*subsubsection [5em] [ | ] []

# Existence and Uniqueness of Exact WKB Solutions
for Second-Order Singularly Perturbed Linear ODEs

Nikita Nikolaev  School of Mathematics and Statistics, University of
Sheffield, United Kingdom

(UPDATED VERSION: )

###### Abstract

We prove an existence and uniqueness theorem for exact WKB solutions of
general singularly perturbed linear second-order ODEs in the complex domain.
These include the one-dimensional time-independent complex Schrödinger
equation. Notably, our results are valid both in the case of generic WKB
trajectories as well as closed WKB trajectories. We also explain in what sense
exact and formal WKB solutions form a basis. As a corollary of the proof, we
establish the Borel summability of formal WKB solutions for a large class of
problems, and derive an explicit formula for the Borel transform.

Keywords: exact WKB analysis, exact WKB method, linear ODEs, Schrödinger
equation, singular perturbation theory, exact perturbation theory, Borel
resummation, Borel-Laplace theory, asymptotic analysis, exponential
asymptotics, Gevrey asymptotics, resurgence

2020 MSC: 34M60 (primary); 34E10, 34E20, 40G10, 34M03, 34M25

###### Contents

  1. 1 Introduction
  2. 2 Setting
    1. 2.1 Background Assumptions
    2. 2.2 Examples
    3. 2.3 Formal and Exact Solutions
  3. 3 Formal WKB Solutions
    1. 3.1 The Semiclassical Limit
    2. 3.2 Existence and Uniqueness
    3. 3.3 Remarks on Formal WKB Solutions
  4. 4 WKB Geometry
    1. 4.1 The Liouville Transformation
    2. 4.2 WKB Trajectories and Rays
    3. 4.3 Closed and generic WKB trajectories
    4. 4.4 WKB Strips, Halfstrips, and Ring Domains
  5. 5 Exact WKB Solutions
    1. 5.1 Existence and Uniqueness of Exact WKB Solutions
    2. 5.2 Borel Summability of WKB Solutions
    3. 5.3 Explicit Formula for the Borel Transform
    4. 5.4 Notable Special Cases and Examples
      1. Closed WKB Trajectories
      2. Generic WKB Trajectories
    5. 5.5 Relation to Previous Work
  6. A Basics of Usual Asymptotics
    1. A.1 Poincaré Asymptotics in One Dimension
    2. A.2 Uniform Poincaré Asymptotics
    3. A.3 Gevrey Asymptotics
  7. B Basics of Exponential Asymptotics
    1. B.1 Exponential Asymptotics in One Dimension
    2. B.2 Uniform Exponential Asymptotics
  8. C Basics of the Borel-Laplace Theory
    1. C.1 The Laplace Transform
    2. C.2 The Borel Transform
    3. C.3 Borel Resummation
  9. D Proofs
    1. C.1 § 3.2: Existence and Uniqueness of Formal WKB Solutions
    2. C.2 Existence and Uniqueness of Exact Characteristic Solutions
    3. C.3 Theorem 5.2: Existence and Uniqueness of Exact WKB Solutions

##  § 1. Introduction

Consider a singularly perturbed $2^{\text{nd}}$-order linear ordinary
differential equation

$\hbar^{2}\partial_{x}^{2}\psi+p\hbar\partial_{x}\psi+q\psi=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(1)

where $x$ is a complex variable, $\hbar$ is a small complex perturbation
parameter, and the coefficients $p,q$ are holomorphic functions of $(x,\hbar)$
in some domain in $\mathbb{C}^{2}_{x\hbar}$. The question we study is a
quintessential problem in singular perturbation theory. Namely, in this paper
we search for solutions of (1) that are holomorphic in both variables $x$ and
$\hbar$ and admit well-defined asymptotics as $\hbar\to 0$ in a specified
sector.

##### 1.1. Results.

The main result of this paper (Theorem 5.2) establishes precise general
conditions for the existence and uniqueness of exact WKB solutions. These are
holomorphic solutions that are canonically specified (in a precise sense via
Borel resummation) by their exponential asymptotic expansions as $\hbar\to 0$
in a halfplane. They are constructed by means of the Borel-Laplace method for
the associated singularly perturbed Riccati equation which we investigated in
[Nik20].

Our approach yields a general result (Theorem 5.7) about the Borel summability
of formal WKB solutions. These are exponential formal $\hbar$-power series
solutions that assume the role of asymptotic expansions as $\hbar\to 0$ of
exact WKB solutions within appropriate domains in $\mathbb{C}_{x}$. We also
prove an existence and uniqueness result (§ 3.2) for formal WKB solutions
that in particular clarifies precisely in what sense they form a basis of
formal solutions.

The construction of exact WKB solutions involves the geometry of certain real
curves in $\mathbb{C}_{x}$ (the WKB trajectories) traced out using a type of
Liouville transformation. Two special classes of this geometry (closed and
generic WKB trajectories) are especially important because they appear in wide
a variety of applications. Notably, our existence and uniqueness and the Borel
summability results remain valid in both of these situations. Thus, we
construct an exact WKB basis both along a closed WKB trajectory (§ 5.4) as
well as a generic WKB trajectory (§ 5.4).

The explicit nature of our approach yields refined information about the Borel
transform of WKB solutions. This includes an explicit recursive formula (Part
 5.10) which we hope will facilitate the analysis of the singularity
structure in the Borel plane and perhaps lead to a fuller understanding of the
resurgent properties of WKB solutions in a large class of problems (see Part
 5.10).

##### 1.2. Brief literature review.

The WKB approximation method was established in the mathematical context by
Jeffreys [Jef24] and independently in the analysis of the Schrödinger equation
in quantum mechanics by Wentzel [Wen26], Kramers [Kra26], and Brillouin
[Bri26]. However, it has a very long history that goes further back to at
least Carlini (1817), Liouville (1837), and Green (1837); for an in-depth
historical overview, see for example the books of Heading [Hea62, Ch.I],
Fröman and Fröman [FF65, Ch.1], and Dingle [Din73, Ch.XIII]. A clear
exposition of the asymptotic theory of the WKB approximation can be found in
the remarkable textbook of Bender and Orszag [BO99, Part III]. The
relationship between the asymptotic properties of the WKB approximation and
the geometry of WKB trajectories was comprehensively examined by Evgrafov and
Fedoryuk [EF66, Fed93].

In the early 1980s, influenced by the earlier work of Balian and Bloch [BB74],
a groundbreaking advancement was made by Voros [Vor81, Vor83a] who lay the
foundations for upgrading the WKB approximation method to an exact method,
dubbed the exact WKB method. Although the value of considering the all-orders
WKB expansions was suggested earlier by Dunham [Dun32], Bender and Wu [BW69],
Dingle [Din73], and later by ’t Hooft [tH79] in a purely physics context,
Voros was the first to introduce in a more systematic fashion techniques from
the theory of Borel-Laplace transformations. Crucial early contributions to
the development of the general theory of exact WKB analysis for second-order
linear ODEs include works of Leray [Ler57], Boutet de Monvel and Krée
[BdMK67], Silverstone [Sil85], Aoki, Sato, Kashiwara, Kawai, Takei, and
Yoshida [SKK73, AKT91, AY93, AKT93], Delabaere, Dillinger, and Pham [DDP93,
DDP97, DP99, Pha00], Dunster, Lutz, and Schäfke [DLS93], Écalle [Éca94], and
Koike [Koi99, Koi00a, Koi00b]. For a survey of early work in exact WKB
analysis, we recommend the excellent book of Kawai and Takei [KT05], as well
as the comprehensive review article by Voros [Vor12].

Later developments focused mainly on understanding WKB-theoretic
transformation series (first introduced in [AKT91]) that transform a given
differential equation in a suitable neighbourhood of critical WKB trajectories
(or Stokes lines) to one in standard form whose WKB-theoretic properties are
better understood. A very partial list of contributions includes the works by
Aoki, Kamimoto, Kawai, Koike, Sasaki, and Takei, [AKT09, KK11, KKT12, KK13,
Sas13, KKT14a]. Parallel to this activity has been the classification of WKB
geometry (or Stokes graphs), which includes the works by Aoki, Kawai, Takei,
Tanda, [AKT01, Tak07, AT13, Tak17], as well as a detailed analysis of some
WKB-theoretic properties of special classes of equations, which includes the
works by Aoki, Kamimoto, Kawai, Koike, and Takei [KT11, KKKT11, ATT14, KKT14b,
Tan15, KKK16, ATT16, AT16, AIT19].

However, although the existence of exact WKB solutions in classes of examples
has been established, a general existence theorem for second-order linear ODEs
has remained unavailable. Contributions towards such a general theory include
Gerard and Grigis [GG88], Bodine, Dunster, Lutz, and Schäfke [DLS93, Dun01,
BS02], Giller and Milczarski [GM01], Koike and Takei [KT13], Ferreira, López,
and Sinusía [FLPS14, FLS15], as well as most recently by Nemes [Nem21] whose
preprint appeared at roughly the same time as our previous work [Nik20] that
underpins our results here. Our paper contributes to this long line of work by
establishing a general theory of existence and uniqueness of exact WKB
solutions, which generalises the relevant results from the aforementioned
works (see § 5.5 for a discussion).

The need for a general existence result for exact WKB solutions of equations
of the form (1) is evident from a recent surge of scientific advances that
rely upon it. For example, this includes works in cluster algebras and
character varieties [IN14, Kid17, All19, Kuw20, AB20], stability conditions
and Donaldson-Thomas invariants [All18, Bri19, All21], high energy physics
[KPT15, HK18, HN20, ST21, GHN21], Gromov-Witten theory [FIMS19], as well as
further developments in WKB analysis [Tan15, AT16, AIT19]. Some of the results
in these references specifically rely on a statement of Borel summability of
formal WKB solutions presented in [IN14, Theorem 2.17]. This statement is
drawn from an unpublished work of Koike and Schäfke on the Borel summability
of WKB solutions of Schrödinger equations with polynomial potentials (see
[Tak17, §3.1] for a brief account of Koike-Schäfke’s ideas). As explained in
Part  5.13, this statement is a special case of our main theorem. Therefore,
our paper provides a rigorous proof of Koike-Schäfke’s assertion.

Acknowledgements. I want to expresses special gratitude to Marco Gualtieri,
Marta Mazzocco, and Jörg Teschner for their encouragement to finish this
project and write this paper. I am thankful to Francis Bischoff, Marco
Gualtieri, and Kento Osuga for very helpful suggestions for improving the
draft of this paper. I also thank André Voros for very useful comments on the
first preprint version of this paper, especially regarding the literature
review. I benefited from discussions with Anton Alekseev, Dylan Allegretti,
Francis Bischoff, Tom Bridgeland, Marco Gualtieri, Kohei Iwaki, Omar Kidwai,
Andrew Neitzke, Gergő Nemes, Kento Osuga, and Shinji Sasaki. This work was
supported by the NCCR SwissMAP of the SNSF, as well as the EPSRC Programme
Grant Enhancing RNG.

##  § 2. Setting

In this section, we describe our general setup, give a few examples, and
define the notion of formal and exact solutions that are sought for in this
paper.

###  § 2.1. Background Assumptions

##### 2.1.

Fix a complex plane $\mathbb{C}_{x}$ with coordinate $x$ and another complex
plane $\mathbb{C}_{\hbar}$ with coordinate $\hbar$. Fix a domain
$X\subset\mathbb{C}_{x}$ and a sectorial domain $S\subset\mathbb{C}_{\hbar}$
at the origin with opening arc
$A=(\vartheta_{-},\vartheta_{+})\subset{\mathbb{R}}$ and opening angle
$\pi\leqslant|A|\leqslant 2\pi$. See 1(a).

We consider the following differential equation for a scalar function
$\psi=\psi(x,\hbar)$:

$\hbar^{2}\partial^{2}_{x}\psi+p\hbar\partial_{x}\psi+q\psi=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(2)

where $p,q$ are holomorphic functions of $(x,\hbar)\in X\times S$ which admit
locally uniform Gevrey asymptotic expansions
$\widehat{p},\widehat{q}\in{\mathcal{O}}(X)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
with holomorphic coefficients along the closed arc $\overline{A}$:

$\begin{gathered}p(x,\hbar)\simeq\widehat{p}(x,\hbar)\mathrel{\mathop{:}}=\sum_{k=0}^{\infty}p_{k}(x)\hbar^{k}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}\\\
q(x,\hbar)\simeq\widehat{q}(x,\hbar)\mathrel{\mathop{:}}=\sum_{k=0}^{\infty}q_{k}(x)\hbar^{k}\end{gathered}\quad\quad\quad\text{as
$\hbar\to 0$ along $\overline{A}$, loc.unif. $\forall x\in X$\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}
missing}$ (3)

##### 2.2.

Basic notions from asymptotic analysis as well as our notation and conventions
are summarised in Appendix A. Explicitly, assumption (3) for, say, $q$ means
the following. For every $x_{0}\in X$, there is a neighbourhood $U_{0}\subset
X$ of $x_{0}$, a sectorial subdomain $S_{0}\subset S$ with the same opening
$A$ (see 1(b)), and constants $\mathnormal{C},\mathnormal{M}>0$ such that for
all $n\geqslant 0$, all $x\in U_{0}$, and all $\hbar\in S_{0}$,

$\mathopen{}\mathclose{{}\left|q(x,\hbar)-\sum_{k=0}^{n-1}q_{k}(x)\hbar^{k}}\right|\leqslant\mathnormal{C}\mathnormal{M}^{n}n!|\hbar|^{n}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(4)

(The sum for $n=0$ is empty.) In particular, this means that each $q_{k}$ is
bounded on $U_{0}$, and $q$ is bounded on $U_{0}\times S_{0}$. Let us also
stress that (3) is stronger than the usual notion of Gevrey asymptotics in
that we require the above bounds to hold $\hbar\to 0$ uniformly in all
directions within $S$ (see Part  A.4). This stronger asymptotic assumption
plays a crucial role in our analysis by allowing us to draw uniqueness
conclusions with the help of a theorem of Nevanlinna [Nev18, Sok80] (see
Theorem C.8; see also [Nik20, Theorem B.11] where we present a detailed
proof).

(a) Sectorial domain $S$ with opening $A=(\vartheta_{-},\vartheta_{+})$.

(b) Sectorial subdomain $S_{0}$ with the same opening $A$.

Figure 1:

##### 2.3.

A typical way to ensure the asymptotic condition (3) is to start with
holomorphic functions $p,q$, defined for $\hbar$ in a strictly larger
sectorial domain $\tilde{S}\supset S$ with a strictly larger opening
$\tilde{A}$ such that $\overline{A}\subset\tilde{A}$, which admit Gevrey
asymptotics as $\hbar\to 0$ along the open arc $\tilde{A}$. Then the
restrictions of $p,q$ to $S$ necessarily satisfy (3).

In particular, if $p,q$ are actually holomorphic at $\hbar=0$, then the
asymptotic condition (3) is automatically satisfied. In this case, the power
series $\widehat{p},\widehat{q}$ are nothing but the convergent Taylor series
expansions in $\hbar$ of $p,q$ at $\hbar=0$.

###  § 2.2. Examples

######  2.3 Example (Classical differential equations).

The simplest interesting situation is when the coefficients $p,q$ are
polynomial functions of $x$ only. In this case, $X=\mathbb{C}_{x}$ and $S$ is
ordinarily taken to be the right halfplane
$\mathopen{}\mathclose{{}\left\\{\operatorname{Re}(\hbar)>0}\right\\}$ where
one demands asymptotic control on solutions as $\hbar\to 0$. The most famous
example of this situation is the $\hbar$-Airy equation:

$\hbar^{2}\partial_{x}^{2}\psi-x\psi=0\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(5)

Another example is the $\hbar$-Weber equation (sometimes also known as the
quantum harmonic oscillator):

$\hbar^{2}\partial_{x}^{2}\psi-\big{(}x^{2}-a\big{)}\psi=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(6)

where $a$ is any complex number.

More general examples are provided by famous classical differential equations
(with $\partial_{x}$ replaced by $\hbar\partial_{x}$) for which the
coefficients $p,q$ are rational functions of $x$ only. This includes the Gauss
hypergeometric equation, as well as Bessel, Heun, Hermite, and many others. In
these cases, $X$ is the complement of finitely many points in
$\mathbb{C}_{x}$.

######  2.3 Example (The Mathieu equation).

All equations in the previous example extend to second-order equations on the
Riemann sphere with a pole at infinity (beware, however, that this extension
is not unique). A famous example where this is not the case is the Mathieu
equation, sometimes written as

$\hbar^{2}\partial_{x}^{2}\psi-2\big{(}\cos(x)-\mathnormal{E}\,\big{)}\psi=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(7)

where $\mathnormal{E}$ is a complex number. This equation has an essential
singularity at infinity, yet the methods in this paper are still directly
applicable to this equation.

######  2.3 Example (Mildly deformed coefficients).

More generally, the coefficients $p,q$ can be polynomials in $\hbar$ with
coefficients which are rational or more general meromorphic functions of $x$.
Such examples appear, in Hermitian matrix models of the Gaussian potential
[BE17, §6.7, equation (6.86)] and more generally in the study of quantum
curves, where the following deformation of the $\hbar$-Weber equation (6) is
encountered:

$\hbar^{2}\partial_{x}^{2}\psi-\big{(}x^{2}-4+2\hbar\big{)}\psi=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

######  2.3 Example (A nontrivially deformed $\hbar$-Airy equation).

Our methods are applicable to classical differential equations with much more
sophisticated $\hbar$-dependence. For example, let $X=\mathbb{C}^{\ast}_{x}$,
$S=\mathopen{}\mathclose{{}\left\\{\operatorname{Re}(\hbar)>0}\right\\}$, and
consider the following nontrivial deformation of the $\hbar$-Airy equation
(5):

$\hbar^{2}\partial^{2}_{x}\psi-\big{(}x+\mathnormal{E}(x,\hbar)\big{)}\psi=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(8)

where

$\mathnormal{E}(x,\hbar)\mathrel{\mathop{:}}=\int_{0}^{+\infty}\frac{e^{-\xi/\hbar}}{x+\xi}\mathinner{{\mathrm{d}}\xi}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(9)

The function $\mathnormal{E}(x,\hbar)$ is not holomorphic at $\hbar=0$, but
one can verify that it admits locally uniform Gevrey asymptotics as $\hbar\to
0$ along the closed arc $\overline{A}=[-\tfrac{\pi}{2},+\tfrac{\pi}{2}]$.

######  2.3 Example (The Schrödinger equation).

The most famous special class of equations (2) is the complex one-dimensional
time-independent Schrödinger equation

$\Big{(}\hbar^{2}\partial_{x}^{2}-\mathnormal{Q}(x,\hbar)\Big{)}\varphi(x,\hbar)=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(10)

In fact, any second-order equation (2) can be put into the Schrödinger form
(10) by means of the following transformation of the unknown variable:

$\varphi(x,\hbar)=\exp\mathopen{}\mathclose{{}\left(\frac{1}{2\hbar}\int\nolimits^{x}_{x_{0}}p(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\psi(x,\hbar)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(11)

where $x_{0}$ is a suitably chosen basepoint. In terms of the coefficients of
(2), the resulting potential is
$\mathnormal{Q}=\tfrac{1}{4}p^{2}+\tfrac{1}{2}\hbar\partial_{x}p-q$.

However, in this paper we prefer not to use the transformation (11) and
instead continue to work with equation (2). We have two main reasons for this
preference. Firstly, the transformation (11) develops essential singularities
wherever $p$ has singularities, which is typically on the boundary of $X$.
Secondly, and perhaps most importantly from the geometric point of view, the
Schrödinger form (10) is not a coordinate-independent expression unless this
differential equation is posed not on functions but on sections of a specific
line bundle over a Riemann surface (the square-root anti-canonical bundle).
These details will be explained in [Nik21a].

###  § 2.3. Formal and Exact Solutions

Our goal is to construct holomorphic solutions of (2) with prescribed
asymptotic behaviour as $\hbar\to 0$. Because of the way our linear equation
is perturbed (i.e., differentiation $\partial_{x}$ is multiplied by a single
power of $\hbar$), it turns out that the correct notion of asymptotics is
exponential asymptotics. This notion is briefly recalled in Appendix B.
Indeed, one can easily verify that nonzero holomorphic solutions generically
cannot admit a usual power series asymptotic expansion at $\hbar=0$. The
following definition gives the precise class of solutions we seek in this
paper.

######  2.3 Definition ( ).

A weakly-exact solution of (2) on an open subset $U\subset X$ is a holomorphic
solution $\psi$ defined on $U\times S^{\prime}\subset X\times S$ for some
sectorial subdomain $S^{\prime}\subset S$ with nonempty opening
$A^{\prime}\subset A$, which admits locally uniform exponential asymptotics as
$\hbar\to 0$ along $A^{\prime}$. If $A^{\prime}=A$, then we call $\psi$ a
strongly-exact solution on $(U;A)$, or simply exact solution.

##### 2.4.

Existence of weakly-exact solutions is a classical fact in the theory of
differential equations (see, e.g., [Was76, Theorem 26.2]). However, weakly-
exact solutions constructed by usual methods are inherently non-unique and in
general there is no control on the size of the opening $A^{\prime}$ (see,
e.g., the remark in [Was76, p.144], immediately following Theorem 26.1). Our
attention in this paper is instead focused on the strongly-exact solutions,
which from this point of view form a more restricted class of solutions. A
priori, these may not exist even if weakly-exact solutions are abundant. The
problem of finding strongly-exact solutions is a nontrivial sharpening of the
more classical problem of finding weakly-exact solutions.

In this paper, we will in fact construct a special kind of exact solutions,
called exact WKB solutions. Their distinguishing property is that they are
canonically associated (in a precise sense) with their exponential
asymptotics.

##### 2.5. The space of exact solutions.

We denote by $\mathbb{S}(U\times S)$ the space of all holomorphic solutions of
(2) defined on the domain $U\times S$ (without any asymptotic restrictions).
Our differential equation is linear of order two, so $\mathbb{S}(U\times S)$
is a rank-two module over the ring ${\mathcal{O}}(S)$ of holomorphic functions
on $S$. More generally, we can consider the space $\mathbb{S}(U;A)$ of
semisectorial germs (see Part  A.6) of holomorphic solutions defined on the
pair $(U;A)$. Again, $\mathbb{S}(U;A)$ is a rank-two module for the ring
${\mathcal{O}}(A)$ of sectorial germs (without asymptotic restrictions).

Let $\mathbb{ES}(U;A)\subset\mathbb{S}(U;A)$ be the subset of exact solutions
on $(U;A)$ in the sense of § 2.3. This is a module over the ring
${\mathcal{A}}^{\exp}(A)$ of holomorphic sectorial germs that admit
exponential asymptotics along $A$. We also let
$\mathbb{GES}(U;A)\subset\mathbb{ES}(U;A)$ be the subset of exact solutions on
$(U;A)$ which admit locally uniform exponential Gevrey asymptotics along $A$.
Similarly, it is a module over the ring ${\mathcal{G}}^{\exp}(A)$ of
holomorphic sectorial germs that admit exponential asymptotics along $A$.
These are the spaces in which we search for (and find!) solutions of (2).

##### 2.6. Formal solutions.

If $\psi$ is an exact solution of (2), its exponential asymptotic expansion
$\widehat{\psi}$ formally satisfies the asymptotic analogue of the
differential equation (2) in which the coefficients $p,q$ are replaced by
their asymptotic power series $\widehat{p},\widehat{q}$:

$\hbar^{2}\partial^{2}_{x}\widehat{\psi}+\widehat{p}\>\hbar\partial_{x}\widehat{\psi}+\widehat{q}\;\widehat{\psi}=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(12)

######  2.6 Definition ( ).

An exponential power series solution on a domain $U\subset X$ is an
exponential power series

$\widehat{\psi}=e^{-\Phi/\hbar}\,\widehat{\Psi}~{}\in~{}{\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(13)

where $\Phi\in{\mathcal{O}}(U)[\hbar^{-1}]$ and
$\widehat{\Psi}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$,
that formally satisfies (12) for all $x\in U$. More generally, a formal
solution on $U$ is an exponential transseries
$\widehat{\psi}\in{\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
that formally satisfies (12); i.e., $\widehat{\psi}$ is a finite combination
of exponential power series:

$\widehat{\psi}=\sum_{\alpha}^{\textup{\tiny{finite}}}e^{-\Phi_{\alpha}/\hbar}\,\widehat{\Psi}_{\alpha}~{}\in~{}{\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
(14)

where $\Phi_{\alpha}\in{\mathcal{O}}(U)[\hbar^{-1}]$ and
$\widehat{\Psi}_{\alpha}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.

A brief account of exponential power series and transseries can be found in
Appendix B. Note also that we have introduced a negative sign in the exponent
in (13) for future convenience.

##### 2.7. The space of formal solutions.

We denote the subset of
${\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
consisting of all formal solutions on $U$ by
$\widehat{\mathbb{S}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.
It is a module over the ring of exponential transseries
$\mathbb{C}^{\exp}\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.
Evidently, the asymptotic expansion defines a map
$\text{\ae}:\mathbb{ES}(U;A)\to\widehat{\mathbb{S}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$,
but it is not a homomorphism because of complicated dominance relations for
exponential prefactors (see Appendix B for a comment).

##  § 3. Formal WKB Solutions

In this section, we analyse the differential equation (2) in a purely formal
setting where we ignore all analytic questions with respect to $\hbar$. As a
notational mnemonic used throughout the paper, objects decorated with a hat
are formal.

##### 3.1. Formal setup.

Thus, we consider a general formal second-order differential equation of the
following form:

$\hbar^{2}\partial^{2}_{x}\widehat{\psi}+\widehat{p}\>\hbar\partial_{x}\widehat{\psi}+\widehat{q}\;\widehat{\psi}=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(15)

where the coefficients
$\widehat{p},\widehat{q}\in{\mathcal{O}}(X)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
are formal power series in $\hbar$ with holomorphic coefficients on $X$:

$\widehat{p}(x,\hbar)\mathrel{\mathop{:}}=\sum_{k=0}^{\infty}p_{k}(x)\hbar^{k}\qquad\text{and}\qquad\widehat{q}(x,\hbar)\mathrel{\mathop{:}}=\sum_{k=0}^{\infty}q_{k}(x)\hbar^{k}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(16)

We search for formal solutions in the sense of Part  2.6.

###  § 3.1. The Semiclassical Limit

##### 3.2.

A standard technique in solving linear ODEs is to introduce the characteristic
equation. We consider only the leading-order characteristic equation, which in
for the second-order equation at hand is the following quadratic equation with
holomorphic coefficients $p_{0},q_{0}\in{\mathcal{O}}(X)$ for a holomorphic
function $\lambda=\lambda(x)$:

$\lambda^{2}-p_{0}\lambda+q_{0}=0\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(17)

We refer to its discriminant as the leading-order characteristic discriminant:

$\mathnormal{D}_{0}\mathrel{\mathop{:}}=p_{0}^{2}-4q_{0}~{}\in{\mathcal{O}}(X)$
(18)

We always assume that $\mathnormal{D}_{0}$ is not identically zero. The zeros
of $\mathnormal{D}_{0}$ are called turning points, and all other points in $X$
are called regular points. If $x_{0}\in X$ is a regular point, then (17) has
two distinct holomorphic solutions. We call them the leading-order
characteristic roots. Upon fixing a local square-root branch
$\sqrt{\mathnormal{D}_{0}}$ near $x_{0}$, we will always label them as
follows:

$\lambda_{\pm}\mathrel{\mathop{:}}=\frac{p_{0}\pm\sqrt{\mathnormal{D}_{0}}}{2}\qquad\text{so
that}\qquad\sqrt{\mathnormal{D}_{0}}=\lambda_{+}-\lambda_{-}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(19)

######  3.2 Example (The Schrödinger equation).

For the Schrödinger equation (10), the leading-order characteristic
discriminant is $\mathnormal{D}_{0}=4\mathnormal{Q}_{0}$, and the leading-
order characteristic roots are $\lambda_{\pm}=\pm\sqrt{\mathnormal{Q}_{0}}$.

######  3.2 Remark ( ).

In analysis, equation (17) is often called the semiclassical limit of the
second-order differential operator in (2). It can be obtained as the vanishing
constraint of the following limit:

$\lim_{\begin{subarray}{c}\hbar\to 0\\\ \hbar\in
S\end{subarray}}\mathopen{}\mathclose{{}\left[e^{+\frac{1}{\hbar}\int\nolimits^{x}\lambda(t)\mathinner{{\mathrm{d}}t}}\Big{(}\hbar^{2}\partial^{2}_{x}+p(x,\hbar)\hbar\partial_{x}+q(x,\hbar)\Big{)}e^{-\frac{1}{\hbar}\int\nolimits^{x}\lambda(t)\mathinner{{\mathrm{d}}t}}}\right]=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

###  § 3.2. Existence and Uniqueness

Existence of formal solutions of (15) away from turning points is well-known.
For Schrödinger equations (i.e., with $\widehat{p}=0$), they are often called
(formal) WKB solutions. However, uniqueness statements are not usually made
completely explicit. It is also sometimes said that ‘formal WKB solutions are
linearly independent and form a basis’, but the space they generate is again
not usually made explicit. The purpose of the following proposition is to make
these statements precise and explicit.

######  3.2 Proposition (Existence and Uniqueness of Formal WKB Solutions).

Let $x_{0}\in X$ be a regular point. If $U\subset X$ is any simply connected
neighbourhood of $x_{0}$ free of turning points, then (15) has precisely two
nonzero exponential power series solutions
$\widehat{\psi}_{+},\widehat{\psi}_{-}\in{\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
on $U$ normalised at $x_{0}$ by $\widehat{\psi}_{\pm}(x_{0},\hbar)=1$. They
form a basis for the space
$\widehat{\mathbb{S}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
of all formal solutions. Moreover, once a square-root branch
$\sqrt{\mathnormal{D}_{0}}$ near $x_{0}$ has been chosen, they can be labelled
and expressed as follows:

$\widehat{\psi}_{\pm}(x,\hbar)\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int_{x_{0}}^{x}\widehat{s}_{\pm}(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(19)

where
$\widehat{s}_{+},\widehat{s}_{-}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
are the two unique formal solutions with leading-order terms
$\lambda_{+},\lambda_{-}$ of the formal singularly perturbed Riccati equation

$\hbar\partial_{x}\widehat{s}=\widehat{s}^{\,2}-\widehat{p}\,\widehat{s}+\widehat{q}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(20)

In fact, $\widehat{\psi}_{\pm}$ are the unique formal solutions on $U$
satisfying the following initial conditions:

$\widehat{\psi}_{\pm}(x_{0},\hbar)=1\qquad\text{and}\qquad{\hbar\partial_{x}\widehat{\psi}_{\pm}}{\raisebox{-2.0pt}{$\big{\rvert}_{(x,\hbar)=(x_{0},0)}$}}=\lambda_{\pm}(x_{0})\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(21)

The proof of this theorem is essentially a computation, presented in § C.1.

######  3.2 Definition ( ).

The two formal solutions $\widehat{\psi}_{\pm}$ from § 3.2 are called the
formal WKB solutions normalised at the regular point $x_{0}\in U$. The basis
$\mathopen{}\mathclose{{}\left\\{\smash{\widehat{\psi}_{+},\widehat{\psi}_{-}}}\right\\}$
is called the formal WKB basis normalised at $x_{0}$. We will also refer to
the two formal solutions $\widehat{s}_{\pm}$ as the formal characteristic
roots.

##### 3.3.

Thus, § 3.2 explains that the formal WKB solutions are uniquely specified
either by the normalisation condition and the requirement that they be
exponential power series (i.e., having only one exponential prefactor) or
equivalently by the two initial conditions (21). Note that the second initial
condition is nothing but a way to select the correct exponential prefactor.

##### 3.4. WKB recursion.

A practical advantage of formal WKB solutions is that the formal
characteristic roots can be computed very explicitly by solving a recursive
tower of linear algebraic equations. The following lemma is a direct
consequence of the proof of § 3.2.

######  3.4 Lemma ( ).

The coefficients $s^{\text{\tiny$(k)$}}_{\pm}\in{\mathcal{O}}(U)$ of the
formal characteristic roots

$\widehat{s}_{\pm}(x,\hbar)=\sum_{k=0}^{\infty}s_{\pm}^{\text{\tiny$(k)$}}(x)\hbar^{k}$
(22)

are given by the following recursive formula:
$s_{\pm}^{\text{\tiny$(0)$}}=\lambda_{\pm}=\tfrac{1}{2}(p_{0}\pm\sqrt{\mathnormal{D}_{0}})$,
and for $k\geqslant 1$,

$s_{\pm}^{\text{\tiny$(k)$}}=\frac{\pm
1}{\sqrt{\mathnormal{D}_{0}}}\mathopen{}\mathclose{{}\left(\partial_{x}s_{\pm}^{\text{\tiny$(k-1)$}}-\sum_{k_{1}+k_{2}=k}^{k_{1},k_{2}\neq
k}s_{\pm}^{\text{\tiny$(k_{1})$}}s_{\pm}^{\text{\tiny$(k_{2})$}}+\sum_{k_{1}+k_{2}=k}^{k_{2}\neq
k}p_{k_{1}}s_{\pm}^{\text{\tiny$(k_{2})$}}-q_{k}}\right)\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(23)

Explicitly, these formulas for low values of $k$ are:

$\displaystyle s_{\pm}^{\text{\tiny$(1)$}}$ $\displaystyle=\frac{\pm
1}{\sqrt{\mathnormal{D}_{0}}}\Big{(}\partial_{x}\lambda_{\pm}+p_{1}\lambda_{\pm}-q_{1}\Big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(24) $\displaystyle s_{\pm}^{\text{\tiny$(2)$}}$ $\displaystyle=\frac{\pm
1}{\sqrt{\mathnormal{D}_{0}}}\Big{(}\partial_{x}s_{\pm}^{\text{\tiny$(1)$}}-(s_{\pm}^{\text{\tiny$(1)$}})^{2}+p_{1}s_{\pm}^{\text{\tiny$(1)$}}+p_{2}\lambda_{\pm}-q_{2}\Big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(25) $\displaystyle s_{\pm}^{\text{\tiny$(3)$}}$ $\displaystyle=\frac{\pm
1}{\sqrt{\mathnormal{D}_{0}}}\Big{(}\partial_{x}s_{\pm}^{\text{\tiny$(2)$}}-2s_{\pm}^{\text{\tiny$(1)$}}s_{\pm}^{\text{\tiny$(2)$}}+p_{1}s_{\pm}^{\text{\tiny$(2)$}}+p_{2}s_{\pm}^{\text{\tiny$(1)$}}+p_{3}\lambda_{\pm}-q_{3}\Big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(26) $\displaystyle s_{\pm}^{\text{\tiny$(4)$}}$ $\displaystyle=\frac{\pm
1}{\sqrt{\mathnormal{D}_{0}}}\Big{(}\partial_{x}s_{\pm}^{\text{\tiny$(3)$}}-2s_{\pm}^{\text{\tiny$(1)$}}s_{\pm}^{\text{\tiny$(3)$}}-(s_{\pm}^{\text{\tiny$(2)$}})^{2}+p_{1}s_{\pm}^{\text{\tiny$(3)$}}+p_{2}s_{\pm}^{\text{\tiny$(2)$}}+p_{3}s_{\pm}^{\text{\tiny$(1)$}}+p_{4}\lambda_{\pm}-q_{4}\Big{)}{\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}.}$
(27)

######  3.4 Example (unperturbed coefficients).

In the simplest but very common situation where the coefficients $p,q$ are
independent of $\hbar$, identities (23) simplify as follows:

$s_{\pm}^{\text{\tiny$(k)$}}=\pm\frac{1}{\sqrt{\mathnormal{D}_{0}}}\mathopen{}\mathclose{{}\left(\partial_{x}s_{\pm}^{\text{\tiny$(k)$}}-\sum_{k_{1}+k_{2}=k}^{k_{1},k_{2}\neq
k}s_{\pm}^{\text{\tiny$(k_{1})$}}s_{\pm}^{\text{\tiny$(k_{2})$}}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(28)

For low values of $k$, these are:

$\begin{gathered}s_{\pm}^{\text{\tiny$(1)$}}=\pm\frac{\partial_{x}\lambda_{\pm}}{\sqrt{\mathnormal{D}_{0}}},\quad
s_{\pm}^{\text{\tiny$(2)$}}=\pm\frac{\partial_{x}s_{\pm}^{\text{\tiny$(1)$}}-\big{(}s_{\pm}^{\text{\tiny$(1)$}}\big{)}^{2}}{\sqrt{\mathnormal{D}_{0}}},\quad
s_{\pm}^{\text{\tiny$(3)$}}=\pm\frac{\partial_{x}s_{\pm}^{\text{\tiny$(2)$}}-2s_{\pm}^{\text{\tiny$(1)$}}s_{\pm}^{\text{\tiny$(2)$}}}{\sqrt{\mathnormal{D}_{0}}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}\\\
s_{\pm}^{\text{\tiny$(4)$}}=\frac{\pm
1}{\sqrt{\mathnormal{D}_{0}}}\Big{(}\partial_{x}s_{\pm}^{\text{\tiny$(3)$}}-2s_{\pm}^{\text{\tiny$(1)$}}s_{\pm}^{\text{\tiny$(3)$}}-(s_{\pm}^{\text{\tiny$(2)$}})^{2}\Big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}\end{gathered}$
(29)

######  3.4 Example (Schrödinger equation).

For the Schrödinger equation (10), the leading-order characteristic roots are
$\lambda_{\pm}=\pm\sqrt{\mathnormal{Q}_{0}}$. Formula (23) for the
coefficients reduces to the following:

$s^{\text{\tiny$(k)$}}_{\pm}=\pm\frac{1}{2\sqrt{\mathnormal{Q}_{0}}}\mathopen{}\mathclose{{}\left(\partial_{x}s^{\text{\tiny$(k-1)$}}_{\pm}-\sum_{k_{1}+k_{2}=k}^{k_{1},k_{2}\neq
k}s^{\text{\tiny$(k_{1})$}}_{\pm}s^{\text{\tiny$(k_{2})$}}_{\pm}+\mathnormal{Q}_{k}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(30)

If, furthermore, the potential $\mathnormal{Q}$ is independent of $\hbar$
(i.e., $\mathnormal{Q}=\mathnormal{Q}_{0}$), then every $\mathnormal{Q}_{k}$
in the recursive formula (30) is $0$. In this case, for low values of $k$:

$s^{\text{\tiny$(1)$}}_{\pm}=\frac{1}{4}\frac{\mathnormal{Q}^{\prime}}{\mathnormal{Q}},\quad
s^{\text{\tiny$(2)$}}_{\pm}=\pm\frac{1}{8}\frac{\mathnormal{Q}^{\prime\prime}}{\mathnormal{Q}^{3/2}}\mp\frac{5}{32}\frac{(\mathnormal{Q}^{\prime})^{2}}{\mathnormal{Q}^{5/2}},\quad
s^{\text{\tiny$(3)$}}_{\pm}=\frac{1}{16}\frac{\mathnormal{Q}^{\prime\prime\prime}}{\mathnormal{Q}^{2}}-\frac{9}{16}\frac{\mathnormal{Q}^{\prime}\mathnormal{Q}^{\prime\prime}}{\mathnormal{Q}^{3}}+\frac{35}{64}\frac{(\mathnormal{Q}^{\prime})^{3}}{\mathnormal{Q}^{4}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

where ′ denotes $\partial_{x}$. So, for instance, for the $\hbar$-Airy
equation (5), $\mathnormal{Q}=x$, so these formulas reduce to:
$\lambda_{\pm}=\pm\sqrt{x}$, $s^{\text{\tiny$(1)$}}_{\pm}=\frac{1}{4}x^{-1}$,
$s^{\text{\tiny$(1)$}}_{\pm}=\mp\frac{5}{32}x^{-5/2}$,
$s^{\text{\tiny$(2)$}}_{\pm}=\frac{35}{64}x^{-4}$.

##### 3.5. WKB exponents.

To express (19) more explicitly as an exponential power series, we separate
out the leading-order part of the formal solutions to the Riccati equation.
Let us define

$\widehat{\SS}_{\pm}(x,\hbar)\mathrel{\mathop{:}}=\sum_{k=0}^{\infty}s^{\text{\tiny$(k+1)$}}_{\pm}(x)\hbar^{k}\qquad\text{so
that}\qquad\widehat{s}_{\pm}=\lambda_{\pm}+\hbar\widehat{\SS}_{\pm}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(30)

Then the formal WKB solutions $\widehat{\psi}_{\pm}$ can be written as
follows:

$\displaystyle\widehat{\psi}_{\pm}(x,\hbar)$
$\displaystyle=e^{-\Phi_{\pm}(x)/\hbar}\;\widehat{\Psi}_{\pm}(x,\hbar)$ (31)
$\displaystyle=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\pm}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\widehat{\SS}_{\pm}(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(32)

where $\Phi_{\pm}\in{\mathcal{O}}(U)$ and
$\widehat{\Psi}_{\pm}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
are defined by the following formulas:

$\displaystyle\Phi_{\pm}(x)$
$\displaystyle=\Phi_{\pm}(x)\mathrel{\mathop{:}}=\int\nolimits_{x_{0}}^{x}\lambda_{\pm}(t)\mathinner{{\mathrm{d}}t}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(33) $\displaystyle\widehat{\Psi}_{\pm}(x,\hbar)$
$\displaystyle=\widehat{\Psi}_{\pm}(x,\hbar)\mathrel{\mathop{:}}=\sum_{n=0}^{\infty}\Psi_{\pm}^{\text{\tiny$(n)$}}(x)\hbar^{n}\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\widehat{\SS}_{\pm}(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(34)

The functions $\Phi_{\pm}$ are sometimes called the WKB exponents. The
integral of $\widehat{\SS}_{\pm}$ in (32) is interpreted as termwise
integration. In principle, the basepoint of integration $x_{0}$ may be taken
on the boundary of $X$ as long as these integrals make sense.

###  § 3.3. Remarks on Formal WKB Solutions

######  3.5 Remark (Behaviour at a turning point).

Let $x_{{\mathrm{tp}}}\in X$ be a turning point of order $m\geqslant 1$, which
means it is an $m$-th order zero of the leading-order characteristic
discriminant $\mathnormal{D}_{0}$. By examining the recursive formula (23),
one can conclude that the coefficients $s^{\text{\tiny$(k)$}}_{\pm}$ of each
formal characteristic root $\widehat{s}_{\pm}$ have the following behaviour
near $x_{{\mathrm{tp}}}$.

When $m$ is odd, the leading-order characteristic roots $\lambda_{\pm}$ have a
square-root branch singularity at $x_{{\mathrm{tp}}}$, but they are bounded as
$x\to x_{{\mathrm{tp}}}$ in sectors. Every subleading-order coefficient
$s^{\text{\tiny$(k)$}}_{\pm}$ has at worst a square-root branch singularity at
$x_{{\mathrm{tp}}}$ but in general it is unbounded as $x\to
x_{{\mathrm{tp}}}$. On the other hand, when $m$ is even, the leading-order
characteristic roots $\lambda_{\pm}$ are holomorphic at $x_{{\mathrm{tp}}}$.
Every subleading-order coefficient $s^{\text{\tiny$(k)$}}_{\pm}$ is single-
valued near $x_{{\mathrm{tp}}}$ but in general it has a pole there.

For either parity of $m$, in generic situation, the $k$-th order coefficient
$s^{\text{\tiny$(k)$}}_{\pm}$ (with $k\geqslant 1$) is bounded below by
$x^{-km/2}$, where $x$ is a coordinate centred at $x_{{\mathrm{tp}}}$.
Therefore, the singular behaviour of the coefficients of $\widehat{s}_{\pm}$
gets progressively worse in higher and higher orders of $\hbar$. The upshot of
this analysis is that in general it is not possible to expect the formal
characteristic roots $\widehat{s}_{\pm}$ (and therefore the corresponding
formal WKB solutions $\widehat{\psi}_{\pm}$) to be the uniform asymptotic
expansions of exact solutions near a turning point. This phenomenon lies at
the heart of the breakdown of singular perturbation theory in the vicinity of
a turning point.

######  3.5 Remark (Alternative expression for formal WKB solutions).

In the literature, a different but closely related expression and
normalisation for the formal WKB solutions is commonly used (see, e.g., [KT05,
equation (2.11)] or [IN14, equation (2.24)]). Consider the following odd and
even parts of the formal characteristic solutions:

$\widehat{s}_{\text{od}}\mathrel{\mathop{:}}=\tfrac{1}{2}\big{(}\widehat{s}_{+}-\widehat{s}_{-}\big{)}\qquad\text{and}\qquad\widehat{s}_{\text{ev}}\mathrel{\mathop{:}}=\tfrac{1}{2}\big{(}\widehat{s}_{-}+\widehat{s}_{+}\big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(35)

They satisfy
$\widehat{s}_{\pm}=\pm\widehat{s}_{\text{od}}+\widehat{s}_{\text{ev}}$. It
follows from the Riccati equation that the even part $\widehat{s}_{\text{ev}}$
can be expressed in terms of the odd part $\widehat{s}_{\text{od}}$ as
follows:

$\widehat{s}_{\text{ev}}=-\tfrac{1}{2}\hbar\partial_{x}\log\widehat{s}_{\text{od}}-\tfrac{1}{2}\widehat{p}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(36)

Substituting these expressions into (19) yields

$\widehat{\psi}_{\pm}(x,\hbar)=\exp\mathopen{}\mathclose{{}\left(\frac{1}{\hbar}\int_{x_{0}}^{x}\Big{(}\pm\widehat{s}_{\text{od}}(t,\hbar)-\tfrac{1}{2}\partial_{t}\log\widehat{s}_{\text{od}}(t,\hbar)-\tfrac{1}{2}\widehat{p}(t,\hbar)\Big{)}\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(37)

To integrate out the term involving the logarithmic derivative of
$\widehat{s}_{\text{od}}$, we must first make sense of choosing a square root
of the formal power series $\widehat{s}_{\text{od}}$. To this end, we write
$\widehat{s}_{\pm}=\lambda_{\pm}+\hbar\widehat{\SS}_{\pm}$ and put
$\widehat{\SS}_{\text{od}}\mathrel{\mathop{:}}=\tfrac{1}{2}\big{(}\widehat{\SS}_{+}-\widehat{\SS}_{-}\big{)}$.
Then using the identity $\sqrt{\mathnormal{D}_{0}}=\lambda_{+}-\lambda_{-}$,
we get

$\widehat{s}_{\text{od}}=\tfrac{1}{2}\sqrt{\mathnormal{D}_{0}}+\hbar\widehat{\SS}_{\text{od}}=\tfrac{1}{2}\sqrt{\mathnormal{D}_{0}}\Big{(}1+\tfrac{2}{\sqrt{\mathnormal{D}_{0}}}\widehat{\SS}_{\text{od}}\hbar\Big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(38)

Upon fixing a square-root branch $\mathnormal{D}_{0}^{\nicefrac{{1}}{{4}}}$ of
$\sqrt{\mathnormal{D}_{0}}$, we let

$\pm\sqrt{\widehat{s}_{\text{od}}(x,\hbar)}\mathrel{\mathop{:}}=\pm\tfrac{1}{\sqrt{2}}\mathnormal{D}_{0}^{\nicefrac{{1}}{{4}}}\sqrt{1+\tfrac{2}{\sqrt{\mathnormal{D}_{0}}}\widehat{\SS}_{\text{od}}\hbar}=\pm\tfrac{1}{\sqrt{2}}\mathnormal{D}_{0}^{\nicefrac{{1}}{{4}}}\sum_{k=1}^{\infty}\mathopen{}\mathclose{{}\left(\tfrac{2}{\sqrt{\mathnormal{D}_{0}}}\widehat{\SS}_{\text{od}}\hbar}\right)^{k}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(39)

Notice that this is an invertible formal power series in $\hbar$ with
holomorphic coefficients. We therefore obtain the following four exponential
power series:

$\widehat{\psi}_{\pm,\epsilon}(x,\hbar)=\frac{1}{\epsilon\sqrt{\widehat{s}_{\text{od}}(x,\hbar)}}\exp\mathopen{}\mathclose{{}\left(\frac{1}{\hbar}\int_{x_{0}}^{x}\Big{(}\pm\widehat{s}_{\text{od}}(t,\hbar)-\tfrac{1}{2}\widehat{p}(t,\hbar)\Big{)}\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(40)

where $\epsilon\in\mathopen{}\mathclose{{}\left\\{+,-}\right\\}$. These are
the expressions that in the literature are often referred to as (formal) WKB
solutions, although the choice of $\epsilon$ is usually made implicitly. Of
course, it is always possible to choose the branch
$\mathnormal{D}_{0}^{\nicefrac{{1}}{{4}}}$ such that
$\widehat{\psi}_{\pm,\epsilon}=\widehat{\psi}_{\pm}$.

######  3.5 Remark (The first-order WKB approximation).

Expression (40) is the more traditional form used to derive the WKB
approximation for Schrödinger equations. Indeed, if $p=0$ and
$q=-\mathnormal{Q}$, then $\mathnormal{D}_{0}=4\mathnormal{Q}_{0}$ and
$\lambda_{\pm}=\pm\sqrt{\mathnormal{Q}_{0}}$, so the leading-order term of
$\widehat{s}_{\text{od}}$ is simply $\sqrt{\mathnormal{Q}_{0}}$. Truncating
$\widehat{s}_{\text{od}}$ at the leading order yields the famous analytic
expressions

$\pm\frac{1}{\sqrt[4]{\mathnormal{Q}_{0}}}\exp\mathopen{}\mathclose{{}\left(\pm\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\sqrt{\mathnormal{Q}_{0}(t)}\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

######  3.5 Remark (Normalisation at a turning point).

Expression (40) is often used to normalise WKB solutions at a turning point.
However, the existence of exact WKB solutions with such normalisation depends
on some more global properties of the differential equation, which is not
guaranteed in general. In practical terms, even if an exact WKB solution
normalised at a regular point exists, the corresponding exact WKB solution
normalised at some nearby turning point may not exist due to the fact that the
change of normalisation constant may not exist or does not have appropriate
asymptotic behaviour as $\hbar\to 0$. For this reason, in this paper we focus
our attention exclusively on WKB solution normalised at regular points. A more
detailed explanation of this phenomenon will appear in [Nik21b].

######  3.5 Remark (Formal characteristic discriminant).

Since the formal characteristic roots $\widehat{s}_{\pm}$ are uniquely
determined, we can introduce the formal characteristic discriminant of the
differential equation (15) by the usual formula for discriminants:

$\widehat{\mathnormal{D}}\mathrel{\mathop{:}}=-\big{(}\widehat{s}_{+}-\widehat{s}_{-}\big{)}\big{(}\widehat{s}_{-}-\widehat{s}_{+}\big{)}=\big{(}\widehat{s}_{+}-\widehat{s}_{-}\big{)}^{2}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(41)

Its leading-order part is the leading-order characteristic discriminant
$\mathnormal{D}_{0}$. If we furthermore define
$\sqrt{\widehat{\mathnormal{D}}}\mathrel{\mathop{:}}=\widehat{s}_{+}-\widehat{s}_{-}$,
so that its leading-order part is just
$\sqrt{\mathnormal{D}_{0}}=\lambda_{+}-\lambda_{-}$, then we obtain the
following relation:

$\widehat{s}_{\text{od}}=\tfrac{1}{2}\sqrt{\widehat{\mathnormal{D}}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(42)

Then (40) can be written as follows:

$\widehat{\psi}_{\pm,\epsilon}(x,\hbar)=\frac{\sqrt{2}}{\epsilon\sqrt[4]{\widehat{\mathnormal{D}}(x,\hbar)}}\exp\mathopen{}\mathclose{{}\left(\frac{1}{2\hbar}\int_{x_{0}}^{x}\Big{(}\pm\sqrt{\widehat{\mathnormal{D}}(t,\hbar)}-\widehat{p}(t,\hbar)\Big{)}\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(43)

##### 3.6. Convergence of the formal Borel transform.

Our final remark about formal WKB solutions in this section is a lemma that
says that if $\widehat{p},\widehat{q}$ are in a certain Gevrey regularity
class, then the formal WKB solutions are in the corresponding exponential
Gevrey regularity class. Gevrey series are briefly reviewed in § A.3.

Let $x_{0}\in X$ be a regular point and $U\subset X$ a simply connected
neighbourhood of $x_{0}$ free of turning points. Let
$\widehat{\psi}_{+},\widehat{\psi}_{-}\in{\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
be the two formal WKB solutions normalised at $x_{0}$. Consider their formal
Borel transform (see Appendix C):

$\widehat{\mathfrak{B}}\big{[}\,\widehat{\psi}_{\pm}\,\big{]}(x,\xi)=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\pm}(t)\mathinner{{\mathrm{d}}t}}\right)\widehat{\mathfrak{B}}\big{[}\,\widehat{\Psi}_{\pm}\,\big{]}(x,\xi)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(44)

It is convenient to also consider the following exponentiated formal Borel
transform of $\widehat{\psi}_{\pm}$ by using the exponential presentation (32)
and applying the ordinary formal Borel transform directly to the exponent:

$\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\psi}_{\pm}\,\big{]}(x,\xi)\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\pm}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\widehat{\mathfrak{B}}\big{[}\,\widehat{\SS}_{\pm}\,\big{]}(t,\xi)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(45)

Note that
$\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\psi}_{\pm}\,\big{]}\neq\widehat{\mathfrak{B}}\big{[}\,\widehat{\psi}_{\pm}\,\big{]}$
because the Borel transform converts multiplication into convolution, but
clearly the convergence properties of one imply the same about the other.
Denote the formal Borel transform of $\widehat{\SS}_{\pm}$ by

$\widehat{\sigma}_{\pm}(x,\xi)\mathrel{\mathop{:}}=\widehat{\mathfrak{B}}\big{[}\,\widehat{\SS}_{\pm}\,\big{]}(x,\xi)=\sum_{k=0}^{\infty}s_{k+2}^{\text{\tiny$(i)$}}(x)\frac{\xi^{k}}{k!}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(46)

Then we have the following assertion.

######  3.6 Proposition ( ).

Suppose the coefficients $\widehat{p},\widehat{q}$ are locally uniformly
Gevrey power series on $U$; in symbols,
$\widehat{p},\widehat{q}\in{\mathcal{G}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.
Then the formal Borel transforms $\widehat{\sigma}_{\pm}$ are locally
uniformly convergent power series and hence $\widehat{\SS}_{\pm}$ are locally
uniformly Gevrey series; in symbols,
$\widehat{\sigma}_{\pm}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\\{\xi}\right\\}$
and
$\widehat{\SS}_{\pm}\in{\mathcal{G}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.
Consequently, exponentiated formal Borel transforms
$\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\Psi}_{\pm}\,\big{]}$ and
hence the ordinary formal Borel transforms
$\widehat{\mathfrak{B}}\big{[}\,\widehat{\Psi}_{\pm}\,\big{]}$ are locally
uniformly convergent power series in $\xi$ on $U$. Thus, the formal WKB
solutions are locally uniformly exponential Gevrey series; in symbols,
$\widehat{\psi}_{\pm}\in{\mathcal{G}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.

This proposition is not necessary for the proof of our main result in this
paper (Theorem 5.2). In fact, on certain subsets $U\subset X$, this
proposition can be seen as a consequence of Theorem 5.2 (or more specifically
of Part  5.8). In more generality, a direct proof can be found in [Nik20,
Lemma 3.11].

######  3.6 Example (mildly perturbed coefficients).

If $\widehat{p},\widehat{q}$ are polynomials in $\hbar$, then necessarily
$\widehat{p},\widehat{q}\in{\mathcal{G}}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$;
i.e., they automatically satisfy the hypothesis of Part  3.6.

Concretely, Part  3.6 says that if the coefficients $p_{k},q_{k}$ of the
power series $\widehat{p},\widehat{q}$ grow no faster than $k!$, then the
power series coefficients $\Psi^{\text{\tiny$(k)$}}_{\pm}$ of the formal WKB
solution $\widehat{\psi}_{\pm}$ given by (34) likewise grow no faster than
$k!$. This is made precise in the following corollary.

######  3.6 Corollary (at most factorial growth).

Let $x_{0}\in X$ be a regular point and let $U\subset X$ be any simply
connected neighbourhood of $x_{0}$ free of turning points. Let
$\widehat{\psi}_{\pm}=e^{-\Phi_{\pm}/\hbar}\widehat{\Psi}_{\pm}$ be the formal
WKB solutions on $U$ normalised at $x_{0}$, written as in (34). Take any pair
of nested compactly contained subsets $U_{0}\Subset U_{1}\Subset U$, and
suppose that there are real constants $\mathnormal{A},\mathnormal{B}>0$ such
that

$\big{|}p_{k}(x)\big{|},\big{|}q_{k}(x)\big{|}\leqslant\mathnormal{A}\mathnormal{B}^{k}k!\quad\quad\quad(\forall
k\geqslant 0,\forall x\in U_{1})\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(47)

Then there are real constants $\mathnormal{C},\mathnormal{M}>0$ such that

$\big{|}\Psi^{\text{\tiny$(k)$}}_{\pm}(x)\big{|}\leqslant\mathnormal{C}\mathnormal{M}^{k}k!\quad\quad\quad\quad\quad(\forall
k\geqslant 0,\forall x\in U_{0})\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(48)

In particular, if $\widehat{p},\widehat{q}$ are polynomials in $\hbar$, then
estimates (48) hold.

##  § 4. WKB Geometry

In this section, we introduce a coordinate transformation which plays a
central role in our construction of exact WKB solutions in § 5. It is used
to determine regions in $\mathbb{C}_{x}$ where the Borel-Laplace method can be
applied to our differential equation.

The material of this section can essentially be found in [Str84, §9-11] (see
also [BS15, §3.4]). These references use the language of foliations given by
quadratic differentials on Riemann surfaces, where the quadratic differential
in question is $\mathnormal{D}_{0}(x)\mathinner{{\mathrm{d}}x}^{2}$. The
reader may be more familiar with the set of critical leaves of this foliation
which is encountered in the literature under various names including Stokes
curves, Stokes graph, spectral network, geodesics, and critical trajectories
[GMN13b, KT05, DDP93, GMN13a, Nik19].

To keep the discussion a little more elementary, we state the relevant
definitions and facts by appealing directly to explicit formulas using the
Liouville transformation (defined below) commonly used in the WKB analysis of
Schrödinger equations111See also [Fen20] for another interesting geometric use
of the Liouville transformation in the context of complex projective
structures..

##### 4.1. Résumé.

The following is a quick summary of our conventions and notations. It is
intended for the reader who is quite familiar with this story and who may
therefore wish to skip the rest of this section and go directly to § 5.

Fix a phase $\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}$, a regular point
$x_{0}\in X$, and a univalued square-root branch $\sqrt{\mathnormal{D}_{0}}$
near $x_{0}$. By a Liouville transformation we mean the following local
coordinate transformation near $x_{0}$:

$z=\Phi(x)\mathrel{\mathop{:}}=\int_{x_{0}}^{x}\sqrt{\mathnormal{D}_{0}(t)}\mathinner{{\mathrm{d}}t}=\int_{x_{0}}^{x}\Big{(}\lambda_{+}(t)-\lambda_{-}(t)\Big{)}\mathinner{{\mathrm{d}}t}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

A WKB $\theta$-trajectory through $x_{0}$ is given locally by the equation
$\operatorname{Im}\big{(}e^{-i\theta}\Phi(x)\big{)}=0$. It is parameterised by
the real number
$\tau(x)\mathrel{\mathop{:}}=\operatorname{Re}\big{(}e^{-i\theta}\Phi(x)\big{)}$.
It is called complete if it exists for all real time $\tau$. A complete WKB
$\theta$-trajectory is mapped by $\Phi$ to the entire straight line
$e^{i\theta}{\mathbb{R}}$.

A WKB $(\theta,\pm)$-ray emanating from $x_{0}$ is the part of the WKB
$\theta$-trajectory corresponding to $\tau\geqslant 0$ or $\tau\leqslant 0$,
respectively. It is called complete if it exists for all nonnegative real time
$\tau\geqslant 0$ or all nonpositive real time $\tau\leqslant 0$,
respectively. In particular, it is not allowed to flow into a finite critical
point (i.e., a turning point or a simple pole of $\mathnormal{D}_{0}$). It is
mapped by $\Phi$ to the ray $e^{i\theta}{\mathbb{R}}_{\pm}$.

Two special classes of complete trajectories are considered: a closed WKB
$\theta$-trajectory which is a closed curve in $X$, and a generic WKB
$\theta$-trajectory which tends at both ends to infinite critical points
(i.e., poles of $\mathnormal{D}_{0}$ of order at least $2$). A generic WKB
$(\theta,\pm)$-ray tends to an infinite critical point at one end.

A WKB $\theta$-strip domain is swept out by nonclosed complete WKB
$\theta$-trajectories, mapped by $\Phi$ to an infinite strip made up of
straight lines parallel to $e^{i\theta}{\mathbb{R}}$. Fact: any generic WKB
curve can always be embedded in a WKB strip domain.

A WKB $\theta$-ring domain is swept out by closed WKB $\theta$-trajectories,
mapped by $\Phi$ to an infinite strip made up of straight lines parallel to
$e^{i\theta}{\mathbb{R}}$. It is homeomorphic to an annulus and $\Phi$ extends
to it as a multivalued local biholomorphism. Fact: any closed WKB trajectory
can always be embedded in a WKB ring domain.

A WKB $(\theta,\pm)$-halfstrip is swept out by WKB $(\theta,\pm)$-rays
emanating from a WKB disc
$\Phi^{-1}\mathopen{}\mathclose{{}\left\\{|z-z_{1}|<\varepsilon}\right\\}$
around a point $x_{1}$ near $x_{0}$, where
$z_{1}\mathrel{\mathop{:}}=\Phi(x_{1})$. It is the preimage under $\Phi$ of a
tubular neighbourhood
$\mathopen{}\mathclose{{}\left\\{z~{}\big{|}~{}\mathop{{\mathrm{dist}}}\nolimits\big{(}z,z_{1}+e^{i\theta}{\mathbb{R}}_{\pm}\big{)}<\varepsilon}\right\\}$.

###  § 4.1. The Liouville Transformation

##### 4.2.

Recall the leading-order characteristic discriminant
$\mathnormal{D}_{0}=p_{0}^{2}-4q_{0}$ which is a holomorphic function on $X$.
Fix a phase $\theta\in{\mathbb{R}}/2\pi{\mathbb{Z}}$, a regular point
$x_{0}\in X$, and a univalued square-root branch $\sqrt{\mathnormal{D}_{0}}$
near $x_{0}$. Consider the following local coordinate transformation near
$x_{0}$, called the Liouville transformation:

$z=\Phi(x)\mathrel{\mathop{:}}=\int_{x_{0}}^{x}\sqrt{\mathnormal{D}_{0}(t)}\mathinner{{\mathrm{d}}t}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(49)

The basepoint of integration $x_{0}$ can in principle be chosen on the
boundary of $X$ or at infinity in $\mathbb{C}_{x}$ provided that this integral
is well-defined. This transformation is encountered in the analysis of the
Schrödinger equation (10) as described for example in Olver’s textbook [Olv97,
§6.1]. However, note that our formula (49) in the special case of the
Schrödinger equation (10) reads

$\Phi(x)=\int\nolimits_{x_{0}}^{x}\sqrt{\mathnormal{D}_{0}(t)}\mathinner{{\mathrm{d}}t}=2\int\nolimits_{x_{0}}^{x}\sqrt{\mathnormal{Q}_{0}(t)}\mathinner{{\mathrm{d}}t}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(50)

which differs from formula (1.05) in [Olv97, §6.1] by a factor of $2$.

##### 4.3.

If $\lambda_{\pm}$ are the two leading-order characteristic roots defined near
$x_{0}$, labelled such that
$\sqrt{\mathnormal{D}_{0}}=\lambda_{+}-\lambda_{-}$, then we have the
following identity relating the Liouville transformation $\Phi$ with the WKB
exponents normalised at $x_{0}$ from (33):

$\Phi=\Phi_{+}-\Phi_{-}=\int_{x_{0}}^{x}\Big{(}\lambda_{+}(t)-\lambda_{-}(t)\Big{)}\mathinner{{\mathrm{d}}t}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(51)

##### 4.4.

If $U\subset X$ is any domain that can support a univalued square-root branch
$\sqrt{\mathnormal{D}_{0}}$ (e.g., if $U$ is simply connected and free of
turning points), then the Liouville transformation defines a (possibly
multivalued) local biholomorphism $\Phi:U\longrightarrow\mathbb{C}_{z}$. The
main utility of the Liouville transformation is that it transforms the
differential operator $\frac{1}{\sqrt{\mathnormal{D}_{0}}}\partial_{x}$ (which
appears prominently in formula (23) for the formal characteristic roots) into
the constant-coefficient differential operator $\partial_{z}$. Using the
language of differential geometry,

$\Phi_{\ast}:\frac{1}{\sqrt{\mathnormal{D}_{0}}}\partial_{x}\longmapsto\partial_{z}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(52)

This straightening-out of the local geometry using the Liouville
transformation (as explained in the next subsection) will be exploited in our
construction of exact WKB solutions in § 5. This point of view also makes
the troublesome nature of turning points more transparent.

###  § 4.2. WKB Trajectories and Rays

##### 4.5.

A WKB $\theta$-trajectory passing through $x_{0}$ is the real $1$-dimensional
smooth curve on $X$ locally determined by the equation

$\operatorname{Im}\big{(}e^{-i\theta}\Phi(x)\big{)}=0\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8};}\hss}}\qquad\text{i.e.,}\qquad\operatorname{Im}\mathopen{}\mathclose{{}\left(\>e^{-i\theta}\int\nolimits_{x_{0}}^{x}\sqrt{\mathnormal{D}_{0}(t)}\mathinner{{\mathrm{d}}t}}\right)=0\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(53)

By definition, WKB trajectories are regarded as being maximal under inclusion.
The Liouville transformation $\Phi$ maps a WKB $\theta$-trajectory to a
connected subset of the straight line
$e^{i\theta}{\mathbb{R}}\subset\mathbb{C}_{z}$. The image is a possibly
unbounded line segment $e^{i\theta}(\tau_{-},\tau_{+})\subset
e^{i\theta}{\mathbb{R}}$ containing the origin $0=\Phi(x_{0})$. Maximality
means that the line segment $e^{i\theta}(\tau_{-},\tau_{+})$ is the largest
possible image.

All other nearby WKB $\theta$-trajectory can be locally described by an
equation of the form $\operatorname{Im}\big{(}e^{-i\theta}\Phi(x)\big{)}=c$
for some $c\in{\mathbb{R}}$. That is, if $V\subset X$ is a simply connected
neighbourhood of $x_{0}$ free of turning points, then a WKB
$\theta$-trajectory intersecting $V$ is locally given by this equation with
$c=\operatorname{Im}e^{-i\theta}\Phi(x_{1})$ for some $x_{1}\in V$. Its image
in $\mathbb{C}_{z}$ under $\Phi$ is an interval of the straight line
$\mathopen{}\mathclose{{}\left\\{z=z_{1}+\xi~{}\big{|}~{}\xi\in
e^{i\theta}{\mathbb{R}}}\right\\}$ containing the point
$z_{1}\mathrel{\mathop{:}}=\Phi(x_{1})$.

##### 4.6.

The chosen square-root $\sqrt{\mathnormal{D}_{0}}$ near $x_{0}$ endows the WKB
$\theta$-trajectory passing through $x_{0}$ with a canonical parameterisation
given by the real number

$\tau(x)\mathrel{\mathop{:}}=\operatorname{Re}\mathopen{}\mathclose{{}\left(e^{-i\theta}\Phi(x)}\right)\in(\tau_{-},\tau_{+})\subset{\mathbb{R}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(54)

We define the WKB $(\theta,\pm)$-ray emanating from $x_{0}$ as the preimage
under $\Phi$ of the line segment $e^{i\theta}[0,\tau_{+})$ or
$e^{i\theta}(\tau_{-},0]$, respectively.

##### 4.7. Complete WKB trajectories and rays.

Suppose that either $\tau_{+}$ or $\tau_{-}$ is finite. As $\tau$ approaches
$\tau_{+}$ or $\tau_{-}$ respectively, the WKB trajectory either tends to a
turning point or escapes to the boundary of $X$ in finite time. If it tends to
a single point on the boundary of $X$, this point is either a turning point or
a simple pole of the discriminant $\mathnormal{D}_{0}$, [Str84, §10.2]. For
this reason, turning points and simple poles are sometimes collectively
referred to as finite critical points. These situations are inadmissible for
the purpose of constructing exact solutions using our methods, so we introduce
the following definitions.

######  4.7 Definition ( ).

A complete WKB $\theta$-trajectory is one for which both $\tau_{+}=+\infty$
and $\tau_{-}=-\infty$; i.e., its image in $\mathbb{C}_{z}$ under the
Liouville transformation $\Phi$ is the entire straight line
$e^{i\theta}{\mathbb{R}}$. A complete WKB $(\theta,\pm)$-ray is one for which
$\tau_{\pm}=\pm\infty$, respectively; i.e, its image under $\Phi$ is the
entire ray $e^{i\theta}{\mathbb{R}}_{+}$ or $e^{i\theta}{\mathbb{R}}_{-}$,
respectively.

###  § 4.3. Closed and generic WKB trajectories

Two classes of complete trajectories are especially important.

##### 4.8.

A closed WKB $\theta$-trajectory is one with the property that there is a
nonzero time $\omega\in{\mathbb{R}}$ such that
$\Phi^{-1}(e^{i\theta}\omega)=\Phi^{-1}(0)$. This only happens when the
Liouville transformation is analytically continued along the trajectory to a
multivalued function (WKB trajectories are smooth so they cannot have self-
intersections). Closed WKB trajectories are necessarily complete and form
closed curves in $\mathbb{C}_{x}$, [Str84, §9.2]. We refer to the smallest
possible positive such $\omega\in{\mathbb{R}}_{+}$ as the trajectory period.

Consider now a complete WKB $(\theta,\pm)$-ray emanating from $x_{0}$ which is
not part of a closed WKB trajectory. Its limit set is by definition the limit
of the set

$\overline{\Phi^{-1}\big{(}e^{i\theta}[\tau,+\infty)\big{)}}\quad\text{as}\quad\tau\to+\infty\qquad\text{or}\qquad\overline{\Phi^{-1}\big{(}e^{i\theta}(-\infty,\tau]\big{)}}\quad\text{as}\quad\tau\to-\infty\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

Obviously, this definition is independent of the chosen basepoint $x_{0}$
along the trajectory. The limit set may be empty or it may contain one or more
points. If it contains a single point $x_{\infty}\in\mathbb{C}_{x}$, then this
point (sometimes called an infinite critical point) is necessarily a pole of
$\mathnormal{D}_{0}$ of order $m\geqslant 2$, [Str84, §10.2]. A generic WKB
ray is one whose limit set is an infinite critical point. A generic WKB
trajectory is one both of whose rays are generic.

###  § 4.4. WKB Strips, Halfstrips, and Ring Domains

##### 4.9.

For us, the model neighbourhood of a complete WKB $\theta$-trajectory is the
preimage under $\Phi$ of an infinite strip in the $z$-plane, which is a subset
of the form
$\mathopen{}\mathclose{{}\left\\{z~{}\big{|}~{}\varepsilon_{-}<\operatorname{Im}(e^{-i\theta}z)<\varepsilon_{+}}\right\\}$.
Such a neighbourhood may not exist, but if it does, it is swept out by
complete WKB $\theta$-trajectories. We consider separately the situations when
these complete trajectories are closed or not.

##### 4.10.

A WKB $\theta$-strip domain is the preimage under $\Phi$ of an infinite strip
swept out by nonclosed trajectories. It is simply connected and $\Phi$ maps it
to an infinite strip biholomorphically. If one WKB trajectory in a WKB strip
domain is generic, then all trajectories sweeping out this strip are generic.
The main fact we need about generic WKB trajectories is that any generic WKB
$\theta$-trajectory can be embedded in a WKB $\theta$-strip domain [Str84,
§10.5].

##### 4.11.

A WKB $\theta$-ring domain is an open subset of $\mathbb{C}_{x}$, homeomorphic
to an annulus, swept out by closed WKB $\theta$-trajectories. The restriction
of the Liouville transformation $\Phi$ is a multivalued holomorphic function.
All closed WKB trajectories sweeping out a WKB ring domain have the same
trajectory period. The main fact we need about closed WKB trajectories is that
any closed WKB $\theta$-trajectory can be embedded in a WKB $\theta$-ring
domain [Str84, §9.3].

##### 4.12.

Similarly, the model neighbourhood of a complete WKB $(\theta,\pm)$-ray is the
preimage under $\Phi$ of a tubular neighbourhood of a ray
$z_{0}+e^{i\theta}{\mathbb{R}}_{\pm}$ for some $z_{0}\in\mathbb{C}_{z}$. Such
a tubular neighbourhood is the subset of the form
$\mathopen{}\mathclose{{}\left\\{z~{}\big{|}~{}\mathop{{\mathrm{dist}}}\nolimits(z,z_{0}+e^{i\theta}{\mathbb{R}}_{\pm})<\varepsilon}\right\\}$.
We refer to its preimage under $\Phi$ as a WKB $(\theta,\pm)$-halfstrip
domain. It is swept out by complete WKB $(\theta,\pm)$-rays emanating from a
WKB disc around $x_{0}$ of radius $\varepsilon$; i.e., the set
$V=\Phi^{-1}\big{(}\mathopen{}\mathclose{{}\left\\{|z|<\varepsilon}\right\\}\big{)}$.
Any generic WKB $(\theta,\pm)$-ray can be embedded in a WKB
$(\theta,\pm)$-halfstrip domain.

(a) The arc $\Theta$ of copolar directions of $A$.

(b) A Borel disc $S_{\theta}$ with opening $A_{\theta}$.

Figure 2:

(a) Domain $U\subset X$ (in blue) is a simply connected neighbourhood of
$x_{0}$ that is free of turning points. Every WKB $(\theta,\alpha)$-ray
emanating from $U$ is complete. A few randomly chosen rays are represented by
green curves, and completeness is represented by having them all flow into
e.g. a pole of $\mathnormal{D}_{0}$ (red dot).

(b) Every $x\in U$ has a neighbourhood $V\subset U$ which flows into the red
dot. The full flow is denoted by $V_{\theta,\alpha}$. The halfplane
$\mathbb{H}_{\theta,\alpha}(\mathnormal{R})$ on the right is drawn in yellow,
and its preimage in $V_{\theta,\alpha}$ is also drawn in yellow and thought of
as a (sectorial) neighbourhood of the limit set.

Figure 3:

##  § 5. Exact WKB Solutions

In this section, we state and prove the main results of this paper.

##### 5.1. Background assumptions.

We remain in the setting of Part  2.1. Throughout this section, we also fix
a regular point $x_{0}\in X$ and a univalued square-root branch
$\sqrt{\mathnormal{D}_{0}}$ near $x_{0}$. Let $\lambda_{+},\lambda_{-}$ be the
two leading-order characteristic roots given by (19) so that
$\sqrt{\mathnormal{D}_{0}}=\lambda_{+}-\lambda_{-}$. Let
$\widehat{\psi}_{+},\widehat{\psi}_{-}$ be the corresponding pair of formal
WKB solutions normalised at $x_{0}$ as guaranteed by § 3.2. Let $z=\Phi(x)$
be the Liouville transformation with basepoint $x_{0}$ given by (49).

In addition, let $\Theta\mathrel{\mathop{:}}=[\theta_{-},\theta_{+}]$ be the
closed arc such that
$A=(\theta_{-}-\tfrac{\pi}{2},\theta_{+}+\tfrac{\pi}{2})$; i.e.,
$\theta_{\pm}\mathrel{\mathop{:}}=\vartheta_{\pm}\mp\tfrac{\pi}{2}$. See 2(a).
This arc $\Theta$ is sometimes called the arc of copolar directions of $A$.
For every $\theta\in\Theta$, let $A_{\theta}\subset A$ be the halfplane arc
bisected by $\theta$, and let $S_{\theta}\subset S$ be a Borel disc bisected
by $\theta$ of some diameter $\delta>0$ (see 2(b)):

$A_{\theta}\mathrel{\mathop{:}}=(\theta-\tfrac{\pi}{2},\theta+\tfrac{\pi}{2})\quad\text{and}\quad
S_{\theta}\mathrel{\mathop{:}}=\mathopen{}\mathclose{{}\left\\{\operatorname{Re}\big{(}\smash{e^{i\theta}}/\hbar\big{)}>1/\delta}\right\\}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(55)

Finally, for any $\mathnormal{R}>0$, $\theta\in\Theta$, let
$\mathbb{H}_{\theta,\pm}(\mathnormal{R})\mathrel{\mathop{:}}=\mathopen{}\mathclose{{}\left\\{\operatorname{Re}(\pm
e^{i\theta}z)>\mathnormal{R}}\right\\}\subset\mathbb{C}_{z}$ (see 3(b)).

###  § 5.1. Existence and Uniqueness of Exact WKB Solutions

First, we investigate our problem for a single fixed direction in $\Theta$.
The main result of this paper is then the following theorem (see Fig. 3 for a
visual).

######  5.2 Theorem (Existence and Uniqueness in a Halfplane).

Fix a sign $\alpha\in\mathopen{}\mathclose{{}\left\\{+,-}\right\\}$ and a
phase $\theta\in\Theta$, and let $U\subset X$ be any simply connected domain
containing $x_{0}$ which is free of turning points and such that every WKB
$(\theta,\alpha)$-ray emanating from $U$ is complete. Assume in addition that
for every point $x\in U$, there is a neighbourhood $V\subset U$ of $x$ and a
sufficiently large number $\mathnormal{R}>0$ such that the following two
conditions are satisfied on the domain
$V_{\theta,\alpha,\mathnormal{R}}\mathrel{\mathop{:}}=V_{\theta,\alpha}\cap\Phi^{-1}\Big{(}\mathbb{H}_{\theta,\alpha}(\mathnormal{R})\Big{)}$,
where $V_{\theta,\alpha}$ is the union of all WKB $(\theta,\alpha)$-rays
emanating from $V$:

  1. (1)

$\frac{1}{\sqrt{\mathnormal{D}_{0}}}\partial_{x}\log\sqrt{\mathnormal{D}_{0}}$
is bounded on $V_{\theta,\alpha,\mathnormal{R}}$;

  2. (2)

$\tfrac{1}{\sqrt{\mathnormal{D}_{0}}}p\simeq\tfrac{1}{\sqrt{\mathnormal{D}_{0}}}\widehat{p}\quad\text{and}\quad\tfrac{1}{\mathnormal{D}_{0}}q\simeq\tfrac{1}{\mathnormal{D}_{0}}\widehat{q}\quad\text{as
$\hbar\to 0$ along $\overline{A}_{\theta}$, unif. $\forall x\in
V_{\theta,\alpha,\mathnormal{R}}$.}$

Then the differential equation (2) has a canonical exact solution
$\psi_{\alpha}$ on $U$ whose exponential asymptotics as $\hbar\to 0$ along
$A_{\theta}$ are given by the formal WKB solution $\widehat{\psi}_{\alpha}$.
Namely, there is a Borel disc $S^{\prime}_{\theta}\subset S_{\theta}$ of
possibly smaller diameter $\delta^{\prime}\in(0,\delta]$ such that (2) has a
unique holomorphic solution $\psi_{\alpha}$ defined on $U\times
S^{\prime}_{\theta}$ that satisfies the following conditions:

$\displaystyle\psi_{\alpha}(x_{0},\hbar)=1\qquad\text{for all $\hbar\in
S^{\prime}_{\theta}$\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8};}\hss}}
}$ (56)
$\displaystyle\psi_{\alpha}(x,\hbar)\simeq\widehat{\psi}_{\alpha}(x,\hbar)\quad\text{as
$\hbar\to 0$ along $\overline{A}_{\theta}$, loc.unif. $\forall x\in U$
\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}
missing}$ (57)

Furthermore, if the hypotheses hold for both choices of the sign $\alpha$,
then $\psi_{+},\psi_{-}$ on $U$ define a basis for the space
$\mathbb{ES}(U;A)$ of all exact solutions on $(U;A)$ as well as for the space
$\mathbb{GES}(U;A)$ of all exact solutions on $(U;A)$ with exponential Gevrey
asymptotics.

The strategy of the proof of this theorem is to reduce the problem to finding
exact solutions to an associated singularly perturbed Riccati equation as
captured by the following lemma.

######  5.2 Lemma (WKB ansatz and the associated Riccati equation).

The solution $\psi_{\alpha}$ from Theorem 5.2 is given by the following
formula: for all $(x,\hbar)\in U\times S^{\prime}_{\theta}$,

$\psi_{\alpha}(x,\hbar)=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}s_{\alpha}(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(58)

where $s_{\alpha}=\lambda_{\alpha}+\hbar\SS_{\alpha}$ is an exact solution of
the singularly perturbed Riccati equation

$\hbar\partial_{x}s=s^{2}-ps+q\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(59)

More precisely, for any compactly contained subset $V\subset U$, there is a
Borel disc $S^{\prime\prime}_{\theta}\subset S^{\prime}_{\theta}$ of possibly
smaller diameter $\delta^{\prime\prime}\in(0,\delta^{\prime}]$ such that
identity (58) holds for all $(x,\hbar)$ in the domain $V\times
S^{\prime\prime}_{\theta}$. Here, $s_{\alpha}$ is the unique holomorphic
solution of the Riccati equation (59) on $V\times S^{\prime\prime}_{\theta}$
which admits the formal characteristic solution $\widehat{s}_{\alpha}$ as its
uniform Gevrey asymptotic expansion along $\overline{A}_{\theta}$:

$s_{\alpha}(x,\hbar)\simeq\widehat{s}_{\alpha}(x,\hbar)\quad\text{as $\hbar\to
0$ along $\overline{A}_{\theta}$, unif. $\forall x\in V$ \text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}
missing}$ (60)

Moreover, $s_{\alpha}$ uniquely extends to a meromorphic function on $U\times
S^{\prime}_{\theta}$ with poles only at the zeros of $\psi_{\alpha}$.

Thus, our proof of Theorem 5.2 mainly rests on the ability to construct a
unique exact solution of the Riccati equation (59). In [Nik20, Theorem 5.17],
we proved a general existence and uniqueness theorem for exact solutions of a
singularly perturbed Riccati equation. This result and the sketch of its proof
(specialised to our situation at hand) is presented in § C.2. The proof of
§ 5.1 and therefore of Theorem 5.2 is then presented in § C.3.

######  5.2 Definition ( ).

The solution $\psi_{\alpha}$ from Theorem 5.2 is called an exact WKB solution
normalised at $x_{0}$. The basis
$\mathopen{}\mathclose{{}\left\\{\psi_{+},\psi_{-}}\right\\}$ is called the
exact WKB basis normalised at $x_{0}$. We will also refer to the exact
solution $s_{\alpha}$ of the Riccati equation (59) as an exact characteristic
root for the differential equation (2).

######  5.2 Example (Mildly deformed coefficients).

If the coefficients $p,q$ of the differential equation (2) are at most
polynomial in $\hbar$ (i.e., if $p,q\in{\mathcal{O}}(X)[\hbar]$) then
condition (2) of Theorem 5.2 simplifies considerably and can be replaced by
the following equivalent condition: for every $k\geqslant 0$, the functions
$p_{k}(x)$ and $q_{k}(x)$ are bounded on $V_{\theta,\alpha,\mathnormal{R}}$
respectively by $\sqrt{\mathnormal{D}_{0}(x)}$ and $\mathnormal{D}_{0}(x)$.
For the Schrödinger equation (10) with $\hbar$-independent potential (i.e.,
with $p=0$ and $q=-\mathnormal{Q}$ where
$\mathnormal{Q}(x,\hbar)=\mathnormal{Q}_{0}(x)$), condition (2) of Theorem 5.2
is vacuous.

##### 5.3. Analytic continuation to larger domains in $\mathbb{C}_{x}$.

Using the usual Parametric Existence and Uniqueness Theorem for linear ODEs
(see, e.g., [Was76, Theorem 24.1]), exact WKB solutions can be analytically
continued anywhere in $X$, yielding the following statement.

######  5.3 Corollary ( ).

For any simply connected domain $X^{\prime}\subset X$ containing $x_{0}$, the
exact WKB solution $\psi_{\alpha}\in\mathbb{ES}(U\times S^{\prime}_{\theta})$
from Theorem 5.2 extends to a holomorphic solution on $X^{\prime}\times
S^{\prime}_{\theta}$. In fact, it is the unique holomorphic solution of (2) on
$X^{\prime}\times S^{\prime}_{\theta}$ satisfying the following initial
conditions for all $\hbar\in S^{\prime}_{\theta}$:

$\psi_{\alpha}(x_{0},\hbar)=1\qquad\text{and}\qquad\hbar\partial_{x}\psi_{\alpha}(x_{0},\hbar)=s_{\alpha}(x_{0},\hbar)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(61)

If $\psi_{+},\psi_{-}\in\mathbb{ES}(U\times S^{\prime}_{\theta})$ is a WKB
basis from Theorem 5.2, it defines a basis for the space
$\mathbb{S}(X^{\prime}\times S^{\prime}_{\theta})$ of all holomorphic
solutions on $X^{\prime}\times S^{\prime}_{\theta}$.

However, beware that the asymptotic property (57) of exact WKB solutions is
not necessarily continued outside the domain $U$, and indeed in general there
are no global exact solutions even if $X$ is simply connected and free of
turning points.

##### 5.4. Existence and uniqueness in wider sectors.

Now we extend Theorem 5.2 to the full arc $A$. Without loss of generality, we
can assume that $S\mathrel{\mathop{:}}=\bigcup_{\theta\in\Theta}S_{\theta}$ is
the union of Borel discs
$S_{\theta}=\mathopen{}\mathclose{{}\left\\{\operatorname{Re}\big{(}\smash{e^{i\theta}}/\hbar\big{)}>1/\delta}\right\\}$,
one for each bisecting direction $\theta\in\Theta$, of some diameter
$\delta>0$ independent of $\theta$.

######  5.5 Theorem ( ).

Fix a sign $\alpha\in\mathopen{}\mathclose{{}\left\\{+,-}\right\\}$, and let
$U\subset X$ be any simply connected domain containing $x_{0}$ which is free
of turning points and has the following property: for all $\theta\in\Theta$,
every WKB $(\theta,\alpha)$-ray emanating from $U$ is complete. Assume in
addition that for every point $x\in U$, there is a neighbourhood $V\subset U$
of $x$ and a sufficiently large number $\mathnormal{R}>0$ such that the
following two conditions are satisfied on the domain
$V_{\Theta,\alpha,\mathnormal{R}}\mathrel{\mathop{:}}=V_{\Theta,\alpha}\cap\Phi^{-1}\Big{(}~{}\bigcup_{\theta\in\Theta}\mathbb{H}_{\theta,\alpha,\mathnormal{R}}\Big{)}$,
where $V_{\Theta,\alpha}$ is the union of all WKB $(\theta,\alpha)$-rays
emanating from $V$ for all $\theta\in\Theta$:

  1. (1)

$\frac{1}{\sqrt{\mathnormal{D}_{0}}}\partial_{x}\log\sqrt{\mathnormal{D}_{0}}$
is bounded on $V_{\Theta,\alpha,\mathnormal{R}}$;

  2. (2)

$\tfrac{1}{\sqrt{\mathnormal{D}_{0}}}p\simeq\tfrac{1}{\sqrt{\mathnormal{D}_{0}}}\widehat{p}\quad\text{and}\quad\tfrac{1}{\mathnormal{D}_{0}}q\simeq\tfrac{1}{\mathnormal{D}_{0}}\widehat{q}\quad\text{as
$\hbar\to 0$ along $\overline{A}$, unif. $\forall x\in
V_{\Theta,\alpha,\mathnormal{R}}$.}$

Then the differential equation (2) has a canonical exact solution
$\psi_{\alpha}$ on $U$ whose exponential asymptotics as $\hbar\to 0$ along $A$
are given by the formal WKB solution $\widehat{\psi}_{\alpha}$. Namely, there
is a sectorial subdomain

$S^{\prime}\mathrel{\mathop{:}}=\bigcup_{\theta\in\Theta}S^{\prime}_{\theta}\subset
S\quad\text{with}\quad
S^{\prime}_{\theta}\mathrel{\mathop{:}}=\mathopen{}\mathclose{{}\left\\{\operatorname{Re}\big{(}\smash{e^{i\theta}}/\hbar\big{)}>1/\delta^{\prime}}\right\\}$

for some $\delta^{\prime}\in(0,\delta]$ such that the differential equation
(2) has a unique holomorphic solution $\psi_{\alpha}$ defined on $U\times
S^{\prime}$ that satisfies the following conditions:

$\displaystyle\psi_{\alpha}(x_{0},\hbar)=1\qquad\text{for all $\hbar\in
S^{\prime}$\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8};}\hss}}
}$ (62)
$\displaystyle\psi_{\alpha}(x,\hbar)\simeq\widehat{\psi}_{\alpha}(x,\hbar)\quad\text{as
$\hbar\to 0$ along $\overline{A}$, loc.unif. $\forall x\in U$ \text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}
missing}$ (63)

Furthermore, $\psi_{\alpha}$ is again given by the formula (58) for all
$(x,\hbar)\in U\times S^{\prime}$ where $s_{\alpha}$ is the unique exact
solution on $U$ of the Riccati equation (59) with leading-order
$\lambda_{\alpha}$.

  * Proof.

Let $V$ be a neighbourhood of $x_{0}$ with all the properties in the
hypothesis. For every $\theta\in\Theta$, let $\psi_{\theta,\alpha}$ be the
exact WKB solution on $V\times S^{\prime}_{\theta}$ guaranteed by Theorem 5.2.
Then we can use the uniqueness property to argue that all
$\psi_{\theta,\alpha}$ ‘glue together’ to the desired exact WKB solution
$\psi_{\alpha}$. Indeed, for any $\theta_{1},\theta_{2}\in\Theta$ with
$|\theta_{1}-\theta_{2}|<\pi$, $S^{\prime}_{\theta_{1}}\cap
S^{\prime}_{\theta_{2}}\neq\varnothing$. By uniqueness,
$\psi_{\theta_{1},\alpha}$ and $\psi_{\theta_{2},\alpha}$ must agree for all
$\hbar\in S^{\prime}_{\theta_{1}}\cap S^{\prime}_{\theta_{2}}\neq\varnothing$
and all $x\in V$, and therefore extend to $V\times(S^{\prime}_{\theta_{1}}\cup
S^{\prime}_{\theta_{2}})$. Since $\Theta$ is closed, each
$S^{\prime}_{\theta}$ can be chosen to have the same diameter
$\delta^{\prime}\in(0,\delta]$. ∎

######  5.5 Remark ( ).

Theorem 5.2 is a special case of Theorem 5.5 with
$\Theta=\mathopen{}\mathclose{{}\left\\{\theta}\right\\}$.

###  § 5.2. Borel Summability of WKB Solutions

In this subsection, we translate Theorem 5.2 and its method of proof into the
language of Borel-Laplace theory, the basics of which are briefly recalled in
Appendix C. Namely, it follows directly from our construction that the exact
WKB solutions are the Borel resummation of the corresponding formal WKB
solutions. In what follows, we make this statement precise and explicit.

##### 5.6.

Recall that we write the formal WKB solution $\widehat{\psi}_{\alpha}$ as

$\displaystyle\widehat{\psi}_{\alpha}(x,\hbar)$
$\displaystyle=e^{-\Phi_{\alpha}(x)/\hbar}\;\widehat{\Psi}_{\alpha}(x,\hbar)$
(64)
$\displaystyle=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\alpha}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\widehat{\SS}_{\alpha}(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(65)

The main result in this subsection is the following theorem.

######  5.7 Theorem ( ).

Assume all the hypotheses of Theorem 5.2. The exact WKB solution
$\psi_{\alpha}$ on $U\times S^{\prime}_{\theta}$ is the locally uniform Borel
resummation in the direction $\theta$ of the formal WKB solution
$\widehat{\psi}_{\alpha}$ on $U$: for all $x\in U$ and all sufficiently small
$\hbar\in S^{\prime}_{\theta}$,

$\displaystyle\psi_{\alpha}(x,\hbar)$
$\displaystyle={\mathcal{S}}_{\theta}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}(x,\hbar)$
(66)
$\displaystyle=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\alpha}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}{\mathcal{S}}_{\theta}\big{[}\,\widehat{\SS}_{\alpha}\,\big{]}(t,\hbar)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

To be more precise, for every compactly contained subset $V\subset U$, there
is a Borel disc $S^{\prime\prime}_{\theta}\subset S^{\prime}_{\theta}$ of
possibly smaller diameter $\delta^{\prime\prime}\in(0,\delta^{\prime}]$ such
that identity (66) is valid uniformly for all $(x,\hbar)\in V\times
S^{\prime\prime}_{\theta}$.

Thus, exact and formal WKB solutions can be thought of as being canonically
specified by their asymptotic expansions, and hence in some sense
‘identified’. This is the reason that the vast majority of literature in WKB
analysis speaks simply of “WKB solutions” without specifying whether the exact
or the formal object is in question. However, it is important to stress that
this ‘identification’ is not global and highly depends on the location in
$\mathbb{C}_{x}$.

Theorem 5.7 packs a lot of information, which we now unpack by breaking it
down into a sequence of four lemmas (Part  5.8, 5.9, 5.9, and 5.9), all of
which follow immediately from the proof of Theorem 5.2.

##### 5.8. The formal Borel transform.

Recall that the formal WKB solution $\widehat{\psi}_{\alpha}$ is an
exponential power series on $U$; in symbols,
$\widehat{\psi}_{\alpha}\in{\mathcal{O}}^{\exp}(U)\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.
Instead of considering directly the Borel transform
$\widehat{\mathfrak{B}}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}$ of
$\widehat{\psi}_{\alpha}$, it is more convenient to consider the following
exponentiated formal Borel transform of $\widehat{\psi}_{\alpha}$, which we
define as

$\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}(x,\xi)\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\alpha}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\widehat{\mathfrak{B}}\big{[}\,\widehat{\SS}_{\alpha}\,\big{]}(t,\xi)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(67)

Note that
$\widehat{\mathfrak{B}}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}\neq\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}$
because the Borel transform converts multiplication into convolution. Denote
the formal Borel transform of $\widehat{\SS}_{\alpha}$ by

$\widehat{\sigma}_{\alpha}(x,\xi)\mathrel{\mathop{:}}=\widehat{\mathfrak{B}}\big{[}\,\widehat{\SS}_{\alpha}\,\big{]}(x,\xi)=\sum_{k=0}^{\infty}s_{k+2}^{\text{\tiny$(i)$}}(x)\frac{\xi^{k}}{k!}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(68)

######  5.8 Lemma (Convergence of the formal Borel transform).

The formal Borel transform $\widehat{\sigma}_{\alpha}$ is a locally uniformly
convergent power series in $\xi$. Consequently, the exponentiated formal Borel
transform
$\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\Psi}_{\alpha}\,\big{]}$ and
hence the ordinary formal Borel transform
$\widehat{\mathfrak{B}}\big{[}\,\widehat{\Psi}_{\alpha}\,\big{]}$ are locally
uniformly convergent power series in $\xi$. In symbols,
$\widehat{\sigma}_{\alpha},\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\Psi}_{\alpha}\,\big{]},\widehat{\mathfrak{B}}\big{[}\,\widehat{\Psi}_{\alpha}\,\big{]}\in{\mathcal{O}}(U)\mathopen{}\mathclose{{}\left\\{\xi}\right\\}$.

##### 5.9. The analytic Borel transform.

Similarly, it is more convenient to consider the exponentiated analytic Borel
transform of the exact WKB solution $\psi_{\alpha}$ in the direction $\theta$,
defined as

$\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}(x,\xi)\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\alpha}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\mathfrak{B}_{\theta}\big{[}\,\SS_{\alpha}\,\big{]}(t,\xi)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(69)

Note again that
$\mathfrak{B}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}\neq\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$
for the same reason as above. However, if
$\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ is holomorphic
at some point $(x,\xi)$ then clearly so is
$\mathfrak{B}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$, and therefore we can
deduce a lot of information about the Borel transform from the exponentiated
Borel transform. Denote the analytic Borel transform of $\SS_{\alpha}$ in the
direction $\theta$ by

$\sigma_{\alpha}(x,\xi)\mathrel{\mathop{:}}=\mathfrak{B}_{\theta}[\,\SS_{\alpha}\,](x,\xi)=\frac{1}{2\pi
i}\oint\nolimits_{\theta}\SS_{\alpha}(x,\hbar)e^{\xi/\hbar}\frac{\mathinner{{\mathrm{d}}\hbar}missing}{\hbar^{2}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(70)

######  5.9 Lemma (Convergence of the analytic Borel transform).

For any compactly contained subset $V\Subset U$, there exists an
$\varepsilon>0$ such that the analytic Borel transform $\sigma_{\alpha}$ is
uniformly convergent for all $(x,\xi)\in V\times\Xi_{\theta}$ where
$\Xi_{\theta}\mathrel{\mathop{:}}=\mathopen{}\mathclose{{}\left\\{\xi~{}\big{|}~{}\mathop{{\mathrm{dist}}}\nolimits\big{(}\xi,e^{i\theta}{\mathbb{R}}_{+}\big{)}<\varepsilon}\right\\}$.
Consequently, the exponentiated analytic Borel transform
$\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ and hence the
analytic Borel transform
$\mathfrak{B}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ are uniformly
convergent for all $(x,\xi)\in V\times\Xi_{\theta}$. In particular,
$\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ and
$\mathfrak{B}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ are convergent for all
$\xi\in e^{i\theta}{\mathbb{R}}_{+}$, locally uniformly for all $x\in U$.

######  5.9 Lemma (Analytic continuation of the formal Borel transform).

The analytic Borel transform $\sigma_{\alpha}$ defines the analytic
continuation of the formal Borel transform $\widehat{\sigma}_{\alpha}$ along
the ray $e^{i\theta}{\mathbb{R}}_{+}\subset\mathbb{C}_{\xi}$. Consequently,
the exponentiated analytic Borel transform
$\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ and hence the
analytic Borel transform
$\mathfrak{B}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}$ define the analytic
continuations along the ray $e^{i\theta}{\mathbb{R}}_{+}$ of
$\widehat{\mathfrak{B}}^{\exp}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}$ and
$\widehat{\mathfrak{B}}\big{[}\,\widehat{\psi}_{\alpha}\,\big{]}$,
respectively. In particular, there are no singularities in the Borel plane
$\mathbb{C}_{\xi}$ along the ray $e^{i\theta}{\mathbb{R}}_{+}$.

Let us define the exponentiated Laplace transform
$\mathfrak{L}^{\exp}_{\theta}$ the same way by applying the ordinary Laplace
transform $\mathfrak{L}_{\theta}$ to the exponent.

######  5.9 Lemma (Borel-Laplace identity for WKB solutions).

For any compactly contained subset $V\subset U$, there is a Borel disc
$S^{\prime\prime}_{\theta}\subset S_{\theta}$ of possibly smaller diameter
$\delta^{\prime\prime}\in(0,\delta^{\prime}]$ such that the Laplace transform
of $\sigma_{\alpha}$ in the direction $\theta$ is uniformly convergent for all
$(x,\hbar)\in V\times S^{\prime\prime}_{\theta}$ and satisfies the following
identity:

$\SS_{\alpha}(x,\hbar)=s_{\alpha}^{\text{\tiny$(1)$}}(x)+\mathfrak{L}_{\theta}[\,\sigma_{\alpha}\,](x,\hbar)=s_{\alpha}^{\text{\tiny$(1)$}}(x)+\int\nolimits_{e^{i\theta}{\mathbb{R}}_{+}}e^{-\xi/\hbar}\sigma_{\alpha}(x,\xi)\mathinner{{\mathrm{d}}\xi}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

Consequently, the exponentiated Laplace transform of
$\mathfrak{B}_{\theta}^{\exp}\big{[}\,\psi_{\alpha}\,\big{]}$ is uniformly
convergent for all $(x,\hbar)\in V\times S^{\prime\prime}_{\theta}$ and
satisfies the following identity:

$\displaystyle\psi_{\alpha}(x,\hbar)$
$\displaystyle=e^{-\Phi_{\alpha}(x)/\hbar}\,e^{-s_{\alpha}^{\text{\tiny$(1)$}}(x)}\,\mathfrak{L}^{\exp}_{\theta}\Big{[}\,\mathfrak{B}_{\theta}^{\exp}\big{[}\,\psi_{\alpha}\,\big{]}\,\Big{]}$
$\displaystyle=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\alpha}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\int\nolimits_{x_{0}}^{x}\SS_{\alpha}(t,\xi)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

In particular,
$\mathfrak{L}_{\theta}\big{[}\,\mathfrak{B}_{\theta}[\,\psi_{\alpha}\,]\,\big{]}$
is uniformly convergent on $V\times S^{\prime\prime}_{\theta}$ and equals
$\psi_{\alpha}$.

###  § 5.3. Explicit Formula for the Borel Transform

Thanks to the explicit nature of our construction of the exact WKB solutions,
we can write down an explicit recursive formula for the analytic continuation
of the exponentiated formal Borel transform (67).

##### 5.10.

To this end, it is convenient to introduce the following expressions. First,
we factorise $\SS_{\alpha}$ as follows:

$\SS_{\alpha}=s_{\alpha}^{\text{\tiny$(1)$}}+\epsilon_{\alpha}\sqrt{\mathnormal{D}_{0}}\mathnormal{T}_{\alpha}\qquad\text{so
that}\qquad\sigma_{\alpha}=\epsilon_{\alpha}\sqrt{\mathnormal{D}_{0}}\tau_{\alpha}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(71)

where $\epsilon_{\pm}\mathrel{\mathop{:}}=\pm 1$,
$\mathnormal{T}_{\alpha}=\mathnormal{T}_{\alpha}(x,\hbar)$ is defined by this
equality, and
$\tau_{\alpha}\mathrel{\mathop{:}}=\mathfrak{B}_{\theta}\big{[}\,\mathnormal{T}_{\alpha}\,\big{]}$.
We also define functions $p_{\ast},q_{\ast}$ of $(x,\hbar)$ using the
identities $p=p_{0}+p_{1}\hbar+p_{\ast}\hbar^{2}$ and
$q=q_{0}+q_{1}\hbar+q_{\ast}\hbar^{2}$. Next, introduce the following
expressions:

$\displaystyle\mathnormal{B}_{0}$
$\displaystyle\mathrel{\mathop{:}}=\frac{q_{\ast}-p_{\ast}(\lambda_{\alpha}+\hbar
s_{\alpha}^{\text{\tiny$(1)$}})-q_{2}+p_{2}\lambda_{\alpha}}{\mathnormal{D}_{0}}\quad\text{and}\quad\mathnormal{B}_{1}\mathrel{\mathop{:}}=\frac{\hbar
p_{\ast}}{\epsilon_{\alpha}\sqrt{\mathnormal{D}_{0}}},$ (72) $\displaystyle
b_{0}$
$\displaystyle\mathrel{\mathop{:}}=\frac{q_{2}-p_{2}\lambda_{\alpha}+(s_{\alpha}^{\text{\tiny$(1)$}})^{2}-\partial_{x}s_{\alpha}^{\text{\tiny$(1)$}}-p_{1}s_{\alpha}^{\text{\tiny$(1)$}}}{\mathnormal{D}_{0}}\quad\text{and}\quad
b_{1}\mathrel{\mathop{:}}=\frac{p_{1}-2s_{\alpha}^{\text{\tiny$(1)$}}-\partial_{x}\log\sqrt{\mathnormal{D}_{0}}}{\epsilon_{\alpha}\sqrt{\mathnormal{D}_{0}}}.$

Notice that $\mathnormal{B}_{0}$ and $\mathnormal{B}_{1}$ are both zero in the
limit as $\hbar\to 0$. An examination of (25) reveals that
$s_{\alpha}^{\text{\tiny$(2)$}}=-\epsilon_{\alpha}\sqrt{\mathnormal{D}_{0}}b_{0}$.
Let
$\beta_{1}\mathrel{\mathop{:}}=\mathfrak{B}_{\theta}\big{[}\,\mathnormal{B}_{1}\,\big{]},\beta_{0}\mathrel{\mathop{:}}=\mathfrak{B}_{\theta}\big{[}\,\mathnormal{B}_{0}\,\big{]}$.

Finally, we introduce two integral operators $\mathnormal{I}_{\pm}$ acting on
holomorphic functions $\beta=\beta(x,\xi)$ by the following formula:

$\mathnormal{I}_{\pm}\big{[}\,\beta\,\big{]}(x,\xi)\mathrel{\mathop{:}}=-\int_{0}^{\xi}\beta\big{(}x_{t}^{\pm},\xi-t\big{)}\mathinner{{\mathrm{d}}t}\qquad\text{where}\qquad
x_{t}^{\pm}\mathrel{\mathop{:}}=\Phi^{-1}\big{(}\Phi(x)\pm t\big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(73)

where the integration path is the straight line segment from $0$ to $\xi$. Let
us fix a compactly contained subset $V\subset U$, and let $V_{\theta,\alpha}$
be the union of all WKB $(\theta,\alpha)$-rays emanating from $V$ (see 3(b)).
Then expression (73) is well-defined for all $x\in V_{\theta,\alpha}$ and all
$\xi\in\mathbb{C}_{\xi}$ provided that $x_{\xi}^{\alpha}\in
V_{\theta,\alpha}$.

Heuristically, this formula should be thought of as integrating along nearby
WKB curves. Indeed, for values of $\xi$ with phase exactly $\theta$, the path
$\mathopen{}\mathclose{{}\left\\{x_{t}^{\alpha}~{}|~{}t\in[0,\xi]}\right\\}$
is nothing but a segment of the WKB $(\theta,\alpha)$-ray emanating from
$x_{0}$. Thus, restricting $\xi$ to the ray
$e^{i\theta}{\mathbb{R}}_{+}\subset\mathbb{C}_{\xi}$, expression (73) is well-
defined for all $(x,\xi)\in V_{\theta,\alpha}\times
e^{i\theta}{\mathbb{R}}_{+}$. In fact, since $V$ is compactly contained in
$U$, there is some tubular neighbourhood $\Xi_{\theta}$ of the ray
$e^{i\theta}{\mathbb{R}}_{+}\subset\mathbb{C}_{\xi}$ such that (73) is well-
defined for all $(x,\xi)\in V_{\theta,\alpha}\times\Xi_{\theta}$. The main
result of this subsection is then the following proposition.

######  5.10 Proposition ( ).

The exponentiated analytic Borel transform
$\mathfrak{B}^{\exp}_{\theta}\big{[}\psi_{\alpha}\big{]}$, which equals the
analytic continuation along the ray $e^{i\theta}{\mathbb{R}}_{+}$ of the
exponentiated formal Borel transform
$\widehat{\mathfrak{B}}^{\exp}\big{[}\widehat{\psi}_{\alpha}\big{]}$, can be
expressed for all $(x,\xi)\in V\times\Xi^{\prime}_{\theta}$ as follows:

$\mathfrak{B}^{\exp}_{\theta}\big{[}\,\psi_{\alpha}\,\big{]}(x,\hbar)=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\int\nolimits_{x_{0}}^{x}\lambda_{\alpha}(t)\mathinner{{\mathrm{d}}t}}\right)\exp\mathopen{}\mathclose{{}\left(-\epsilon_{\alpha}\int\nolimits_{x_{0}}^{x}\sqrt{\mathnormal{D}_{0}(t)}\,\tau_{\alpha}(t,\xi)\mathinner{{\mathrm{d}}t}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(74)

where $\tau_{\alpha}$ is a holomorphic function on
$V\times\Xi^{\prime}_{\theta}$ defined as the following uniformly convergent
infinite series:

$\tau_{\alpha}(x,\xi)=\sum_{n=0}^{\infty}\tau_{\alpha,n}(x,\xi)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(75)

The terms $\tau_{\alpha,n}$ are holomorphic functions given by the following
recursive formula:
$\tau_{\alpha,0}\mathrel{\mathop{:}}=b_{0},\tau_{\alpha,1}\mathrel{\mathop{:}}=\mathnormal{I}_{\alpha}\big{[}\beta_{0}+b_{1}b_{0}\big{]}$,
and, for $n\geqslant 2$,

$\tau_{\alpha,n}\mathrel{\mathop{:}}=\mathnormal{I}_{\alpha}\Bigg{[}b_{1}\tau_{\alpha,n-1}+\beta_{1}\ast\tau_{\alpha,n-2}+\sum_{\begin{subarray}{c}n_{1},n_{2}\geqslant
0\\\
n_{1}+n_{2}=n-2\end{subarray}}\tau_{\alpha,n_{1}}\ast\tau_{\alpha,n_{2}}\Bigg{]}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(76)

  * Proof.

This is a consequence of the proof of Part  C.1. Namely, the recursive
formula (76) is the formula (152) but written in the coordinate $x$ instead of
the coordinate $z$. ∎

######  5.10 Example (undeformed coefficients).

As ever, the above expressions are simplest when the coefficients $p,q$ of our
differential equation are independent of $\hbar$. In this case, formulas (72)
are considerably simplified:

$\mathnormal{B}_{0}=\mathnormal{B}_{1}=0,\quad
b_{0}=\frac{(s_{\alpha}^{\text{\tiny$(1)$}})^{2}-\partial_{x}s_{\alpha}^{\text{\tiny$(1)$}}}{\mathnormal{D}_{0}},\quad
b_{1}=-\frac{2s_{\alpha}^{\text{\tiny$(1)$}}+\partial_{x}\log\sqrt{\mathnormal{D}_{0}}}{\epsilon_{\alpha}\sqrt{\mathnormal{D}_{0}}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(77)

In particular, this means $\beta_{0}=\beta_{1}=0$, and so the recursion (76)
reduces to

$\tau_{\alpha,0}=b_{0},\quad\tau_{\alpha,1}=\mathnormal{I}_{\alpha}\big{[}b_{1}b_{0}\big{]},\quad\tau_{\alpha,n}\mathrel{\mathop{:}}=\mathnormal{I}_{\alpha}\Bigg{[}b_{1}\tau_{\alpha,n-1}+\sum_{\begin{subarray}{c}n_{1},n_{2}\geqslant
0\\\
n_{1}+n_{2}=n-2\end{subarray}}\tau_{\alpha,n_{1}}\ast\tau_{\alpha,n_{2}}\Bigg{]}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(78)

######  5.10 Example (Schrödinger equation).

Formulas (72) are also considerably simplified for the Schrödinger equation:

$\begin{gathered}\mathnormal{B}_{0}=\frac{\mathnormal{Q}_{\ast}-\mathnormal{Q}_{2}}{4\mathnormal{Q}_{0}},\quad\mathnormal{B}_{1}=0,\quad
b_{0}=\frac{\mathnormal{Q}_{2}+(s_{\alpha}^{\text{\tiny$(1)$}})^{2}-\partial_{x}s_{\alpha}^{\text{\tiny$(1)$}}}{4\mathnormal{Q}_{0}},\quad
b_{1}=-\frac{s_{\alpha}^{\text{\tiny$(1)$}}+\tfrac{1}{2}\partial_{x}\log\sqrt{\mathnormal{Q}_{0}}}{\epsilon_{\alpha}\sqrt{\mathnormal{Q}_{0}}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}\end{gathered}$

Thus, $\beta_{1}=0$ but $\beta_{0}$ is not necessarily $0$. Recursion (76) in
this case has exactly the same form as in (78) but with
$\tau_{\alpha,1}=\mathnormal{I}_{\alpha}[\beta_{0}+b_{1}b_{0}]$.

Combining this with the previous example, the formulas in the case of a
Schrödinger equation with $\hbar$-independent potential (i.e.,
$\mathnormal{Q}=\mathnormal{Q}_{0}$) have the simplest possible form:

$\mathnormal{B}_{0}=\mathnormal{B}_{1}=0,\qquad
b_{0}=\frac{5}{64}\frac{(\mathnormal{Q}^{\prime})^{2}}{\mathnormal{Q}^{3}}-\frac{1}{16}\frac{\mathnormal{Q}^{\prime\prime}}{\mathnormal{Q}^{2}},\qquad
b_{1}=-\epsilon_{\alpha}\frac{1}{2}\frac{\mathnormal{Q}^{\prime}}{\mathnormal{Q}^{3/2}}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

In this case, $\beta_{0}=\beta_{1}=0$ so the recursion (76) is again given by
(78).

######  5.10 Remark (Resurgent nature of the exact WKB analysis).

The resurgent property of WKB solutions for Schrödinger equations with
polynomial potential was conjectured by Voros in [Vor83b, Vor83a] and
partially argued by Écalle in the preprint [Éca84, p.40] (see [DP99, Comment
on p.32]). We do not address this point directly in our paper. However, we
believe that formula (74) for the Borel transform is sufficiently simple and
explicit to keep track of the singularities in the Borel plane. We therefore
hope it can yield a full proof of the conjectured resurgence property of WKB
solutions not only for Schrödinger equations with polynomial potential (as
conjectured by Voros), but more generally for all second-order ODEs (2) with
rational dependence on $x$.

###  § 5.4. Notable Special Cases and Examples

In this subsection, we restate the existence and uniqueness results proved in
this paper for two important classes of WKB geometry: closed and generic WKB
trajectories. In both of these cases, the technical assumptions in Theorem 5.2
simplify considerably. Throughout this subsection, we maintain our background
assumptions of Part  5.1.

#### Closed WKB Trajectories

The statement of Theorem 5.2 is simplest for closed trajectories. In this
case, assumptions (1) and (2) are automatic because a closed trajectory can
always be embedded in a WKB ring domain whose closure is a compact subset of
$X$.

######  5.10 Corollary (Existence and uniqueness for closed WKB trajectories).

Let $\theta\in\Theta$ be fixed. Suppose that the WKB $\theta$-trajectory
passing through $x_{0}$ is closed. Let $U\subset X$ be any simply connected
neighbourhood of $x_{0}$ contained in a WKB $\theta$-ring domain $R\subset X$.
Then all the conclusions of Theorem 5.2, § 5.1, Part  5.3, and Theorem 5.7
hold verbatim simultaneously for both $\alpha=\pm$.

##### 5.11. Monodromy of exact WKB solutions on WKB ring domains.

The exact WKB solutions $\psi_{+},\psi_{-}$ from § 5.4 extend to the entire
WKB ring domain $R$ but only as multivalued functions. Thanks to the explicit
formula in § 5.1, their monodromy is easy to calculate.

######  5.11 Proposition ( ).

The exact WKB solutions $\psi_{+},\psi_{-}$ from § 5.4 extend via the
formula (58) to multivalued holomorphic solutions on $R\times
S^{\prime}_{\theta}$ with monodromy

$a_{\pm}(\hbar)\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\oint\nolimits_{\gamma_{\pm}}s_{\pm}(x,\hbar)\mathinner{{\mathrm{d}}x}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(79)

where the integration contour $\gamma_{\pm}$ is any path contained in $R$ and
homotopic to the closed WKB $\theta$-trajectory passing through $x_{0}$ and
with orientation matching the orientation of the WKB $(\theta,\pm)$-ray. The
monodromy $a_{\pm}$ is a holomorphic function of $\hbar\in
S^{\prime}_{\theta}$ which admits exponential Gevrey asymptotics in a
halfplane:

$a_{\pm}(\hbar)\simeq\widehat{a}_{\pm}(\hbar)\mathrel{\mathop{:}}=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\oint\nolimits_{\gamma_{\pm}}\widehat{s}_{\pm}(x,\hbar)\mathinner{{\mathrm{d}}x}}\right)\qquad\text{as
$\hbar\to 0$ along $\overline{A}_{\theta}$}\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(80)

######  5.11 Corollary (Existence and uniqueness in wider sectors).

More generally, suppose that for every $\theta\in\Theta$, the WKB
$\theta$-trajectory passing through $x_{0}$ is closed. Then $U$ can be chosen
sufficiently small such that all the conclusions of Theorem 5.5 hold verbatim,
and the monodromy $a_{\pm}$ extends to a holomorphic function on $S^{\prime}$
with exponential Gevrey asymptotics:
$a_{\pm}(\hbar)\simeq\widehat{a}_{\pm}(\hbar)$ as $\hbar\to 0$ along
$\overline{A}$.

######  5.11 Remark ( ).

We note that upon writing
$s_{\pm}(x,\hbar)=\lambda_{\pm}(x)+\hbar\SS_{\pm}(x,\hbar)$, the monodromy
(79) is expressed as

$a_{\pm}(\hbar)=\exp\mathopen{}\mathclose{{}\left(-\frac{1}{\hbar}\oint\nolimits_{\gamma_{\pm}}\lambda_{\pm}(x)\mathinner{{\mathrm{d}}x}}\right)\exp\mathopen{}\mathclose{{}\left(-\oint\nolimits_{\gamma_{\pm}}\SS_{\pm}(x,\hbar)\mathinner{{\mathrm{d}}x}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(81)

This expression is notable because the complex number
$\oint\nolimits_{\gamma_{\pm}}\lambda_{\pm}(x)\mathinner{{\mathrm{d}}x}$ is a
period of a certain covering Riemann surface (called spectral curve) naturally
associated with our differential equation (namely, the one given by the
leading-order characteristic equation (17)). These numbers, and therefore
functions (81), play pivotal role in the global analysis of such differential
equations and more general meromorphic connections on Riemann surfaces. These
topics are beyond the scope of this paper, but see for example [GMN13b]. More
comments will appear in [Nik21a].

#### Generic WKB Trajectories

For generic WKB rays, condition (1) in Theorem 5.2 is automatically taken care
of by insisting (in the definition of generic rays) that the limiting point
$x_{\infty}$ is an infinite critical point; i.e., a pole of
$\mathnormal{D}_{0}$ of order $m\geqslant 2$. At the same time, condition (2)
must still be imposed but it is somewhat simplified by the fact that
$\mathnormal{D}_{0}$ behaves like $(x-x_{\infty})^{-m}$ near $x_{\infty}$.
Altogether, we have the following statement.

######  5.11 Corollary (Existence and uniqueness for generic WKB rays).

Fix a sign $\alpha\in\mathopen{}\mathclose{{}\left\\{+,-}\right\\}$ and a
phase $\theta\in\Theta$. Suppose that the WKB $(\theta,\alpha)$-ray emanating
from $x_{0}$ is generic. Let $x_{\infty}\in\mathbb{C}_{x}$ be the limiting
infinite critical point of order $m\geqslant 2$. In addition, we make the
following assumption on the coefficients $p,q$:

$\begin{gathered}(x-x_{\infty})^{m/2}p(x,\hbar)\simeq(x-x_{\infty})^{m/2}\widehat{p}(x,\hbar)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}\\\
(x-x_{\infty})^{m}q(x,\hbar)\simeq(x-x_{\infty})^{m}\widehat{q}(x,\hbar)\end{gathered}$
(82)

as $\hbar\to 0$ along $\overline{A}$, uniformly for all $x\in X$ sufficiently
close to $x_{\infty}$. Then $x_{0}$ has a neighbourhood $U\subset X$ such that
all the conclusions of Theorem 5.2, § 5.1, Part  5.3, and Theorem 5.7 hold
verbatim.

Specifically, $U\subset X$ can be chosen to be any simply connected domain
containing $x_{0}$ which is free of turning points and such that every WKB
$(\theta,\alpha)$-ray emanating from $U$ is generic and tends to $x_{\infty}$.
Note that the assumption in § 5.4 that the WKB $(\theta,\alpha)$-ray
emanating from $x_{0}$ is generic guarantees that such a neighbourhood $U$
always exists, see Part  4.12.

######  5.11 Example (Mildly perturbed coefficients).

If the coefficients $p,q$ of the differential equation (2) are at most
polynomial in $\hbar$ (i.e., if $p,q\in{\mathcal{O}}(X)[\hbar]$) then
condition (82) simplifies even further as follows: for every $k\geqslant 0$,

$\textup{(pole order of $p_{k}$ at
$x_{\infty}$)}\leqslant\tfrac{1}{2}m\quad\text{and}\quad\textup{(pole order of
$q_{k}$ at $x_{\infty}$)}\leqslant m\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(83)

######  5.11 Corollary (Existence and uniqueness in wider sectors).

More generally, suppose that for every $\theta\in\Theta$, the WKB
$\theta$-trajectory passing through $x_{0}$ is generic, and that condition
(82) is satisfied at the (necessarily $\theta$-independent) limiting infinite
critical point $x_{\infty}$. Then $U$ can be chosen sufficiently small such
that all the conclusions of Theorem 5.5 hold verbatim.

##### 5.12. The exact WKB basis on a WKB strip domain.

If the WKB trajectory through $x_{0}$ is generic, then § 5.4 yields two
exact WKB solutions, one for each WKB ray emanating from $x_{0}$. These exact
WKB solutions define a basis of exact solutions on any WKB strip domain
containing the WKB trajectory through $x_{0}$. To be precise, we have the
following statement.

######  5.12 Corollary ( ).

Suppose the WKB $\theta$-trajectory passing through $x_{0}$ is generic, and
let $U\subset X$ be any WKB $\theta$-strip containing $x_{0}$. Let
$x_{\pm\infty}\in\mathbb{C}_{x}$ be the two limiting infinite critical points
of order $m_{\pm}\geqslant 2$. In addition, we make the following assumption
on the coefficients $p,q$:

$\begin{gathered}(x-x_{\pm\infty})^{m_{\pm}/2}p(x,\hbar)\simeq(x-x_{\pm\infty})^{m_{\pm}/2}\widehat{p}(x,\hbar)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}\\\
(x-x_{\pm\infty})^{m_{\pm}}q(x,\hbar)\simeq(x-x_{\pm\infty})^{m_{\pm}}\widehat{q}(x,\hbar)\end{gathered}$
(84)

as $\hbar\to 0$ along $\overline{A}_{\theta}$, uniformly for all $x\in X$
sufficiently close to $x_{\pm\infty}$. Then all the conclusions of Theorem
5.2, Theorem 5.2, § 5.1, Part  5.3, and Theorem 5.7 hold verbatim for both
$\alpha=\pm$.

###  § 5.5. Relation to Previous Work

In this final subsection, we explain how our results relate to other works
about the existence of WKB solutions.

######  5.12 Remark (Relation to the work of Nemes).

In the recent paper [Nem21], Nemes considers Schrödinger equations of the
form222Explicitly, the notations compare as follows: equation (1.3) in [Nem21]
is our equation (85) with $u\leftrightarrow\hbar^{-1}$, $\xi\leftrightarrow
z$, $\phi\leftrightarrow a_{1}$, $\psi\leftrightarrow a_{2}$,
$\mathnormal{W}(u,\xi)\leftrightarrow\mathnormal{W}(z,\hbar)$, and
$\mathrm{\mathbf{G}}\leftrightarrow\Omega$.

$\hbar^{2}\partial^{2}_{z}\mathnormal{W}=\big{(}1+a_{1}(z)\hbar+a_{2}(z)\hbar^{2}\big{)}\mathnormal{W}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$
(85)

where $a_{1},a_{2}$ are holomorphic functions on a domain
$\Omega\subset\mathbb{C}_{z}$ which contains an infinite horizontal strip.
Using a Banach fixed-point theorem argument, he shows (see [Nem21, Theorem
1.1]) that under certain boundedness assumptions on the coefficients
$a_{1},a_{2}$ (see [Nem21, Conditions 1.1 and 1.2]), the Schrödinger equation
(85) has (in our terminology) two exact solutions
$\mathnormal{W}^{\pm}=\mathnormal{W}^{\pm}(x,\hbar)$, defined for all
$(z,\hbar)\in\Omega^{\pm}_{0}\times S^{\prime}$ where $S^{\prime}$ is a Borel
disc with opening $A\mathrel{\mathop{:}}=(-\pi/2,+\pi/2)$ and
$\Omega_{0}^{\pm}\Subset\Omega$ are any properly contained horizontal
halfstrips (unbounded respectively as $z\to\pm\infty$). The following
proposition asserts that this existence result is a corollary of our main
theorem.

######  5.12 Proposition ( ).

Theorem 5.2 (or, more specifically, § 5.1) implies Theorem 1.1 and the first
assertion of Theorem 2.1 in [Nem21].

  * Proof.

For the equation (85), the leading-order characteristic discriminant
$\mathnormal{D}_{0}=1$, so condition (1) of (5.2) is vacuously true. The
boundedness Conditions 1.1 and 1.2 in [Nem21] imply in particular that the
coefficients $a_{1},a_{2}$ are bounded on $\Omega^{\pm}_{0}$, which by the
discussion in § 5.1 means condition (2) of Theorem 5.2 is met. So Theorem
1.1 in [Nem21] follows. Finally, the first assertion of Theorem 2.1 in [Nem21]
is a special case of Part  5.8. ∎

##### 5.13.

Equations of the form (85) can be related by means of a Liouville
transformation $z=\Phi(x)$ from (49) to Schrödinger equations of the form (10)
with potentials that are at most quadratic in $\hbar$; i.e.,
$\mathnormal{Q}(x,\hbar)=\mathnormal{Q}_{0}(x)+\hbar\mathnormal{Q}_{1}(x)+\hbar^{2}\mathnormal{Q}_{2}(x)$.
Explicitly, the unknown variables $\psi$ and $\mathnormal{W}$ are related by

$\psi(x,\hbar)=\mathnormal{Q}_{0}^{-1/4}(x)\mathnormal{W}\big{(}\tfrac{1}{2}\Phi(x),\hbar\big{)}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}$

and the coefficients are related by

$a_{1}\big{(}\tfrac{1}{2}\Phi(x)\big{)}\mathrel{\mathop{:}}=\frac{\mathnormal{Q}_{1}(x)}{\mathnormal{Q}_{0}(x)}\quad\text{and}\quad
a_{2}\big{(}\tfrac{1}{2}\Phi(x)\big{)}\mathrel{\mathop{:}}=\frac{\mathnormal{Q}_{2}(x)}{\mathnormal{Q}_{0}(x)}-\frac{1}{\mathnormal{Q}^{3/4}_{0}(x)}\partial_{x}^{2}\mathopen{}\mathclose{{}\left(\frac{1}{\mathnormal{Q}_{0}^{1/4}(x)}}\right)\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$

However, note that this transformation of the unknown variable involves a
choice of a fourth-root branch $\mathnormal{Q}_{0}^{1/4}$ and, more
importantly, even if $\mathnormal{W}=\mathnormal{W}(z,\hbar)$ is a solution of
(85) for $z$ in some domain $\Omega^{\pm}_{0}\subset\Omega$, then
$\psi(x,\hbar)$ is a well-defined solution only if
$\mathnormal{W}(z_{1},\hbar)=\mathnormal{W}(z_{2},\hbar)$ whenever
$\Phi^{-1}(z_{1})=\Phi^{-1}(z_{2})$. The explicit approach pursued in our
paper (aided specifically by the recursive formula of Part  5.10) makes this
verification obvious.

######  5.13 Remark (Relation to the work of Koike-Schäfke).

Some of the results in a number of references mentioned in the introduction
(see paragraph 5 of Part  1.2) rely on the statement of Theorem 2.17
presented in [IN14] from an unpublished work of Koike and Schäfke on the Borel
summability of formal WKB solutions of Schrödinger equations with polynomial
potential. The following proposition asserts that our results imply part (a)
of Theorem 2.17 in [IN14]. Part (b) of Theorem 2.17 in [IN14] will be derived
from a more general result in [Nik21b]. It is also stated in Theorem 2.18 in
[IN14] that Theorem 2.17 in [IN14] holds for any compact Riemann surface: this
theorem will also be derived as a special case of a more general result in
[Nik21a].

######  5.13 Proposition ( ).

§ 5.4 implies part (a) of Theorem 2.17 in [IN14].

  * Proof.

The main assumption for Theorem 2.17 in [IN14] is that the potential
$\mathnormal{Q}$ is a polynomial in $\hbar$ with rational coefficients
$\mathnormal{Q}_{k}$ whose behaviour at the poles is as stated in Assumption
2.5 and the third bullet point of Assumption 2.3 in [IN14]. We claim that
these assumptions are a special case of (83) in § 5.4

First, let us explain how the notation in [IN14] compares with ours. In
[IN14], the equation variable $z$ is the same as our variable $x$, and the
large parameter $\eta$ is our $\hbar^{-1}$. The authors consider Schrödinger
equations of the form (10) but where
$\mathnormal{Q}(x,\hbar)=\mathnormal{Q}_{0}(x)+\mathnormal{Q}_{1}(x)\hbar+\cdots$
is a polynomial in $\hbar$ (cf. [IN14, equation (2.2)]). This is the situation
in § 5.4 with $p_{k}=0$ for all $k$. In the statement of Theorem 2.17 (a) in
[IN14], the chosen point $z\in\mathnormal{D}$ in a Stokes region
$\mathnormal{D}$ ($=$ a maximal WKB strip domain) is our regular basepoint
$x_{0}\in X$.

Let $x_{\infty}$ be the pole in question either on the boundary of $X$ or at
infinity in $\mathbb{C}_{x}$. By the assumptions in § 5.4, $x_{\infty}$ is
an infinite critical point, which means in particular that the pole order of
$\mathnormal{Q}_{0}$ at $x_{\infty}$ is $m\geqslant 2$, which coincides with
the third bullet point of Assumption 2.3 in [IN14]. Parts (i) and (iii) of
Assumption 2.5 in [IN14] are also clearly included in (83). Finally, part (ii)
of Assumption 2.5 in [IN14] is included in (83) because $1+\tfrac{1}{2}m<m$
whenever $m\geqslant 3$. ∎

##  Appendix A Basics of Usual Asymptotics

##### A.1. Sectorial domains.

Fix a circle
$\mathbb{S}^{1}\mathrel{\mathop{:}}={\mathbb{R}}/2\pi{\mathbb{Z}}$ once and
for all. We refer to its points as directions, and we think of it as the set
of directions at the origin in $\mathbb{C}_{\hbar}$ when $\hbar$ is written in
polar coordinates. More precisely, we consider the real-oriented blowup of the
complex plane $\mathbb{C}_{\hbar}$ at the origin, which by definition is the
bordered Riemann surface
$[\mathbb{C}_{\hbar}:0]\mathrel{\mathop{:}}={\mathbb{R}}_{+}\times\mathbb{S}^{1}$
with coordinates $(r,\theta)$, where ${\mathbb{R}}_{+}$ is the nonnegative
reals. The projection $[\mathbb{C}_{\hbar}:0]\to\mathbb{C}_{\hbar}$ sends
$(r,\theta)\mapsto re^{i\theta}$, which is a biholomorphism away from the
circle of directions. See Fig. 4 for an illustration.

Figure 4: Examples of sectorial domains. (a): an arbitrary sectorial domain
with opening $A$. (b): straight sector with opening $A$ and some radius
$\mathnormal{R}>0$. (c): a Borel disc with diameter $\delta>0$.

A sectorial domain near the origin in $\mathbb{C}_{\hbar}$ is a simply
connected domain
$S\subset\mathbb{C}_{\hbar}^{\ast}=\mathbb{C}_{\hbar}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}$
whose closure $\overline{S}$ in $[\mathbb{C}_{\hbar}:0]$ intersects the
boundary circle $\mathbb{S}^{1}$ in a closed arc
$\overline{A}\subset\mathbb{S}^{1}$ with nonzero length. In this case, the
open arc $A$ is called the opening of $S$, and its length $|A|$ is called the
opening angle of $S$. A proper subsectorial domain $S_{0}\subset S$ is one
whose closure $\overline{S}_{0}$ in $[\mathbb{C}_{\hbar}:0]$ is contained in
$S$. This means in particular that the opening $A_{0}$ of $S_{0}$ is compactly
contained in $A$; i.e., $\overline{A}_{0}\subset A$.

The simplest example of a sectorial domain $S$ is of course a straight sector
of radius $\delta>0$ and opening $A$, which is the set of points
$\hbar\in\mathbb{C}_{\hbar}$ satisfying $\arg(\hbar)\in A$ and
$0<|\hbar|<\mathnormal{R}$. The most typical example of a sectorial domain
encountered in this paper is a Borel disc of diameter $\delta>0$:

$S=\mathopen{}\mathclose{{}\left\\{\hbar\in\mathbb{C}_{\hbar}~{}\big{|}~{}\operatorname{Re}(1/\hbar)>1/\delta}\right\\}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(86)

Its opening is $A=(-\pi/2,+\pi/2)$. Notice that any straight sector with
opening $A$ contains a Borel disc, but a Borel disc contains no straight
sectors with opening $A$. It is also not difficult to see that any sectorial
domain with opening $A$ contains a Borel disc. More generally, we will
consider Borel discs bisected by some direction $\theta\in\mathbb{S}^{1}$:

$S_{\theta}=\mathopen{}\mathclose{{}\left\\{\hbar\in\mathbb{C}_{\hbar}~{}\big{|}~{}\operatorname{Re}(e^{i\theta}/\hbar)>1/\delta}\right\\}\text{\hbox
to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}$
(87)

###  § A.1. Poincaré Asymptotics in One Dimension

First, for the benefit of the reader and to fix some notation, let us briefly
recall some basic notions from asymptotic analysis in one complex variable. We
denote by
$\mathbb{C}\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$ the
ring of formal power series in $\hbar$, and by
$\mathbb{C}\mathopen{}\mathclose{{}\left\\{\hbar}\right\\}$ the ring of
convergent power series in $\hbar$. Fix an arc of directions
$A\subset\mathbb{S}^{1}$. We denote by ${\mathcal{O}}(S)$ the ring of
holomorphic functions on a sectorial domain $S\subset\mathbb{C}_{\hbar}$ with
opening $A$.

##### A.2. Sectorial germs.

For the purpose of asymptotic behaviour as $\hbar\to 0$, the actual nonzero
radial size of $S$ is irrelevant. So it is better to consider germs of
holomorphic functions defined on sectorial domains with opening $A$, formally
defined next.

######  A.2 Definition ( ).

A sectorial germ on $A$ is an equivalence class of pairs $(f,S)$ where
$S\subset\mathbb{C}_{\hbar}$ is a sectorial domain with opening $A$ and $f$ is
a holomorphic function on $S$. Two such pairs $(f,S)$ and
$(f^{\prime},S^{\prime})$ are considered equivalent if the intersection
${S\cap S^{\prime}}$ contains a sectorial domain $S^{\prime\prime}$ with
opening $A$ on which $f$ and $f^{\prime}$ are equal.

Sectorial germs on $A$ form a ring which we denote by ${\mathcal{O}}(A)$. For
any sectorial domain $S\subset\mathbb{C}_{\hbar}$ with opening $A$, there is a
map ${\mathcal{O}}(S)\to{\mathcal{O}}(A)$ that sends a holomorphic function
$f$ to the corresponding sectorial germ.

Terminology: For the benefit of the reader who is uncomfortable with the
language of germs, we stress that every sectorial germ on $A$ can be
represented by an actual holomorphic function $f\in{\mathcal{O}}(S)$ on some
sectorial domain $S$. In fact, we normally denote the equivalence class of any
pair $(f,S)$ simply by “$f$” and we often even refer to it as a holomorphic
function on $A$.

##### A.3. Poincaré asymptotics.

Recall that a holomorphic function $f\in{\mathcal{O}}(S)$ defined on a
sectorial domain $S$ is said to admit (Poincaré) asymptotics as $\hbar\to 0$
along $A$ if there is a formal power series
$\widehat{f}(\hbar)\in\mathbb{C}\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$
such that the order-$n$ remainder

$\mathnormal{R}_{n}(\hbar)\mathrel{\mathop{:}}=f(\hbar)-\sum_{k=0}^{n-1}f_{k}\hbar^{k}$
(88)

is bounded by $\hbar^{n}$ for all sufficiently small $\hbar\in S$. That is,
for every ${n\geqslant 0}$, and every compactly contained subarc $A_{0}\Subset
A$, there is a sectorial subdomain $S_{0}\subset S$ with opening $A_{0}$ and a
real constant $\mathnormal{C}_{n,0}>0$ such that

$\big{|}\mathnormal{R}_{n}(\hbar)\big{|}\leqslant\mathnormal{C}_{n,0}|\hbar|^{n}$
(89)

for all $\hbar\in S_{0}$. The constants $\mathnormal{C}_{n,0}$ may depend on
$n$ and the opening $A_{0}$. If this is the case, we write

$f(\hbar)\sim\widehat{f}(\hbar)\quad\quad\quad\quad\text{as $\hbar\to 0$ along
$A$\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}.}\hss}}
missing}$ (90)

Sectorial germs with this property form a subring
${\mathcal{A}}(A)\subset{\mathcal{O}}(A)$, and the asymptotic expansion map
defines a ring homomorphism
$\text{\ae}:{\mathcal{A}}(A)\to\mathbb{C}\mathopen{}\mathclose{{}\left\llbracket\hbar}\right\rrbracket$.

##### A.4. Asymptotics along a closed arc.

Furthermore, we will write

$f(\hbar)\sim\widehat{f}(\hbar)\quad\quad\quad\quad\text{as $\hbar\to 0$ along
$\overline{A}$\text{\hbox to0.0pt{\,
{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8},}\hss}}
}$ (91)