. In short, we consider bell-
shaped functions with some smoothness.

To be more precise, we assume that the function $A$ satisfies the following

  * •

$A(x)>0$ for $x\in\mathbb{R}$

  * •

$A(-x)=A(x)$ for $x\in\mathbb{R}$

  * •

$A$ is in $C^{4}({\mathbb{R}})$ and of class $C^{5}$ in a neighborhood of $0$

  * •

$xA^{\prime}(x)<0$ for $x\in\mathbb{R}\setminus\\{0\\}$

  * •

$A^{\prime\prime}(0)<0$; we set $0<A(0)=:A_{max}$

  * •

there exists $\tau>0$ so that as $x\to\pm\infty$

$\displaystyle A(x)=\mathcal{O}\Big{(}\tfrac{1}{|x|^{1+\tau}}\Big{)}$
$\displaystyle A^{\prime}(x)=\mathcal{O}\Big{(}\tfrac{1}{|x|^{2+\tau}}\Big{)}$
$\displaystyle
A^{\prime\prime}(x)=\mathcal{O}\Big{(}\tfrac{1}{|x|^{3+\tau}}\Big{)}$

Figure 1. A bell-shaped function.

Now let $\mu\in(0,A_{max}]\subset\mathbb{R}_{+}$. Observe that

  * •

for $\mu\in(0,A_{max})$ the equation $A(x)=\mu$ has exactly two solutions
$x_{\pm}$ which of course depend on $\mu$ and by the symmetry of $A$ satisfy
$x_{\mp}=-x_{\pm}$. Furthermore, $A(x)>\mu$ for $x\in(x_{-},x_{+})$ and
$A(x)<\mu$ for $x\in(-\infty,x_{-})\cup(x_{+},+\infty)$.

  * •

when $\mu=A_{max}$ the two points $x_{\pm}$ coalesce into one double at $x=0$.

We believe that neither the evenness assumption, nor the single local maximum
assumption are strictly necessary. If evenness is not imposed, we have what
Klaus & Shaw call a “single lobe” potential (see [11] and [12]). Essentially
our discussion in this article goes through mostly unaltered, since the
results of Klaus & Shaw mentioned below (in §3) are still valid. One thing
that changes is that the norming constants are no more real, but we still have
uniform estimates for them (see Corollary 10.5). If the single local maximum
assumption is dropped we no more expect to have imaginary EVs. But we do
expect to have EVs accumulate along curves as in sections §10, §11 and Bohr-
Sommerfeld conditions to appear. A forthcoming paper will hopefully show how
to handle more general cases.

##  3\. From Dirac to Schrödinger

As stated in the introduction, in this paper we examine the scattering data
for a Dirac operator. We start with an investigation of the EVs. Specifically,
we study the EV problem

$\mathfrak{D}_{\hbar}\mathbf{u}=\lambda\mathbf{u}$ (3.1)

where $\mathfrak{D}_{\hbar}$ is the Dirac operator

$\mathfrak{D}_{\hbar}:=\begin{bmatrix}i\hbar\partial_{x}&-iA\\\
-iA&-i\hbar\partial_{x}\end{bmatrix}$ (3.2)

with $0<\hbar\ll 1$ a small parameter (Planck), $A$ as in §2 and
$\mathbf{u}=\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix}$ is a function from
$\mathbb{R}$ to $\mathbb{C}^{2}$. As usual, $\lambda\in{\mathbb{C}}$ plays the
role of the spectral parameter.

Let us make explicit what we mean when we discuss about the EVs of the
operator in (3.2). We have the following definition.

######  Definition 3.1.

A number $\lambda\in\mathbb{C}$ is an EV of the operator
$\mathfrak{D}_{\hbar}$ if equation (3.1) has a non-trivial solution
$\mathbf{u}\in L^{2}(\mathbb{R};\mathbb{C}^{2})$; that is

$0<\int_{-\infty}^{+\infty}\Big{[}|u_{1}(x)|^{2}+|u_{2}(x)|^{2}\Big{]}dx<+\infty.$

The spectral characteristics of an operator like $\mathfrak{D}_{\hbar}$ having
a potential $A$ satisfying the assumptions of §2 have been established in [11]
and [12] by M. Klaus and J. K. Shaw. More precicely we know that

  * •

the continuous spectrum of $\mathfrak{D}_{\hbar}$ is the whole real line
$\mathbb{R}$ and

  * •

the EVs are simple, purely imaginary and symmetric with respect to the real
axis; their imaginary part lying in $[-A_{max},A_{max}]$

The spectral facts above suggest writing $\lambda=i\mu$ for $0<\mu\leq
A_{max}$ (due to symmetry). Hence, (3.1) is written as

$\hbar\begin{bmatrix}u_{1}^{\prime}(x,\mu,\hbar)\\\
u_{2}^{\prime}(x,\mu,\hbar)\end{bmatrix}=\begin{bmatrix}\mu&A(x)\\\
-A(x)&-\mu\end{bmatrix}\begin{bmatrix}u_{1}(x,\mu,\hbar)\\\
u_{2}(x,\mu,\hbar)\end{bmatrix}$ (3.3)

where the prime denotes differentiation with respect to $x$.

Under the change of variables (cf. equation (4) in [14])

$y_{\pm}=\frac{u_{2}\pm u_{1}}{\sqrt{A\mp\mu}}$ (3.4)

system (3.3) is equivalent to the following two independent equations

$y_{\pm}^{\prime\prime}=\bigg{\\{}\hbar^{-2}[\mu^{2}-A^{2}(x)]+\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)\mp\mu}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)\mp\mu}\bigg{\\}}y_{\pm}$
(3.5)

where we dropped the dependence of $y_{\pm}$ on $(x,\mu,\hbar)$ for notational
simplicity. Again, prime denotes differentiation with respect to $x$.

In (3.5), we will only consider the equation with the lower index because the
term

$\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)+\mu}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)+\mu}$

has no singularities and thus work with the equation

$\frac{d^{2}y}{dx^{2}}=\bigg{\\{}\hbar^{-2}[\mu^{2}-A^{2}(x)]+\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)+\mu}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)+\mu}\bigg{\\}}y.$
(3.6)

Observe that the change of variables (3.4) with the minus choice does not
alter the discrete spectrum. Hence we are led to the following important fact.

######  Proposition 3.2.

Finding the discrete spectrum of $\mathfrak{D}_{\hbar}$ in (3.2) is equivalent
to finding the values of $\mu\in(0,A_{max}]$ for which (3.6) has an
$L^{2}(\mathbb{R})$ solution.

Now, let us choose any $A_{0}$ such that $0<A_{0}<A_{max}$. In equation (3.6)
$x$ runs on the whole real line ${\mathbb{R}}$ and $\mu$ will play the role of
a spectral parameter in $[A_{0},A_{max}]\subset{\mathbb{R}}_{+}$. For
$\mu\in[A_{0},A_{max})$ the function $x\mapsto\mu^{2}-A^{2}(x)$ is non-
vanishing on ${\mathbb{R}}$ except for two distinct simple zeros at $x=x_{-}$
and $x=x_{+}$ with $x_{-}<x_{+}$. In the critical case $\mu=A_{max}$ the
function $x\mapsto A_{max}^{2}-A^{2}(x)$ has a single double zero at $x=0$.
Both $x_{-}$, $x_{+}$ are continuous functions of the parameter $\mu$ and tend
to zero as $\mu\uparrow A_{max}$.

$A(x)$$x=a$$a_{0}$$y=\mu$$A_{max}$$A_{0}$ Figure 2. The relationship between
parameters $\mu$ and $a$.

We introduce a change of variables for the spectral parameter $\mu$ in order
to rely on the results from [16]. For this, we first define the function $B$
to be the restriction of $A$ on $[0,+\infty)$, i.e.
$B={\left.\kern-1.2ptA\vphantom{\big{|}}\right|_{[0,+\infty)}}$ and note that,
by the assumptions on $A$, the function $B$ is invertible. We set

$a=x_{+}$

so

$A(x_{+})=\mu\Leftrightarrow B(a)=\mu\Leftrightarrow a=B^{-1}(\mu).$

Since $B$ is a decreasing function, so is $B^{-1}$ and we get

$a\in B^{-1}\Big{(}[A_{0},A_{max}]\Big{)}=[0,B^{-1}(A_{0})]=:[0,a_{0}].$

Thus, the zeros of $x\mapsto A^{2}(a)-A^{2}(x)$ are located at $x_{\pm}=\pm
a$. Furthermore, the critical value of $a$ is now zero and $a$ ranges over the
compact interval $[0,a_{0}]$ (we should add that $A_{0}$ and consequently
$a_{0}$ may be allowed to depend on $\hbar$ as is the case in §11).

With this new parameter, equation (3.6) is replaced by

$\frac{d^{2}y}{dx^{2}}=[\hbar^{-2}f(x,a)+g(x,a)]y$ (3.7)

where $f$ and $g$ satisfy

$f(x,a)=A^{2}(a)-A^{2}(x)$ (3.8)

and

$g(x,a)=\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)+A(a)}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)+A(a)}.$
(3.9)

We close this section with an important definition concerning the zeros of $f$
in (3.8).

######  Definition 3.3.

The zeros (with respect to $x$) of the function

$f(x,a)=A^{2}(a)-A^{2}(x)$

are called turning ponts (or transition points) of the equation (3.7).

Hence our equation facilitates two turning points at $x=\pm a$.

##  4\. The Liouville Transformation

In this section we introduce new variables $Y$ and $\zeta$ according to the
Liouville transform

$Y=\dot{x}^{-\frac{1}{2}}y$

where the dot signifies differentiation with respect to $\zeta$. Equation
(3.7) becomes

$\frac{d^{2}Y}{d\zeta^{2}}=\Big{[}\hbar^{-2}\dot{x}^{2}f(x,a)+\dot{x}^{2}g(x,a)+\dot{x}^{\frac{1}{2}}\frac{d^{2}}{d\zeta^{2}}(\dot{x}^{-\frac{1}{2}})\Big{]}Y.$
(4.1)

In our case, $f(\cdot,a)$ is negative in $(-a,a)$ and positive in
$(-\infty,-a)\cup(a,+\infty)$. Hence we prescribe

$\dot{x}^{2}f(x,a)=\zeta^{2}-\alpha^{2}$ (4.2)

where $\alpha\geq 0$ is chosen in such a way that $x=-a$ corresponds to
$\zeta=-\alpha$ and $x=a$ to $\zeta=\alpha$ accordingly. Indeed, after
integration, (4.2) yields

$\int_{-a}^{x}[-f(t,a)]^{\frac{1}{2}}dt=\int_{-\alpha}^{\zeta}(\alpha^{2}-\tau^{2})^{\frac{1}{2}}d\tau$
(4.3)

provided that $-a\leq x\leq a$. Notice that by taking these integration
limits, $-a$ corresponds to $-\alpha$. For the remaining correspondence, we
require

$\int_{-a}^{a}[-f(t,a)]^{\frac{1}{2}}dt=\int_{-\alpha}^{\alpha}(\alpha^{2}-\tau^{2})^{\frac{1}{2}}d\tau$

and hence

$\alpha^{2}=\frac{2}{\pi}\int_{-a}^{a}[-f(t,a)]^{\frac{1}{2}}dt.$ (4.4)

For every fixed value of $\hbar$, relation (4.4) defines $\alpha$ as a
continuous and increasing function of $a$ which vanishes as $a\downarrow 0$
and equals $\alpha_{0}$ when $a=a_{0}$; so $\alpha\in[0,\alpha_{0}]$.

Next, from (4.3) we find

$\int_{-a}^{x}[-f(t,a)]^{\frac{1}{2}}dt=\frac{1}{2}\alpha^{2}\arccos\Big{(}-\frac{\zeta}{\alpha}\Big{)}+\frac{1}{2}\zeta\big{(}\alpha^{2}-\zeta^{2}\big{)}^{\frac{1}{2}}\quad\text{for}\quad-a\leq
x\leq a$ (4.5)

with the principal value choice for the inverse cosine taking values in
$[0,\pi]$. For the remaining $x$-intervals, we integrate (4.2) to obtain

$\int_{x}^{-a}[f(t,a)]^{\frac{1}{2}}dt=-\frac{1}{2}\alpha^{2}\operatorname{arcosh}\Big{(}-\frac{\zeta}{\alpha}\Big{)}-\frac{1}{2}\zeta\big{(}\zeta^{2}-\alpha^{2}\big{)}^{\frac{1}{2}}\quad\text{for}\quad
x\leq-a$ (4.6)

and

$\int_{a}^{x}[f(t,a)]^{\frac{1}{2}}dt=-\frac{1}{2}\alpha^{2}\operatorname{arcosh}\Big{(}\frac{\zeta}{\alpha}\Big{)}+\frac{1}{2}\zeta\big{(}\zeta^{2}-\alpha^{2}\big{)}^{\frac{1}{2}}\quad\text{for}\quad
x\geq a$ (4.7)

with $\operatorname{arcosh}(x)=\ln\big{(}x+\sqrt{x^{2}-1}\big{)}$ for $x\geq
1$.

Equations (4.5), (4.6) and (4.7) show that $\zeta$ is a continuous and
increasing function of $x$ in ${\mathbb{R}}$. Moreover, this shows that there
is a one-to-one correspondence between these two variables. Finally, we
substitute (4.2) in (4.1) and obtain

$\frac{d^{2}Y}{d\zeta^{2}}=\big{[}\hbar^{-2}(\zeta^{2}-\alpha^{2})+\psi(\zeta,\alpha)\big{]}Y$
(4.8)

where the error term $\psi(\zeta,\alpha)$ is

$\psi(\zeta,\alpha)=\dot{x}^{2}g(x,a)+\dot{x}^{\frac{1}{2}}\frac{d^{2}}{d\zeta^{2}}(\dot{x}^{-\frac{1}{2}})$
(4.9)

or equivalently

$\psi(\zeta,\alpha)=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}}{(\zeta^{2}-\alpha^{2})^{2}}+\frac{1}{16}\frac{\zeta^{2}-\alpha^{2}}{f^{3}(x,a)}\Big{\\{}4f(x,a)f^{\prime\prime}(x,a)-5[f^{\prime}(x,a)]^{2}\Big{\\}}\\\
+(\zeta^{2}-\alpha^{2})\frac{g(x,a)}{f(x,a)}$ (4.10)

where prime denotes differentiation with respect to $x$.

In the critical case in which the two (simple) turning points coalesce into
one double point, we get a limit of the above transformation with $a=0$. So

$\int_{x}^{0}[f(t,0)]^{\frac{1}{2}}dt=\frac{1}{2}\zeta^{2}\quad\text{for}\quad
x\leq 0$ (4.11)
$\int_{0}^{x}[f(t,0)]^{\frac{1}{2}}dt=\frac{1}{2}\zeta^{2}\quad\text{for}\quad
x\geq 0$ (4.12)

and equations (4.8), (4.9) and (4.10) apply with $a=\alpha=0$.

##  5\. Two useful lemmas

In this section we prove two helpful assertions that will be facilitated in
the following sections. First, that the error function $\psi(\zeta,\alpha)$
resulting from the Liouville transformation of §4, is continuous in $\alpha$
and $\zeta$; a fact that will be used in §6 to prove the existence of
approximate solutions of equation (4.8). Secondly, we give asymptotics of $x$
for big values of $\zeta$.

The first lemma that concerns the error term in (4.8) is the following.

######  Lemma 5.1.

The function $\psi(\zeta,\alpha)$ in equation (4.8) as defined in (4.9) is
continuous in $\zeta$ and $\alpha$ in the region
$(-\infty,+\infty)\times[0,\alpha_{0}]$ of the $(\zeta,\alpha)$-plane.

###### Proof.

First we introduce an auxilliary function $p$. We define it by

$f(x,a)=(x^{2}-a^{2})p(x,a)$ (5.1)

where

$p(\pm a,a)=\mp\frac{A(a)A^{\prime}(\pm a)}{a}>0\quad\text{for}\quad
a\in(0,a_{0}]$

and

$p(0,0)=-A_{max}A^{\prime\prime}(0)>0.$

The functions $f$, $g$ and $p$ defined by (3.8), (3.9) and (5.1) respectively
satisfy the following properties

  * (i)

$p,\frac{\partial p}{\partial x},\frac{\partial^{2}p}{\partial x^{2}}$ and $g$
are continuous functions of $x$ and $a$ (this means in $x$ and $a$
simultaneously and not separately) in the region ${\mathbb{R}}\times[0,a_{0}]$

  * (ii)

$p$ is positive throughout the same region

  * (iii)

$|\frac{\partial^{3}p}{\partial x^{3}}|$ is bounded in a neighborhood of the
point $(x,a)=(0,0)$ in the same region and

  * (iv)

$f$ is a non-increasing function of $a$ when $x\in[-a,a]$ and $a\in[0,a_{0}]$.

Indeed, (i) and (iii) follow from (5.1) and the fact that $A$ is in $C^{4}$
and of class $C^{5}$ in some neighborhood of $0$ (see §2). For (ii) recall the
sign of $f$ using (3.8). Finally (iv) is a consequence of the monotonicity of
$A$ in $[0,+\infty)$ (again cf. §2). By Lemma I in Olver’s paper [16], the
function $\psi$ defined by (4.9) (or (4.10)) is continuous in the
corresponding region of the $(\zeta,\alpha)$-plane. ∎

Finally, recall 4.7). It shows that $x\to+\infty$ as $\zeta\to+\infty$. The
lemma below deals with the asymptotic behavior of $x$ as
$\zeta\rightarrow+\infty$.

######  Lemma 5.2.

Considering $x$ as a function of $\zeta$ we see that

$x=\frac{\zeta^{2}}{2A(a)}\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\uparrow+\infty$ (5.2)

uniformly with respect to $a\in[0,a_{0}]$.

###### Proof.

By (3.8) and (4.2) we have

$\Big{(}\frac{dx}{d\zeta}\Big{)}^{2}[A^{2}(a)-A^{2}(x)]=\begin{cases}\alpha^{2}-\zeta^{2},\quad
0\leq\zeta\leq\alpha\\\ \zeta^{2}-\alpha^{2},\quad\zeta>\alpha\end{cases}$

Choosing $x_{0}$ to satisfy $\zeta(x_{0})=0$, we have

$\int_{x_{0}}^{x}dt=\frac{1}{A(a)}\bigg{(}\int_{0}^{\alpha}\sqrt{\alpha^{2}-\eta^{2}}d\eta+\int_{\alpha}^{\zeta}\sqrt{\eta^{2}-\alpha^{2}}d\eta\bigg{)}\quad\text{as}\hskip
7.0pt\zeta\rightarrow+\infty.$

We obtain

$x-x_{0}=\frac{\pi\alpha^{2}}{4A(a)}+\frac{\zeta^{2}}{2A(a)}\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\rightarrow+\infty$

from which the desired result follows. ∎

##  6\. Approximate Solutions

In this section we exploit the tools assembled in the previous sections. Here,
we state a theorem concerning approximate solutions of equation (4.8), i.e.

$\frac{d^{2}Y}{d\zeta^{2}}=\big{[}\hbar^{-2}(\zeta^{2}-\alpha^{2})+\psi(\zeta,\alpha)\big{]}Y$
(6.1)

where the error term $\psi(\zeta,\alpha)$ is

$\psi(\zeta,\alpha)=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}}{(\zeta^{2}-\alpha^{2})^{2}}+\frac{1}{16}\frac{\zeta^{2}-\alpha^{2}}{f^{3}(x,a)}\Big{\\{}4f(x,a)f^{\prime\prime}(x,a)-5[f^{\prime}(x,a)]^{2}\Big{\\}}\\\
+(\zeta^{2}-\alpha^{2})\frac{g(x,a)}{f(x,a)}.$ (6.2)

To this goal, we need a way to assess the error. So we introduce an error-
control function $H$ along with a balancing function $\Omega$ 222For $\Omega$
we can actually choose any continuous function of the real variable $x$ which
is positive (except possibly at $x=0$) and satisfies the asymptotics
$\Omega(x)=\mathcal{O}(|x|^{\frac{1}{3}})\quad\text{as}\quad x\to\pm\infty$. .

######  Definition 6.1.

Define the function $\Omega$ by

$\Omega(x)=1+|x|^{\frac{1}{3}}.$ (6.3)

As an error-control function $H(\zeta,\alpha,\hbar)$ of equation (6.1) we
consider any primitive of the function

$\frac{\psi(\zeta,\alpha)}{\Omega(\zeta\sqrt{2\hbar^{-1}})}.$

Furthermore, we need the notion of the variation of the error-control function
$H$. We have

######  Definition 6.2.

Take $\zeta_{1}<\zeta_{2}$ for $\zeta_{1}\in[0,+\infty)$ and
$\zeta_{2}\in(0,+\infty)\cup\\{+\infty\\}$. The variation
$\mathcal{V}_{\zeta_{1},\zeta_{2}}[H]$ in the interval $(\zeta_{1},\zeta_{2})$
of the error-control function $H$ of equation (6.1) is defined by

$\mathcal{V}_{\zeta_{1},\zeta_{2}}[H](\alpha,\hbar)=\int_{\zeta_{1}}^{\zeta_{2}}\frac{|\psi(t,\alpha)|}{\Omega(t\sqrt{2\hbar^{-1}})}dt.$

Before stating the main theorem, we also need to define an auxiliary function
that shows up in the error estimates of the approximate solutions. So for any
$b\leq 0$ we set

$l(b):=\sup_{x\in(0,+\infty)}\bigg{\\{}\Omega(x)\frac{\mathsf{M}(x,b)^{2}}{\Gamma(\tfrac{1}{2}-b)}\bigg{\\}}$
(6.4)

where $\mathsf{M}$ is a function defined in terms of Parabolic Cylinder
Functions in section B of the appendix and $\Gamma$ denotes the Gamma
function. We note that the above supremum is finite for each value of $b$.
This fact is a consequence of (6.3) and the first relation in (B.9).
Furthermore, because the relations (B.9) hold uniformly in compact intervals
of the parameter $b$, the function $l$ is continuous.

We are now ready for the main theorem of this section.

######  Theorem 6.3.

For each value of $\hbar$, the equation (6.1) has in the region
$[0,+\infty)\times[0,\alpha_{0}]$ of the $(\zeta,\alpha)$-plane two solutions
$Y_{1}$ and $Y_{2}$ satisfying

$\displaystyle
Y_{1}(\zeta,\alpha,\hbar)=U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})+\varepsilon_{1}(\zeta,\alpha,\hbar)$
(6.5) $\displaystyle
Y_{2}(\zeta,\alpha,\hbar)=\overline{U}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})+\varepsilon_{2}(\zeta,\alpha,\hbar)$
(6.6)

where $U$, $\overline{U}$ are the PCFs defined in appendix B. These two
solutions $Y_{1}$, $Y_{2}$ are continuous and have continuous first and second
partial $\zeta$-derivatives. The errors $\varepsilon_{1}$, $\varepsilon_{2}$
in the relations above satisfy the estimates

$\frac{|\varepsilon_{1}(\zeta,\alpha,\hbar)|}{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})},\frac{\Big{|}\frac{\partial\varepsilon_{1}}{\partial\zeta}(\zeta,\alpha,\hbar)\Big{|}}{\sqrt{2\hbar^{-1}}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\\\
\leq\frac{1}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\Big{(}\exp\big{\\{}\tfrac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{V}_{\zeta,+\infty}[H](\alpha,\hbar)\big{\\}}-1\Big{)}$
(6.7)

and

$\frac{|\varepsilon_{2}(\zeta,\alpha,\hbar)|}{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})},\frac{\Big{|}\frac{\partial\varepsilon_{2}}{\partial\zeta}(\zeta,\alpha,\hbar)\Big{|}}{\sqrt{2\hbar^{-1}}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\\\
\leq\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\Big{(}\exp\big{\\{}\tfrac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{V}_{0,\zeta}[H](\alpha,\hbar)\big{\\}}-1\Big{)}$
(6.8)

###### Proof.

By Theorem C.2 (cf. Theorem I in [16]), it suffices to prove that

  * •

the function $\psi$ is continuous in the region
$[0,+\infty)\times[0,\alpha_{0}]$ and

  * •

the integral

$\mathcal{V}_{0,+\infty}[H](\alpha,\hbar)=\int_{0}^{+\infty}\frac{|\psi(t,\alpha)|}{\Omega(t\sqrt{2\hbar^{-1}})}dt$
(6.9)

converges uniformly in $\alpha$.

The first assertion has already been proven in Lemma 5.1. For the second, we
argue as follows. Using (3.8), (3.9) and (4.10) we find

$\displaystyle\frac{\psi(\zeta,\alpha)}{\zeta^{1/3}}$
$\displaystyle=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}}{\zeta^{1/3}(\zeta^{2}-\alpha^{2})^{2}}$
$\displaystyle+\frac{1}{16}\frac{\zeta^{2}-\alpha^{2}}{\zeta^{1/3}[A^{2}(a)-A^{2}(x)]^{3}}\cdot$
$\displaystyle\hskip
34.14322pt\big{\\{}-8A^{2}(a)[A^{\prime}(x)^{2}+A(x)A^{\prime\prime}(x)]-12A^{2}(x)A^{\prime}(x)^{2}+8A^{3}(x)A^{\prime\prime}(x)\big{\\}}$
$\displaystyle+\frac{\zeta^{2}-\alpha^{2}}{\zeta^{1/3}[A^{2}(a)-A^{2}(x)]}\Bigg{\\{}\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)+A(a)}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)+A(a)}\Bigg{\\}}.$

This in addition to the asymptotics for $A$ in §2 and (5.2), implies that
$|\psi(\zeta,\alpha)|/\zeta^{\frac{1}{3}}$ is integrable at $\zeta=+\infty$
and hence the variation (6.9) is finite; in fact uniformly bounded in
$\alpha$. ∎

##  7\. An Example

In this section we illustrate the theory developed so far to the special case
of the potential $A(x)=\tfrac{1}{1+x^{2}}$, $x\in{\mathbb{R}}$. First, observe
that this particular potential $A$ satisfies the assumptions of §2; indeed

  * •

it is always positive, even and smooth,

  * •

it is increasing in $(-\infty,0]$ and decreasing in $[0,+\infty)$,

  * •

it has a maximum at $x=0$, namely $A_{max}=A(0)=1$,

  * •

$\|A\|_{L^{1}({\mathbb{R}})}=\pi$ and

  * •

if $\mu\in(0,1)$ the equation $A(x)=\mu$ gives the two zeros
$x_{\pm}=\pm\sqrt{\mu^{-1}-1}$ while for $\mu=1$ we get a double solution
$x=0$.

When $\mu\in[A_{0},1]$ for $A_{0}>0$ ($\mu=1$ corresponding to the critical
case), the parameter $a=x_{+}=\sqrt{\mu^{-1}-1}$ ranges over $[0,a_{0}]$ where
$a_{0}=\sqrt{A_{0}^{-1}-1}$ (the criticality now being $a=0$). The equation in
question is

$\frac{d^{2}y}{dx^{2}}=[\hbar^{-2}f(x,a)+g(x,a)]y$ (7.1)

where $f$ and $g$ satisfy

$f(x,a)=A^{2}(a)-A^{2}(x)=\frac{(x^{2}-a^{2})(x^{2}+a^{2}+2)}{[(1+a^{2})(1+x^{2})]^{2}}$
(7.2)

and

$\displaystyle g(x,a)$
$\displaystyle=\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)+A(a)}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)+A(a)}$
$\displaystyle=\frac{(1+a^{2})(-3x^{4}-2x^{2}+a^{2}+2)}{[(1+x^{2})(x^{2}+a^{2}+2)]^{2}}.$
(7.3)

The function $p$ that satisfies $f(x,a)=(x^{2}-a^{2})p(x,a)$ is

$p(x,a)=\frac{x^{2}+a^{2}+2}{[(1+a^{2})(1+x^{2})]^{2}}.$ (7.4)

For the non-critical case, applying the Liouville transform

$Y=\dot{x}^{-\frac{1}{2}}y,\quad\dot{x}^{2}f(x,a)=\zeta^{2}-\alpha^{2},$

where $\alpha\in(0,\alpha_{0}]$ in which $\alpha_{0}>0$ satisfies

$\displaystyle\alpha_{0}^{2}$
$\displaystyle=\frac{2}{\pi}\int_{-a_{0}}^{a_{0}}[-f(t,a_{0})]^{\frac{1}{2}}dt$
$\displaystyle=\frac{4}{\pi(1+a_{0}^{2})}\int_{0}^{a_{0}}\frac{\sqrt{(a_{0}^{2}-t^{2})(t^{2}+a_{0}^{2}+2)}}{1+t^{2}}dt,$

(cf. (4.4)) we get

$\frac{1}{1+a^{2}}\int_{x}^{-a}\frac{\sqrt{(t^{2}-a^{2})(t^{2}+a^{2}+2)}}{1+t^{2}}dt=\\\
-\frac{1}{2}\alpha^{2}\operatorname{arcosh}\Big{(}-\frac{\zeta}{\alpha}\Big{)}-\frac{1}{2}\zeta\big{(}\zeta^{2}-\alpha^{2}\big{)}^{\frac{1}{2}}\quad\text{for}\quad
x\leq-a$

(cf. (4.6)) and

$\frac{1}{1+a^{2}}\int_{-a}^{x}\frac{\sqrt{(a^{2}-t^{2})(t^{2}+a^{2}+2)}}{1+t^{2}}dt=\\\
\frac{1}{2}\alpha^{2}\arccos\Big{(}-\frac{\zeta}{\alpha}\Big{)}+\frac{1}{2}\zeta\big{(}\alpha^{2}-\zeta^{2}\big{)}^{\frac{1}{2}}\quad\text{for}\quad-a\leq
x\leq a$

(cf. (4.5)), in which the inverse cosine takes its principal value (i.e. the
value in $[0,\pi]$) and

$\frac{1}{1+a^{2}}\int_{a}^{x}\frac{\sqrt{(t^{2}-a^{2})(t^{2}+a^{2}+2)}}{1+t^{2}}dt=\\\
-\frac{1}{2}\alpha^{2}\operatorname{arcosh}\Big{(}\frac{\zeta}{\alpha}\Big{)}+\frac{1}{2}\zeta\big{(}\zeta^{2}-\alpha^{2}\big{)}^{\frac{1}{2}}\quad\text{for}\quad
x\geq a$

(cf. (4.7)). Additionally, equation (7.1) is transformed to

$\frac{d^{2}Y}{d\zeta^{2}}=\big{[}\hbar^{-2}(\zeta^{2}-\alpha^{2})+\psi(\zeta,\alpha)\big{]}Y$

where

$\psi(\zeta,\alpha)=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}}{(\zeta^{2}-\alpha^{2})^{2}}-(1+a^{2})^{4}(\zeta^{2}-\alpha^{2})\frac{5x^{6}+9x^{4}+3x^{2}+a^{4}+2a^{2}}{[(x^{2}-a^{2})(x^{2}+a^{2}+2)]^{3}}\\\
+(1+a^{2})^{3}(\zeta^{2}-\alpha^{2})\frac{-3x^{4}-2x^{2}+a^{2}+2}{(x^{2}-a^{2})(x^{2}+a^{2}+2)^{3}}.$
(7.5)

In the critical case ($a=\alpha=0$) we have

$\int_{0}^{x}\frac{t\sqrt{2+t^{2}}}{1+t^{2}}dt=\frac{1}{2}\zeta^{2}\quad\text{for}\quad
x\in{\mathbb{R}}$

(cf. (4.11) and (4.12)) and

$\psi(\zeta,0)=\frac{3}{4}\frac{1}{\zeta^{2}}-\zeta^{2}\frac{3x^{6}+7x^{4}+7x^{2}+3}{x^{4}(x^{2}+2)^{3}}.$
(7.6)

We note that all the integrals above can be explicitly evaluated in terms of
elliptic integrals.

From (7.4) we have

$\displaystyle\frac{\partial p}{\partial x}(x,a)$
$\displaystyle=\frac{2x(-x^{2}-2a^{2}-3)}{(1+a^{2})^{2}(1+x^{2})^{3}}$
$\displaystyle\frac{\partial^{2}p}{\partial x^{2}}(x,a)$
$\displaystyle=\frac{2(3x^{4}+10a^{2}x^{2}+12x^{2}-2a^{2}-3)}{(1+a^{2})^{2}(1+x^{2})^{4}}$
$\displaystyle\frac{\partial^{3}p}{\partial x^{3}}(x,a)$
$\displaystyle=\frac{24x(-x^{4}-5a^{2}x^{2}-5x^{2}+3a^{2}+4)}{(1+a^{2})^{2}(1+x^{2})^{5}}$

and from (7.2) we have

$\frac{\partial f}{\partial a}(x,a)=-\frac{4a}{(1+a^{2})^{3}}.$

Hence, these last observations about $f$ and $p$ along with (7) clearly show
that

  * (i)

$p,\frac{\partial p}{\partial x},\frac{\partial^{2}p}{\partial x^{2}}$ and $g$
are continuous functions in the region ${\mathbb{R}}\times[0,a_{0}]$

  * (ii)

$p$ is positive in ${\mathbb{R}}\times[0,a_{0}]$

  * (iii)

$|\frac{\partial^{3}p}{\partial x^{3}}|(0,0)=0$ and

  * (iv)

$\frac{\partial f}{\partial a}<0$ when $x\in[-a,a]$ and $a\in(0,a_{0}]$.

As argued in §5, these four properties imply that $\psi$ is continuous in the
region ${\mathbb{R}}\times[0,\alpha_{0}]$ of the $(\zeta,\alpha)$-plane.

Now, the variation integral

$\int_{0}^{+\infty}\frac{|\psi(t,\alpha)|}{\Omega(t\sqrt{2\hbar^{-1}})}dt$

where $\psi$ is given by (7.5) or (7.6) and $\Omega(x)=1+|x|^{\frac{1}{3}}$,
converges uniformly for $\alpha\in[0,\alpha_{0}]$ as $\hbar\downarrow 0$. So
we can obtain the two specific approximate solutions guaranteed by Theorem
6.3.

##  8\. Asymptotic Behavior of Solutions

In order to deduce the asymptotic behavior of the solutions
$Y_{1}(\zeta,\alpha,\hbar),Y_{2}(\zeta,\alpha,\hbar)$ when $\hbar\downarrow
0$, we need to determine the asymptotic form as $\hbar\downarrow 0$ of the
error bounds (6.7), (6.8) examining closely
$l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$ and
$\mathcal{V}_{0,+\infty}[H](\alpha,\hbar)$ for $\alpha\in[0,\alpha_{0}]$.

We start 333 The subsequent analysis follows the idea found in §6.2 of [16].
by investigating $l(b)$ as in (6.4) for $b\downarrow-\infty$. Take $\nu\geq 1$
to be a large positive number and set $b=-\frac{1}{2}\nu^{2}$ and $x=\nu
y\sqrt{2}$. Then by (B.8), (B.5) and (B.6) the quantity

$\frac{\mathsf{M}(\nu
y\sqrt{2},-\frac{1}{2}\nu^{2})^{2}}{\Gamma(\tfrac{1}{2}+\frac{1}{2}\nu^{2})}$
(8.1)

is equal to

$\sqrt{\frac{16\pi\nu^{-\frac{2}{3}}\eta}{y^{2}-1}}\cdot\begin{cases}\Big{[}\operatorname{Ai}^{2}(\nu^{\frac{4}{3}}\eta)+\operatorname{Bi}^{2}(\nu^{\frac{4}{3}}\eta)+E^{2}(\nu^{\frac{4}{3}}\eta)M^{2}(\nu^{\frac{4}{3}}\eta)\mathcal{O}(\nu^{-2})\Big{]},0\leq
y\leq\tfrac{\rho(-\frac{1}{2}\nu^{2})}{\nu\sqrt{2}}\\\ \\\
\Big{[}\operatorname{Ai}(\nu^{\frac{4}{3}}\eta)\operatorname{Bi}(\nu^{\frac{4}{3}}\eta)+M^{2}(\nu^{\frac{4}{3}}\eta)\mathcal{O}(\nu^{-2})\Big{]},y\geq\tfrac{\rho(-\frac{1}{2}\nu^{2})}{\nu\sqrt{2}}\end{cases}$

where in each case, the estimate $\mathcal{O}(\nu^{-2})$ is uniform with
respect to $y$. Using (B.7) we see that

$\frac{\rho(-\frac{1}{2}\nu^{2})}{\nu\sqrt{2}}=1+2^{-\frac{1}{3}}c_{*}\nu^{-\frac{4}{3}}+\mathcal{O}(\nu^{-\frac{8}{3}})\quad\text{as}\quad\nu\to+\infty$

and consequently
$\nu^{\frac{4}{3}}\eta=c_{*}+\mathcal{O}(\nu^{-\frac{4}{3}})$. But $E$ is
bounded in $[0,c_{*}+\mathcal{O}(\nu^{-\frac{4}{3}})]$. Hence we may write
(8.1) as

$\frac{4\sqrt{\pi}}{\nu^{\frac{1}{3}}}\Big{(}\frac{\eta}{y^{2}-1}\Big{)}^{\frac{1}{2}}\cdot\begin{cases}\Big{[}\operatorname{Ai}^{2}(\nu^{\frac{4}{3}}\eta)+\operatorname{Bi}^{2}(\nu^{\frac{4}{3}}\eta)+M^{2}(\nu^{\frac{4}{3}}\eta)\mathcal{O}(\nu^{-2})\Big{]},\hskip
3.0pt0\leq y\leq\frac{\rho(-\frac{1}{2}\nu^{2})}{\nu\sqrt{2}}\\\ \\\
\Big{[}\operatorname{Ai}(\nu^{\frac{4}{3}}\eta)\operatorname{Bi}(\nu^{\frac{4}{3}}\eta)+M^{2}(\nu^{\frac{4}{3}}\eta)\mathcal{O}(\nu^{-2})\Big{]},\hskip
3.0pty\geq\frac{\rho(-\frac{1}{2}\nu^{2})}{\nu\sqrt{2}}\end{cases}$

where the $\mathcal{O}$-terms are again uniform in $y$.

Next, we employ the asymptotic approximations for the functions
$\operatorname{Ai},\operatorname{Bi}$ and $M$ (cf. section A in appendix) so
that for $y\geq 1$ we obtain

$\frac{\mathsf{M}(\nu
y\sqrt{2},-\frac{1}{2}\nu^{2})^{2}}{\Gamma(\tfrac{1}{2}+\frac{1}{2}\nu^{2})}\leq\frac{C}{\nu^{\frac{1}{3}}}\Big{(}\frac{\eta}{y^{2}-1}\Big{)}^{\frac{1}{2}}\frac{1}{1+\nu^{\frac{2}{3}}\eta^{\frac{1}{2}}}$
(8.2)

where $C$ denotes a positive constant, used generically in what follows. By
(B.4) we have $\eta\sim(\tfrac{3}{4})^{\frac{2}{3}}y^{\frac{4}{3}}$ as
$y\to+\infty$, whence for $y\geq 0$ the estimate

$\Big{(}\frac{\eta}{y^{2}-1}\Big{)}^{\frac{1}{2}}\leq\frac{C}{1+\eta^{\frac{1}{4}}}.$
(8.3)

Also, from (6.3) we have

$\Omega(\nu y\sqrt{2})\leq C(1+\nu^{\frac{1}{3}}y^{\frac{1}{3}})\leq
C\nu^{\frac{1}{3}}(1+\eta^{\frac{1}{4}}).$ (8.4)

Finally, combining (8.2), (8.3) and (8.4) we get

$\Omega(\nu y\sqrt{2})\frac{\mathsf{M}(\nu
y\sqrt{2},-\frac{1}{2}\nu^{2})^{2}}{\Gamma(\tfrac{1}{2}+\frac{1}{2}\nu^{2})}\leq
C\frac{1}{1+\nu^{\frac{2}{3}}\eta^{\frac{1}{2}}}\leq C$

implying that

$l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})=\mathcal{O}(1)\quad\text{as}\quad\hbar\downarrow
0.$ (8.5)

Next, we examine $\mathcal{V}_{0,+\infty}[H](\alpha,\hbar)$. In the proof of
Theorem 6.3 we showed that $|\psi(\zeta,\alpha)|/\zeta^{\frac{1}{3}}$ is
integrable at $\zeta=+\infty$ uniformly with respect to $\alpha$. Thus

$\displaystyle\mathcal{V}_{0,+\infty}[H](\alpha,\hbar)\leq C\int_{0}^{1}$
$\displaystyle\frac{dt}{1+(t\sqrt{2\hbar^{-1}})^{\frac{1}{3}}}$ (8.6)
$\displaystyle+\Big{(}\frac{\hbar}{2}\Big{)}^{\frac{1}{6}}\int_{1}^{+\infty}\frac{|\psi(t,\alpha)|}{t^{\frac{1}{3}}}dt=\mathcal{O}(\hbar^{1/6})\quad\text{as}\quad\hbar\downarrow
0.$

The last two relations applied to (6.7) and (6.8) supply us with the required
results 444 Observe that since $\psi(\zeta,\alpha)$ is integrable at
$\zeta=+\infty$ the same results could be achieved by demanding $\Omega(x)=1$
for all $x$. We chose to present the general case since it is more broadly
applicable.

$\displaystyle\varepsilon_{1}(\zeta,\alpha,\hbar)$
$\displaystyle=\frac{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\mathcal{O}(\hbar^{\frac{2}{3}})$
(8.7) $\displaystyle\varepsilon_{2}(\zeta,\alpha,\hbar)$
$\displaystyle=\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\frac{\partial\varepsilon_{1}}{\partial\zeta}(\zeta,\alpha,\hbar)$
$\displaystyle=\frac{\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\mathcal{O}(\hbar^{\frac{1}{6}})$
$\displaystyle\frac{\partial\varepsilon_{2}}{\partial\zeta}(\zeta,\alpha,\hbar)$
$\displaystyle=\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{O}(\hbar^{\frac{1}{6}})$

as $\hbar\downarrow 0$ uniformly for $\zeta\geq 0$ and
$\alpha\in(0,\alpha_{0}]$.

######  Remark 8.1.

The special case $\alpha=0$ (i.e. when equation (4.8) has a double turning
point at $\zeta=0$) satisfies the same estimates. Indeed, as in the proof of
Theorem 6.3, $|\psi(\zeta,0)|/\zeta^{\frac{1}{3}}$ is integrable at
$\zeta=+\infty$. Furthermore $l(0)$ is independent of $\hbar$ and from the
formula $\Omega(x)=1+|x|^{\frac{1}{3}}$ we see that the estimates above remain
unchanged.

##  9\. Connection Formulae

The results obtained so far are somewhat inadequate because Theorem 6.3
defines the character of solutions of equation (4.8) only for non-negative
values of $\zeta$. Indeed, we are incapable of constructing error bounds like
those ones in (6.7) and (6.8) for negative $\zeta$, a drawback pertinent to
the nature of parabolic cylinder functions (cf. Miller’s [13]).

Consider $Y_{1}$ for example 555 Similar thinking can be argued for $Y_{2}$ as
well.  . As $\hbar\downarrow 0$ in a continuous manner, the asymptotic
behavior of its approximant
$U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$ at
$\zeta=-\infty$, changes abruptly as $\hbar^{-1}\alpha^{2}$ goes through odd
positive integers (cf. exceptional values in appendix B). $Y_{1}$ on the other
hand is not expected to exhibit the same change at exactly the same values of
$\hbar^{-1}\alpha^{2}$.

But we can determine the asymptotic behavior of $Y_{1},Y_{2}$ for small
$\hbar>0$ and $\zeta<0$ by establishing appropriate connection formulae. Since
$|\psi(\zeta,\alpha)|/|\zeta|^{\frac{1}{3}}$ is integrable at
$\zeta=\pm\infty$ uniformly with respect to $\alpha$, we can replace $\zeta$
by $-\zeta$ and appeal to Theorem 6.3 to ensure two more solutions
$Y_{3},Y_{4}$ of equation (4.8) satisfying

$\displaystyle Y_{3}(\zeta,\alpha,\hbar)$
$\displaystyle=U(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})+\frac{\mathsf{M}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\mathcal{O}(\hbar^{\frac{2}{3}})$
(9.1) $\displaystyle Y_{4}(\zeta,\alpha,\hbar)$
$\displaystyle=\overline{U}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})+$
$\displaystyle\hskip
56.9055pt\mathsf{E}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathsf{M}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{O}(\hbar^{\frac{2}{3}})$
(9.2)

as $\hbar\downarrow 0$ uniformly for $\zeta\leq 0$ and
$\alpha\in[0,\alpha_{0}]$. We express $Y_{1},Y_{2}$ in terms of $Y_{3},Y_{4}$
and write

$\displaystyle Y_{1}(\zeta,\alpha,\hbar)$
$\displaystyle=\sigma_{1}^{1}Y_{3}(\zeta,\alpha,\hbar)+\sigma_{1}^{2}Y_{4}(\zeta,\alpha,\hbar)$
(9.3) $\displaystyle Y_{2}(\zeta,\alpha,\hbar)$
$\displaystyle=\sigma_{2}^{1}Y_{3}(\zeta,\alpha,\hbar)+\sigma_{2}^{2}Y_{4}(\zeta,\alpha,\hbar).$
(9.4)

The connection will become clear once we find approximations for the
coefficients $\sigma_{i}^{j}$, $i,j=1,2$ in the linear relations (9.3) and
(9.4).

Evaluating at $\zeta=0$ both equations (9.3) and (9.4) and their derivatives,
after algebraic manipulations we obtain

$\sigma_{i}^{j}=(-1)^{j+1}\frac{\mathcal{W}[Y_{i}(\cdot,\alpha,\hbar),Y_{5-j}(\cdot,\alpha,\hbar)](0)}{\mathcal{W}[Y_{3}(\cdot,\alpha,\hbar),Y_{4}(\cdot,\alpha,\hbar)](0)}\quad\text{for}\quad
i,j=1,2.$ (9.5)

But using the results and properties of Parabolic Cylinder Functions and their
auxiliary functions from section B in the appendix, we find

$\displaystyle Y_{1}(0,\alpha,\hbar)$
$\displaystyle=\mathsf{M}(0)[\sin\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle Y_{2}(0,\alpha,\hbar)$
$\displaystyle=\mathsf{M}(0)[\cos\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle Y_{3}(0,\alpha,\hbar)$
$\displaystyle=\mathsf{M}(0)[\sin\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle Y_{4}(0,\alpha,\hbar)$
$\displaystyle=\mathsf{M}(0)[\cos\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle\frac{\partial Y_{1}}{\partial\zeta}(0,\alpha,\hbar)$
$\displaystyle=-\sqrt{2\hbar^{-1}}\mathsf{N}(0)[\cos\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle\frac{\partial Y_{2}}{\partial\zeta}(0,\alpha,\hbar)$
$\displaystyle=\sqrt{2\hbar^{-1}}\mathsf{N}(0)[\sin\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle\frac{\partial Y_{3}}{\partial\zeta}(0,\alpha,\hbar)$
$\displaystyle=\sqrt{2\hbar^{-1}}\mathsf{N}(0)[\cos\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$
$\displaystyle\frac{\partial Y_{4}}{\partial\zeta}(0,\alpha,\hbar)$
$\displaystyle=-\sqrt{2\hbar^{-1}}\mathsf{N}(0)[\sin\varphi+\mathcal{O}(\hbar^{\frac{2}{3}})]$

as $\hbar\downarrow 0$ where $\varphi=(1+\hbar^{-1}\alpha^{2})\frac{\pi}{4}$.
Finally substituting these estimates in (9.5) we obtain

$\displaystyle\sigma_{1}^{1}$
$\displaystyle=\sin(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2})+\mathcal{O}(\hbar^{\frac{2}{3}})$
(9.6) $\displaystyle\sigma_{1}^{2}$
$\displaystyle=\cos(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2})+\mathcal{O}(\hbar^{\frac{2}{3}})$
(9.7) $\displaystyle\sigma_{2}^{1}$
$\displaystyle=\cos(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2})+\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\sigma_{2}^{2}$
$\displaystyle=-\sin(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2})+\mathcal{O}(\hbar^{\frac{2}{3}})$

as $\hbar\downarrow 0$ uniformly for $\alpha\in[0,\alpha_{0}]$.

##  10\. A Quantization Condition for Eigenvalues

In this section, we will derive information about the EVs of (3.1) by
assembling the results of the previous paragraphs. This process will be
facilitated by the equivalent equation (4.8) where EVs appear for those values
of $\alpha$ for which there exists a solution that is decaying at both
$\zeta=-\infty$ and $\zeta=+\infty$ of the real line. In the end, this
approach will help us establish a quantization condition for the EVs of the
Dirac operator $\mathfrak{D}_{\hbar}$ and their corresponding norming
constants. We have the following theorem.

######  Theorem 10.1.

Suppose that $\lambda=i\mu\in\\{i\kappa\mid\kappa\in[A_{0},A_{max}]\\}$ is an
EV of the operator $\mathfrak{D}_{\hbar}$ (see (3.2)) and consider the $a>0$
such that $\mu=A(a)$. There exists a non-negative integer $n$ (depending both
on $\mu$ and $\hbar$) for which the Bohr-Sommerfeld quantization condition is
satisfied, i.e.

$\displaystyle\int_{-a}^{a}\big{[}A^{2}(x)-\mu^{2}\big{]}^{1/2}dx=\pi\Big{(}n+\tfrac{1}{2}\Big{)}\hbar+\mathcal{O}(\hbar^{\frac{5}{3}})\quad\text{as}\quad\hbar\downarrow
0.$ (10.1)

Conversely, for every non-negative integer $n$ such that
$\pi(n+\frac{1}{2})\hbar\in[0,\tfrac{\pi}{2}\alpha_{0}^{2}]$ there exists a
unique $\mu$ and consequently an $a>0$ with $\mu=A(a)$ (where both $\mu$ and
$a$ depend on $n,\hbar$) satisying

$\Bigg{|}\displaystyle\int_{-a}^{a}\big{[}A^{2}(x)-\mu^{2}\big{]}^{1/2}dx-\pi\Big{(}n+\frac{1}{2}\Big{)}\hbar\Bigg{|}\leq
C\hbar^{\frac{5}{3}}$

with a constant $C$ depending neither on $n$ nor on $\hbar$; whence
$\lambda=i\mu$ is an EV of $\mathfrak{D}_{\hbar}$.

###### Proof.

For the first part of the theorem, we observe the following. By referring to
the asymptotic form of $Y_{1}(\zeta,\alpha,\hbar)$ as $\zeta\to+\infty$ and
the asymptotics for $Y_{3}(\zeta,\alpha,\hbar)$ and
$Y_{4}(\zeta,\alpha,\hbar)$ as $\zeta\to-\infty$ (see (6.5), (8.7), (9.1) and
(9.2)), equation (9.3) implies that in the presense of an EV, the coefficient
$\sigma_{1}^{2}$ has to be zero. Accordingly, by (9.7) we have

$\cos(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2})=\mathcal{O}(\hbar^{\frac{2}{3}})\quad\text{as}\quad\hbar\downarrow
0$

or equivalently, there is a non-negative integer such that

$\alpha^{2}=(2n+1)\hbar+\mathcal{O}(\hbar^{\frac{5}{3}})\quad\text{as}\quad\hbar\downarrow
0.$ (10.2)

In view of (4.4), this is exactly what we wanted.

To the converse now 666Here we follow Yafaev’s idea found in §4 of [20].. Let
us first deal with the existence. Define the map
$\mathsf{\Phi}:[0,a_{0}]\rightarrow{\mathbb{R}}$ by

$\mathsf{\Phi}(a):=\frac{\pi}{2}\alpha^{2}(a)=\int_{-a}^{a}[-f(t,a)]^{\frac{1}{2}}dt$
(10.3)

(cf. (4.4) and/or the LHS of (10.1)). Fix a non-negative integer $n$ such that
$\pi(n+1/2)\hbar$ belongs to a neighborhood of $\mathsf{\Phi}(\tilde{a})$
where $A(\tilde{a})=\tilde{\mu}$ and $\alpha(\tilde{a})=\tilde{\alpha}$. From
$(\ref{U-asympt})$ we know that the functions $Y_{1}$, $Y_{3}$ belong to
$L^{2}(\mathbb{R}_{+})$ and $L^{2}(\mathbb{R}_{-})$ respectively. Define the
function

$\sigma(a,\hbar):=\sigma_{1}^{2}(\alpha(a),\hbar).$ (10.4)

It is enough to show that $\sigma$ vanishes for some $a_{n}(\hbar)$ satisfying

$\Big{|}\mathsf{\Phi}\big{(}a_{n}(\hbar)\big{)}-\pi\Big{(}n+\frac{1}{2}\Big{)}\hbar\Big{|}\leq
C\hbar^{\frac{5}{3}}.$

and set

$\alpha(a_{n}(\hbar))=\alpha_{n}(\hbar).$

Using (4.4) and Leibniz’s rule we have

$\frac{d\mathsf{\Phi}}{da}(a)=-A(a)A^{\prime}(a)\int_{-a}^{a}\big{[}A^{2}(t)-A^{2}(a)\big{]}^{-1/2}dt>0.$
(10.5)

This result tells us that $\mathsf{\Phi}$ maps a neighborhood $(a_{1},a_{2})$
of $\tilde{a}$ in a one-to-one way onto the neighborhood
$(\mathsf{\Phi}(a_{1}),\mathsf{\Phi}(a_{2}))$ of $\mathsf{\Phi}(\tilde{a})$.
Let $\mathsf{X}=\mathsf{\Phi}(a)$, $a\in[0,a_{0}]$,
$\tilde{\mathsf{X}}=\mathsf{\Phi}(\tilde{a})$ and set

$\chi(\mathsf{X},\hbar):=\sigma\big{(}\mathsf{\Phi}^{-1}(\mathsf{X}),\hbar\big{)}-\cos(\hbar^{-1}\mathsf{X}),\quad\mathsf{X}\in\mathsf{\Phi}\big{(}[0,a_{0}]\big{)}.$

By definition (10.4) of $\sigma$ and (9.7) we have

$|\chi(\mathsf{X},\hbar)|\leq C\hbar^{\frac{2}{3}}$

for $\mathsf{X}$ in a neighborhood of $\tilde{\mathsf{X}}$, where once more
the constant $C$ is independent of $\hbar$ and $\mathsf{X}$.

With the above definitions, our equation now reads

$\displaystyle 0$ $\displaystyle=\sigma(a,\hbar)$
$\displaystyle=\chi(\mathsf{X},\hbar)+\cos(\hbar^{-1}\mathsf{X}).$

So this equation has to have a solution $\mathsf{X}_{n}(\hbar)$ satisfying the
estimate

$\Big{|}\mathsf{X}_{n}(\hbar)-\pi\Big{(}n+\frac{1}{2}\Big{)}\hbar\Big{|}\leq
C\hbar^{\frac{5}{3}}.$

A change of variables $s=\hbar^{-1}\mathsf{X}$ transforms our problem to the
equivalent assertion that equation

$\chi(\hbar s,\hbar)+\cos s=0$ (10.6)

has to have a solution with respect to $s$, namely $s_{n}(\hbar)$, such that

$\Big{|}s_{n}(\hbar)-\pi\Big{(}n+\frac{1}{2}\Big{)}\Big{|}\leq
C\hbar^{\frac{2}{3}}.$ (10.7)

But this is true because

$\chi(\hbar
s,\hbar)=\mathcal{O}(\hbar^{\frac{2}{3}})\quad\text{as}\quad\hbar\downarrow
0.$

To complete the proof of the theorem, we need uniqueness as well. Once again
fix $n\in\mathbb{Z}$. We have just proved that for this $n$, equation (10.6)
has a solution obeying (10.7). We shall employ reductio ad absurdum. Suppose,
on the contrary, that there are $s_{1},s_{2}$ \- with $s_{1}<s_{2}$ \-
satisfying (10.7) so that the function

$g(s):=\chi(\hbar s,\hbar)+\cos s,\quad s\in[s_{1},s_{2}]$

is zero; $g(s_{1})=g(s_{2})=0$. Furthermore, $g$ is continuous in
$[s_{1},s_{2}]$ and differentiable in $(s_{1},s_{2})$ with

$g^{\prime}(s)=\hbar\frac{\partial\chi}{\partial\mathsf{X}}(\hbar
s,\hbar)-\sin s,\quad s\in(s_{1},s_{2}).$

By Rolle’s theorem there is $\tilde{s}\in(s_{1},s_{2})$ such that

$\displaystyle 0$ $\displaystyle=g^{\prime}(\tilde{s})$
$\displaystyle=\hbar\frac{\partial\chi}{\partial\mathsf{X}}(\hbar\tilde{s},\hbar)-\sin\tilde{s}.$

Recapping, we have found

  * •

$\tilde{s}\in(s_{1},s_{2})$ which says that $\tilde{s}$ satisfies (10.7) too;
namely

$\tilde{s}=\pi\Big{(}n+\frac{1}{2}\Big{)}+\mathcal{O}(\hbar^{\frac{2}{3}})\quad\text{as}\quad\hbar\downarrow
0$ (10.8)

  * •

$\tilde{s}$ is a root of the equation

$\sin s=\hbar\frac{\partial\chi}{\partial\mathsf{X}}(\hbar\tilde{s},\hbar).$
(10.9)

Using (10.8), the left-hand side of (10.9) is seen to be $(-1)^{n}$ as
$\hbar\downarrow 0$. Now, using (9.7) observe that

$\frac{\partial\sigma}{\partial
a}(a,\hbar)=-\hbar^{-1}\mathsf{\Phi}^{\prime}(a)\sin\Big{[}\hbar^{-1}\mathsf{\Phi}(a)\Big{]}+\mathcal{O}(1)\quad\text{as}\quad\hbar\downarrow
0$

which eventually leads to

$\frac{\partial\chi}{\partial\mathsf{X}}(\mathsf{X},\hbar)=\mathcal{O}(1)\quad\text{as}\quad\hbar\downarrow
0.$

Hence the right-hand side of (10.9) is $\mathcal{O}(\hbar)$ as
$\hbar\downarrow 0$ which is a contradiction. Thus, there is only one such
eigenvalue. ∎

######  Remark 10.2.

A result like equation (10.1) can also be found in [20] for the Schrödinger
operator, with the slightly better asymptotic estimate of order $\hbar^{2}$.
Although the result we provide here is only
$\mathcal{O}(\hbar^{\frac{5}{3}})$, it has the additional advantage of holding
for the critical case of a double turning point as well.

The following corollary is a straightforward application of the Theorem 10.1
giving the number of EVs of the Dirac operator $\mathfrak{D}_{\hbar}$ in a
fixed (independent of $\hbar$) interval not containing 0, on the imaginary
axis.

######  Corollary 10.3.

Consider an interval $(\mu_{1},\mu_{2})\subset[A_{0},A_{max}]$ and take
$a_{j}$, $j=1,2$ such that $A(a_{j})=\mu_{j}$ for $j=1,2$. Then the total
number $\mathcal{N}_{\hbar}$ of eigenvalues $\lambda=i\mu$ of the Dirac
operator $\mathfrak{D}_{\hbar}$ lying in the set
$\\{i\mu\mid\mu\in(\mu_{1},\mu_{2})\\}\subset\mathbb{C}$ is equal to

$\mathcal{N}_{\hbar}=\pi^{-1}\big{[}\mathsf{\Phi}(a_{1})-\mathsf{\Phi}(a_{2})\big{]}\hbar^{-1}+R(\hbar)$
(10.10)

where $|R(\hbar)|\leq 1$ for sufficiently small $\hbar$.

$\mu$-space$\mu_{2}$$\mu_{1}$$\mu$$a_{1}$$a_{2}$$a$$a$-space$\mathsf{X}$-space$\mathsf{\Phi}(a_{1})$$\mathsf{\Phi}(a_{2})$$\pi(n+\tfrac{1}{2})\hbar$$\mathsf{\Phi}$
Figure 3. Counting eigenvalues using the Bohr-Sommerfeld condition.

###### Proof.

Observe that

$\mathcal{N}_{\hbar}=\\#\big{\\{}a\in(a_{1},a_{2})\mid\text{equation
(\ref{schrodi_final-version}) with}\hskip 4.26773pt\mu=A(a)\hskip
4.26773pt\text{has an}\hskip 4.26773ptL^{2}({\mathbb{R}})\hskip
4.26773pt\text{solution}\big{\\}}$

By Theorem 10.1, there is only one $a$-eigenvalue in a neighborhood of length
$C\hbar^{\frac{5}{3}}$ of every point
$\mathsf{\Phi}^{-1}\big{(}\pi(n+1/2)\hbar\big{)}$. For sufficiently small
$\hbar$ these neighborhoods are mutually disjoint. But this means that the
number $\mathcal{N}_{\hbar}$ is equal to the number of the points
$\pi(n+1/2)\hbar$ that lie in the interval
$\big{(}\mathsf{\Phi}(a_{2}),\mathsf{\Phi}(a_{1})\big{)}$, i.e.

$\mathcal{N}_{\hbar}=\\#\big{\\{}n\in{\mathbb{Z}}\big{|}\pi(n+1/2)\hbar\in\big{(}\mathsf{\Phi}(a_{2}),\mathsf{\Phi}(a_{1})\big{)}\big{\\}}$

for sufficiently small $\hbar$. And this number is exactly
$\pi^{-1}\big{[}\mathsf{\Phi}(a_{1})-\mathsf{\Phi}(a_{2})\big{]}\hbar^{-1}+R(\hbar)$
with $|R(\hbar)|\leq 1$. ∎

######  Remark 10.4.

(Weyl’s formula) Using the definition (10.3), we can write $\mathsf{\Phi}$ in
a different way. Indeed, we have

$\displaystyle\mathsf{\Phi}(a)$
$\displaystyle=\int_{-a}^{a}\Big{[}A^{2}(x)-\mu^{2}\Big{]}^{1/2}dx=\frac{1}{2}\int_{-a}^{a}2\Big{[}A^{2}(x)-\mu^{2}\Big{]}^{1/2}dx=$
$\displaystyle=\frac{1}{2}\iint_{A^{2}(x)-k^{2}\geq\mu^{2}}dkdx$

With the help of this last equality, the difference
$\mathsf{\Phi}(a_{1})-\mathsf{\Phi}(a_{2})$ in (10.10) can be equivalently
written as:

$\displaystyle\mathsf{\Phi}(a_{1})-\mathsf{\Phi}(a_{2})$
$\displaystyle=\frac{1}{2}\iint_{A^{2}(x)-k^{2}\geq\mu_{1}^{2}}dkdx-\frac{1}{2}\iint_{A^{2}(x)-k^{2}\geq\mu_{2}^{2}}dkdx$
$\displaystyle=\frac{1}{2}\iint_{\mu_{1}^{2}\leq
A^{2}(x)-k^{2}\leq\mu_{2}^{2}}dkdx$ $\displaystyle=\frac{1}{2}\cdot
Area\Big{(}\\{(x,k)\in\mathbb{R}^{2}\big{|}\mu_{1}^{2}\leq
A^{2}(x)-k^{2}\leq\mu_{2}^{2}\\}\Big{)}$

which means that the asymptotic coefficient in (10.10) is the area of a region
in the phase space. Consequently, relation (10.10) is the WKB analogue of
Weyl’s formula with a strong estimate on the remainder.

Another straightforward application of Theorem 10.1 allows us to express the
norming constants of the Dirac operator $\mathfrak{D}_{\hbar}$. In particular
we see that the asymptotics obtained agree with the known fact that (because
of the symmetry of the potential, see Chapter 3 of [9]) the corresponding
norming constant is exactly $(-1)^{n}$ for some integer $n$. But of course,
our method is easily extensible to the non-symmetric case, which we will in
fact consider in a sequel to this paper. We have the following corollary.

######  Corollary 10.5.

Suppose that $\lambda(\hbar)$ is an EV of $\mathfrak{D}_{\hbar}$. Then there
is a non-negative integer $n$ (depending both on $\lambda$ and $\hbar$) such
that the corresponding norming constant has asymptotics

$(-1)^{n}+\mathcal{O}(\hbar^{\frac{2}{3}})\quad\text{as}\quad\hbar\downarrow
0.$

###### Proof.

If $\lambda(\hbar)=i\mu(\hbar)$ then by Theorem 10.1, (10.2) and (9.6) there
is a non-negative integer $n$ such that

$\sigma_{1}^{1}=(-1)^{n}+\mathcal{O}(\hbar^{\frac{2}{3}})\quad\text{as}\quad\hbar\downarrow
0$

where $n$ is the same as in (10.2). Thus for the corresponding
$\alpha$-eigenvalue, namely $\alpha(\hbar)$, we have

$Y_{1}(\zeta,\alpha(\hbar),\hbar)=\big{[}(-1)^{n}+\mathcal{O}(\hbar^{\frac{2}{3}})\big{]}Y_{3}(\zeta,\alpha(\hbar),\hbar)\quad\text{as}\quad\hbar\downarrow
0$

and this proves the assertion. ∎

##  11\. Eigenvalues Near Zero

It is important for the applications to the semiclassical theory of the
focusing NLS equation, to understand the behavior of the EVs near 0\. As done
in [6] we will examine here two specific -but quite inclusive- families of
data $A$: the asymptotically rational case and the asymptotically exponential
case. We do note however that the formula (11.3) we have for the function
$\psi$, makes it very easy to check if the behavior of the EVs near zero is
good enough for any family of $A$, defined by explicit prescribed asymptotics
at infinity 777 Indeed, apart from the the asymptotically rational case and
the asymptotically exponential case presented here, we have also done so for
functions $A$ with asymptotics $x^{n}\exp(-|x|^{d})$ and
$(\log|x|)^{n}|x|^{-d-1}$ where $n\geq 0$ is an integer and
$d\in{\mathbb{R}}_{+}$.  .

To begin with, we choose $b>0$ independent of $\hbar$ and set
$\mu(\hbar)=\hbar^{b}$. Consider the equation $\mu(\hbar)=A(a(\hbar))$.
Observe that $\mu(\hbar)\downarrow 0$ as $\hbar\downarrow 0$ while
$a(\hbar)\uparrow+\infty$ as $\hbar\downarrow 0$. Furthermore, (recalling the
notation of section §3) $0<a_{0}(\hbar)=B(\hbar^{b})$ where $B$ is the inverse
of $A\big{|}_{[0,+\infty)}$. Consequently, if we use the abbreviation

$\alpha(\hbar)\equiv\alpha(a(\hbar))$

the above can be translated in the $\alpha$-space as
$0<\alpha_{0}(\hbar)\equiv\alpha(a_{0}(\hbar))<+\infty$; observe that
$B(\hbar^{b})\uparrow+\infty$ as $\hbar\downarrow 0$ and hence by the
definition of $\alpha$ (see (4.4)) we obtain

$\alpha(\hbar)\uparrow\big{(}\tfrac{2}{\pi}\|A\|_{L^{1}({\mathbb{R}})}\big{)}^{\frac{1}{2}}\quad\text{as}\quad\hbar\downarrow
0.$

In this setting, $a$ depends on $\hbar$ and using (3.8), (3.9) and (5.1) we
have

$f\big{(}x,a(\hbar)\big{)}=A^{2}\big{(}a(\hbar)\big{)}-A^{2}(x)$ (11.1)
$g\big{(}x,a(\hbar)\big{)}=\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)+A\big{(}a(\hbar)\big{)}}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)+A\big{(}a(\hbar)\big{)}}$
(11.2)

and

$f(x,a(\hbar))=[x^{2}-a^{2}(\hbar)]p(x,a(\hbar)).$

It is easy to see that for each value of $\hbar$ the functions $f$, $g$ and
$p$ satisfy properties $(i)$ through $(iv)$ of Lemma 5.1 in §5. This implies
-again with the help of Lemma I in [16]\- that for each $\hbar$ the function

$\psi(\zeta,\alpha(\hbar))=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}(\hbar)}{[\zeta^{2}-\alpha^{2}(\hbar)]^{2}}+\frac{1}{16}\frac{\zeta^{2}-\alpha^{2}(\hbar)}{f^{3}(x,a(\hbar))}\\\
\cdot\Big{\\{}4f(x,a(\hbar))f^{\prime\prime}(x,a(\hbar))-5[f^{\prime}(x,a(\hbar))]^{2}\Big{\\}}+[\zeta^{2}-\alpha^{2}(\hbar)]\frac{g(x,a(\hbar))}{f(x,a(\hbar))}$
(11.3)

is continuous in the corresponding region of the $(\zeta,\alpha)$-plane.

So in order to have a conclusion such as Theorem 6.3 -and eventually a result
like Theorem 10.1\- we need to investigate the convergence of the integral in
(6.9) but now with an $\alpha$ that depends on $\hbar$, i.e.

$\int_{0}^{+\infty}\frac{|\psi(t,\alpha(\hbar))|}{\Omega(t\sqrt{2\hbar^{-1}})}dt.$
(11.4)

Before considering our two specific families of data, we would like to remind
the reader of Lemma 5.2 and especially the formula (5.2); an asymptotic
relation which now reads

$x=\frac{\zeta^{2}}{2A(a(\hbar))}\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\uparrow+\infty.$ (11.5)

It shall be used to allow us understand the nature of $\psi$ for big $\zeta$.

###  11.1. The Rational Case

For the moment, assume for simplicity that in addition to the assumptions of
paragragh §2, $A$ also satisfies

$A(x)=\frac{1}{|x|^{d}}\quad\text{for}\quad|x|\geq 1$ (11.6)

where $d>1$. In this case, using (11.5) we get

$x=\frac{1}{2}a^{d}(\hbar)\zeta^{2}\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\uparrow+\infty.$ (11.7)

Using (11.1), (11.2), (11.3), (11.6) and (11.7) we arrive at

$\psi(\zeta,\alpha(\hbar))=\psi_{1}(\zeta,\alpha(\hbar))\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\uparrow+\infty$ (11.8)

uniformly in $\alpha$ and consequently in $\hbar$, where

$\displaystyle\psi_{1}(\zeta,\alpha(\hbar))$
$\displaystyle=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}(\hbar)}{[\zeta^{2}-\alpha^{2}(\hbar)]^{2}}$
$\displaystyle-\frac{d}{4^{d}}a^{2d^{2}+2d}(\hbar)\zeta^{4d-4}[\zeta^{2}-\alpha^{2}(\hbar)]\frac{\frac{2(2d+1)}{4^{d}}a^{2d^{2}}(\hbar)\zeta^{4d}+(d-2)a^{2d}(\hbar)}{[\frac{1}{4^{d}}a^{2d^{2}}(\hbar)\zeta^{4d}-a^{2d}(\hbar)]^{3}}$
$\displaystyle+\frac{d}{4^{d}}a^{2d^{2}+d}(\hbar)\zeta^{4d-4}[\zeta^{2}-\alpha^{2}(\hbar)]\frac{-\frac{d+1}{2^{d-1}}a^{d^{2}}(\hbar)\zeta^{2d}+(d-2)a^{d}(\hbar)}{[\frac{a^{d^{2}}(\hbar)}{2^{d}}\zeta^{2d}-a^{d}(\hbar)][\frac{a^{d^{2}}(\hbar)}{2^{d}}\zeta^{2d}+a^{d}(\hbar)]^{3}}.$

Similar considerations to the ones just presented can be applied to a more
general $A$ and the result still remains the same. Namely, we can take a
function $A$ satisfying the assumptions of §2 and such that

$A(x)=C\frac{1}{|x|^{d}}+r(x)\quad\text{for}\quad|x|\geq 1$ (11.9)

where $C$ is a positive constant, $d>1$ and $r$ is a function satisfying the
assumptions of §2 along with the asymptotics $r(x)=o(|x|^{-d})$,
$r^{\prime}(x)=o(|x|^{-d-1})$ and $r^{\prime\prime}(x)=o(|x|^{-d-2})$ as
$x\to\pm\infty$ (observe that the potential in the example treated in
paragraph §7 belongs to this case). Hence from now on in this subsection, when
we write $A$ we mean the special one from (11.9).

The asymptotics above imply that for each $\hbar$, the integral in (11.4)
converges; furthermore, this convergence is uniform in $a(\hbar)$. This means
that a variation of Theorem 6.3 can be applied to guarantee the existence of
approximate functions in this case too. Indeed, Theorem C.2 comes into play
and guarantees that everything remains unchanged; for each value of $\hbar$,
equation (4.8), i.e.

$\frac{d^{2}Y}{d\zeta^{2}}=\big{[}\hbar^{-2}\big{(}\zeta^{2}-\alpha^{2}(\hbar)\big{)}+\psi(\zeta,\alpha(\hbar))\big{]}Y$
(11.10)

has in the region $[0,+\infty)\times[0,\alpha_{0}(\hbar)]$ of the
$(\zeta,\alpha)$-plane solutions $Y_{+}$ and $\overline{Y}_{+}$ (being
extensions in $\alpha$ of $Y_{1}$ and $Y_{2}$ respectively, cf. (6.5), (6.6))
which are continuous, have continuous first and second partial
$\zeta$-derivatives, and are given by

$\displaystyle
Y_{+}(\zeta,\alpha(\hbar),\hbar)=U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))+\varepsilon(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle\overline{Y}_{+}(\zeta,\alpha(\hbar),\hbar)=\overline{U}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))+\overline{\varepsilon}(\zeta,\alpha(\hbar),\hbar)$

(cf. (6.5), (6.6)) where for the remainders we have the relations

$\frac{|\varepsilon(\zeta,\alpha(\hbar),\hbar)|}{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))},\frac{\Big{|}\frac{\partial\varepsilon}{\partial\zeta}(\zeta,\alpha(\hbar),\hbar)\Big{|}}{\sqrt{2\hbar^{-1}}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}\\\
\leq\frac{1}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}\Big{(}\exp\big{\\{}\tfrac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathcal{V}_{\zeta,+\infty}[H](\alpha(\hbar),\hbar)\big{\\}}-1\Big{)}$
(11.11)

and

$\frac{|\overline{\varepsilon}(\zeta,\alpha(\hbar),\hbar)|}{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))},\frac{\Big{|}\frac{\partial\overline{\varepsilon}}{\partial\zeta}(\zeta,\alpha(\hbar),\hbar)\Big{|}}{\sqrt{2\hbar^{-1}}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}\\\
\leq\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\Big{(}\exp\big{\\{}\tfrac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathcal{V}_{0,\zeta}[H](\alpha(\hbar),\hbar)\big{\\}}-1\Big{)}$
(11.12)

(analogous to (6.7), (6.8)).

Additionally, $l$ and $\mathcal{V}_{0,+\infty}[H]$ satisfy the same
asymptotics as before (cf. (8.5), (8.6)) and consequently one obtains the same
asymptotic behavior of solutions as in §8; namely

$\displaystyle\varepsilon(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\frac{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\overline{\varepsilon}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\frac{\partial\varepsilon}{\partial\zeta}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\frac{\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}\mathcal{O}(\hbar^{\frac{1}{6}})$
$\displaystyle\frac{\partial\overline{\varepsilon}}{\partial\zeta}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathcal{O}(\hbar^{\frac{1}{6}})$

as $\hbar\downarrow 0$ uniformly for $\zeta\geq 0$ and
$\alpha(\hbar)\in[0,\alpha_{0}(\hbar)]$.

Arguing as in §9, we obtain two more solutions of (11.10), namely $Y_{-}$ and
$\overline{Y}_{-}$ (the equivalent of $Y_{3}$ and $Y_{4}$ correspondingly),
satisfying

$\displaystyle Y_{-}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=U(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))+\frac{\mathsf{M}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}{\mathsf{E}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))}\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\overline{Y}_{-}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\overline{U}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))+$
$\displaystyle\hskip
56.9055pt\mathsf{E}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathsf{M}(-\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}(\hbar))\mathcal{O}(\hbar^{\frac{2}{3}})$

as $\hbar\downarrow 0$ uniformly for $\zeta\leq 0$ and
$\alpha(\hbar)\in[0,\alpha_{0}(\hbar)]$.

Consequently we have the same connection formulae (all the results of §9 are
not altered at all). Indeed, expressing $Y_{+}$, $\overline{Y}_{+}$ in terms
of $Y_{-}$, $\overline{Y}_{-}$ and writing

$\displaystyle Y_{+}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\tau_{1}^{1}Y_{-}(\zeta,\alpha(\hbar),\hbar)+\tau_{1}^{2}\overline{Y}_{-}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle\overline{Y}_{+}(\zeta,\alpha(\hbar),\hbar)$
$\displaystyle=\tau_{2}^{1}Y_{-}(\zeta,\alpha(\hbar),\hbar)+\tau_{2}^{2}\overline{Y}_{-}(\zeta,\alpha(\hbar),\hbar)$

(confer (9.3), (9.4)) in the same way we find that

$\displaystyle\tau_{1}^{1}$
$\displaystyle=\sin(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2}(\hbar))+\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\tau_{1}^{2}$
$\displaystyle=\cos(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2}(\hbar))+\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\tau_{2}^{1}$
$\displaystyle=\cos(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2}(\hbar))+\mathcal{O}(\hbar^{\frac{2}{3}})$
$\displaystyle\tau_{2}^{2}$
$\displaystyle=-\sin(\tfrac{1}{2}\pi\hbar^{-1}\alpha^{2}(\hbar))+\mathcal{O}(\hbar^{\frac{2}{3}})$

(like (9.6), (9.7)) as $\hbar\downarrow 0$ uniformly for
$\alpha(\hbar)\in[0,\alpha_{0}(\hbar)]$.

Eventually, this means that the results of §10 for the EVs remain the same
(eg. Theorem 10.1 but this time for $\mu(\hbar)\in[\hbar^{b},A_{max}]$).
Hence, we arrive at the following theorem the proof of which has been already
provided in the previous paragraph (cf. Theorem 10.1).

######  Theorem 11.1.

Suppose that $\lambda(\hbar)=i\mu(\hbar)$, where
$\mu(\hbar)\in[\hbar^{b},A_{max}]$ for an arbitrary $\hbar$-independent
positive constant $0<b<\tfrac{5}{3}$, is an EV of the operator
$\mathfrak{D}_{\hbar}$ (see (3.2)). Consider $a(\hbar)$ such that
$\mu(\hbar)=A[a(\hbar)]$. Then there exists a non-negative integer $n$ for
which

$\displaystyle\int_{-a(\hbar)}^{a(\hbar)}\big{[}A^{2}(x)-\mu^{2}(\hbar)\big{]}^{1/2}dx=\pi\Big{(}n+\tfrac{1}{2}\Big{)}\hbar+\mathcal{O}(\hbar^{\frac{5}{3}})\quad\text{as}\quad\hbar\downarrow
0.$ (11.13)

Conversely, for every non-negative integer $n$ such that
$\pi(n+\frac{1}{2})\hbar\in$ $[0,\tfrac{\pi}{2}\alpha_{0}^{2}(\hbar)]$ there
exists a unique $\mu_{n}(\hbar)=A[a_{n}(\hbar)]$ satisying

$\Bigg{|}\displaystyle\int_{-a_{n}(\hbar)}^{a_{n}(\hbar)}\big{[}A^{2}(x)-\mu_{n}(\hbar)^{2}\big{]}^{1/2}dx-\pi\Big{(}n+\frac{1}{2}\Big{)}\hbar\Bigg{|}\leq
C\hbar^{\frac{5}{3}}$

with a constant $C$ depending neither on $n$ nor on $\hbar$.

Now, let us define $a_{n}^{WKB}(\hbar)$ such that

$\displaystyle\int_{-a_{n}^{WKB}(\hbar)}^{a_{n}^{WKB}(\hbar)}\big{[}A^{2}(x)-A^{2}(a_{n}^{WKB}(\hbar))\big{]}^{1/2}dx=\pi\Big{(}n+\tfrac{1}{2}\Big{)}\hbar$
(11.14)

and set

$\mu_{n}^{WKB}(\hbar):=A(a_{n}^{WKB}(\hbar))$ (11.15)

to be the WKB-approximant to an actual $\mu$-eigenvalue $\mu_{n}(\hbar)$
satisfying

$\mu_{n}(\hbar)=A(a_{n}(\hbar)).$

We have the following corollary.

######  Corollary 11.2.

Let $0<b<\frac{5}{3}$ (independent of $\hbar$) and consider a function $A$
satisfying the assumptions of §2 and (11.9). Then for every non-negative
integer $n$ such that $\pi(n+\frac{1}{2})\hbar$ belongs to
$(0,\tfrac{\pi}{2}\alpha_{0}^{2}(\hbar))$, there exists a unique
$\mu_{n}(\hbar)$ satisfying

$|\mu_{n}(\hbar)-\mu_{n}^{WKB}(\hbar)|=\mathcal{O}(\hbar^{\frac{5}{3}+\frac{b}{d}})\quad\text{as}\quad\hbar\downarrow
0$

uniformly for $\mu_{n}(\hbar)$ in $[\hbar^{b},A_{max}]$.

###### Proof.

Using the result of the previous Theorem, (11.9) and (10.5) we have

$\displaystyle|\mu_{n}(\hbar)-\mu_{n}^{WKB}(\hbar)|$
$\displaystyle=|A(a_{n}(\hbar))-A(a_{n}^{WKB}(\hbar))|$ $\displaystyle\leq
C|A^{\prime}(a)|[\mathsf{\Phi}^{\prime}(a)]^{-1}\hbar^{\frac{5}{3}}\quad(\text{for
some $a$ between}\hskip 2.0pta_{n}(\hbar),a_{n}^{WKB}(\hbar))$
$\displaystyle\leq Ca^{-d-1}a^{d}\hbar^{\frac{5}{3}}$ $\displaystyle\leq
C\hbar^{\frac{5}{3}+\frac{b}{d}}\quad(\text{since}\hskip
4.0pta^{-1}\sim\hbar^{\frac{b}{d}})$

where as usual $C$ denotes a generic constant. ∎

###  11.2. The Exponential Case

In this subsection we start with a function $A$ that satisfies the assumptions
of §2 and furthermore

$A(x)=e^{-|x|^{\delta}},\quad\text{for}\quad|x|\geq 1$ (11.16)

where $\delta>0$. Now using (11.5) we get

$x=\frac{1}{2}\zeta^{2}\exp(a^{\delta}(\hbar))\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\uparrow+\infty.$ (11.17)

To simplify notation, we set

$\Lambda(\hbar)\equiv\Lambda(a(\hbar)):=\frac{1}{2}\exp(a^{\delta}(\hbar))$
(11.18)

Using (11.1), (11.2), (11.3), (11.16), (11.17) and (11.18) we obtain

$\psi(\zeta,\alpha(\hbar))=\psi_{2}(\zeta,\alpha(\hbar))\Big{[}1+\mathcal{O}\Big{(}\tfrac{\log\zeta}{\zeta^{2}}\Big{)}\Big{]}\quad\text{as}\hskip
7.0pt\zeta\uparrow+\infty$ (11.19)

uniformly in $\alpha$ and consequently in $\hbar$, where

$\displaystyle\psi_{2}(\zeta,\alpha(\hbar))$
$\displaystyle=\frac{1}{4}\frac{3\zeta^{2}+2\alpha^{2}(\hbar)}{[\zeta^{2}-\alpha^{2}(\hbar)]^{2}}$
$\displaystyle+\frac{\delta}{4}\zeta^{2\delta-4}[\zeta^{2}-\alpha^{2}(\hbar)]\frac{\Lambda^{\delta-2}(\hbar)\exp\\{-2\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}}{[\Lambda^{-2}(\hbar)-\exp\\{-2\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}]^{3}}\cdot$
$\displaystyle\hskip
28.45274pt[-\delta\Lambda^{\delta}(\hbar)\zeta^{2\delta}\exp\\{-2\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}-2(\delta-1)\exp\\{-2\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}$
$\displaystyle\hskip
156.49014pt-4\delta\Lambda^{\delta-2}(\hbar)\zeta^{2\delta}+2(\delta-1)\Lambda^{-2}(\hbar)]$
$\displaystyle+\frac{\delta}{4}\zeta^{2\delta-4}[\zeta^{2}-\alpha^{2}(\hbar)]\cdot$
$\displaystyle\hskip
28.45274pt\frac{\Lambda^{\delta-2}(\hbar)\exp\\{-\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}}{[\Lambda^{-1}(\hbar)-\exp\\{-\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}][\Lambda^{-1}(\hbar)+\exp\\{-\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}]^{3}}\cdot$
$\displaystyle\hskip
56.9055pt[\delta\Lambda^{\delta}(\hbar)\zeta^{2\delta}\exp\\{-\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}+2(\delta-1)\exp\\{-\Lambda^{\delta}(\hbar)\zeta^{2\delta}\\}$
$\displaystyle\hskip
156.49014pt-2\delta\Lambda^{\delta-1}(\hbar)\zeta^{2\delta}+2(\delta-1)\Lambda^{-1}(\hbar)].$

Similar arguments can be applied to a more general function $A$ satisfying the
assumptions of §2 and such that

$A(x)=Ce^{-|x|^{\delta}}+r(x)\quad\text{for}\quad|x|\geq 1$ (11.20)

where $C$ is a positive constant and $r$ is a function satisfying the
assumptions of §2 along with the asymptotics $r(x)=o(e^{-|x|^{\delta}})$,
$r^{\prime}(x)=o(|x|^{\delta-1}e^{-|x|^{\delta}})$ and
$r^{\prime\prime}(x)=o(|x|^{2\delta-2}e^{-|x|^{\delta}})$ as $x\to\pm\infty$.
The rest of this subsection applies to all $A$ satisfying (11.20).

This result above implies that for each $\hbar$, the integral in (11.4)
converges; furthermore the convergence is uniform in $a(\hbar)$. As in the
rational case, this means that a variation of Theorem 6.3 can be applied to
guarantee the existence of approximate functions in this case as well. The
same analysis as in the previous subsection leads to a corollary about the EVs
that lie close to $0$ (here we use once again the symbolism of (11.14),
(11.15) but with the $A$ from (11.20)).

######  Corollary 11.3.

Let $0<b<\frac{5}{3}$ (independent of $\hbar$) and consider a function $A$
satisfying the assumptions of §2 and (11.20). Then for every non-negative
integer $n$ such that $\pi(n+\frac{1}{2})\hbar$ belongs to
$(0,\tfrac{\pi}{2}\alpha_{0}^{2}(\hbar))$ there exists a unique
$\mu_{n}(\hbar)$ satisfying

$|\mu_{n}(\hbar)-\mu_{n}^{WKB}(\hbar)|=\begin{cases}\mathcal{O}\big{(}\frac{\hbar^{5/3}}{\log\hbar}\big{)},\quad\text{if}\quad
0<\delta<1\\\ \mathcal{O}(\hbar^{5/3}),\quad\text{if}\quad\delta\geq
1\end{cases}\quad\text{as}\quad\hbar\downarrow 0$

uniformly for $\mu_{n}(\hbar)$ in $[\hbar^{b},A_{max}]$.

###### Proof.

Using Theorem 11.1 (applied to our current case), (11.20) and (10.5) we get

$\displaystyle|\mu_{n}(\hbar)-\mu_{n}^{WKB}(\hbar)|$
$\displaystyle=|A(a_{n}(\hbar))-A(a_{n}^{WKB}(\hbar))|$ $\displaystyle\leq
C|A^{\prime}(a)|[\mathsf{\Phi}^{\prime}(a)]^{-1}\hbar^{\frac{5}{3}}\quad(\text{for
some $a$ between}\hskip 2.0pta_{n}(\hbar),a_{n}^{WKB}(\hbar))$
$\displaystyle\leq
C\cdot\begin{cases}a^{\delta-1}e^{-a^{\delta}}a^{-2\delta+1}e^{a^{\delta}}\hbar^{\frac{5}{3}},\quad\text{if}\quad
0<\delta<1\\\
a^{\delta-1}e^{-a^{\delta}}a^{-\delta+1}e^{a^{\delta}}\hbar^{\frac{5}{3}},\quad\text{if}\quad\delta\geq
1\end{cases}$ $\displaystyle\leq
C\cdot\begin{cases}\frac{\hbar^{5/3}}{\log\hbar^{-b}},\quad\text{if}\quad
0<\delta<1\quad(\text{since}\hskip
4.0pta^{-\delta}\sim\frac{1}{\log\hbar^{-b}})\\\
\hbar^{5/3},\quad\text{if}\quad\delta\geq 1\end{cases}$

where as usual $C$ denotes a generic constant. ∎

######  Remark 11.4.

Having reached close enough to 0, at a distance $\hbar^{b}$ with $b>1$, it is
possible to show that even in the remaining interval
$(0,i\hbar^{b}]\subset\mathbb{C}$ the absolute difference
$|\lambda_{n}(\hbar)-\lambda_{n}^{WKB}(\hbar)|$ is bounded by $\hbar^{b}$,
where $\lambda_{n}(\hbar)=i\mu_{n}(\hbar)$ and
$\lambda_{n}^{WKB}(\hbar)=i\mu_{n}^{WKB}(\hbar)$. The argument relies on the
fact that there exists a very accurate semiclassical estimate of the total
number of EVs due to Klaus & Shaw (see e.g. section VI of [6]). Since
neighboring Bohr-Sommerfeld approximations are at distance $O(\hbar)$ from
each other asymptotically, it follows that there is at most one such in the
interval $(0,i\hbar^{b}]$. Because of the previous corollaries and the
established 1-1 correspondence in $(\hbar^{b},A_{max})$, it is clear that
there is also at most one EV in the interval $(0,i\hbar^{b}]$ and the absolute
difference $|\lambda_{n}(\hbar)-\lambda_{n}^{WKB}(\hbar)|$ is indeed bounded
by $\hbar^{b}$.

##  12\. Scattering Coefficients

In this section we will consider the scattering coefficients for our Dirac
operator (3.2) 888In this part, we work using as guide ideas presented in
section IV of [6]. . As mentioned in §3, the continuous spectrum of our Dirac
operator is the whole real line. So in this section we are considering
$\lambda\in{\mathbb{R}}$.

We begin with the case where this $\lambda$ is idependent of $\hbar$. Under
the change of variables

$y_{\pm}=\frac{u_{2}\pm u_{1}}{\sqrt{A\pm i\lambda}}$

equation (3.1) -with the help of (3.2)- is transformed to the following two
independent equations

$y_{\pm}^{\prime\prime}(x,\lambda,\hbar)=\bigg{\\{}\hbar^{-2}[-A^{2}(x)-\lambda^{2}]+\frac{3}{4}\Big{[}\frac{A^{\prime}(x)}{A(x)\pm
i\lambda}\Big{]}^{2}-\frac{1}{2}\frac{A^{\prime\prime}(x)}{A(x)\pm
i\lambda}\bigg{\\}}y_{\pm}(x,\lambda,\hbar).$

Again we only consider the lower index, so from now on we drop all the indices
and work with the equation

$\frac{d^{2}y}{dx^{2}}=[-\hbar^{-2}\tilde{f}(x,\lambda)+\tilde{g}(x,\lambda)]y$
(12.1)

where $\tilde{f}$ and $\tilde{g}$ satisfy

$\tilde{f}(x,\lambda)=A^{2}(x)+\lambda^{2}$ (12.2)

and

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

We have the following definitions.

######  Definition 12.1.

We define the error-control function $\tilde{H}(x,\lambda)$ of equation (12.1)
to be a primitive of

$\frac{1}{\tilde{f}^{\frac{1}{4}}(x,\lambda)}\frac{\partial^{2}}{\partial
x^{2}}\Big{(}\frac{1}{\tilde{f}^{\frac{1}{4}}}\Big{)}(x,\lambda)-\frac{\tilde{g}(x,\lambda)}{\tilde{f}^{\frac{1}{2}}(x,\lambda)}.$
(12.3)

######  Definition 12.2.

For $x_{1}<x_{2}$, where $x_{1}\in[0,+\infty)$ and
$x_{2}\in(0,+\infty)\cup\\{+\infty\\}$, we define the variation of $\tilde{H}$
in the interval $(x_{1},x_{2})$ to be

$\mathfrak{V}_{x_{1},x_{2}}[\tilde{H}](\lambda)=\int_{x_{1}}^{x_{2}}\bigg{|}\frac{1}{\tilde{f}^{\frac{1}{4}}(t,\lambda)}\frac{\partial^{2}}{\partial
t^{2}}\Big{(}\frac{1}{\tilde{f}^{\frac{1}{4}}}\Big{)}(t,\lambda)-\frac{\tilde{g}(t,\lambda)}{\tilde{f}^{\frac{1}{2}}(t,\lambda)}\bigg{|}dt.$
(12.4)

Observe that (12.2) implies $\tilde{f}>0$ in ${\mathbb{R}}$. Consequently,
equation (12.1) has no turning points. Furthermore, notice that

  * •

$\tilde{g}$ is complex-valued

  * •

$\tilde{f}$ is twice continuously differentiable with respect to $x$ (a fact
that comes from the properties of $A$ found in §2) and

  * •

$\tilde{g}$ is continuous.

These properties allow one (cf. Theorem $2.2$ of §$2.4$ from chapter 6 of
[17], along with the remarks from §5.1 of the same chapter) to state that for
$x$ in the (finite or infinite) interval $(x_{1},x_{2})\subseteq{\mathbb{R}}$
and $\kappa$ an arbitrary finite or infinite point in the closure of
$(x_{1},x_{2})$, equation (12.1) has twice continuously differentiable
solutions 999Since $\tilde{g}$ is not real, we cannot expect these solutions
to be complex conjugates. $w_{\pm}$ with

$w_{\pm}(x,\hbar)=\tilde{f}^{-\frac{1}{4}}(x,\lambda)\exp\Big{\\{}\pm\frac{i}{\hbar}\int\tilde{f}^{\frac{1}{2}}(t,\lambda)dt\Big{\\}}(1+\epsilon_{\pm}(x,\hbar))$

where

$|\epsilon_{\pm}(x,\hbar)|\hskip 5.0pt,\hskip
5.0pt\hbar\tilde{f}^{-\frac{1}{2}}(x,\hbar)\Big{|}\frac{\partial\epsilon_{\pm}}{\partial
x}(x,\hbar)\Big{|}\leq\exp\\{\hbar\mathfrak{V}_{\kappa,x}[\tilde{H}](\lambda)\\}-1$
(12.5)

provided that $\mathfrak{V}_{\kappa,x}(\tilde{H})<+\infty$ . As usual, the
symbol $\int\tilde{f}^{\frac{1}{2}}(t,\lambda)dt$ denotes any primitive of
$\tilde{f}^{\frac{1}{2}}(t,\lambda)$. It follows that

  * •

$\epsilon_{\pm}(x;\hbar)\rightarrow 0$ as $x\to\kappa$ and

  * •

$\hbar\tilde{f}^{-\frac{1}{2}}(x,\hbar)\frac{\partial\epsilon_{\pm}}{\partial
x}(x,\hbar)\rightarrow 0$ as $x\to\kappa$.

Notice from (12.3) that $\tilde{H}$ is independent of $\hbar$ whence the
right-hand side of (12.5) is $\mathcal{O}(\hbar)$ as $\hbar\downarrow 0$ and
fixed $x$. But $\mathfrak{V}_{x_{1},x_{2}}[\tilde{H}](\lambda)<+\infty$ which
implies that this $\mathcal{O}$-term is uniform with respect to $x$ since
$\mathfrak{V}_{\kappa,x}[\tilde{H}](\lambda)\leq\mathfrak{V}_{x_{1},x_{2}}[\tilde{H}](\lambda)$.
Hence

$w_{\pm}(x,\hbar)\sim\tilde{f}^{-\frac{1}{4}}(x,\lambda)\exp\Big{\\{}\pm\frac{i}{\hbar}\int\tilde{f}^{\frac{1}{2}}(t,\lambda)dt\Big{\\}}\quad\text{as}\quad\hbar\downarrow
0$

uniformly in $(x_{1},x_{2})$.

Next we define the Jost solutions. Equation (12.1) can be put in the form

$-\frac{d^{2}y}{dx^{2}}+[-\hbar^{-2}A^{2}(x)+\tilde{g}(x,\lambda)]y=\Big{(}\frac{\lambda}{\hbar}\Big{)}^{2}y.$

This is the Schrödinger equation with momentum $\frac{\lambda}{\hbar}$, energy
$(\frac{\lambda}{\hbar})^{2}$ and a complex potential. The Jost solutions are
defined as the components of the bases $\\{J_{-}^{l},J_{+}^{l}\\}$ and
$\\{J_{-}^{r},J_{+}^{r}\\}$ of the two-dimensional linear space of solutions
of equation (12.1), which satisfy the asymptotic conditions

$\displaystyle J_{\pm}^{l}(x,\lambda)$ $\displaystyle\sim\exp\big{\\{}\pm
i\frac{\lambda}{\hbar}x\big{\\}}\quad\text{as}\quad x\to-\infty$
$\displaystyle J_{\pm}^{r}(x,\lambda)$ $\displaystyle\sim\exp\big{\\{}\pm
i\frac{\lambda}{\hbar}x\big{\\}}\quad\text{as}\quad x\to+\infty.$

From scattering theory, we know that the reflection $R(\lambda,\hbar)$ and
transmition $T(\lambda,\hbar)$ coefficients for the waves incident on the
potential from the right, can be expressed in terms of wronskians of the Jost
solutions. More presicely, we have

$\displaystyle R(\lambda,\hbar)$
$\displaystyle=\frac{\mathcal{W}[J_{-}^{l},J_{-}^{r}]}{\mathcal{W}[J_{+}^{r},J_{-}^{l}]}$
(12.6) $\displaystyle T(\lambda,\hbar)$
$\displaystyle=\frac{\mathcal{W}[J_{+}^{r},J_{-}^{r}]}{\mathcal{W}[J_{+}^{r},J_{-}^{l}]}.$
(12.7)

The next step is to construct the Jost solutions as WKB solutions. For this,
we define the following four WKB solutions

$\displaystyle\bar{w}_{\pm}^{l}(x,\hbar)=\tilde{f}^{-\frac{1}{4}}(x,\lambda)\exp\Big{\\{}\pm\frac{i}{\hbar}\Big{(}\lambda
x+\int_{-\infty}^{x}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt\Big{)}\Big{\\}}(1+\bar{\epsilon}_{\pm}^{l}(x,\hbar))$
$\displaystyle\bar{w}_{\pm}^{r}(x,\hbar)=\tilde{f}^{-\frac{1}{4}}(x,\lambda)\exp\Big{\\{}\pm\frac{i}{\hbar}\Big{(}\lambda
x+\int_{+\infty}^{x}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt\Big{)}\Big{\\}}(1+\bar{\epsilon}_{\pm}^{r}(x,\hbar))$

which we are going to modify slightly in a while. If we take the limits as
$x\to\pm\infty$ of the above, we instantly notice the following relations
between $\bar{w}_{\pm}^{l},\bar{w}_{\pm}^{l}$ and the Jost solutions
$J_{\pm}^{l},J_{\pm}^{r}$; we have

$\displaystyle J_{\pm}^{l}$
$\displaystyle=\lambda^{\frac{1}{2}}\bar{w}_{\pm}^{l}$ $\displaystyle
J_{\pm}^{r}$ $\displaystyle=\lambda^{\frac{1}{2}}\bar{w}_{\pm}^{r}.$

Let now $w_{\pm}^{l},w_{\pm}^{r}$ be four WKB solutions satisfying

$\displaystyle
w_{\pm}^{l}(x,\hbar)=\tilde{f}^{-\frac{1}{4}}(x,\lambda)\exp\Big{\\{}\pm\frac{i}{\hbar}\int_{0}^{x}\tilde{f}^{\frac{1}{2}}(t,\lambda)dt\Big{\\}}(1+\epsilon_{\pm}^{l}(x,\hbar))$
(12.8) $\displaystyle
w_{\pm}^{r}(x,\hbar)=\tilde{f}^{-\frac{1}{4}}(x,\lambda)\exp\Big{\\{}\pm\frac{i}{\hbar}\int_{0}^{x}\tilde{f}^{\frac{1}{2}}(t,\lambda)dt\Big{\\}}(1+\epsilon_{\pm}^{r}(x,\hbar)).$
(12.9)

Once again, the connnection between $w_{\pm}^{l},w_{\pm}^{r}$ and
$\bar{w}_{\pm}^{l},\bar{w}_{\pm}^{r}$ is evident. It is

$\displaystyle\bar{w}_{\pm}^{l}$
$\displaystyle=\exp\Big{\\{}\pm\frac{i}{\hbar}\int_{-\infty}^{0}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt\Big{\\}}w_{\pm}^{l}$
$\displaystyle\bar{w}_{\pm}^{r}$
$\displaystyle=\exp\Big{\\{}\mp\frac{i}{\hbar}\int_{0}^{+\infty}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt\Big{\\}}w_{\pm}^{r}.$

Subsequently, for the Jost solutions we have

$\displaystyle J_{\pm}^{l}$
$\displaystyle=\lambda^{\frac{1}{2}}\exp\Big{\\{}\pm\frac{i}{\hbar}\int_{-\infty}^{0}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt\Big{\\}}w_{\pm}^{l}$
(12.10) $\displaystyle J_{\pm}^{r}$
$\displaystyle=\lambda^{\frac{1}{2}}\exp\Big{\\{}\mp\frac{i}{\hbar}\int_{0}^{+\infty}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt\Big{\\}}w_{\pm}^{r}.$
(12.11)

Remember from §2 that the properties of $A$ show that the function
$t\mapsto\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda$ is in
$L^{1}({\mathbb{R}})$. Furthermore, we have

$\int_{-\infty}^{0}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt=\int_{0}^{+\infty}[\tilde{f}^{\frac{1}{2}}(t,\lambda)-\lambda]dt=\frac{1}{2}\|\tilde{f}^{\frac{1}{2}}(\cdot,\lambda)-\lambda\|_{L^{1}({\mathbb{R}})}$
(12.12)

and we define

$\sigma(\lambda):=\|\tilde{f}^{\frac{1}{2}}(\cdot,\lambda)-\lambda\|_{L^{1}({\mathbb{R}})}$
(12.13)

Substituting (12.10), (12.11), (12.12) and (12.13) in (12.6), (12.7) and using

$\mathcal{W}[J_{+}^{r},J_{-}^{r}]=-2i\frac{\lambda}{\hbar}$

we have

$\displaystyle R(\lambda,\hbar)$
$\displaystyle=e^{i\frac{\sigma(\lambda)}{\hbar}}\frac{\mathcal{W}[w_{-}^{l},w_{-}^{r}]}{\mathcal{W}[w_{+}^{r},w_{-}^{l}]}$
(12.14) $\displaystyle T(\lambda,\hbar)$
$\displaystyle=-\frac{2i}{\hbar}e^{i\frac{\sigma(\lambda)}{\hbar}}\frac{1}{\mathcal{W}[w_{+}^{r},w_{-}^{l}]}.$
(12.15)

Finally, using (12.8), (12.9) and (12.5) we find that

  * •

$\mathcal{W}[w_{-}^{l},w_{-}^{r}]=\mathcal{O}(1)$ as $\hbar\downarrow 0$ and

  * •

$\mathcal{W}[w_{+}^{r},w_{-}^{l}]=-\frac{2i}{\hbar}[1+\mathcal{O}(\hbar)]$ as
$\hbar\downarrow 0$.

Substituting these last results in (12.14), (12.15) we get that

$\displaystyle R(\lambda,\hbar)$
$\displaystyle=\frac{i\hbar}{2}e^{i\frac{\sigma(\lambda)}{\hbar}}\mathcal{O}(1)\quad\text{as}\quad\hbar\downarrow
0$ $\displaystyle T(\lambda,\hbar)$
$\displaystyle=e^{i\frac{\sigma(\lambda)}{\hbar}}[1+\mathcal{O}(\hbar)]\quad\text{as}\quad\hbar\downarrow
0.$

Hence we have just prooved that

######  Theorem 12.3.

Let $A$ satisfy the assumptions of §2 and define $\sigma$ by (12.13). The
scattering coefficients of equation (12.1) as defined by (12.6) and (12.7)
respectively, satisfy

$\displaystyle R(\lambda,\hbar)$
$\displaystyle=\frac{i\hbar}{2}e^{i\frac{\sigma(\lambda)}{\hbar}}\mathcal{O}(1)\quad\text{as}\quad\hbar\downarrow
0$ (12.16) $\displaystyle T(\lambda,\hbar)$
$\displaystyle=e^{i\frac{\sigma(\lambda)}{\hbar}}[1+\mathcal{O}(\hbar)]\quad\text{as}\quad\hbar\downarrow
0$ (12.17)

uniformly for $|\lambda|\geq\delta>0$.

Now we turn to the case where $\lambda$ depends on $\hbar$
($\lambda=\lambda(\hbar)$) and particularly we let $\lambda$ approach $0$ like
$\hbar^{b}$ for an $\hbar$-independent positive constant $b$. After using
(12.4) and the asymptotic properties of $A$ and its derivatives from §2, we
find that

$\mathfrak{V}_{0,+\infty}[\tilde{H}](\lambda(\hbar))=\mathcal{O}(\hbar^{-5b})\quad\text{as}\quad\hbar\downarrow
0.$

Hence for $\lambda(\hbar)\in[\hbar^{b},+\infty)$, in use of (12.8) and (12.9)
we get

  * •

$\mathcal{W}[w_{-}^{l},w_{-}^{r}]=\mathcal{O}(\hbar^{-5b})$ as
$\hbar\downarrow 0$ and

  * •

$\mathcal{W}[w_{+}^{r},w_{-}^{l}]=-\frac{2i}{\hbar}[1+\mathcal{O}(\hbar^{1-5b})]$
as $\hbar\downarrow 0$.

Substituting these last results in (12.14), (12.15) we finally obtain that

$\displaystyle R(\lambda(\hbar),\hbar)$
$\displaystyle=\frac{i\hbar}{2}e^{i\frac{\sigma(\lambda(\hbar))}{\hbar}}\mathcal{O}(\hbar^{-5b})\quad\text{as}\quad\hbar\downarrow
0$ $\displaystyle T(\lambda(\hbar),\hbar)$
$\displaystyle=e^{i\frac{\sigma(\lambda(\hbar))}{\hbar}}[1+\mathcal{O}(\hbar^{1-5b})]\quad\text{as}\quad\hbar\downarrow
0.$

So, we have showed the following

######  Theorem 12.4.

Let $A$ satisfy the assumptions of §2, take $0<b<\frac{1}{5}$ (independent of
$\hbar$) and define $\sigma$ by (12.13). Then the reflection coefficient and
the transmission coefficient of equation (12.1) as defined by (12.6) and
(12.7) respectively, satisfy

$\displaystyle R(\lambda(\hbar),\hbar)$
$\displaystyle=\frac{i\hbar}{2}e^{i\frac{\sigma(\lambda(\hbar))}{\hbar}}\mathcal{O}(\hbar^{-5b})\quad\text{as}\quad\hbar\downarrow
0$ (12.18) $\displaystyle T(\lambda(\hbar),\hbar)$
$\displaystyle=e^{i\frac{\sigma(\lambda(\hbar))}{\hbar}}[1+\mathcal{O}(\hbar^{1-5b})]\quad\text{as}\quad\hbar\downarrow
0$ (12.19)

uniformly for $\lambda(\hbar)$ in any closed interval of
$[\hbar^{b},+\infty)$.

######  Remark 12.5.

We check that

$|T(\lambda(\hbar),\hbar)|^{2}-|R(\lambda(\hbar),\hbar)|^{2}=1+\mathcal{O}(\hbar^{1-5b})\quad\text{as}\quad\hbar\downarrow
0$

as of course it should be the case.

######  Remark 12.6.

The result (12.18) above only guarantees asymptotics of order
$O(\hbar^{1-\epsilon})$ for small positive $\epsilon$, as $\hbar\downarrow 0$,
while the results provided by Theorems 2.1 and 2.4 in [6], where the potential
is real-analytic, actually imply exponential decay as $\hbar\downarrow 0$.
Still this is good enough for the applications to the theory of focusing NLS.

##  13\. Conclusion

The results in the last three paragraphs are stronger than those of [6] in the
sense that they cover analytic bell-shaped potentials as well as non-analytic
potentials with a certain smoothness. The reflection coefficient estimate is
weaker but this does not affect the results and proofs pertaining to the
applications to the semiclassical limit of the NLS equation. On the other
hand, the more important Bohr-Sommerfeld estimate is stronger. We refer to [6]
for the actual statements of the precise results; the detailed proofs are now
much more straightforward. Indeed, the proof of Proposition 6.1 in [6] is now
trivial in view of the uniform validity of the Bohr-Sommerfeld estimate near 0
and the remaining propositions in paragraph 6 (statements and proofs) are
unchanged.

##  Appendix A Airy Functions

In this section, some basic properties of Airy functions are presented. For
further reading one may consult [17].

Figure 4. The Airy functions $Ai$, $Bi$ on the real line.

Consider the Airy equation

$-\dfrac{d^{2}w}{dt^{2}}(t)+t\cdot w(t)=0,\quad t\in\mathbb{R}$

We denote by $Ai$ and $Bi$ its two linearly independent solutions having the
asymptotics

$Ai(t)=\frac{1}{2\sqrt{\pi}}t^{-\frac{1}{4}}\exp\\{-\tfrac{2}{3}t^{\frac{3}{2}}\\}\big{[}1+O\big{(}t^{-\frac{3}{2}}\big{)}\big{]}\quad\text{as}\quad
t\rightarrow+\infty$ (A.1)

and

$Bi(t)=-\frac{1}{\sqrt{\pi}}|t|^{-\frac{1}{4}}\sin\Big{(}\tfrac{2}{3}|t|^{\frac{3}{2}}-\tfrac{\pi}{4}\Big{)}+O\big{(}|t|^{-\frac{7}{4}}\big{)}\quad\text{as}\quad
t\rightarrow-\infty$ (A.2)

Their behavior on the opposite side of the real line is known to be

$Ai(t)=\frac{1}{\sqrt{\pi}}|t|^{-\frac{1}{4}}\sin\Big{(}\tfrac{2}{3}|t|^{\frac{3}{2}}+\tfrac{\pi}{4}\Big{)}+O\big{(}|t|^{-\frac{7}{4}}\big{)}\quad\text{as}\quad
t\rightarrow-\infty$ (A.3)

and

$Bi(t)\leq C(1+t)^{-\frac{1}{4}}\exp\\{\tfrac{2}{3}t^{\frac{3}{2}}\\},\quad
t\geq 0$

where $C$ is a positive constant. Observe that as $t\to-\infty$, $Ai$ and $Bi$
only differ by a phase shift. Also $Ai(t),Bi(t)>0$ for all $t\geq 0$. Note
that all asymptotic relations (A.1), (A.2) and (A.3) can be differentiated in
$t$; for example

$Ai^{\prime}(t)=-\frac{1}{\sqrt{\pi}}|t|^{\frac{1}{4}}\cos\Big{(}\tfrac{2}{3}|t|^{\frac{3}{2}}+\tfrac{\pi}{4}\Big{)}+O\big{(}|t|^{-\frac{5}{4}}\big{)}\quad\text{as}\quad
t\rightarrow-\infty$

and

$Ai^{\prime}(t)=-\frac{1}{2\sqrt{\pi}}t^{\frac{1}{4}}\exp\\{-\tfrac{2}{3}t^{\frac{3}{2}}\\}\big{[}1+O\big{(}t^{-\frac{3}{2}}\big{)}\big{]}\quad\text{as}\quad
t\rightarrow+\infty.$

Another property says that

$|Ai(t)|\leq C(1+|t|)^{-\frac{1}{4}},\qquad t\in\mathbb{R}$

where $C$ is a positive constant. The wronskian of $Ai$, $Bi$ satisfies

$\mathcal{W}\big{[}Ai,Bi\big{]}(t):=Ai(t)Bi^{\prime}(t)-Ai^{\prime}(t)Bi(t)=\frac{1}{\pi},\quad
t\in\mathbb{R}.$

In order to have a convenient way of assessing the magnitudes of $Ai$ and $Bi$
we introduce a modulus function $M$, a phase function $\vartheta$ and a weight
function $E$ related by

$E(x)Ai(x)=M(x)\sin\vartheta(x),\quad\frac{1}{E(x)}Bi(x)=M(x)\cos\vartheta(x),\quad
x\in\mathbb{R}.$

Actually, we choose $E$ as follows. Denote by $c_{*}$ the biggest negative
root of the equation $Ai(x)=Bi(x)$ (numerical calculations show that
$c_{*}=-0.36605$ correct up to five decimal places); then define

$E(x)=\begin{cases}1,\quad x\leq c_{*}\\\ \sqrt{\frac{Bi(x)}{Ai(x)}},\quad
x>c_{*}\end{cases}$

With this choice in mind, $M$, $\theta$ become

$M(x)=\begin{cases}\sqrt{Ai^{2}(x)+Bi^{2}(x)},\quad x\leq c_{*}\\\
\sqrt{2Ai(x)Bi(x)},\quad
x>c_{*}\end{cases}\text{and}\quad\vartheta(x)=\begin{cases}\arctan\Big{[}\frac{Ai(x)}{Bi(x)}\Big{]},\quad
x\leq c_{*}\\\ \frac{\pi}{4},\quad x>c_{*}\end{cases}$

where the branch of the inverse tangent is continuous and equal to
$\frac{\pi}{4}$ at $x=c_{*}$. For these functions the asymptotics for large
$|x|$ read

$\displaystyle E(x)\sim$ $\displaystyle\begin{cases}1,\quad
x\rightarrow-\infty\\\ \sqrt{2}\exp\\{\tfrac{2}{3}t^{\frac{3}{2}}\\},\quad
x\rightarrow+\infty\end{cases}$ $\displaystyle M(x)\sim$
$\displaystyle\frac{1}{\sqrt{\pi}}|x|^{-\frac{1}{4}},\quad|x|\rightarrow+\infty$
$\displaystyle\vartheta(x)=$
$\displaystyle\begin{cases}\frac{2}{3}|x|^{\frac{3}{2}}+\frac{\pi}{4}+\mathcal{O}\big{(}\frac{3}{2}|x|^{-\frac{3}{2}}\big{)},\quad
x\rightarrow-\infty\\\ \frac{\pi}{4},\quad x\rightarrow+\infty\end{cases}$

##  Appendix B Parabolic Cylinder Functions

The result of the main theorem found in section §6, involves parabolic
cylinder functions (cf. [1]). So in this section we state a few properties
which will be in heavy use, especially about their asymptotic character,
wronskians and zeros. We prove none of them. For a rigorous exposition on
parabolic cylinder functions, one may consult §5 of [16] or §12 of [18] and
the references therein.

Consider Weber’s equation

$\frac{d^{2}w}{dx^{2}}=\Big{(}\tfrac{1}{4}x^{2}+b\Big{)}w.$ (B.1)

The behavior of its solutions depends on the sign of $b$. When $b$ is negative
then there exist two turning points $\pm 2\sqrt{-b}$. The solutions are of
oscillatory type in the interval between these points but not in the exterior
intervals. When $b>0$ there are no real turning points and there are no
oscillations at all. Since only the case $b\leq 0$ will be of interest to us,
from now on we seldom mention properties having to do with the other case.

Standard solutions of (B.1) are $U(\pm x,b)$ and $\overline{U}(\pm x,b)$
defined by

$U(\pm
x,b)=\frac{\pi^{\frac{1}{2}}2^{-\frac{1}{4}(2b+1)}}{\Gamma(\frac{3}{4}+\frac{1}{2}b)}e^{-\frac{1}{4}x^{2}}\Hypergeometric{1}{1}{\tfrac{1}{4}+\tfrac{1}{2}b}{\tfrac{1}{2}}{\tfrac{1}{2}x^{2}}\\\
\mp\frac{\pi^{\frac{1}{2}}2^{-\frac{1}{4}(2b-1)}}{\Gamma(\frac{1}{4}+\frac{1}{2}b)}xe^{-\frac{1}{4}x^{2}}\Hypergeometric{1}{1}{\tfrac{3}{4}+\tfrac{1}{2}b}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}}$
$\overline{U}(\pm
x,b)=\pi^{-\frac{1}{2}}2^{-\frac{1}{4}(2b+1)}\Gamma(\tfrac{1}{4}-\tfrac{1}{2}b)\sin(\tfrac{3}{4}\pi-\tfrac{1}{2}b\pi)e^{-\frac{1}{4}x^{2}}\Hypergeometric{1}{1}{\tfrac{1}{4}+\tfrac{1}{2}b}{\tfrac{1}{2}}{\tfrac{1}{2}x^{2}}\\\
\mp\pi^{-\frac{1}{2}}2^{-\frac{1}{4}(2b-1)}\Gamma(\tfrac{3}{4}-\tfrac{1}{2}b)\sin(\tfrac{5}{4}\pi-\tfrac{1}{2}b\pi)xe^{-\frac{1}{4}x^{2}}\Hypergeometric{1}{1}{\tfrac{3}{4}+\tfrac{1}{2}b}{\tfrac{3}{2}}{\tfrac{1}{2}x^{2}}$

where ${}_{1}F_{1}$ denotes the confluent hypergeometric function (again cf.
[1]). The pair $U(x,b),\overline{U}(x,b)$ is a numerically satisfactory set of
solutions (in the sense of [13]) when $x\geq 0$ and $b\leq 0$; both are
continuous in $x$ and $b$ in this region.

For $b\in{\mathbb{R}}$, their values at $x=0$ obey

$\displaystyle U(0,b)$
$\displaystyle=\pi^{-\frac{1}{2}}2^{-\frac{1}{4}(2b+1)}\Gamma(\tfrac{1}{4}-\tfrac{1}{2}b)\sin(\tfrac{\pi}{4}-\tfrac{1}{2}b\pi)$
$\displaystyle U^{\prime}(0,b)$
$\displaystyle=-\pi^{-\frac{1}{2}}2^{-\frac{1}{4}(2b-1)}\Gamma(\tfrac{3}{4}-\tfrac{1}{2}b)\sin(\tfrac{3\pi}{4}-\tfrac{1}{2}b\pi)$
$\displaystyle\overline{U}(0,b)$
$\displaystyle=\pi^{-\frac{1}{2}}2^{-\frac{1}{4}(2b+1)}\Gamma(\tfrac{1}{4}-\tfrac{1}{2}b)\sin(\tfrac{3\pi}{4}-\tfrac{1}{2}b\pi)$
$\displaystyle\overline{U}^{\prime}(0,b)$
$\displaystyle=-\pi^{-\frac{1}{2}}2^{-\frac{1}{4}(2b-1)}\Gamma(\tfrac{3}{4}-\tfrac{1}{2}b)\sin(\tfrac{5\pi}{4}-\tfrac{1}{2}b\pi).$

Those values of $b$ that make the Gamma functions in the definitions of $U$
and $\overline{U}$ infinite (the Gamma function has simple poles at the non-
positive integers), are called exceptional values. For a fixed
$b\in{\mathbb{R}}$ other than an exceptional value, the behaviors of $U$ and
$\overline{U}$ as $x\to+\infty$ satisfy

$\displaystyle U(x,b)$ $\displaystyle\sim
x^{-b-\frac{1}{2}}e^{-\frac{1}{4}x^{2}}$ (B.2) $\displaystyle U^{\prime}(x,b)$
$\displaystyle\sim-\frac{1}{2}x^{-b+\frac{1}{2}}e^{-\frac{1}{4}x^{2}}$
$\displaystyle\overline{U}(x,b)$
$\displaystyle\sim\sqrt{\frac{2}{\pi}}\Gamma(\tfrac{1}{2}-b)x^{b-\frac{1}{2}}e^{\frac{1}{4}x^{2}}$
$\displaystyle\overline{U}^{\prime}(x,b)$
$\displaystyle\sim\frac{1}{\sqrt{2\pi}}\Gamma(\tfrac{1}{2}-b)x^{b+\frac{1}{2}}e^{\frac{1}{4}x^{2}}.$

These estimates are uniform in $b$ when $b$ takes values over a fixed compact
interval not containing exceptional values.

For the wronskian of $U(\cdot,b),\overline{U}(\cdot,b)$ we have

$\mathcal{W}[U(\cdot,b),\overline{U}(\cdot,b)](x)=\sqrt{\frac{2}{\pi}}\Gamma\Big{(}\frac{1}{2}-b\Big{)},\quad
x\in{\mathbb{R}}.$ (B.3)

When $b=0$ the standard solutions of equation (B.1) are related to the
modified Bessel functions $K_{\frac{1}{4}}$ and $I_{\frac{1}{4}}$ in the
following way. For $x\geq 0$ we have

$\displaystyle U(x,0)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}x^{\frac{1}{2}}K_{\frac{1}{4}}(\tfrac{1}{4}x^{2})$
$\displaystyle\overline{U}(x,0)$
$\displaystyle=\sqrt{\pi}x^{\frac{1}{2}}I_{\frac{1}{4}}(\tfrac{1}{4}x^{2})+\frac{1}{\sqrt{2\pi}}x^{\frac{1}{2}}K_{\frac{1}{4}}(\tfrac{1}{4}x^{2}).$

In order to express the character of these standard solutions for large (in
absolute value) negative $b$, we need some preparations first. Take $\nu\gg 1$
to be a large positive number and set $b=-\frac{1}{2}\nu^{2}$ and $x=\nu
y\sqrt{2}$ where $y\geq 0$. If we consider the fuction $\eta$ to be

$\eta(y)=\begin{cases}-[\frac{3}{2}\int_{y}^{1}\sqrt{1-s^{2}}ds]^{\frac{2}{3}},\quad
0\leq y\leq 1\\\ [\frac{3}{2}\int_{1}^{y}\sqrt{s^{2}-1}ds]^{\frac{2}{3}},\quad
y\geq 1\end{cases}$ (B.4)

then as $\nu\to+\infty$ we have

$\displaystyle U(\nu y\sqrt{2},-\tfrac{1}{2}\nu^{2})$
$\displaystyle=\frac{2^{\frac{1}{2}}\pi^{\frac{1}{4}}\Gamma(\tfrac{1}{2}+\frac{1}{2}\nu^{2})^{\frac{1}{2}}}{\nu^{\frac{1}{6}}}\Big{(}\frac{\eta}{y^{2}-1}\Big{)}^{\frac{1}{4}}\Big{[}\operatorname{Ai}(\nu^{\frac{4}{3}}\eta)+\frac{M(\nu^{\frac{4}{3}}\eta)}{E(\nu^{\frac{4}{3}}\eta)}\mathcal{O}(\nu^{-2})\Big{]}$
(B.5) $\displaystyle\overline{U}(\nu y\sqrt{2},-\tfrac{1}{2}\nu^{2})$
$\displaystyle=\frac{2^{\frac{1}{2}}\pi^{\frac{1}{4}}\Gamma(\tfrac{1}{2}+\frac{1}{2}\nu^{2})^{\frac{1}{2}}\eta^{\frac{1}{4}}}{\nu^{\frac{1}{6}}(y^{2}-1)^{\frac{1}{4}}}\Big{[}\operatorname{Bi}(\nu^{\frac{4}{3}}\eta)+M(\nu^{\frac{4}{3}}\eta)E(\nu^{\frac{4}{3}}\eta)\mathcal{O}(\nu^{-2})\Big{]}$
(B.6)

where $\operatorname{Ai}$, $\operatorname{Bi}$, $E$ and $M$ are the standard
Airy functions’ terminology (cf. section A in the appendix).

For $b\leq 0$, the number of zeros of $U(\cdot,b)$ in the interval
$[0,+\infty)$ is $\lfloor\tfrac{1}{4}-\frac{1}{2}b\rfloor$ while
$\overline{U}(\cdot,b)$ has $\lfloor\tfrac{3}{4}-\frac{1}{2}b\rfloor$ zeros in
$[0,+\infty)$. Actually, the zeros of $U(\cdot,b)$ and $\overline{U}(\cdot,b)$
do not cross each other. They interlace, with the largest one belonging to
$\overline{U}(\cdot,b)$. For sufficiently large $|b|$, all the real zeros of
these two functions lie to the left of $2\sqrt{-b}$, the positive turning
point of Weber’s equation 101010 For $U(\cdot,b)$, this result holds for all
$b\leq 0$.  .

To express the errors for the approximations of our problem, we need to define
some auxiliary functions having to do with the nature of $U(\cdot,b)$ and
$\overline{U}(\cdot,b)$ for negative $b$. In this case the character of each
is partly oscillatory and partly exponential, so we introduce one weight
function $\mathsf{E}$, two modulus functions $\mathsf{M}$ and $\mathsf{N}$,
and finally two phase functions $\theta$ and $\omega$.

We denote by $\rho(b)$ the largest real root of the equation

$U(x,b)=\overline{U}(x,b).$

We know (cf. §13 of [18] and the references therein) that $\rho(0)=0$ and
$\rho(b)>0$ for $b<0$. Also, $\rho$ is continuous when $b\in(-\infty,0]$. An
asymptotic estimate for large negative $b$ is

$\rho(b)=2\sqrt{-b}+c_{*}(-b)^{-\frac{1}{6}}+\mathcal{O}\big{(}b^{-\frac{5}{6}}\big{)}\quad\text{as}\quad
b\to-\infty$ (B.7)

where $c_{*}$ ($\approx-0.36605$) is the smallest in absolute value root of
the equation $\operatorname{Ai}(x)=\operatorname{Bi}(x)$.

For $b\leq 0$ we define

$\mathsf{E}(x,b)=\begin{cases}1,\quad 0\leq x\leq\rho(b)\\\
\Big{[}\frac{\overline{U}(x,b)}{U(x,b)}\Big{]}^{1/2},\quad
x>\rho(b).\end{cases}$

It is seen that $\mathsf{E}$ is continuous in the region
$[0,+\infty)\times(-\infty,0]$ of the $(x,b)$-plane and for $b\leq 0$ fixed,
$\mathsf{E}(\cdot,b)$ is non-decreasing in the interval $[0,+\infty)$. Again
for $b\leq 0$ and $x\geq 0$ we set

$U(x,b)=\frac{1}{\mathsf{E}(x,b)}\mathsf{M}(x,b)\sin\theta(x,b),\quad\overline{U}(x,b)=\mathsf{E}(x,b)\mathsf{M}(x,b)\cos\theta(x,b)$

and

$U^{\prime}(x,b)=\frac{1}{\mathsf{E}(x,b)}\mathsf{N}(x,b)\sin\omega(x,b),\quad\overline{U}^{\prime}(x,b)=\mathsf{E}(x,b)\mathsf{N}(x,b)\cos\omega(x,b).$

Thus

$\mathsf{M}(x,b)=\begin{cases}\big{[}U(x,b)^{2}+\overline{U}(x,b)^{2}\big{]}^{1/2},\quad
0\leq x\leq\rho(b)\\\ \big{[}2U(x,b)\overline{U}(x,b)\big{]}^{1/2},\quad
x>\rho(b)\end{cases}$ (B.8)

and

$\theta(x,b)=\begin{cases}\arctan\Big{[}\frac{U(x,b)}{\overline{U}(x,b)}\Big{]},\quad
0\leq x\leq\rho(b)\\\ \frac{\pi}{4},\quad x>\rho(b)\end{cases}$

where the branch of the inverse tangent is continuous and equal to
$\frac{\pi}{4}$ at $x=\rho(b)$.

Similarly

$\mathsf{N}(x,b)=\begin{cases}\Big{[}U^{\prime}(x,b)^{2}+\overline{U}^{\prime}(x,b)^{2}\Big{]}^{1/2},\quad
0\leq x\leq\rho(b)\\\ \\\
\bigg{[}\frac{U^{\prime}(x,b)^{2}\overline{U}(x,b)^{2}+\overline{U}^{\prime}(x,b)^{2}U(x,b)^{2}}{U(x,b)\overline{U}(x,b)}\bigg{]}^{1/2},\quad
x>\rho(b)\end{cases}$

and

$\omega(x,b)=\begin{cases}\arctan\Big{[}\frac{U^{\prime}(x,b)}{\overline{U}^{\prime}(x,b)}\Big{]},\quad
0\leq x\leq\rho(b)\\\ \\\
\arctan\Big{[}\frac{U^{\prime}(x,b)\overline{U}(x,b)}{\overline{U}^{\prime}(x,b)U(x,b)}\Big{]},\quad\quad
x>\rho(b)\end{cases}$

where the branches of the inverse tangents are chosen to be continuous and
fixed by $\omega(x,b)\rightarrow-\frac{\pi}{4}$ as $x\rightarrow+\infty$.

For large $x$ we have

$\mathsf{E}(x,b)\sim\Big{(}\frac{2}{\pi}\Big{)}^{\frac{1}{4}}\Gamma(\tfrac{1}{2}-b)^{\frac{1}{2}}x^{b}e^{\frac{1}{4}x^{2}}$

and

$\mathsf{M}(x,b)\sim\Big{(}\frac{8}{\pi}\Big{)}^{\frac{1}{4}}\frac{\Gamma(\tfrac{1}{2}-b)^{\frac{1}{2}}}{x^{\frac{1}{2}}},\quad\mathsf{N}(x,b)\sim\frac{\Gamma(\tfrac{1}{2}-b)^{\frac{1}{2}}}{(2\pi)^{\frac{1}{4}}}x^{\frac{1}{2}}.$
(B.9)

Both of these hold for fixed $b$ and are also uniform for $b$ ranging over any
compact interval in $(-\infty,0]$.

##  Appendix C A Theorem on Integral Equations

The proofs of theorems about WKB approximation when there is an absence of
turning points (like Theorems 2.1 and 2.2 in chapter 6 of [17]), may be
adapted to other types of approximate solutions of linear differential
equations where turning points may be present. For second-order equations the
basic steps consist of

  * (i)

construction of a Volterra integral equation for the error term -say $h$\- of
the solution, by the method of variation of parameters

  * (ii)

construction of the Liouville-Neumann expansion (a uniformly convergent
series) for the solution $h$ of the integral equation in (i) by Picard’s
method of successive approximations

  * (iii)

confirmation that $h$ is twice differentiable by construction of similar
series for $h^{\prime}$ and $h^{\prime\prime}$

  * (iv)

production of bounds for $h$ and $h^{\prime}$ by majoring the Liouville-
Neumann expansion.

It would be tedious to carry out all these steps in every case. But we have
the following general theorem which automatically provides (ii), (iii) and
(iv) in problems relevant to us.

######  Theorem C.1.

111111 This is Theorem 10.2 found in chapter 6 of [17]. It is a variant of
Theorem 10.1 from the same reference.

Consider the equation

$h(\zeta)=\int_{\beta}^{\zeta}\mathsf{K}(\zeta,t)\phi(t)\\{J(t)+h(t)\\}dt$
(C.1)

for the function $h$ accompanied by the following assumptions

  * •

the path of integration consists of a segment $[\beta,\zeta]$ of the real
axis, finite or infinite where $\beta\leq t\leq\zeta\leq\gamma$

  * •

the real functions $J$ and $\phi$ are continuous in $(\beta,\gamma)$ except
for a finite number of discontinuities and infinities

  * •

the real kernel $\mathsf{K}$ and its first two partial derivatives with
respect to $\zeta$ are continuous functions of both variables when
$\zeta,t\in(\beta,\gamma)$

  * •

$\mathsf{K}(\zeta,\zeta)=0,\quad\zeta\in(\beta,\gamma)$

  * •

when $\zeta\in(\beta,\gamma)$ and $t\in(\beta,\zeta]$ we have

$|\mathsf{K}(\zeta,t)|\leq
P_{0}(\zeta)Q(t),\qquad\Big{|}\frac{\partial\mathsf{K}(\zeta,t)}{\partial\zeta}\Big{|}\leq
P_{1}(\zeta)Q(t),\qquad\Big{|}\frac{\partial^{2}\mathsf{K}(\zeta,t)}{\partial\zeta^{2}}\Big{|}\leq
P_{2}(\zeta)Q(t)$

where the $P_{j},j=0,1,2$ and $Q$ are continuous real functions, the
$P_{j},j=0,1,2$ being positive.

  * •

when $\zeta\in(\beta,\gamma)$, the integral

$\Phi(\zeta)=\int_{\beta}^{\zeta}|\phi(t)|dt$

converges and the following suprema

$\kappa=\sup_{\zeta\in(\beta,\gamma)}\\{Q(\zeta)|J(\zeta)|\\},\qquad\kappa_{0}=\sup_{\zeta\in(\beta,\gamma)}\\{P_{0}(\zeta)Q(\zeta)\\}$

are finite.

Under these assumptions, equation (C.1) has a unique solution $h$ which is
continuously differentiable in $(\beta,\gamma)$ and satisfies

$\frac{h(\zeta)}{P_{0}(\zeta)}\to
0\qquad\frac{h^{\prime}(\zeta)}{P_{1}(\zeta)}\to
0\qquad\text{as}\quad\zeta\downarrow\beta.$

Furthermore,

$\frac{|h(\zeta)|}{P_{0}(\zeta)},\frac{|h^{\prime}(\zeta)|}{P_{1}(\zeta)}\leq\frac{\kappa}{\kappa_{0}}[\exp\\{\kappa_{0}\Phi(\zeta)\\}-1]$

and $h^{\prime\prime}$ is continuous except at the discontinuities -if any- of
$\phi,J$.

###### Proof.

The proof is a slight variation of that for Theorem 10.1 of chapter 6 in [17].
∎

We are going to use this theorem to prove the existence and behavior of
approximate solutions of the equation

$\frac{d^{2}\mathcal{Y}}{d\zeta^{2}}=\big{[}\hbar^{-2}(\zeta^{2}-\alpha^{2})+\psi(\zeta,\hbar,\alpha)\big{]}\mathcal{Y}.$
(C.2)

We have the following

######  Theorem C.2.

For each value of $\hbar$, assume that the function $\psi(\zeta,\hbar,\alpha)$
is continuous in the region $[0,Z)\times[0,\delta]$ of the
$(\zeta,\alpha)$-plane 121212 Here $Z$ is always positive and may depend
continuously on $\alpha$, or be infinite. Also, $\delta$ is a positive finite
constant. , take $\Omega$ as in (6.3) and consider that

$\mathcal{V}_{0,Z}[H](\alpha,\hbar)=\int_{0}^{Z}\frac{|\psi(t,\alpha)|}{\Omega(t\sqrt{2\hbar^{-1}})}dt$

converges uniformly with respect to $\alpha$. Then in this region, equation
(C.2) has solutions $\mathcal{Y}_{1}$ and $\mathcal{Y}_{2}$ which are
continuous, have continuous first and second partial $\zeta$-derivatives and
are given by

$\displaystyle\mathcal{Y}_{1}(\zeta,\alpha,\hbar)=U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})+\epsilon_{1}(\zeta,\alpha,\hbar)$
$\displaystyle\mathcal{Y}_{2}(\zeta,\alpha,\hbar)=\overline{U}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})+\epsilon_{2}(\zeta,\alpha,\hbar)$

where

$\frac{|\epsilon_{1}(\zeta,\alpha,\hbar)|}{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})},\frac{\Big{|}\frac{\partial\epsilon_{1}}{\partial\zeta}(\zeta,\alpha,\hbar)\Big{|}}{\sqrt{2\hbar^{-1}}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\\\
\leq\frac{1}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\Big{(}\exp\big{\\{}\tfrac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{V}_{\zeta,Z}[H](\alpha,\hbar)\big{\\}}-1\Big{)}$
(C.3)

and

$\frac{|\epsilon_{2}(\zeta,\alpha,\hbar)|}{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})},\frac{\Big{|}\frac{\partial\epsilon_{2}}{\partial\zeta}(\zeta,\alpha,\hbar)\Big{|}}{\sqrt{2\hbar^{-1}}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\\\
\leq\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\Big{(}\exp\big{\\{}\tfrac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathcal{V}_{0,\zeta}[H](\alpha,\hbar)\big{\\}}-1\Big{)}.$
(C.4)

###### Proof.

We will prove the theorem only for the first solution since the proof for the
second follows mutatis mutandis. Observe that the approximating function
$U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$ satisfies
$\frac{d^{2}U}{d\zeta^{2}}=\hbar^{-2}(\zeta^{2}-\alpha^{2})U$. If we subtract
this from (C.2) we obtain the following differential equation for the error
term

$\frac{d^{2}\epsilon_{1}}{d\zeta^{2}}-\hbar^{-2}(\zeta^{2}-\alpha^{2})\epsilon_{1}=\psi(\zeta,\alpha,\hbar)\big{[}\epsilon_{1}+U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\big{]}.$

By use of the method of variation of parameters and also (B.3) one arrives at
the integral equation

$\epsilon_{1}(\zeta,\alpha,\hbar)=\frac{1}{2}\frac{(\pi\hbar)^{\frac{1}{2}}}{\Gamma\big{(}\frac{1}{2}+\frac{1}{2}\hbar^{-1}\alpha^{2}\big{)}}\int_{\zeta}^{Z}\mathcal{K}(\zeta,t)\psi(t,\alpha,\hbar)\big{[}\epsilon_{1}(t,\alpha,\hbar)+U(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\big{]}dt$

in which

$\mathcal{K}(\zeta,t)=U(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\overline{U}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\\\
-U(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\overline{U}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2}).$

Bounds for the kernel $\mathcal{K}$ and its first two partial derivatives
(with respect to $\zeta$) are expressible in terms of the auxiliary functions
$\mathsf{E},\mathsf{M}$ and $\mathsf{N}$. We have

$\displaystyle|\mathcal{K}(\zeta,t)|$
$\displaystyle\leq\frac{\mathsf{E}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathsf{M}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$
$\displaystyle\bigg{|}\frac{\partial\mathcal{K}}{\partial\zeta}(\zeta,t)\bigg{|}$
$\displaystyle\leq\sqrt{2\hbar^{-1}}\frac{\mathsf{E}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathsf{M}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$

and similarly

$\frac{\partial^{2}\mathcal{K}}{\partial\zeta^{2}}(\zeta,t)=(2\hbar^{-1})^{\frac{3}{2}}\zeta\mathsf{K}(\zeta,t).$

All these estimates allow us to solve the equation (C.2) by applying Theorem
C.1. Using the notation of that theorem we have

$\displaystyle\phi(t)$
$\displaystyle=\frac{\psi(\zeta,\alpha,\hbar)}{\Omega(\zeta\sqrt{2\hbar^{-1}})}$
$\displaystyle\psi_{1}(t)$ $\displaystyle=0$ $\displaystyle J(t)$
$\displaystyle=U(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$
$\displaystyle\mathsf{K}(\zeta,t)$
$\displaystyle=-\frac{1}{2}\frac{(\pi\hbar)^{\frac{1}{2}}}{\Gamma\big{(}\frac{1}{2}+\frac{1}{2}\hbar^{-1}\alpha^{2}\big{)}}\Omega(t\sqrt{2\hbar^{-1}})\mathcal{K}(\zeta,t)$
$\displaystyle Q(t)$
$\displaystyle=\frac{1}{2}\frac{(\pi\hbar)^{\frac{1}{2}}}{\Gamma\big{(}\frac{1}{2}+\frac{1}{2}\hbar^{-1}\alpha^{2}\big{)}}\Omega(t\sqrt{2\hbar^{-1}})\mathsf{E}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})\mathsf{M}(t\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$
$\displaystyle P_{0}(\zeta)$
$\displaystyle=\frac{\mathsf{M}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}$
$\displaystyle P_{1}(\zeta)$
$\displaystyle=\sqrt{2\hbar^{-1}}\frac{\mathsf{N}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}{\mathsf{E}(\zeta\sqrt{2\hbar^{-1}},-\tfrac{1}{2}\hbar^{-1}\alpha^{2})}$
$\displaystyle\Phi(\zeta)$
$\displaystyle=\mathcal{V}_{0,\zeta}[H](\alpha,\hbar)$
$\displaystyle\kappa_{0}$
$\displaystyle\leq\frac{1}{2}(\pi\hbar)^{\frac{1}{2}}l(-\tfrac{1}{2}\hbar^{-1}\alpha^{2})$

where the role of $\beta$ is played here by $Z$ and $\kappa$ is replaced for
simplicity by the upper bound $\kappa_{0}$. Then the bounds (C.3) and (C.4)
follow from Theorem C.1.

Finally, observe that all the integrals which occur in the analysis above,
converge uniformly when $\alpha\in[0,\delta]$ and $\zeta$ lies in any compact
interval of $[0,Z)$; allowing us to state that $\epsilon_{1}$ and its first
two partial $\zeta$-derivatives are continuous in $\alpha$ and $\zeta$.
Consequently, the same stands for $\mathcal{Y}_{1}$ which signifies the end of
the proof. ∎

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## Acknowledgements

We are grateful to a referee for insisting on the clarification of the results
and proofs of section §11. The first author acknowledges the support of the
Institute of Applied and Computational Mathematics of the Foundation of
Research and Technology - Hellas (FORTH), via grant MIS 5002358. Also, the
first author expresses his sincere gratitude to the Independent Power
Transmission Operator (IPTO) for a scholarship through the School of Sciences
and Engineering of the University of Crete.

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