2.1 Background assumptions

Throughout the paper, we fix a complex plane ${\mathbb {C}}_x$ with coordinate x and another complex plane ${\mathbb {C}}_{\hbar }$ with coordinate $\hbar $ . Let $\mathsf{X}$ be a domain in ${\mathbb {C}}_x$ or indeed a coordinate chart on a Riemann surface. Let $\mathsf{S} \subset {\mathbb {C}}_{\hbar }$ be a sectorial domain at the origin with opening arc $\mathsf{A}$ . We assume that $0 < |\mathsf{A}| \leq 2\pi $ .

Consider the Riccati equation

(5) $$ \begin{align} \hbar \partial_x f = a f^2 + b f + c \end{align} $$

whose coefficients $a,b,c$ are holomorphic functions of $(x,\hbar ) \in \mathsf{X} \times \mathsf{S}$ admitting locally uniform asymptotic expansions with holomorphic coefficients as $\hbar \to 0$ along $\mathsf{A}$ :

(6) $$ \begin{align} \begin{aligned} a (x, \hbar) &\sim \hat{a} (x, \hbar) {\mathrel{\mathop:}=} \sum_{n = 0}^{\infty} a_n (x) \hbar^n {,}\\ b (x, \hbar) &\sim \hat{b} (x, \hbar) {\mathrel{\mathop:}=} \sum_{n = 0}^{\infty} b_n (x) \hbar^n {,} \\ c (x, \hbar) &\sim \hat{c} (x, \hbar) {\mathrel{\mathop:}=} \sum_{n = 0}^{\infty} c_n (x) \hbar^n. \end{aligned} \qquad\qquad \text{as } \hbar \to 0 \text{ along } \mathsf{A}, \text{ loc.unif. } \forall x \in \mathsf{X} {,} \end{align} $$

The main problem we pose in this article is to find canonical exact solutions of the Riccati equation (5) in the following sense.

2.2 Examples

The following is a list, included here for illustrative purposes only, containing a few explicit examples of Riccati equations to which the main results in this paper can be applied.

The most typical situation is one where the coefficients $a,b,c$ of the Riccati equation (5) are polynomials in $\hbar $ with coefficients which are rational functions of x. In this case, $\mathsf{X}$ is the complement of the poles in ${\mathbb {C}}_x$ , and the sectorial domain $\mathsf{S}$ can be taken to be the whole open right half-plane ${\left \{ {\operatorname {Re}} (\hbar )> 0 \right \}}$ . The simplest example is

1. (1) ${\hbar \partial _x f = f^2 - x}$ .

This Riccati equation is examined in great detail in §6.2. It arises in the exact WKB analysis of the Airy equation $\hbar ^{2} \partial _{x}^{2} \psi (x, \hbar ) = x \psi (x, \hbar )$ (see [Reference Nikolaev12]). In this case, $\mathsf{X} = {\mathbb {C}}_x$ and the sectorial domain $\mathsf{S}$ is the open right half-plane ${\left \{ {\operatorname {Re}} (\hbar )> 0 \right \}}$ . If $\mathsf{U}$ is any of the three sectorial domains in ${\mathbb {C}}_x$ given by ${\left \{ 0 < \arg (x) < +4\pi /3 \right \}}$ , or ${\left \{ +2\pi /3 < \arg (x) < +2\pi \right \}}$ , or ${\left \{ -2\pi /3 < \arg (x) < +2\pi /3 \right \}}$ , then on each of these domains, the main existence and uniqueness result in this paper produces a pair of canonical exact solutions. More generally,

1. (2) ${\hbar \partial _x f = f^2 + q(x)}$ ,

where $q(x)$ is any polynomial or a rational function with poles of order $2$ or higher. In this case, $\mathsf{S}$ can again be arranged to be the right half-plane, and $\mathsf{U}$ is a sectorial domain near a pole of $q(x)$ of order $2$ or higher.

Many Riccati equations arise from the WKB analysis of classical second-order differential equation. For example, the following Riccati equation appears in the WKB analysis of the Gauss hypergeometric equation:

1. (3) ${\hbar \partial _x f = f^2 + \frac {\gamma - (\alpha + \beta + 1) x}{x(x-1)} f + \frac {\alpha \beta }{x (x-1)}}$ for any $\alpha , \beta , \gamma \in {\mathbb {C}}^{\ast }$ .

Riccati equations also arise in the analysis of singularly perturbed second-order systems. For example, the Riccati equation

1. (4) ${\hbar \partial _x f = \hbar f^2 + (1-x) f + x \hbar }$

arises in the analysis of the system $\hbar \partial _{x} \psi + \begin {bmatrix} 1 & - \hbar \\ x\hbar & x\end {bmatrix} \psi = 0$ . See [Reference Nikolaev13].

Our methods also apply to the following nontrivial deformation of example (1):

1. (5) ${\hbar \partial _x f = f^2 - x + x { {E}} (x, \hbar ) \quad \text {where} \quad { {E}} (x, \hbar ) {\mathrel {\mathop :}=} \int _0^{+\infty } \frac {e^{- \xi /\hbar }}{x + \xi } {\mathrm {d}}{\xi }}$ .

The sectorial domain $\mathsf{S}$ in this case is the open right half-plane. The function ${ {E}} $ is holomorphic in $\hbar \in \mathsf{S}$ , and it admits a locally uniform asymptotic expansion as $\hbar \to 0$ in the right half-plane. Notice, however, that ${ {E}} $ is not holomorphic at $\hbar = 0$ , and it also has nonisolated singularities along the negative real axis in ${\mathbb {C}}_x$ . Nevertheless, if $\mathsf{U}$ is the domain given by ${\left \{ -2\pi /3 < \arg (x) < +2\pi /3 \right \}}$ or by ${\left \{ 0 < \arg (x) < +\pi \right \}}$ or ${\left \{ +\pi < \arg (x) < +2\pi \right \}}$ , then our method yields canonical exact solutions on $\mathsf{U}$ .

3.1 Leading-order solutions

Consider the leading-order equation corresponding to (7):

(8) $$ \begin{align} a_0 f_0^2 + b_0 f_0 + c_0 = 0. \end{align} $$

It is a quadratic equation in the unknown variable $f_0$ , and we refer to its solutions as leading-order solutions of the Riccati equation. Generically, they are locally holomorphic, but may have poles and branch-point singularities.

The discriminant of (8),

(9) $$ \begin{align} {{D}}_0 {\mathrel{\mathop:}=} b_0^2 - 4 a_0 c_0, \end{align} $$

which we call the leading-order discriminant of the Riccati equation, is a holomorphic function on $\mathsf{X}$ . We always assume that ${ {D}}_0$ is not identically zero. The zeros of ${ {D}}_0$ are called turning points of the Riccati equation, and all other points in $\mathsf{X}$ are called regular points. Locally, away from turning points, there is at least one holomorphic leading-order solution. For reference, we state the following elementary lemma.

We will always label the leading-order solutions as follows:

(10a) $$ \begin{align} f_0^{\pm} {\mathrel{\mathop:}=} \frac{-b_0 \pm \sqrt{{{D}}_0}}{2 a_0} &\qquad\text{if } a_0 \not\equiv 0, \end{align} $$

(10b) $$ \begin{align} f_0^+ {\mathrel{\mathop:}=} - c_0 / b_0 &\qquad\text{if } a_0 \equiv 0.\\[-9pt]\nonumber\end{align} $$

This choice of labels yields the following relations:

(11) $$ \begin{align} \pm \sqrt{{{D}}_0} = 2a_0 f_0^{\pm} + b_0 \qquad \text{and} \qquad \sqrt{{{D}}_0} = a_0 \big( f_0^+ - f_0^- \big).\\[-15pt]\nonumber \end{align} $$

Thus, if $a_0$ is nonvanishing, then both $f_0^{\pm }$ from (10a) are holomorphic functions on $\mathsf{U}$ . If $a_0$ has zeros in $\mathsf{U}$ , then $f_0^+$ from (10a) remains holomorphic on $\mathsf{U}$ , but $f_0^-$ has poles where $a_0$ has zeros. If $a_0 \equiv 0$ , then $f_0^+$ from (10b) is a holomorphic function on $\mathsf{U}$ .

3.2 Existence and uniqueness of formal solutions

The following elementary theorem says that a formal Riccati equation (7) always has at least one local solution away from turning points, and it is uniquely specified in the leading-order.

Theorem 3.1 (Formal existence and uniqueness theorem).

Consider the formal Riccati equation (7). Assume that its leading-order discriminant ${ {D}}_0$ is not identically zero. Let $\mathsf{U} \subset \mathsf{X}$ be a domain free of turning points that supports a univalued square-root branch $\sqrt {{ {D}}_0}$ .

1. (1) If $a_0 \equiv 0$ , then (7) has a unique formal solution $\hat {f}_+$ on $\mathsf{U}$ . Its leading-order term is $f_0^+$ from (10b).

2. (2) If $a_0 \not \equiv 0$ and nonvanishing on $\mathsf{U}$ , then (7) has exactly two distinct formal solutions $\hat {f}_{\pm }$ on $\mathsf{U}$ . Their leading-order terms $f^{\pm }_0$ are given by (10a).

3. (3) If $a_0 \not \equiv 0$ but has zeros in $\mathsf{U}$ , then (7) has a unique formal solution $\hat {f}_+$ on $\mathsf{U}$ . Its leading-order term $f_0^+$ is the unique holomorphic leading-order solution on $\mathsf{U}$ given by (10a).

Moreover, the coefficients $f_k^{\pm }$ of $\hat {f}_{\pm }$ for $k \geq 1$ are given by the following recursive formula:

(12) $$ \begin{align} f_k^{\pm} = \pm \frac{1}{\sqrt{ {{D}}_0 }} \partial_x f_{k-1}^{\pm} \mp \frac{1}{\sqrt{ {{D}}_0 }} \left(\sum_{k_1 + k_2 + k_3 = k}^{k_2, k_3 \neq k} a_{k_1} f_{k_2}^{\pm} f_{k_3}^{\pm} + \sum_{k_1 + k_2 = k}^{k_2 \neq k} b_{k_1} f_{k_2}^{\pm} + c_k \right).\\[-9pt]\nonumber \end{align} $$

Proof. We expand the formal Riccati equation (7) order-by-order in $\hbar $ :

(13) $$ \begin{align} \hbar^0 ~\big|~ ~ & 0 = a_0 f_0^2 + b_0 f_0 + c_0;\phantom{+ b_1 f_0 + c_1;+ b_1 f_0 + c_1 a_2 f_0^2 . .b_1 f_1 + b_2 f_0 + c_2;} \end{align} $$

(14) $$ \begin{align} \hbar^1 ~\big|~ ~ &\partial_x f_0 = (2a_0 f_0 + b_0) f_1 + a_1 f_0^2 + b_1 f_0 + c_1;\phantom{+ a_2 f_0^2 . .b_1 f_1 + b_2 f_0 + c_2;} \end{align} $$

(15) $$ \begin{align} \hbar^2 ~\big|~ ~ &\partial_x f_1 = (2a_0 f_0 + b_0) f_2 + a_0 f_1^2 + 2 a_1 f_0 f_1 + a_2 f_0^2 + b_1 f_1 + b_2 f_0 + c_2; \end{align} $$

(16) $$ \begin{align} \nonumber \vdots \phantom{~~\big|~ ~}\nonumber \\ \hbar^k ~\big|~ ~ &\partial_x f_{k-1} = (2a_0 f_0 + b_0) f_k + \sum_{k_1 + k_2 + k_3 = k}^{k_2, k_3 \neq k} a_{k_1} f_{k_2} f_{k_3} + \sum_{k_1 + k_2 = k}^{k_2 \neq k} b_{k_1} f_{k_2} + c_k.\\ \nonumber \vdots \phantom{~~\big|~ ~} \end{align} $$

Observe that these are no longer differential equations because the derivative term at each order depends only on the solutions from previous orders. If we fix a leading-order solution $f_0^{\pm }$ , then the expression $(2 a_0 f_0^{\pm } + b_0)$ , appearing as a factor in front of $f^{\pm }_k$ in each equation (16), is simply $\pm \sqrt {{ {D}}_0}$ . From the assumption that ${ {D}}_0 \not \equiv 0$ , it follows that at each order in $\hbar $ , we can uniquely solve for $f_k^{\pm }$ . This establishes the formula (12), from which the other statements readily follow.

Remark 3.2 (Formal discriminant).

Since in the generic situation the Riccati equation has precisely two formal solutions $\hat {f}_+, \hat {f}_-$ , we can introduce a notion of discriminant for the Riccati equation analogous to the discriminant of a quadratic equation by simply mimicking the formula.

Thus, let $\mathsf{U} \subset \mathsf{X}$ be a domain free of turning points that supports a univalued square-root branch $\sqrt {{ {D}}_0}$ , and suppose that $a_0$ is nonvanishing on $\mathsf{U}$ . We define the formal discriminant of the Riccati equation (7) by the following formula:

(17) $$ \begin{align} \hat{{{D}}} {\mathrel{\mathop:}=} \hat{a}^2 \big( \hat{f}_+ - \hat{f}_- \big) \big( \hat{f}_- - \hat{f}_+ \big). \end{align} $$

It is a formal power series with holomorphic coefficients on $\mathsf{U}$ , and its leading-order term is precisely the leading-order discriminant ${ {D}}_0$ . This quantity plays an important role in addressing global questions in the WKB analysis that will be studied elsewhere.

3.3 Gevrey regularity of formal solutions

In this subsection, we prove the following general result about the regularity of formal solutions, which generalizes Proposition A.1.1 in [Reference Aoki, Kawai and Takei1, p. 19] (see also [Reference Voros20, p. 252]), where it is assumed that $\hat {a} = -1, \hat {b} = 0$ , and $\hat {c}$ is an entire holomorphic function of x only (i.e., $\hat {c} (x, \hbar ) = c_0 (x)$ ).

Concretely, Proposition 3.1 says that if the coefficients $a_k, b_k, c_k$ of the power series $\hat {a},\hat {b},\hat {c}$ grow at most like $k!$ , then the coefficients $f_k$ of any formal solution $\hat {f}$ likewise grow at most like $k!$ . This is made precise in the following corollary.

Corollary 3.1 (At most factorial growth).

Consider a formal Riccati equation (7) on $\mathsf{X}$ with leading-order discriminant ${ {D}}_0 \not \equiv 0$ . Let $\mathsf{U} \subset \mathsf{X}$ be any domain free of turning points that supports a univalued square-root branch $\sqrt {{ {D}}_0}$ , and let $\hat {f}$ be a formal solution on $\mathsf{U}$ . Take any pair of nested compactly contained subsets $\mathsf{U}_0 \Subset \mathsf{U}_1 \Subset \mathsf{U}$ , and suppose that there are real constants ${ {A}}, { {B}}> 0$ such that

(18) $$ \begin{align} \big| a_k (x) \big|,\big| b_k (x) \big|, \big| c_k (x) \big| \leq {{A}} {{B}}^{k} k! \qquad\quad (\forall k \geq 0, \forall x \in \mathsf{U}_1). \end{align} $$

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

(19) $$ \begin{align} \big| f_k (x) \big| \leq {{C}} {{M}}^{k} k! \qquad\qquad\qquad (\forall k \geq 0, \forall x \in \mathsf{U}_0). \end{align} $$

Proof of Proposition 3.1

Let ${\mathbb {D}}_{{ {R}}} \subset \mathsf{U}$ be any sufficiently small disk of some radius ${ {R}}> 0$ on which $\hat {a}, \hat {b}, \hat {c}$ are uniformly Gevrey and $\sqrt {{ {D}}_0}$ is bounded both above and below by a nonzero constant. Thus, there are real constants ${ {A}}, { {B}}> 0$ , which give the following uniform bounds:

(20) $$ \begin{align} \big| a_k (x) \big|, \big| b_k (x) \big|, \big| c_k (x) \big| \leq {{A}} {{B}}^{k} k! \qquad \text{and}\qquad {{A}}^{-1} \leq \big| \sqrt{{{D}}_0 (x)} \big| \leq {{A}} \end{align} $$

for all integers $k \geq 0$ and all $x \in {\mathbb {D}}_{{ {R}}}$ . It will be convenient for us to assume without loss of generality that ${ {A}} \geq 3$ and ${ {R}} < 1$ . We will prove that $\hat {f}$ is a uniformly Gevrey power series on any compactly contained subset of ${\mathbb {D}}_{{ {R}}}$ . In fact, we will prove something a little bit stronger as follows. For any $r \in (0, { {R}})$ , denote by ${\mathbb {D}}_r \subset {\mathbb {D}}_{{ {R}}}$ the concentric subdisk of radius r. Then Proposition 3.1 follows from the following claim.

Claim 3.1. There exist real constants ${ {C}}, { {M}}> 0$ such that, for any $r \in (0,{ {R}})$ ,

(21) $$ \begin{align} \big| f_k (x) \big| \leq {{C}} {{M}}^k \delta^{-k} k! \end{align} $$

for all integers $k \geq 0$ and uniformly for all $x \in {\mathbb {D}}_r$ , where $\delta {\mathrel {\mathop :}=} { {R}} - r$ . (The constants ${ {C}}, { {M}}$ are independent of $r, x, k$ , but may depend on ${ {R}}, { {A}}, { {B}}$ .) In particular, for any $r \in (0,{ {R}})$ , the power series $\hat {f}$ is Gevrey uniformly for all $x \in {\mathbb {D}}_r$ .

Proof. First, it is easy to find a constant ${ {C}}> 0$ (independent of r) such that

(22) $$ \begin{align} \big| f_0 (x) \big| \leq {{C}} \end{align} $$

uniformly for all $x \in {\mathbb {D}}_{{ {R}}}$ (see Lemma 3.1). Without loss of generality, assume that

(23) $$ \begin{align} {{C}} \geq {{A}} \geq 3. \end{align} $$

Then the bound (21) will be demonstrated in two main steps. First, we will recursively construct a sequence $({ {M}}_k)_{k=0}^{\infty }$ of positive real numbers such that, for all $k \geq 0$ and all $r \in (0,{ {R}})$ , we have the following uniform bound for all $x \in {\mathbb {D}}_r$ :

(24) $$ \begin{align} \big| f_k (x) \big| \leq {{C}} {{M}}_k \delta^{-k} k!. \end{align} $$

Then we will show that there is a constant ${ {M}}> 0$ (independent of r) such that ${ {M}}_k \leq { {M}}^k$ for all k.

Construction of $({ {M}}_k)_{k = 0}^{\infty }$ . The bound (24) for $k = 0$ is just the bound (22) if we put ${ {M}}_0 {\mathrel {\mathop :}=} 1$ . Now, we use induction on k and formula (12). Assume that we have already constructed positive real numbers ${ {M}}_0, \ldots , { {M}}_{k-1}$ such that, for all $i = 0, \ldots , k-1$ , all $r \in (0,{ {R}})$ , and all $x \in {\mathbb {D}}_r$ , we have the bound

(25) $$ \begin{align} \big| f_i (x) \big| \leq {{C}} {{M}}_i \delta^{-i} i!. \end{align} $$

In order to derive an estimate for $f_k$ , we first need to estimate the derivative term $\partial _x f_{k-1}$ , for which we use Cauchy estimates as follows.

Sub-Claim. For all $r \in (0,{ {R}})$ and all $x \in {\mathbb {D}}_r$ ,

(26) $$ \begin{align} \big| \partial_x f_{k-1} (x) \big| \leq {{C}}^2 {{M}}_{k-1} \delta^{-k} k!. \end{align} $$

Proof of Sub-Claim

For every $r \in (0,{ {R}})$ , define

$$ \begin{align*} \delta_k {\mathrel{\mathop:}=} \delta \frac{k}{k+1} \qquad\text{and}\qquad r_k {\mathrel{\mathop:}=} {{R}} - \delta_k. \end{align*} $$

Inequality (25) holds in particular for $i = k-1$ , $r = r_k$ . So for all $x \in {\mathbb {D}}_{r_k}$ , we find

$$ \begin{align*} \big| f_{k-1} (x) \big|\leq {{C}} {{M}}_{k-1} \delta_k^{1-k} (k-1)! \leq {{C}}^2 {{M}}_{k-1} \delta^{-k} k! \tfrac{\delta}{k+1}. \end{align*} $$

Here, we have used the estimate $( 1 + 1/k )^{k-1} \leq e \leq { {C}}$ . Finally, notice that for every $x \in {\mathbb {D}}_r$ , the closed disk around x of radius $r_k - r = \delta - \delta _k = \frac {\delta }{k+1}$ is contained inside the disk ${\mathbb {D}}_{r_k}$ . Therefore, Cauchy estimates imply (26).

Using (20), (23), (25), (26), and the fact that $\delta < 1$ , we can now estimate $f_k$ :

$$ \begin{align*} \big| f_k \big| &\leq {{C}} \left(\big| \partial_x f_{k-1} \big| + \sum_{k_1 + k_2 + k_3 = k}^{k_2, k_3 \neq k} \big| a_{k_1} \big| \cdot \big| f_{k_2} \big| \cdot \big| f_{k_3} \big| + \sum_{k_1 + k_2 = k}^{k_2 \neq k} \big| b_{k_1} \big| \cdot \big| f_{k_2} \big| + \big| c_k \big|\right)\\ &\leq {{C}}\left({{C}}^2 {{M}}_{k-1} \delta^{-k} k! + \delta^{-k} {{C}}^3 k! \sum_{k_1 + k_2 + k_3 = k}^{k_2, k_3 \neq k} {{B}}^{k_1} {{M}}_{k_2} {{M}}_{k_3} \right.\\ &\hspace{3.65cm}\left.+ \delta^{-k} {{C}}^2 k! \sum_{k_1 + k_2 = k}^{k_2 \neq k} {{B}}^{k_1} {{M}}_{k_2} + {{C}} {{B}}^k k! \right)\\ &\leq {{C}}^4 \left({{M}}_{k-1} + \!\!\!\sum_{k_1 + k_2 + k_3 = k}^{k_2, k_3 \neq k} {{B}}^{k_1}{{M}}_{k_2} {{M}}_{k_3} + \sum_{k_1 + k_2 = k}^{k_2 \neq k} {{B}}^{k_1} {{M}}_{k_2} + {{B}}^k\right) \delta^{-k} k!. \end{align*} $$

We can therefore define, for $k \geq 1$ ,

(27) $$ \begin{align} {{M}}_k {\mathrel{\mathop:}=} {{C}}^3 \left( {{M}}_{k-1} + \!\!\!\! \sum_{k_1 + k_2 + k_3 = k}^{k_2, k_3 \neq k} {{B}}^{k_1} {{M}}_{k_2} {{M}}_{k_3} + \sum_{k_1 + k_2 = k}^{k_2 \neq k} {{B}}^{k_1} {{M}}_{k_2} + {{B}}^k \right). \end{align} $$

Construction of ${ {M}}$ . To see that ${ {M}}_k \leq { {M}}^k$ for some ${ {M}}> 0$ , we argue as follows. Consider the following power series in an abstract variable t:

$$ \begin{align*} \hat{p} (t) {\mathrel{\mathop:}=} \sum_{k=0}^{\infty} {{M}}_k t^k \qquad\text{and}\qquad q (t) {\mathrel{\mathop:}=} \sum_{k=1}^{\infty} {{B}}^k t^k \quad \in \quad {\mathbb{C}} {\left[\![ t \right]\!]}. \end{align*} $$

Note that $\hat {p} (0) = { {M}}_0 = 1$ and $q (0) = 0$ , and notice that $q (t)$ is convergent. We will show that $\hat {p} (t)$ is also convergent. The key is the observation that they satisfy the following algebraic equation:

(28) $$ \begin{align} \hat{p} (t) - 1 = {{C}}^3 \Big( t \hat{p} (t) + q(t) \hat{p}(t)^2 + \big(\hat{p} (t) - 1\big)^2 + q(t) \hat{p}(t) + q(t) \Big). \end{align} $$

This equation was found by trial and error, and it is straightforward to verify directly by substituting the power series $\hat {p}(t), q(t)$ and comparing the coefficients of $t^k$ using the defining formula (27) for ${ {M}}_k$ . Namely, the terms ${ {M}}_{k-1}$ and ${ {B}}^k$ in (27) correspond, respectively, to the terms $t \hat {p} (t)$ and $q(t)$ in (28), whereas the first and second sums correspond, respectively, to $q(t) \hat {p}(t)^2 + \big (\hat {p} (t) - 1\big )^2$ and $q(t) \hat {p}(t)$ .

Now, consider the following holomorphic function in two complex variables $(t,p)$ :

$$ \begin{align*} {{F}} (t,p) {\mathrel{\mathop:}=} -p + 1 + {{C}}^3 \Big( t p + q(t) p^2 + (p-1)^2 + q(t) p + q(t) \Big). \end{align*} $$

It has the following properties:

$$ \begin{align*} {{F}} (0, 1) = 0 \qquad\text{and}\qquad \left. {\frac{\partial {{F}}}{\partial p} }\right|_{ (t,p) = (0,1) } = -1 \neq 0. \end{align*} $$

By the Holomorphic Implicit Function Theorem, there exists a unique holomorphic function $p (t)$ near $t = 0$ such that $p (0) = 1$ and ${ {F}} \big (t, p (t)\big ) \big ) = 0$ . Thus, $\hat {p}(t)$ must be its convergent Taylor series expansion at $t = 0$ and its coefficients grow at most exponentially: that is, there is a constant ${ {M}}> 0$ such that ${ {M}}_k \leq { {M}}^k$ . This completes the proof of the Claim and hence of Proposition 3.1.

4.1 The Liouville transformation

Throughout this section, we remain in the background setting of §2.1. Recall the leading-order discriminant ${ {D}}_0 = b_0^2 - 4 a_0 c_0$ , which is a holomorphic function on $\mathsf{X}$ , assumed not identically zero. Fix a phase $\theta \in {\mathbb {R}} / 2\pi {\mathbb {Z}}$ , a basepoint $x_0 \in \mathsf{X}$ , and a univalued square-root branch $\sqrt {{ {D}}_0}$ near $x_0$ (i.e., either in a disk or a sectorial neighborhood of $x_0$ ). Consider the following local coordinate transformation near $x_0$ , called the Liouville transformation:

(29) $$ \begin{align} z = \Phi (x) {\mathrel{\mathop:}=} \int_{x_0}^x \sqrt{ {{D}}_0 (t) } {\mathrm{d}}{t}. \end{align} $$

Let $\mathsf{U} \subset \mathsf{X}$ be any domain which is free of turning points, supports a univalued square-root branch $\sqrt {{ {D}}_0}$ (e.g., $\mathsf{U}$ is simply connected), and contains $x_0$ in the interior or on the boundary. Then the Liouville transformation defines a (possibly multivalued) local biholomorphism $\Phi : \mathsf{U} {\longrightarrow } {\mathbb {C}}_z$ . Notice that turning points are precisely the locations in $\mathsf{X}$ where $\Phi $ fails to be conformal.

Remark 4.1. The basepoint of integration $x_0$ can in principle be chosen even on the boundary of $\mathsf{X}$ or at infinity in ${\mathbb {C}}_x$ provided that the integral is well defined. Liouville transformations such as (29) are encountered in the analysis of the Schrödinger equation $\hbar ^2 \partial _x^2 \psi - q (x) \psi = 0$ as described, for example, in Olver’s textbook [Reference Olver14, §6.1]. However, note that our formula (29) in the special case of the Schrödinger equation reads

(30) $$ \begin{align} \Phi (x) = \int\nolimits_{x_0}^x \sqrt{{{D}}_0 (t)} {\mathrm{d}}{t} = 2 \int\nolimits_{x_0}^x \sqrt{q (t)} {\mathrm{d}}{t}, \end{align} $$

which differs from formula (1.05) in [Reference Olver14, §6.1] by a factor of $2$ .

4.2 WKB trajectories

Let $x_0 \in \mathsf{X}$ be a regular point, and consider the Liouville transformation (29). A WKB $\theta $ -trajectory through $x_0$ is the real one-dimensional smooth curve $\Gamma _{\theta }$ on $\mathsf{X}$ locally determined by the following equation:

(31) $$ \begin{align} \Gamma_{\theta} = \Gamma_{\theta} (x_0) \quad:\quad {\operatorname{Im}} \big( e^{-i\theta} \Phi (x) \big) = 0. \end{align} $$

A WKB $\theta $ -trajectory $\pm $ -ray (or simply a WKB ray if the context is clear) emanating from $x_0$ is the component $\Gamma ^{\pm }_{\theta }$ of $\Gamma _{\theta }$ given, respectively, by

(32) $$ \begin{align} \Gamma^{\pm}_{\theta} = \Gamma_{\theta}^{\pm} (x_0) \quad:\quad \pm {\operatorname{Re}} \big( e^{-i\theta} \Phi (x) \big) \geq 0; \end{align} $$

WKB trajectories are regarded by definition as being maximal under inclusion. Explicitly, (31) and (32) read

(33) $$ \begin{align} {\operatorname{Im}} \left( \: e^{-i\theta} \int\nolimits_{x_0}^x \sqrt{{{D}}_0 (t)} {\mathrm{d}}{t} \right) = 0 \quad \text{and}\quad\pm {\operatorname{Re}} \left( \: e^{-i\theta} \int\nolimits_{x_0}^x \sqrt{{{D}}_0 (t)} {\mathrm{d}}{t} \right) \geq 0. \end{align} $$

The Liouville transformation $\Phi $ with basepoint $x_0$ maps the WKB $\theta $ -trajectory $\Gamma _{\theta } (x_0)$ to a possibly infinite straight line segment $(\tau _{-} e^{i\theta }, \tau _{+} e^{i\theta }) \subset e^{i\theta } {\mathbb {R}} \subset {\mathbb {C}}_z$ containing the origin $0 = \Phi (x_0)$ , that is, with $\tau _- < 0 < \tau _+$ . Maximality means that this line segment is the largest possible image. The image of the WKB $\theta $ -trajectory $\pm $ -ray emanating from $x_0$ is then, respectively, the line segment $[0, \tau _{+}e^{i\theta })$ or $(\tau _-e^{i\theta }, 0]$ .

All other nearby WKB $\theta $ -trajectories 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 $\mathsf{U}_0 \subset \mathsf{X}$ is a simply connected neighborhood of $x_0$ free of turning points, then any WKB $\theta $ -trajectory $\Gamma ^{\prime }_{\theta }$ intersecting $\mathsf{U}_0$ is locally given by this equation with $c = {\operatorname {Im}} e^{-i\theta } \Phi (x^{\prime }_0)$ for some $x^{\prime }_0 \in \mathsf{U}_0$ . Its image in ${\mathbb {C}}_z$ under $\Phi $ is an interval on the parallel line containing $z^{\prime }_0 {\mathrel {\mathop :}=} \Phi (x^{\prime }_0)$ :

$$ \begin{align*} z^{\prime}_0 + e^{i\theta} {\mathbb{R}} {\mathrel{\mathop:}=} {\left\{ z = z^{\prime}_0 + \xi ~\big|~ \xi \in e^{i\theta}{\mathbb{R}} \right\}}. \end{align*} $$

Our primary focus is infinite WKB rays $\Gamma _{\theta }^{\pm }$ , defined as having $|\tau _{\pm }| = \infty $ , respectively. An infinite WKB trajectory is one with at least one infinite ray. A generic WKB trajectory is one with both rays being infinite.

An infinite WKB trajectory may be a closed WKB trajectory if it is a simple closed curve in the complement of the turning points. A closed WKB $\theta $ -trajectory has the property that there is a nonzero time $\omega \in {\mathbb {R}}$ such that $\Phi ^{-1} (e^{i\theta }\omega ) = \Phi ^{-1} (0)$ (see [Reference Strebel18, §9.2]). This only happens when the Liouville transformation is analytically continued along the trajectory to a multivalued function. We refer to the smallest possible positive such $\omega \in {\mathbb {R}}_+$ as the WKB trajectory period. It follows from general considerations (see [Reference Strebel18, §9]) that if the WKB $\theta $ -trajectory through $x_0$ is a closed trajectory, then all nearby WKB $\theta $ -trajectories are also closed with the same period.

A nonclosed infinite WKB ray may tend to a single point, limit to a dense subset of $\mathsf{X}$ , or escape $\mathsf{X}$ altogether. Formally, the limit of an infinite WKB ray $\Gamma _{\theta }^{\pm }$ by definition, respectively, is the limit set

$$ \begin{align*} \overline{\Phi^{-1} \Big( [\tau e^{i\theta},+\infty \cdot e^{i\theta}) \Big)} \quad\text{as}\quad \tau \to +\infty \quad\text{or}\quad \overline{\Phi^{-1} \Big( (-\infty \cdot e^{i\theta},\tau e^{i\theta}] \Big)} \quad\text{as}\quad \tau \to -\infty. \end{align*} $$

Obviously, this definition is independent of the chosen basepoint $x_0$ along the trajectory. If the limit is a single point $x_{\infty } \in {\mathbb {C}}_x$ , then this point (sometimes called an infinite critical point) is necessarily a pole of ${ {D}}_0$ of order $m \geq 2$ (see [Reference Strebel18, §10.2]). Given $\alpha \in {\left \{ +, - \right \}}$ , it also follows from general considerations that if the WKB $\theta $ -trajectory $\alpha $ -ray emanating from $x_0$ tends to an infinite critical point, then $x_0$ has a disk neighborhood $\mathsf{U}_0$ such that every WKB $\theta $ -trajectory $\alpha $ -ray emanating from $\mathsf{U}_0$ tends to the same infinite critical point.

Finite WKB rays—those with finite $\tau _+$ or $\tau _-$ —are inadmissible for our construction of exact solutions in §5. As $\tau $ approaches $\tau _{+}$ or $\tau _{-}$ , respectively, such a WKB trajectory either tends to a turning point or escapes to the boundary of $\mathsf{X}$ in finite time. If it tends to a single point on the boundary of $\mathsf{X}$ , this point is either a turning point or a simple pole of the discriminant ${ {D}}_0$ (see [Reference Strebel18, §10.2]). For this reason, turning points and simple poles are sometimes collectively referred to as finite critical points. A singular WKB ray is one that approaches a finite critical point. They are important in the global analysis of exact solutions, which will be discussed in detail elsewhere.

A WKB $\theta $ -strip domain containing $x_0$ is any domain neighborhood of $x_0$ which is swept out by generic WKB $\theta $ -trajectories. It necessarily has the form

(34) $$ \begin{align} \begin{aligned} \mathsf{W}_{\theta} &= \mathsf{W}_{\theta} (x_1, r) {\mathrel{\mathop:}=} \Phi^{-1} \big( \mathsf{H}_{\theta} \big) \subset {\mathbb{C}}_x, \\ \text{where}\quad \mathsf{H}_{\theta} &= \mathsf{H}_{\theta} (z_1, r) {\mathrel{\mathop:}=} {\left\{z ~\Big|~ {\mathop{\mathrm{dist}}\nolimits} (z, z_1 + e^{i\theta} {\mathbb{R}}) < r \right\}} \subset {\mathbb{C}}_z, \end{aligned} \end{align} $$

for some $r> 0$ and some $x_1 = \Phi ^{-1} (z_1) \in X$ . Similarly, a WKB $(\theta ,\pm )$ -half-strip domain containing $x_0$ is any domain neighborhood of $x_0$ which is swept out by infinite WKB $\theta $ -trajectory $\pm $ -rays. It necessarily has the form

(35) $$ \begin{align} \begin{aligned} \mathsf{W}^{\pm}_{\theta} &= \mathsf{W}^{\pm}_{\theta} (x_1, r) {\mathrel{\mathop:}=} \Phi^{-1} \big( \mathsf{H}^{\pm}_{\theta} \big) \subset {\mathbb{C}}_x, \\ \text{where}\quad \mathsf{H}^{\pm}_{\theta} &= \mathsf{H}^{\pm}_{\theta} (z_1, r) {\mathrel{\mathop:}=} {\left\{ z ~\Big|~ {\mathop{\mathrm{dist}}\nolimits} (z, z_1 + e^{i\theta} {\mathbb{R}}_{\pm}) < r \right\}} \subset {\mathbb{C}}_z. \end{aligned} \end{align} $$

Note that we obviously have $\mathsf{W}_{\theta } = \mathsf{W}_{\theta }^- \cup \mathsf{W}_{\theta }^+$ . The intersection $\mathsf{W}_{\theta }^- \cap \mathsf{W}_{\theta }^+ = \Phi ^{-1} \big (\! {\left \{ |z {\,-\,} z_1| {\,<\,} r \right \}}\! \big )$ may be called a WKB disk around $x_1$ , and it is clearly independent of $\theta $ .

If the WKB $\theta $ -trajectory through $x_0$ is not closed, then WKB $\theta $ -strip $\mathsf{W}_{\theta }$ is a simply connected domain conformally equivalent to the infinite strip $\mathsf{H}_{\theta }$ via the Liouville transformation .

On the other hand, if the WKB $\theta $ -trajectory through $x_0$ is closed, then $\mathsf{W}_{\theta }$ is swept out by closed WKB $\theta $ -trajectories, so $\mathsf{W}_{\theta }$ has the topology of an annulus. In this case, $\mathsf{W}_{\theta }$ is sometimes called a WKB $\theta $ -ring domain. The Liouville transformation $\Phi $ is a multivalued holomorphic function on $\mathsf{W}_{\theta }$ , but the inverse $\Phi ^{-1} : \mathsf{H}_{\theta } \to \mathsf{W}_{\theta }$ is still necessarily a local biholomorphism.

5.1 Existence and uniqueness of local exact solutions

The main result of this paper is the following theorem.

Theorem 5.1 (Main exact existence and uniqueness theorem).

Consider the Riccati equation

(36) $$ \begin{align} \hbar \partial_x f = a f^2 + b f + c \end{align} $$

whose coefficients $a,b,c$ are holomorphic functions of $(x,\hbar ) \in \mathsf{X} \times \mathsf{S}$ admitting locally uniform asymptotic expansions $\hat {a}, \hat {b}, \hat {c}$ as $\hbar \to 0$ along $\mathsf{A}$ . Assume that ${ {D}}_0 = b_0^2 - 4 a_0 c_0 \not \equiv 0$ . Fix a regular point $x_0 \in \mathsf{X}$ , a square-root branch $\sqrt {{ {D}}_0}$ near $x_0$ , and a sign $\alpha \in {\left \{ +, - \right \}}$ . In addition, we assume the following hypotheses:

1. (1) There is a WKB $(\theta , \alpha )$ -half-strip domain $\mathsf{W} = \mathsf{W}_{\theta }^{\alpha } \subset \mathsf{X}$ containing $x_0$ ; assume in addition that $a_0$ is nonvanishing on $\mathsf{W}$ if $\alpha = -$ .

2. (2) The asymptotic expansions of the coefficients $a,b,c$ are valid with Gevrey bounds as $\hbar \to 0$ along the closed arc $\bar {\mathsf{A}}_{\theta } = [\theta - \tfrac {\pi }{2}, \theta + \tfrac {\pi }{2}]$ , with respect to the asymptotic scale $\sqrt {{ {D}}_0}$ , uniformly for all $x \in \mathsf{W}$ :

   (37) $$ \begin{align} a \simeq \hat{a}, \quad b \simeq \hat{b}, \quad c \simeq \hat{c} \qquad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ wrt } \sqrt{{{D}}_0}, \text{ unif. } \forall x \in \mathsf{W}. \end{align} $$

Then the Riccati equation has a canonical local exact solution $f^{\theta }_{\alpha }$ near $x_0$ which is asymptotic to the formal solution $\hat {f}_{\alpha }$ as $\hbar \to 0$ in the direction $\theta $ . Namely, for any compactly contained domain $\mathsf{U}_0 \Subset \mathsf{W}$ , there is a sectorial domain $\mathsf{S}_0 \subset \mathsf{S}$ with the same opening $\mathsf{A}_{\theta }$ such that the Riccati equation has a unique holomorphic solution $f^{\theta }_{\alpha }$ on $\mathsf{U}_0 \times \mathsf{S}_0$ which is Gevrey asymptotic to $\hat {f}_{\alpha }$ as $\hbar \to 0$ along the closed arc $\bar {\mathsf{A}}_{\theta }$ uniformly for all $x \in \mathsf{U}_0$ :

(38) $$ \begin{align} f^{\theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ unif. } \forall x \in \mathsf{U}_0. \end{align} $$

We will prove this theorem in §5.1.2. First, let us make some remarks.

Remark 5.1. (1) For the reader’s convenience, we recall here that hypothesis (2) in Theorem 5.1 explicitly means that there are real constants ${ {A}}, { {B}}> 0$ such that for all $n \geq 0$ , all $x \in \mathsf{W}$ , and all sufficiently small $\hbar \in \mathsf{S}$ ,

(39) $$ \begin{align} \left| a (x, \hbar) - \sum_{k=0}^{n-1} a_k (x) \hbar^k \right| \leq \Big| \sqrt{{{D}}_0 (x)} \Big| {{A}} {{B}}^n n! |\hbar|^n, \end{align} $$

and similarly for b and c. Likewise, the asymptotic condition (38) reads explicitly as follows: there are real constants ${ {C}}, { {M}}> 0$ such that for all $n \geq 0$ , all $x \in \mathsf{U}_0$ , and all sufficiently small $\hbar \in \mathsf{S}_0$ ,

(40) $$ \begin{align} \left| f_{\alpha}^{\theta} (x, \hbar) - \sum_{k=0}^{n-1} f^{\alpha}_k (x) \hbar^k \right| \leq {{C}} {{M}}^n n! |\hbar|^n. \end{align} $$

(2) If all the hypotheses of Theorem 5.1 are satisfied for both signs $\alpha = +, -$ , then we obviously obtain a pair of distinct canonical exact solutions $f^{\theta }_+, f^{\theta }_-$ near $x_0$ . Let us also note that the solution $f_{\alpha }^{\theta }$ obviously does not depend in any serious way on the chosen basepoint $x_0$ .

(3) In many applications, including the exact WKB analysis of Schrödinger, the coefficients of the Riccati equation (36) do not satisfy hypothesis (2) in Theorem 5.1 on the nose and we have to do an additional transformation in order to apply our theorem. This is discussed in §5.4.

(4) The somewhat abstract general hypotheses of this theorem get significantly simplified in many notable situations which are discussed in §6.

(5) The sectorial domain $\mathsf{S}_0$ in the conclusion of Theorem 5.1 can be chosen to be a Borel disK ${\left \{ \hbar ~\big |~ {\operatorname {Re}} (e^{i\theta }/\hbar )> 1/d_0 \right \}}$ bisected by the direction $\theta $ of sufficiently small diameter $d_0> 0$ .

We have the following immediate corollary of the uniqueness property of canonical exact solutions.

Corollary 5.1 (Extension to larger domains).

Let $\mathsf{U} \subset \mathsf{X}$ be a domain free of turning points that supports a univalued square-root branch $\sqrt {{ {D}}_0}$ . Fix a sign $\alpha \in {\left \{ +, - \right \}}$ such that the leading-order solution $f_0^{\alpha }$ is holomorphic on $\mathsf{U}$ . In addition, assume that hypotheses (1)–(3) in Theorem 5.1 are satisfied for every point $x_0 \in \mathsf{U}$ .

Then the Riccati equation (36) has a canonical exact solution $f^{\theta }_{\alpha }$ on $\mathsf{U}$ asymptotic to the formal solution $\hat {f}_{\alpha }$ as $\hbar \to 0$ in the direction $\theta $ . Namely, there is a domain $\mathbb {U} \subset \mathsf{U} \times \mathsf{S}$ and a holomorphic solution $f^{\theta }_{\alpha }$ defined on $\mathbb {U}$ with the following property: for every point $x_0 \in \mathsf{U}$ , there is a neighborhood $\mathsf{U}_0 \subset \mathsf{U}$ of $x_0$ and a sectorial domain $\mathsf{S}_0 \subset \mathsf{S}$ with the same opening $\mathsf{A}_{\theta }$ such that $\mathsf{U}_0 \times \mathsf{S}_0 \subset \mathbb {U}$ and $f_{\alpha }^{\theta }$ is the unique holomorphic solution on $\mathsf{U}_0 \times \mathsf{S}_0$ satisfying (38). In particular, $f_{\alpha }^{\theta }$ is the unique solution on ${\mathbb {U}}$ with the following locally uniform Gevrey asymptotics:

(41) $$ \begin{align} f^{\theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ loc.unif. } \forall x \in \mathsf{U}. \end{align} $$

In particular, the domain $\mathsf{U} \subset \mathsf{X}$ in Corollary 5.1 can be a union of WKB half-strips. In fact, examining the proof of Theorem 5.1 more closely, it is readily seen that on any WKB half-strip, we can state the asymptotic property of canonical exact solutions more precisely as follows.

Proposition 5.1 (Asymptotics on WKB half-strips).

Assume all the hypotheses of Theorem 5.1. Then the canonical local exact solution $f_{\alpha }^{\theta }$ uniquely extends to an exact solution on $\mathsf{W}$ with the following locally uniform Gevrey asymptotics:

(42) $$ \begin{align} f^{\theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ loc.unif. } \forall x \in \mathsf{W}. \end{align} $$

In fact, even more is true. Let $r> 0$ be such that $\mathsf{W} = \mathsf{W}_{\theta }^{\alpha } (x^{\prime }_0, r)$ . For any $r_0 \in (0, r)$ , let $\mathsf{W}_0 {\mathrel {\mathop :}=} \mathsf{W}_{\theta }^{\alpha } (x^{\prime }_0, r_0)$ . Then there is a sectorial domain $\mathsf{S}_0 \subset \mathsf{S}$ with the same opening $\mathsf{A}_{\theta }$ such that the canonical exact solution $f_{\alpha }^{\theta }$ extends to a holomorphic solution on $\mathsf{W}_0 \times \mathsf{S}_0$ with the following uniform Gevrey asymptotic property:

(43) $$ \begin{align} f^{\theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ unif. } \forall x \in \mathsf{W}_0. \end{align} $$

5.1.1 Riccati equations on horizontal half-strips

The strategy of the proof of Theorem 5.1 is to use the Liouville transformation $\Phi _{\alpha }$ to transform the Riccati equation into one in standard form over a horizontal half-strip in the z-space, and then apply the Borel–Laplace method. First, we give a general description of this standard form of the Riccati equation and prove the corresponding version of the exact existence and uniqueness theorem (Lemma 5.1). Then we will show that any Riccati equation satisfying the assumptions of Theorem 5.1 can be put into this standard form, thereby deducing our claims.

Let $\mathsf{H}_+ \subset {\mathbb {C}}_z$ be a horizontal half-strip around the positive real axis ${\mathbb {R}}_+ \subset {\mathbb {C}}_z$ of some radius $r> 0$ , and let $\mathsf{S}_+ \subset {\mathbb {C}}_{\hbar }$ a Borel disk bisected by the positive real axis of some diameter $d> 0$ :

(44) $$ \begin{align} \mathsf{H}_+ {\mathrel{\mathop:}=} {\left\{ z ~\big|~ {\mathop{\mathrm{dist}}\nolimits} (z, {\mathbb{R}}_+) < r \right\}} \quad\text{and}\quad \mathsf{S}_+ {\mathrel{\mathop:}=} {\left\{ \hbar ~\big|~ {\operatorname{Re}} (1/\hbar)> 1/d \right\}}. \end{align} $$

The opening of $\mathsf{S}_+$ is the semicircular arc $\mathsf{A}_+ {\mathrel {\mathop :}=} (-\frac {\pi }{2},+\frac {\pi }{2})$ . Consider the following singularly perturbed Riccati equation on $\mathsf{H}_+ \times \mathsf{S}_+$ :

(45) $$ \begin{align} \hbar \partial_z {{F}} = {{F}} + \hbar \big( {{A}}_2 {{F}}^2 + {{A}}_1 {{F}} + {{A}}_0 \big), \end{align} $$

where ${ {A}}_i$ are holomorphic functions of $(z, \hbar ) \in \mathsf{H}_+ \times \mathsf{S}_+$ which admit uniform Gevrey asymptotic expansions as $\hbar \to 0$ along the closed arc $\bar {\mathsf{A}}_+$ :

(46) $$ \begin{align} {{A}}_i \simeq \hat{{{A}}}_i, \qquad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_+, \text{ unif. } \forall z \in \mathsf{H}_+. \end{align} $$

Denote their leading-order parts by $a_i = a_i (z)$ . The corresponding leading-order equation is simply ${ {F}}_0 = 0$ . By Theorem 3.1(1), this Riccati equation has a unique formal solution $\hat {{ {F}}}_+$ on $\mathsf{H}_+$ , and its leading order part is ${ {F}}_0^+ = 0$ and its next-to-leading order part is ${ {F}}_1^+ = - a_0$ .

Lemma 5.1 (Main Lemma).

For every $r_0 \in (0, r)$ , there is $d_0 \in (0, d]$ such that the Riccati equation (45) has a canonical exact solution ${ {F}}_+$ defined on

(47) $$ \begin{align} \mathsf{H}^+_0 \times \mathsf{S}^+_0 {\mathrel{\mathop:}=} {\left\{ z ~\big|~ {\mathop{\mathrm{dist}}\nolimits} (z, {\mathbb{R}}_+) < r_0 \right\}} \times {\left\{ \hbar ~\big|~ {\operatorname{Re}} (1/\hbar)> 1/d_0 \right\}} \subset \mathsf{H}_+ \times \mathsf{S}_+.\end{align} $$

Namely, ${ {F}}_+$ is the unique holomorphic solution on $\mathsf{H}^+_0 \times \mathsf{S}^+_0$ , which admits the formal solution $\hat {{ {F}}}_+$ as its uniform Gevrey asymptotic expansion along $\bar {\mathsf{A}}_+$ :

(48) $$ \begin{align} {{F}}_+ (z, \hbar) \simeq \hat{{{F}}}_+ (z, \hbar) \quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_+, \text{ unif. } \forall z \in \mathsf{H}^+_0. \end{align} $$

Moreover, it has the following properties.

1. (P1) The formal Borel transform

   $$ \begin{align*} \hat{\varphi}_+ (x, \xi) = \hat{{\mathfrak{B}}} [ \, \hat{{{F}}}_+ \, ] (x, \xi) {\mathrel{\mathop:}=} \sum_{n=0}^{\infty} \varphi_n^+ (x) \xi^n \quad\text{where}\quad \varphi_k^+ (x) {\mathrel{\mathop:}=} \tfrac{1}{k!} {{F}}^+_{k+1} (x) \end{align*} $$

   converges uniformly on $\mathsf{H}^+_0$ .

2. (P2) For any $\epsilon \in (0, r - r_0)$ , let $\Xi _+ {\mathrel {\mathop :}=} {\left \{ \xi ~\big |~ {\mathop {\mathrm {dist}}\nolimits } (\xi , {\mathbb {R}}_+) < \epsilon \right \}}$ . Then the analytic Borel transform

   $$ \begin{align*} \varphi_+ (z, \xi) = {\mathfrak{B}}_+ [\, {{F}}_+ \,] (z, \xi){\mathrel{\mathop:}=} \frac{1}{2\pi i} \oint\nolimits {{F}}_+(x, \hbar) e^{\xi / \hbar} \frac{{\mathrm{d}}{\hbar}}{\hbar^2} \end{align*} $$

   is uniformly convergent for all $(z, \xi ) \in \mathsf{H}^+_0 \times \Xi _+$ . Here, the integral “ $\oint $ ” is done along the boundary of any Borel disk $\mathsf{S}^{\prime }_+ \subsetneq \mathsf{S}_+$ of strictly smaller diameter $d' < d$ , traversed anticlockwise.

3. (P3) The Laplace transform $\mathfrak {L}_+ \big [ \, \varphi _+ \, \big ] (z, \hbar )$ is uniformly convergent for all $(z, \hbar ) \in \mathsf{H}^+_0 \times \mathsf{S}^+_0$ and satisfies ${ {F}}_+ (z, \hbar ) = \mathfrak {L}_+ \big [ \, \varphi _+ \, \big ] (z, \hbar )$ .

4. (P4) Therefore, ${ {F}}_+$ is the uniform Borel resummation of its asymptotic power series $\hat {{ {F}}}_+$ ; that is, ${ {F}}_+ (z, \hbar ) = \mathcal {S}_+ \big [ \, \hat {{ {F}}}_+ \, \big ] (z, \hbar )$ for all $(z, \hbar ) \in \mathsf{H}^+_0 \times \mathsf{S}^+_0$ .

5. (P5) If the coefficients ${ {A}}_0, { {A}}_1, { {A}}_2$ are periodic in z with period $\omega \in {\mathbb {C}}$ , then so is ${ {F}}_+$ .

Proof. We use the Borel–Laplace method to construct the exact solution ${ {F}}_+$ . Namely, we first apply the Borel transform to obtain a first-order nonlinear PDE which is easy to rewrite as an integral equation. Then most of the heavy lifting is constrained to solving this integral equation, which we do via the method of successive approximations. The desired solution ${ {F}}_+$ is then obtained by applying the Laplace transform.

Uniqueness.

Suppose ${ {F}}_+,{ {F}}^{\prime }_+$ are two such exact solutions defined on $\mathsf{H}^+_0 \times \mathsf{S}^+_0$ . Their difference ${ {F}}_+ - { {F}}^{\prime }_+$ is a holomorphic function on $\mathsf{H}^+_0 \times \mathsf{S}^+_0$ which is uniformly Gevrey asymptotic to $0$ as $\hbar \to 0$ along $\bar {\mathsf{A}}_+$ . By Nevanlinna’s theorem [Reference Loday-Richaud9, Theorem 5.3.9], there can be only one holomorphic function on $\mathsf{S}^+_0$ (namely, the constant function $0$ ), which is Gevrey asymptotic to $0$ as $\hbar \to 0$ along $\bar {\mathsf{A}}_+$ . Thus, ${ {F}}_+ - { {F}}^{\prime }_+$ is must be the zero function.

Step 1: The analytic Borel transform.

It follows from Nevanlinna’s theorem [Reference Loday-Richaud9, Theorem 5.3.9] that there is some tubular neighborhood $\Xi _+ {\mathrel {\mathop :}=} {\left \{ \xi \big | {\mathop {\mathrm {dist}}\nolimits } (\xi , {\mathbb {R}}_+) < \epsilon \right \}}$ for some $\epsilon> 0$ such that the analytic Borel transforms $\alpha _i (z, \xi ) {\mathrel {\mathop :}=} {\mathfrak {B}}_+ [ \: { {A}}_i \: ] (z, \xi )$ are holomorphic functions on $\mathsf{H}_+ \times \Xi _+ \subset {\mathbb {C}}^2_{z\xi }$ with uniformly at most exponential growth as $|\xi | \to + \infty $ . Moreover, ${ {A}}_i (z, \hbar ) = a_i (z) + \mathfrak {L}_+ [ \: \alpha _i \: ]$ for all $(z,\xi ) \in \mathsf{H}_+ \times \Xi _+$ .

Dividing (45) through by $\hbar $ and applying the analytic Borel transform ${\mathfrak {B}}_+$ , we obtain the following PDE with convolution product:

(49) $$ \begin{align} \partial_z \phi - \partial_{\xi} \phi = \alpha_0 + a_1 \phi + \alpha_1 \ast \phi + a_2 \varphi \ast \phi + \alpha_2 \ast \phi \ast \phi, \end{align} $$

where the unknown variables $\phi $ and ${ {F}}$ are related by $\phi = {\mathfrak {B}}_+ [\, { {F}} \,]$ and ${ {F}} = \mathfrak {L}_+ [\, \phi \,]$ .

Step 2: The integral equation.

The principal part of the PDE (49) has constant coefficients, so it is easy to rewrite it as an equivalent integral equation as follows. Consider the holomorphic change of variables $ {T} : (z, \xi ) \longmapsto (w, t) = (z + \xi , \xi )$ . For any function $\alpha = \alpha (z, \xi )$ of two variables, introduce the following notation:

$$ \begin{align*} {T}^{\ast} \alpha (z, \xi) {\mathrel{\mathop:}=} \alpha \big( {T} (z, \xi) \big) = \alpha (z + \xi, \xi) \quad\text{and}\quad {T}_{\ast} \alpha (w, t) {\mathrel{\mathop:}=} \alpha \big( {T}^{-1} (w, t) \big) = \alpha (w - t, t). \end{align*} $$

Note that $ {T}^{\ast } {T}_{\ast } \alpha = \alpha $ . Under this change of coordinates, the differential operator $\partial _z - \partial _{\xi }$ transforms into $- \partial _t$ , and so the left-hand side of (49) becomes $- \partial _t \big ( {T}_{\ast } \phi )$ . Integrating from $0$ to t, and imposing the initial condition $\phi (z, 0) = \phi _0 (z) {\mathrel {\mathop :}=} a_0 (z)$ , the left-hand side of the PDE (49) becomes $- {T}_{\ast } \phi $ . Applying $ {T}^{\ast }$ , we therefore obtain the following integral equation for $\phi = \phi (z, \xi )$ :

(50) $$ \begin{align} \phi = \phi_0 - {T}^{\ast} \int\nolimits_0^t {T}_{\ast} \Big( \alpha_0 + a_1 \phi + \alpha_1 \ast \phi + a_2 \phi \ast \phi + \alpha_2 \ast \phi \ast \phi\Big) {\mathrm{d}}{u}. \end{align} $$

Introduce the following notation: for any function $\alpha = \alpha (z, \xi )$ of two variables,

(51) $$ \begin{align} {I}_+ \big[ \alpha \big] (z, \xi) {\mathrel{\mathop:}=} - {T}^{\ast} \int\nolimits_0^t {T}_{\ast} \alpha {\mathrm{d}}{u} = - \int\nolimits_0^{\xi} \alpha (z + \xi - u, u) {\mathrm{d}}{u} = \int\nolimits_0^{\xi} \alpha (z + t, \xi - t) {\mathrm{d}}{t}{,}\\[-15pt]\nonumber \end{align} $$

where the integration path is the straight line segment connecting $0$ to $\xi $ . Then the integral equation (50) can be written more succinctly as

(52) $$ \begin{align} \phi = \phi_0 + {I}_+ \Big[ \, \alpha_0 + a_1 \phi + \alpha_1 \ast \phi + a_2 \phi \ast \phi + \alpha_2 \ast \phi \ast \phi \, \Big].\\[-15pt]\nonumber\end{align} $$

Step 3: Method of successive approximations.

To solve (52), we use the method of successive approximations. Consider a sequence of functions ${\left \{ \phi _n \right \}}_{n=0}^{\infty }$ defined recursively by $\phi _0 = a_0$ , $\phi _1 {\mathrel {\mathop :}=} {I}_+ \big [ \alpha _0 + a_1 \phi _0 \big ]$ , and for $n \geq 2$ by the following formula:

(53) $$ \begin{align} \phi_n {\mathrel{\mathop:}=} {I}_+ \left[ a_1 \phi_{n-1} + \alpha_1 \ast \phi_{n-2} + \sum_{\substack{i,j \geq 0 \\ i + j = n-2}} a_2 \phi_i \ast \phi_j + \sum_{\substack{i,j \geq 0 \\ i + j = n-3}} \alpha_2 \ast \phi_i \ast \phi_j \right].\\[-15pt]\nonumber \end{align} $$

Claim 5.1 (Main Claim).

Let $\epsilon $ be so small that $\epsilon < r - r_0$ . Then the infinite series

(54) $$ \begin{align} \phi_+ (z, \xi) {\mathrel{\mathop:}=} \sum_{n=0}^{\infty} \phi_n (z, \xi) \end{align} $$

defines a holomorphic solution of the integral equation (52) on the domain

$$ \begin{align*} \mathbf{H}_+ {\mathrel{\mathop:}=} {\left\{ (z, \xi) \in \mathsf{H}_+ \times \Xi_+ ~\big|~ z + \xi \in \mathsf{H}_+ \right\}} \end{align*} $$

with at most exponential growth at infinity in $\xi $ ; more precisely, it satisfies the following uniform exponential bound: there are real constants ${ {A}}, {K}> 0$ such that

(55) $$ \begin{align} \big| \phi_+ (z, \xi) \big| \leq {{A}} e^{{K} |\xi|} \qquad\qquad \forall (z, \xi) \in \mathbf{H}_+. \end{align} $$

In particular, $\phi _+$ is a well-defined holomorphic solution on $\mathsf{H}^+_0 \times \Xi _+ \subset \mathbf {H}_+$ where it satisfies the exponential estimate above.

Assuming this claim, only one step remains in order to complete the proof of Lemma 5.1, which is to take the Laplace transform of $\phi _+$ .

Step 4: The Laplace transform.

Let

(56) $$ \begin{align} {{F}}_+ (z, \hbar) {\mathrel{\mathop:}=} \mathfrak{L}_+ \big[ \phi_+ \big] (z, \hbar) = \int\nolimits_{0}^{+\infty} e^{- \xi / \hbar} \phi (z, \xi) {\mathrm{d}}{\xi}. \end{align} $$

This integral is uniformly convergent for all $z \in \mathsf{H}^+_0$ provided that ${{\operatorname {Re}} (\hbar ^{-1})> { {C}}_2}$ . Thus, if we take $d_0 \in (0, d]$ strictly smaller than $1/{ {C}}_2$ , then formula (56) defines a holomorphic solution of the Riccati equation (45) on the domain $\mathsf{H}^+_0 \times \mathsf{S}^+_0$ where $\mathsf{S}^+_0 {\mathrel {\mathop :}=} {\left \{ \hbar ~\big |~ {\operatorname {Re}} (\hbar ^{-1})> 1 / d_0 \right \}}$ . Furthermore, Nevanlinna’s theorem implies that ${ {F}}_+$ admits a uniform Gevrey asymptotic expansion on $\mathsf{H}^+_0$ as $\hbar \to 0$ along $\bar {\mathsf{A}}_+$ , and this asymptotic expansion is necessarily the formal solution $\hat {{ {F}}}_+$ .

Proof of Claim 5.1.

If we assume for the moment that the series (54) is uniformly convergent on $\mathbf {H}_+$ , it is easy to check by direct substitution that it satisfies the integral equation (52). To demonstrate uniform convergence of the series $\phi _+$ , we first note the following exponential estimates on the coefficients of (52): there are constants ${ {C}}, {L}> 0$ such that for each $i = 0,1,2$ and for all $(z, \xi ) \in \mathbf {H}_+$ ,

(57) $$ \begin{align} |a_i| \leq {{C}} \qquad\text{and}\qquad |\alpha_i| \leq {{C}} e^{{L} |\xi|}.\\[-9pt]\nonumber \end{align} $$

We prove the Main Technical Claim by showing that there are constants ${ {A}}, { {M}}> 0$ such that for all n and for all $(z, \xi ) \in \mathbf {H}_+$ ,

(58) $$ \begin{align} \big| \phi_n (z, \xi) \big| \leq {{A}} {{M}}^n \frac{|\xi|^n}{n!} e^{{L} |\xi|}.\\[-9pt]\nonumber \end{align} $$

This is enough to deduce the uniform convergence of the infinite series $\phi _+$ as well as the exponential estimate (55) by taking $ {K} {\mathrel {\mathop :}=} { {M}} + {L}$ because

$$ \begin{align*} \big| \phi_+ (z, \xi) \big| \leq \sum_{n=0}^{\infty} |\phi_n| \leq \sum_{n=0}^{\infty} {{A}} {{M}}^n \frac{|\xi|^n}{n!} e^{{L} |\xi|} = {{A}} e^{ ({{M}} + {L}) |\xi|}.\\[-9pt] \end{align*} $$

To show (58), we will first recursively construct a sequence of positive real numbers $({ {M}}_n)_{n=0}^{\infty }$ such that for all n and for all $(z, \xi ) \in \mathbf {H}_+$ ,

(59) $$ \begin{align} \big| \phi_n (z, \xi) \big| \leq {{M}}_n \frac{|\xi|^n}{n!} e^{{L} |\xi|}.\\[-9pt]\nonumber \end{align} $$

We will then show that there are ${ {A}}, { {M}}> 0$ such that ${ {M}}_n \leq { {A}} { {M}}^n$ for all n.

Construction of ${ {M}}_0, { {M}}_1$ . We can take ${ {M}}_0 {\mathrel {\mathop :}=} { {C}}$ because $\big | \phi _0 (z, \xi ) \big | = \big | a_0 (z) \big | \leq { {C}}$ . We can take ${ {M}}_1 {\mathrel {\mathop :}=} { {C}} ( 1 + { {C}})$ because Lemma A.1 gives the estimate

$$ \begin{align*} \big| \phi_1 (z, \xi) \big|\leq \int_0^{\xi} |\alpha_0| |{\mathrm{d}}{u}| + \int_0^{\xi} |a_1| |\phi_0| |{\mathrm{d}}{u}| \leq {{C}} ( 1 + {{C}}) \int_0^{|\xi|} e^{{L} r} {\mathrm{d}}{r} \leq {{C}} ( 1 + {{C}}) |\xi| e^{{L} |\xi|}.\\[-9pt] \end{align*} $$

Construction of ${ {M}}_n$ for $n \geq 2$ . We assume that the estimate (59) holds for $\phi _0, \ldots , \phi _{n-1}$ and derive an estimate for $\phi _n$ . Using Lemmas A.1–A.3, we obtain the following bounds on the terms in the recursive formula (53):

$$ \begin{align*} \Big| {I}_+ \big[ a_1 \phi_{n-1} \big] \Big| &\leq {{C}} {{M}}_{n-1} \frac{|\xi|^n}{n!} e^{{L} |\xi|},\\ \Big| {I}_+ \big[ \alpha_1 \ast \phi_{n-2} \big] \Big| &\leq {{C}} {{M}}_{n-2} \frac{|\xi|^n}{n!} e^{{L} |\xi|},\\ \Big| {I}_+ \big[ a_2 \phi_i \ast \phi_j \big] \Big| &\leq {{C}} {{M}}_i {{M}}_j \frac{|\xi|^n}{n!} e^{{L} |\xi|} \quad \text{if } i + j = n-2, \\ \Big| {I}_+ \big[ \alpha_2 \ast \phi_i \ast \phi_j \big] \Big| &\leq {{C}} {{M}}_i {{M}}_j \frac{|\xi|^n}{n!} e^{{L} |\xi|} \quad \text{if } i + j = n-3. \end{align*} $$

Using these estimates in (53), we find

$$ \begin{align*} \big| \phi_n \big| &\leq\Big| {I}_+ \big[ a_1 \phi_{n-1} \big] \Big| + \Big| {I}_+ \big[ \alpha_1 \ast \phi_{n-2} \big] \Big| \\ &\phantom{\leq \Big| {I} \big[ a_1 \phi_{n-1} \big] \Big|~} + \sum_{\substack{i,j \geq 0 \\ i + j = n-2}} \Big| {I}_+ \big[ a_2 \phi_i \ast \phi_j \big] \Big| + \sum_{\substack{i,j \geq 0 \\ i + j = n-3}} \Big| {I}_+ \big[ \alpha_2 \ast \phi_i \ast \phi_j \big] \Big|\\ &\leq {{C}} \left( {{M}}_{n-1} + {{M}}_{n-2}+ \sum_{\substack{i,j \geq 0 \\ i + j = n-2}} {{M}}_i {{M}}_j + \sum_{\substack{i,j \geq 0 \\ i + j = n-3}} {{M}}_i {{M}}_j\right) \frac{|\xi|^n}{n!} e^{{L} |\xi|}. \end{align*} $$

We can therefore define, for all $n \geq 2$ ,

(60) $$ \begin{align} {{M}}_n {\mathrel{\mathop:}=} {{C}} \left( {{M}}_{n-1} + {{M}}_{n-2} + \sum_{\substack{i,j \geq 0 \\ i + j = n-2}} {{M}}_i {{M}}_j + \sum_{\substack{i,j \geq 0 \\ i + j = n-3}} {{M}}_i {{M}}_j \right). \end{align} $$

Bounds on ${ {M}}_n$ . Consider the following power series in an abstract variable t:

$$ \begin{align*} \hat{p} (t) {\mathrel{\mathop:}=} \sum_{n=0}^{\infty} {{M}}_n t^n \in {\mathbb{C}} {\left[\![ t \right]\!]}. \end{align*} $$

We will show that $\hat {p} (t)$ is in fact a convergent power series. First, we observe that $\hat {p} (0) = { {M}}_0 = { {C}}$ and that $\hat {p} (t)$ satisfies the following algebraic equation:

(61) $$ \begin{align} \hat{p} = {{C}} \Big( 1 + t + \hat{p} t + \hat{p} t^2 + \hat{p}^2 t^2 + \hat{p}^2 t^3 \Big), \end{align} $$

which can be seen by expanding and comparing the coefficients using the defining formula (60) for ${ {M}}_n$ . Consider the holomorphic function $ {G} = {G} (p,t)$ of two variables, defined by

$$ \begin{align*} {G} (p,t) {\mathrel{\mathop:}=} - p + {{C}} \Big( 1 + t + pt + pt^2 + p^2t^2 + p^2t^3 \Big). \end{align*} $$

It has the following properties:

$$ \begin{align*} {G} ({{C}},0) = 0, \qquad \left. { {\frac{\partial {G}}{\partial p}} }\right|_{ (p,t) = ({{C}},0)} = -1 \neq 0. \end{align*} $$

Thus, by the Holomorphic Implicit Function Theorem, there exists a function $p (t)$ , holomorphic at $t = 0$ , satisfying $p (0) = { {C}}$ and $ {G} \big (p (t), t \big ) = 0$ for all t sufficiently close to $0$ . Since $\hat {p} (0) = { {C}}$ and $ {G} \big (\hat {p} (t), t \big ) = 0$ , the power series $\hat {p} (t)$ must be the Taylor expansion of $p (t)$ at $t = 0$ . As a result, $\hat {p} (t)$ is in fact a convergent power series, which means its coefficients grow at most exponentially: there are constants ${ {A}}, { {M}}> 0$ such that ${ {M}}_n \leq { {A}} { {M}}^n$ for all n. This completes the proof of Claim 5.1 and therefore of Lemma 5.1.

5.1.2 Proof of Theorem 5.1

We can now finish the proof of the main result in this paper.

Proof of Theorem 5.1

We immediately restrict our attention to a Borel disk in the $\hbar $ -plane of some diameter $d> 0$ bisected by the direction $\theta $ ; that is, without loss of generality, assume that $\mathsf{S} = {\left \{ \hbar ~\big |~ {\operatorname {Re}} (e^{i\theta }/\hbar )> 1/ d \right \}}$ . Note that the rotation $\hbar \mapsto e^{-i\theta } \hbar $ sends $\mathsf{S}$ to the Borel disk $\mathsf{S}_+ = {\left \{ \hbar ~\big |~ {\operatorname {Re}} (1/\hbar )> 1/ d \right \}}$ from (44).

Next, let $r> 0$ be such that $\mathsf{W} {\mathrel {\mathop :}=} \mathsf{W}^{\alpha }_{\theta } = W^{\alpha }_{\theta } (x_0; r) = \Phi ^{-1} ( \mathsf{H} )$ is the WKB half-strip from the hypothesis, where $\mathsf{H} {\mathrel {\mathop :}=} \mathsf{H}^{\alpha }_{\theta } = \mathsf{H}^{\alpha }_{\theta } (0, r) = {\left \{ z ~\big |~ {\mathop {\mathrm {dist}}\nolimits } (z, e^{i\theta } {\mathbb {R}}_{\alpha }) < r \right \}}$ . Recall that $\Phi ^{-1} : \mathsf{H} \to \mathsf{W}$ is a local biholomorphism. Put $\mathsf{H}_+ {\mathrel {\mathop :}=} {\left \{ z ~\big |~ {\mathop {\mathrm {dist}}\nolimits } (z, {\mathbb {R}}_+) < r \right \}}$ and $\Phi _{\alpha }^{\theta } {\mathrel {\mathop :}=} \varepsilon _{\alpha } e^{-i \theta } \Phi $ , so that $\mathsf{W} = (\Phi _{\alpha }^{\theta })^{-1} (\mathsf{H}_+)$ . Furthermore, $\Phi _{\alpha }^{\theta }$ transforms the differential operator $\frac {\varepsilon _{\alpha }}{\sqrt {{ {D}}_0}} \partial _x$ into $e^{-i\theta } \partial _z$ .

Consider now holomorphic functions of $(x, \hbar ) \in \mathsf{W} \times \mathsf{S}$ denoted by $a_{\ast }, a_{\ast \ast }$ , $b_{\ast }, b_{\ast \ast }$ , $c_{\ast }, c_{\ast \ast }$ , obtained from $a,b,c$ by removing the leading and the next-to-leading order terms, respectively; that is, they are defined by the following relations:

(62) $$ \begin{align} \begin{aligned} a &= a_0 + \hbar a_{\ast} &\text{and}\quad a_{\ast} &= a_1 + \hbar a_{\ast\ast}, \\ b &= b_0 + \hbar b_{\ast} &\text{and}\quad b_{\ast} &= b_1 + \hbar b_{\ast\ast}, \\ c &= c_0 + \hbar c_{\ast} &\text{and}\quad c_{\ast} &= c_1 + \hbar c_{\ast\ast}. \end{aligned} \end{align} $$

Recall that the leading and the next-to-leading orders $f_0^{\alpha }, f_1^{\alpha }$ of the formal solution $\hat {f}_{\alpha }$ are holomorphic functions on $\mathsf{W}$ that satisfy the following identities:

(63) $$ \begin{align} \begin{gathered} a_0 (f_0^{\alpha})^2 + b_0 f_0^{\alpha} + c_0 = 0 \qquad\text{and}\qquad \varepsilon_{\alpha} \sqrt{{{D}}_0} = 2a_0 f_0^{\alpha} + b_0, \\ \partial_x f_0^{\alpha} = \varepsilon_{\alpha} \sqrt{{{D}}_0} f_1^{\alpha} + a_1 (f_0^{\alpha})^2 + b_1 f_0^{\alpha} + c_1, \end{gathered} \end{align} $$

where $\varepsilon _{\pm } = \pm 1$ . Using these expressions, a straightforward calculation shows that the change of the unknown variable $f \mapsto \tilde {f}$ given by $f = f_0^{\alpha } + \hbar (f_1^{\alpha } + \tilde {f})$ transforms the Riccati equation (36) into the following Riccati equation on $\mathsf{W} \times \mathsf{S}$ :

(64) $$ \begin{align} \varepsilon_{\alpha} \frac{\hbar}{\sqrt{{{D}}_0}} \partial_x \tilde{f} - \tilde{f} = \hbar \big( \tilde{a} \tilde{f}^2 + \tilde{b} \tilde{f} + \tilde{c} \big), \end{align} $$

where

(65) $$ \begin{align} \begin{gathered} \tilde{a} {\mathrel{\mathop:}=} \frac{\varepsilon_{\alpha}}{\sqrt{{{D}}_0}} a {,} \qquad\quad \tilde{b} {\mathrel{\mathop:}=} \frac{\varepsilon_{\alpha}}{\sqrt{{{D}}_0}} \Big( b_{\ast} + 2a f_1^{\alpha} + 2 a_{\ast} f_0^{\alpha} \Big),\\ \tilde{c} {\mathrel{\mathop:}=} \frac{\varepsilon_{\alpha}}{\sqrt{{{D}}_0}} \bigg( - \partial_x f_1^{\alpha} + a (f_1^{\alpha})^2 + \big(2 a_{\ast} f_0^{\alpha} + b_{\ast} \big) f_1^{\alpha} + \big( a_{\ast\ast} (f_0^{\alpha})^2 + b_{\ast\ast} f_0^{\alpha} + c_{\ast\ast} \big) \bigg). \end{gathered} \end{align} $$

Finally, transforming equation (64) by $\Phi _{\alpha }^{\theta }$ and applying a rotation $\hbar \mapsto e^{-i\theta } \hbar $ , we obtain a Riccati equation on $\mathsf{H}_+ \times \mathsf{S}_+$ of the form (45) where the coefficients ${ {A}}_0, { {A}}_1, { {A}}_2$ are given by

(66) $$ \begin{align} {{A}}_2 (z, \hbar) {\mathrel{\mathop:}=} \tilde{a} \big( x(z), e^{i\theta} \hbar \big),\quad {{A}}_1 (z, \hbar) {\mathrel{\mathop:}=} \tilde{b} \big( x(z), e^{i\theta} \hbar \big),\quad {{A}}_0 (z, \hbar) {\mathrel{\mathop:}=} \tilde{c} \big( x(z), e^{i\theta} \hbar \big), \end{align} $$

where $x(z) = (\Phi ^{\theta }_{\alpha })^{-1} (z)$ and the unknown variables $\tilde {f}$ and ${ {F}}$ are related by

(67) $$ \begin{align} {{F}} (z, \hbar) = \tilde{f} \big( x(z), e^{i\theta} \hbar \big). \end{align} $$

Theorem 5.1 now follows from Lemma 5.1.

5.2 Borel summability of formal solutions

In this subsection, we translate Theorem 5.1 and its method of proof into the language of Borel–Laplace theory. Namely, it follows directly from our construction that the canonical exact solutions are the Borel resummation of the corresponding formal solutions. Let us make this statement precise and explicit. The following theorem is a direct consequence of the proof of Theorem 5.1.

Theorem 5.2 (Borel summability of formal solutions).

Assume all the hypotheses of Theorem 5.1. Then the local formal solution $\hat {f}_{\alpha }$ is Borel summable in the direction $\theta $ uniformly near $x_0$ . Namely, the canonical local exact solution $f_{\alpha }^{\theta }$ is the uniform Borel resummation of $\hat {f}_{\alpha }$ in the direction $\theta $ : for all $\hbar \in S_0$ and uniformly for all $x \in \mathsf{U}_0$ ,

(68) $$ \begin{align} f_{\alpha}^{\theta} (x, \hbar) = f^{\alpha}_0 (x) + \mathcal{S}_{\theta} \big[ \: \hat{f}_{\alpha} \: \big] (x, \hbar). \end{align} $$

A lot of information is packed into Theorem 5.2. Let us unpack it into the following series of explicit statements, all of which are deduced immediately from the proof of Theorem 5.1.

The fact that identity (68) holds uniformly for all $x \in \mathsf{U}_0$ means in particular that operations such as differentiation and integration with respect to x can be exchanged with the operation of Borel resummation. Thus, we have the following corollary.

Corollary 5.2. Assume all the hypotheses of Theorem 5.1, and let $f_{\alpha }^{\theta }$ be the canonical exact solution defined on $\mathsf{U}_0 \times \mathsf{S}_0$ . Then the formal power series on $\mathsf{U}_0$ given by the derivative $\partial _x \hat {f}_{\alpha }$ and the integral $\int _{x^{\prime }_0}^x \hat {f}_{\alpha }$ from any basepoint $x^{\prime }_0 \in \mathsf{U}_0$ are uniformly Borel summable on $\mathsf{U}_0$ , and the following identities hold uniformly for all $x \in \mathsf{U}_0$ :

(73) $$ \begin{align} \partial_x f_{\alpha}^{\theta} (x, \hbar) &= \partial_x f_0^{\alpha} (x) + \mathcal{S}_{\theta} \big[ \: \partial_x \hat{f}_{\alpha} \: \big] (x, \hbar), \end{align} $$

(74) $$ \begin{align} \int\nolimits_{x^{\prime}_0}^x f_{\alpha}^{\theta} (t, \hbar) {\mathrm{d}}{t} &= \int\nolimits_{x^{\prime}_0}^x f^{\alpha}_0 (t) {\mathrm{d}}{t}+ \mathcal{S}_{\theta} \left[ \: \int\nolimits_{x^{\prime}_0}^x \hat{f}_{\alpha} {\mathrm{d}}t \: \right] (x, \hbar). \end{align} $$

Thanks to the tighter control on the asymptotics of canonical exact solutions on WKB half-strips, all of the above statements extend uniformly over strictly smaller WKB halfstrips. Explicitly, we have the following corollary.

5.3 Explicit recursion for the Borel transform

The analytic Borel transform $\varphi _{\alpha }^{\theta }$ can be given a reasonably explicit presentation as follows. Define an integral operator $ {I}$ acting on holomorphic functions $\varphi = \varphi (x, \xi )$ by the following formula, wherever it makes sense:

(75) $$ \begin{align} {I} \big[ \, \varphi \, \big] (x, \xi){\mathrel{\mathop:}=} \int\nolimits_0^{\xi} \varphi ( x_t, \xi - t ) {\mathrm{d}}{t} \qquad\text{where} \qquad x_t {\mathrel{\mathop:}=} \Phi^{-1} \big( \Phi (x) + t \big). \end{align} $$

and the integration contour is the straight line segment from $0$ to $\xi \in {\mathbb {C}}$ . In particular, for any $x \in \mathsf{U}_0$ and any sufficiently small $\xi \in e^{i\theta } {\mathbb {R}}_{\alpha }$ , the path ${\left \{ x_t \right \}}_{t=0}^{\xi }$ is a segment of the WKB $(\theta , \alpha )$ -ray emanating from x.

Recall functions $\tilde {a}, \tilde {b}, \tilde {c}$ defined by the identities (65). Their leading-order parts in $\hbar $ are, respectively,

(76) $$ \begin{align} \tilde{a}_0 = \frac{\varepsilon_{\alpha}}{\sqrt{{{D}}_0}} a_0 {,} \qquad \tilde{b}_0 = \frac{\varepsilon_{\alpha}}{\sqrt{{{D}}_0}} \Big( b_1 + 2a_0 f_1^{\alpha} + 2 a_1 f_0^{\alpha} \Big) {,} \qquad \tilde{c}_0 = f_2^{\alpha}, \end{align} $$

where last identity was obtained by comparing with (15). Finally, denote their analytic Borel transforms in direction $\theta $ as follows:

$$ \begin{align*} \begin{aligned} \beta_0 &{\mathrel{\mathop:}=} {\mathfrak{B}}_{\theta} [ \, \tilde{c} \, ],\\ \beta_1 &{\mathrel{\mathop:}=} {\mathfrak{B}}_{\theta} [ \, \tilde{b} \, ],\\ \beta_2 &{\mathrel{\mathop:}=} {\mathfrak{B}}_{\theta} [ \, \tilde{a} \, ], \end{aligned} \quad\qquad \text{so that}\qquad\quad \begin{aligned} {\tilde{a}} &= \tilde{a}_0 + \mathfrak{L}_{\theta} [\, \beta_2 \,], \\ {\tilde{b}} &= \tilde{b}_0 + \mathfrak{L}_{\theta} [\, \beta_1 \,],\\ {\tilde{c}} &= \tilde{c}_0 + \mathfrak{L}_{\theta} [\, \beta_0 \,]. \end{aligned} \end{align*} $$

Proposition 5.3 (Recursive formula for the Borel transform).

Assume all the hypotheses of Theorem 5.1, and let $r> 0$ be such that $\mathsf{W} = \mathsf{W}_{\theta }^{\alpha } (x_0; r)$ . For any $r_0 \in (0, r)$ and any $\epsilon \in (0, r - r_0)$ , let $\mathsf{W}_0 {\mathrel {\mathop :}=} \mathsf{W}_{\theta }^{\alpha } (x_0; r_0)$ and $\Xi _{\theta } {\mathrel {\mathop :}=} {\left \{ \xi ~\big |~ {\mathop {\mathrm {dist}}\nolimits } (\xi , e^{i\theta } {\mathbb {R}}_+) < \epsilon \right \}}$ . Then the analytic Borel transform $\varphi _{\alpha }^{\theta }$ can be expressed more explicitly as follows: uniformly for all $(x, \xi ) \in \mathsf{W}_0 \times \Xi _{\theta }$ ,

(77) $$ \begin{align} \varphi_{\alpha}^{\theta} (x, \xi) = f^{\alpha}_1 (x) + \int\nolimits_0^{\xi} \sigma_{\alpha}^{\theta} (x, t) {\mathrm{d}}{t} \qquad\text{with}\qquad \sigma_{\alpha}^{\theta} (x, \xi) {\mathrel{\mathop:}=} \sum_{k=0}^{\infty} \sigma_k (x, \xi), \end{align} $$

where $\sigma _0 {\mathrel {\mathop :}=} \tilde {c}_0 = f_2^{\alpha }$ , $\sigma _1 {\mathrel {\mathop :}=} {I} \big [ \beta _0 + \tilde {b}_0 \sigma _0 \big ]$ , and for $k \geq 2$ ,

(78) $$ \begin{align} \sigma_k {\mathrel{\mathop:}=} {I} \left[ \tilde{b}_0 \sigma_{k-1} + \beta_1 \ast \sigma_{k-2} + \sum_{\substack{i,j \geq 0 \\ i + j = k-2}} \tilde{a}_0 \sigma_i \ast \sigma_j + \sum_{\substack{i,j \geq 0 \\ i + j = k-3}} \beta_2 \ast \sigma_i \ast \sigma_j \right]. \end{align} $$

5.4 Monic Riccati equations

In many situations, such as those arising in the context of the exact WKB analysis of second-order ODEs, the coefficients of the Riccati equation satisfy hypothesis (2) in Theorem 5.1 only after an additional transformation.

Example 5.1. For example, consider the Riccati equation $\hbar \partial _x f = f^2 - x$ , which is encountered in the WKB analysis of the deformed Airy differential equation. The coefficients are $a = 1, b = 0, c = -x$ , and the leading-order discriminant ${ {D}}_0$ is $4x$ . In this case, a WKB half-strip $\mathsf{W}$ is necessarily an unbounded domain, and the asymptotic condition (37) reduces to requiring that $c = -x$ is bounded on $\mathsf{W}$ by $\sqrt {{ {D}}_0} = 2\sqrt {x}$ , which is not the case. Therefore, Theorem 5.1 cannot be applied to this Riccati equation directly.

However, this can be remedied by making a change of the unknown variable $f \mapsto g$ given by $f = \sqrt {{ {D}}_0} g$ for $x \in \mathsf{W}$ . It transforms the Riccati equation $\hbar \partial _x f = f^2 - x$ into

(79) $$ \begin{align} \hbar \partial_x g = 2\sqrt{x} g^2 - \tfrac{1}{2}\hbar x^{-1} g + \tfrac{1}{2} \sqrt{x}. \end{align} $$

Notice that the leading-order discriminant of this Riccati equation remains ${ {D}}_0 = 4x$ and that its coefficients are now bounded at infinity by $\sqrt {{ {D}}_0} = 2 \sqrt {x}$ . Therefore Theorem 5.1 can be applied to (79).

More generally, this transformation is necessary when dealing with monic Riccati equations, that is, whenever the coefficient $\mathsf{A}$ of the Riccati equation is identically $1$ . This is always the case in the exact WKB analysis of second-order ODEs [Reference Nikolaev12]. Spelled out, we have the following version of our results.

Theorem 5.3 (Exact existence and uniqueness for monic equations).

Consider the following monic Riccati equation:

(80) $$ \begin{align} \hbar \partial_x f = f^2 + pf + q, \end{align} $$

where $p, q$ are holomorphic functions of $(x, \hbar ) \in \mathsf{X} \times \mathsf{S}$ which admit locally uniform asymptotic expansions $\hat {p}, \hat {q}$ as $\hbar \to 0$ along $\mathsf{A}$ . Assume that ${ {D}}_0 = p_0^2 - 4q_0 \not \equiv 0$ . Fix a regular point $x_0 \in \mathsf{X}$ , a square-root branch $\sqrt {{ {D}}_0}$ near $x_0$ , and a sign $\alpha \in {\left \{ +, - \right \}}$ . In addition, we assume the following hypotheses:

1. (1) There is a WKB $(\theta , \alpha )$ -half-strip domain $\mathsf{W} = \mathsf{W}_{\theta }^{\alpha } \subset \mathsf{X}$ containing $x_0$ .

2. (2) The asymptotic expansions of the coefficients $p,q$ are valid with Gevrey bounds as $\hbar \to 0$ along the closed arc $\bar {\mathsf{A}}_{\theta } = [\theta - \tfrac {\pi }{2}, \theta + \tfrac {\pi }{2}]$ , with respect to the asymptotic scales $\sqrt {{ {D}}_0}$ and ${ {D}}_0$ , respectively, uniformly for all $x \in \mathsf{W}$ :

   (81) $$ \begin{align} \hspace{13pt}p &\simeq \hat{p} \qquad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ wrt } \sqrt{{{D}}_0}, \text{ unif. } \forall x \in \mathsf{W}, \end{align} $$
   (82) $$ \begin{align} q &\simeq \hat{q} \qquad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ wrt } {{D}}_0, \text{ unif. } \forall x \in \mathsf{W}. \end{align} $$

3. (3) The logarithmic derivative $\partial _x \log { {D}}_0$ is bounded by ${ {D}}_0$ on $\mathsf{W}$ .

Then all the conclusions of Theorem 5.1, as well as Theorem 5.2, Lemma 5.2, and Corollary 5.2 hold verbatim. Furthermore, all the conclusions of Proposition 5.1 hold verbatim with the only exception that the asymptotic statement (43) must be replaced with the following asymptotic statement with respect to the asymptotic scale $\sqrt {{ {D}}_0}$ :

(83) $$ \begin{align} f^{\theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ wrt } \sqrt{{{D}}_0}, \text{ unif. } \forall x \in \mathsf{W}_0. \end{align} $$

Proof. The change of the unknown variable $f \mapsto g$ given by $f = \sqrt {{ {D}}_0} g$ transforms (80) into

(84) $$ \begin{align} \hbar \partial_x g = \sqrt{{{D}}_0} g^2 + \big( p - \hbar \partial_x \log \sqrt{{{D}}_0} \big) g + \tfrac{1}{\sqrt{{{D}}_0}} q. \end{align} $$

This transformation is well defined for all $x \in \mathsf{W}$ because $\mathsf{W}$ necessarily supports the univalued square-root branch $\sqrt {{ {D}}_0}$ . Notice that the leading-order discriminant of this Riccati equation remains ${ {D}}_0$ . Note that $\partial _x \log \sqrt {{ {D}}_0}$ is bounded by $\sqrt {{ {D}}_0}$ if and only if $\partial _x \log { {D}}_0$ is bounded by ${ {D}}_0$ , as provided by hypothesis (3). Now, it is obvious that the hypotheses of Theorem 5.3 imply that the Riccati equation (84) satisfies all the hypotheses of Theorem 5.1. It yields the canonical local exact solution $g_{\alpha }^{\theta }$ near $x_0$ , and therefore the canonical local exact solution $f_{\alpha }^{\theta } = \sqrt {{ {D}}_0} g_{\alpha }^{\theta }$ near $x_0$ .

Of course, a general Riccati equation (36) can always be put into the monic form (5.3) via the change of the unknown variable $f \mapsto g = af$ , which yields

(85) $$ \begin{align} \hbar \partial_x g = g^2 + (b + \hbar \partial_x \log a) g + ac. \end{align} $$

If a is nowhere-vanishing, then this transformation makes sense globally on $\mathsf{X}$ .

Likewise, Corollary 5.1 is also true for the monic Riccati equation (5.3), though with slightly simplified hypotheses as follows.

However, Propositions 5.2 and 5.3 are no longer true for the canonical exact solutions of the monic Riccati equation (5.3). In this case, one needs either to factorize $\sqrt {{ {D}}_0}$ out of $f_{\alpha }^{\theta }$ and apply these propositions to the regularized Riccati equation (84), or identify and remove the principal part of $f_{\alpha }^{\theta }$ in the limit along the WKB rays. The latter procedure can be explicitly formalized when the WKB rays limit to a pole of ${ {D}}_0$ ; this will be explained in detail elsewhere.

Remark 5.2. The same trick as above can help us tackle more general situations as follows. If $\chi = \chi (x)$ is any holomorphic function, then the change of the unknown variable $f \mapsto g$ given by $f = \chi g$ transforms the Riccati equation (36) into

(86) $$ \begin{align} \hbar \partial_x g = a' g^2 + b' g + c', \end{align} $$

where

(87) $$ \begin{align} a' {\mathrel{\mathop:}=} \chi a \qquad b' {\mathrel{\mathop:}=} b - \hbar \partial_x \log \chi\qquad c' {\mathrel{\mathop:}=} \chi^{-1} c. \end{align} $$

Note that since $\chi $ is independent of $\hbar $ , the leading-order discriminant remains unchanged: ${ {D}}^{\prime }_0 = (b^{\prime }_0)^2 - 4a^{\prime }_0 c^{\prime }_0 = b_0^2 - 4a_0c_0 = { {D}}_0$ . In summary, we have the following proposition, which in particular recovers Theorem 5.1 by taking $\chi = 1$ and Theorem 5.3 by taking $\chi = \sqrt {{ {D}}_0}$ .

Proposition 5.4. Assume all the hypotheses of Theorem 5.1, except hypothesis (2) is replaced with the following:

• (2 $'$ ) There is a nonvanishing holomorphic function $\chi = \chi (x)$ on $\mathsf{W}$ such that $a',b',c'$ defined by (87) admit Gevrey asymptotics as $\hbar \to 0$ along the closed arc $\bar {\mathsf{A}}_{\theta } = [\theta - \tfrac {\pi }{2}, \theta + \tfrac {\pi }{2}]$ , with respect to the asymptotic scale $\sqrt {{ {D}}_0}$ , uniformly for all $x \in \mathsf{W}$ :

(88) $$ \begin{align} a' \simeq \hat{a}',\quad b' \simeq \hat{b}',\quad c' \simeq \hat{c}' \qquad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ wrt } \chi, \text{ unif. }\forall x \in \mathsf{W}. \end{align} $$

Then all the conclusions of Theorem 5.1, as well as Theorem 5.2, Lemma 5.2, and Proposition 5.2, hold verbatim. Furthermore, all the conclusions of Proposition 5.1 hold verbatim with the only exception that the asymptotic statement (43) must be replaced with the following asymptotic statement with respect to the asymptotic scale $\chi $ :

(89) $$ \begin{align} f^{\theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\quad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}_{\theta}, \text{ wrt } \chi, \text{ unif. } \forall x \in \mathsf{W}_0. \end{align} $$

5.5 Exact solutions in Wider sectors

The last general result we prove is about extending canonical exact solutions to sectors with wider openings. However, we not address here the question of extending canonical exact solutions to radially larger sectorial domains in ${\mathbb {C}}_{\hbar }$ or discussing the relationship between unequal canonical exact solutions for different values of $\theta $ . These questions will be examined in detail elsewhere.

Thus, suppose that $\pi \leq |\mathsf{A}| \leq 2\pi $ . Let $\Theta {\mathrel {\mathop :}=} [\theta _-, \theta _+]$ be the closed arc such that $\mathsf{A} = (\theta _- - \tfrac {\pi }{2}, \theta _+ + \tfrac {\pi }{2})$ ; that is, $\theta _{\pm } {\mathrel {\mathop :}=} \vartheta _{\pm } \mp \tfrac {\pi }{2}$ . See Figure 1a. This arc $\Theta $ is sometimes called the arc of copolar directions of $\mathsf{A}$ . For every $\theta \in \Theta $ , let $\mathsf{A}_{\theta } {\mathrel {\mathop :}=} (\theta -\tfrac {\pi }{2}, \theta +\tfrac {\pi }{2}) \subset \mathsf{A}$ be the semicircular arc bisected by $\theta $ . See Figure 1b. Note that $\mathsf{A} = \Cup _{\theta \in \Theta } \mathsf{A}_{\theta }$ .

Figure 1

Proposition 5.5. Consider the Riccati equation (36) whose coefficients $a,b,c$ are holomorphic functions of $(x,\hbar ) \in \mathsf{X} \times \mathsf{S}$ admitting locally uniform asymptotic expansions $\hat {a}, \hat {b}, \hat {c}$ as $\hbar \to 0$ along $\mathsf{A}$ . Assume that the leading-order discriminant ${ {D}}_0 = b_0^2 - 4 a_0 c_0$ is not identically zero. Fix a regular point $x_0 \in \mathsf{X}$ , a square-root branch $\sqrt {{ {D}}_0}$ near $x_0$ , and a sign $\alpha \in {\left \{ +, - \right \}}$ . Fix a regular point $x_0 \in \mathsf{X}$ , a square-root branch $\sqrt {{ {D}}_0}$ near $x_0$ , and a sign $\alpha \in {\left \{ +, - \right \}}$ . In addition, assume that hypotheses (1) and (2) in Theorem 5.1 are satisfied for every $\theta \in \Theta $ .

Then the Riccati equation (36) has a canonical local exact solution $f^{\Theta }_{\alpha }$ near $x_0$ which is asymptotic to the formal solution $\hat {f}_{\alpha }$ as $\hbar \to 0$ in every direction $\theta \in \Theta $ . Namely, there is a neighborhood $\mathsf{U}_0 \subset \mathsf{X}$ of $x_0$ and a sectorial domain $\mathsf{S}_0 \subset \mathsf{S}$ with the same opening $\mathsf{A}$ such that the Riccati equation (36) has a unique holomorphic solution $f^{\Theta }_{\alpha }$ on $\mathsf{U}_0 \times \mathsf{S}_0$ which is Gevrey asymptotic to $\hat {f}_{\alpha }$ as $\hbar \to 0$ along the closed arc $\bar {\mathsf{A}}$ uniformly for all $x \in \mathsf{U}_0$ :

(90) $$ \begin{align} f^{\Theta}_{\alpha} \simeq \hat{f}_{\alpha} \qquad\qquad \text{as } \hbar \to 0 \text{ along } \bar{\mathsf{A}}, \text{ unif. } \forall x \in \mathsf{U}_0. \end{align} $$

Proof. By Theorem 5.2 (or more specifically by part (4) of Lemma 5.2), for every $\theta \in \Theta $ , the canonical exact solution in the direction $\theta $ exists and can be written as

$$ \begin{align*} f_{\alpha}^{\theta} (x, \hbar) = f_0^{\alpha} (x) + \mathfrak{L}_{\theta} \big[\: \varphi_{\alpha}^{\theta} \:\big] (x, \hbar) = f_0^{\alpha} (x) + \int\nolimits_{e^{i\theta} {\mathbb{R}}_+} e^{-\xi/\hbar} \varphi_{\alpha}^{\theta} (x, \xi) {\mathrm{d}}{\xi}, \end{align*} $$

where $\varphi _{\alpha }^{\theta }$ is the analytic Borel transform of $f_{\alpha }^{\theta }$ in the direction $\theta $ . The explicit formula from Proposition 5.3 reveals that these analytic Borel transforms $\varphi _{\alpha }^{\theta }$ for each $\theta \in \Theta $ together define a holomorphic function $\varphi _{\alpha }^{\Theta }$ on an $\epsilon $ -neighborhood $\Xi _{\Theta } = {\left \{ \xi ~\big |~ {\mathop {\mathrm {dist}}\nolimits } (\xi , \Sigma _{\Theta }) < \epsilon \right \}}$ of the sector $\Sigma _{\Theta } {\mathrel {\mathop :}=} {\left \{ \xi ~\big |~ \arg (\xi ) \in \Theta \right \}}$ for a sufficiently small $\epsilon> 0$ . Furthermore, $\varphi _{\alpha }^{\Theta }$ has at most exponential growth at infinity in $\Xi _{\Theta }$ , which means that Cauchy–Goursat’s theorem yields the identity $\mathfrak {L}_{\theta _+} \big [ \varphi _{\alpha }^{\theta _+} \big ] = \mathfrak {L}_{\theta _+} \big [ \varphi _{\alpha }^{\Theta } \big ] = \mathfrak {L}_{\theta _-} \big [ \varphi _{\alpha }^{\Theta } \big ] = \mathfrak {L}_{\theta _-} \big [ \varphi _{\alpha }^{\theta _-} \big ]$ .

6.1 Equations with mildly deformed coefficients

6.2 The simplest explicit example: Deformed Airy

In this subsection, we illustrate the main constructions in this paper in the following explicit example. Consider the following Riccati equation on the domain ${\mathbb {C}}_x \times {\mathbb {C}}_{\hbar }$ :

(91) $$ \begin{align} \hbar \partial_x f = f^2 - x. \end{align} $$

Thus, in this example, $a = 1, b = 0, c = -x$ , and $\mathsf{X} = {\mathbb {C}}_x$ . Let us fix $\theta = 0$ , so we will search for canonical exact solutions of (91) with prescribed asymptotics as $\hbar \to 0$ along the positive real axis ${\mathbb {R}}_+ \subset {\mathbb {C}}_{\hbar }$ . Then the sectorial domain $\mathsf{S}$ can be taken as the complement of the negative real axis ${\mathbb {R}}_- \subset {\mathbb {C}}_{\hbar }$ .

This Riccati equation arises in the WKB analysis of the Airy differential equation $\hbar ^2 \partial ^2_x \psi (x, \hbar ) = x \psi (x, \hbar )$ upon considering the WKB ansatz $\psi = \exp \big ( - \int f / \hbar \big )$ (see [Reference Nikolaev12] for more details). For this Riccati equation, it is known that exact solutions exist (see, e.g., [Reference Kawai and Takei8, §2.2]). There, instead of solving the Riccati equation directly, the Borel–Laplace method is applied to the Airy differential equation. Here, we take a different approach by solving the Riccati equation directly. We have included this example because it is the simplest and most explicit standard example that nicely illustrates most constructions encountered in our paper.

6.2.1 Leading-order analysis.

Following §3.1, the leading-order equation (8) for the Riccati equation (91) is simply $f_0^2 - x = 0$ . The leading-order discriminant given by formula (9) is ${ {D}}_0 (x) = 4x$ . There is a single turning point at $x = 0$ . Let $\sqrt {x}$ be the principal square-root branch (i.e., positive on the positive real axis) in the complement of a branch cut along, say, the negative real axis. Label the two leading-order solutions as

(92) $$ \begin{align} f^{\pm}_0 (x) {\mathrel{\mathop:}=} \pm \sqrt{x} \qquad\text{so that}\qquad \sqrt{{{D}}_0} = 2 \sqrt{x}. \end{align} $$

The leading-order solutions $f^{\pm }_0$ are holomorphic on any simply connected domain $\mathsf{U} \subset {\mathbb {C}}_x^{\ast }$ . However, $f^{\pm }_0$ are unbounded if $\mathsf{U}$ is unbounded: the coefficient $c = -x$ of (91) is not bounded by $\sqrt {{ {D}}_0} = 2 \sqrt {x}$ , so not all the hypotheses of Lemma 3.1 are satisfied. This will be rectified later by regularizing the coefficients following §5.4.

6.2.2 Formal perturbation theory.

Now, we study the formal aspects of this Riccati equation following §3.2. By Theorem 3.1, the Riccati equation (91) has a pair of formal solutions $\hat {f}_{\pm }$ with leading-order terms $f_0^{\pm }$ . Their coefficients $f_k^{\pm }$ for $k \geq 1$ are given by the recursive formula (12), which in this example reduces to

(93) $$ \begin{align} f_k^{\pm} = \pm \tfrac{1}{2 \sqrt{x}} \partial_x f_{k-1}^{\pm} \mp \tfrac{1}{2 \sqrt{x}} \sum_{i+j = k}^{i,j\neq k} f_i^{\pm} f_j^{\pm}. \end{align} $$

The first few coefficients are

(94)

In fact, if we set $d_0^{\pm } = \pm 1$ , it is easy to show by induction that for all $k \geq 1$ ,

(95)

Note that all $d^{\pm }_k$ are rational numbers.

6.2.3 The Liouville transformation and WKB trajectories.

Following §4, let us describe the geometry of WKB trajectories on ${\mathbb {C}}_x$ emerging from this Riccati equation. For any basepoint $x_0 \in {\mathbb {C}}_x$ , the Liouville transformation is given, on the complement of the branch cut, by the simple formula

(96)

It follows that, for example, the WKB $(0,+)$ -ray emanating from any point $x_0$ with $\arg (x_0) \neq \pm 3\pi /2$ is complete. If, on the other hand, $\arg (x_0) = \pm 3\pi /2$ , then the WKB $(0,+)$ -ray emanating from $x_0$ hits the turning point in finite time. Likewise, the WKB $(0,-)$ -ray of every point $x_0$ with $\arg (x_0) \neq 0$ is complete. See Figure. 2. We focus our attention now on the domain

(97) $$ \begin{align} \mathsf{U} {\mathrel{\mathop:}=} {\left\{ x ~\big|~ 0 < \arg (x) < +3\pi/2 \right\}}. \end{align} $$

Its image under the Liouville transformation is the upper half-plane $\mathsf{H} = {\left \{ z ~\big |~ {\operatorname {Im}} (z)> 0 \right \}}$ . Clearly, the domain $\mathsf{U}$ is swept out by complete WKB trajectories and every point is contained in a WKB strip. Thus, for example, let us take $x_0 {\mathrel {\mathop :}=} \tfrac {3}{4} e^{i\pi /3}$ , so $\Phi _0 (x_0) = i$ . Let $\mathsf{U}_0$ be the preimage under $\Phi _0$ of the horizontal strip $\mathsf{H}_0 {\mathrel {\mathop :}=} {\left \{ z ~\big |~ \tfrac {1}{2} < {\operatorname {Im}} (z) < \tfrac {3}{2} \right \}}$ . The Liouville transformation based at $x_0$ is simply $\Phi = \Phi _0 - i$ , and the image of $\mathsf{U}_0$ is the WKB strip ${\left \{ z ~\big |~ {\mathop {\mathrm {dist}}\nolimits } (z, {\mathbb {R}}) < \tfrac {1}{2} \right \}}$ . However, in this example, it is more convenient to work with the Liouville transformation $\Phi _0$ .

Figure 2 Pictured are the complex planes ${\mathbb {C}}_x$ (left) and ${\mathbb {C}}_z$ (right) with $\Phi $ being the Liouville transformation with basepoint $x_0 = 0$ . In ${\mathbb {C}}_x$ , there is a turning point at the origin, indicated by a red circled cross. A few complete WKB trajectories on ${\mathbb {C}}_x$ are drawn in green, with arrows indicating the orientation with respect to the chosen square-root branch $\sqrt {{ {D}}_0} = 2 \sqrt {x}$ , for which the branch cut is taken along the negative real axis. There are two special trajectories, indicated in red, which are not complete: they flow into the turning point in finite time. The domain $\mathsf{U}$ from (97) is shaded in blue.

6.2.4 Regularizing the coefficients.

The first observation is that the Riccati equation (91) does not satisfy hypothesis (2) of Theorem 5.1. This is because the rescaled coefficients $a/\sqrt {{ {D}}_0}$ , $b/\sqrt {{ {D}}_0}$ , and $c/\sqrt {{ {D}}_0}$ are, respectively, , and , which are unbounded on $\mathsf{U}$ . This unboundedness is caused by two separate problems: one is that is unbounded at infinity, and the other is that is unbounded near the turning point at the origin. The latter problem is remedied by restricting to $\mathsf{U}_0$ . In order to remedy the first problem and proceed according to our method, it is necessary to regularize the coefficients of this Riccati equation as in §5.4.

If we make a change of the unknown variable over $\mathsf{U}$ , then the Riccati equation (91) gets transformed into

(98) $$ \begin{align} \hbar \partial_x g = \sqrt{x} g^2 - \tfrac{\hbar}{2x} g - \sqrt{x}. \end{align} $$

This is equation (86) from §5.4 with

$$ \begin{align*} a' = \sqrt{x},\qquad b' = \tfrac{1}{2} \hbar x^{-1}, \qquad c' = - \sqrt{x},\qquad \chi = \sqrt{x} = \tfrac{1}{2} \sqrt{{{D}}}. \end{align*} $$

The Riccati equation (98) now satisfies hypothesis (3’) from Proposition 5.4 with regularizing factor $\chi = \sqrt {x}$ .

The coefficients of the formal solutions $\hat {g}_{\pm }$ of (98) are given by

. Explicitly, the first few coefficients are

We now follow the step-by-step procedure in the proof of Theorem 5.1 in §5.1.

Step 0: Preliminary transformation.

We begin by performing the preliminary transformations (one for each of $\pm $ ) of the unknown variable $g \mapsto \tilde {g}$ given by

(99)

They transform the regularized Riccati equation (98) into a pair of Riccati equations

(100) $$ \begin{align} \tfrac{\hbar}{2\sqrt{x}} \partial_x \tilde{g} - \tilde{g} = \hbar \big( \tfrac{1}{2} \tilde{g}^2 - d_2^{\pm} x^{-3} \big) = \hbar \big( \tfrac{1}{2} \tilde{g}^2 \pm \tfrac{5}{32} x^{-3} \big). \end{align} $$

This is equation (64) from §5.1 with $\tilde {a} = \tfrac {1}{2}$ , $\tilde {b} = 0$ , and $\tilde {c} = \pm \tfrac {5}{32} x^{-3}$ . Applying the Liouville transformation $\Phi _0$ to the Riccati equations (100), we get

(101) $$ \begin{align} \hbar \partial_z {{F}} - {{F}} = \hbar \Big( \tfrac{1}{2} {{F}}^2 - d_2^{\pm} \big( 3 z /4 \big)^{-2} \Big) = \hbar \Big( \tfrac{1}{2} {{F}}^2 \pm \tfrac{5}{18} z^{-2} \Big). \end{align} $$

The unknown variables $\tilde {g}$ and ${ {F}}$ are related by . Equation (101) is equation (45) from §5.1 with ${ {A}}_0 = \pm \tfrac {5}{18} z^{-2}$ , ${ {A}}_1 = 0$ , and ${ {A}}_2 = \tfrac {1}{2}$ .

Step 1: The analytic Borel transform.

Since ${ {A}}_i$ are independent of $\hbar $ , it follows that their Borel transforms $\alpha _i$ are zero, and so the Borel transform of (101) is the following PDE:

(102) $$ \begin{align} \partial_z \phi - \partial_{\xi} \phi = \tfrac{1}{2} \phi \ast \phi. \end{align} $$

This is equation (49) from §5.1 with $\alpha _0 = \alpha _1 = \alpha _2 = 0$ , $a_0 = \pm \tfrac {5}{18} z^{-2}$ , $a_1 = 0$ , and $a_2 = \tfrac {1}{2}$ . In this case, the tubular neighborhood $\Xi _+$ can be taken arbitrarily large.

Step 2: The integral equation.

The PDE (102) is easy to transform into an integral equation:

(103) $$ \begin{align} \phi (z, \xi) = \phi_0^{\pm} (z) + \frac{1}{2} \int\nolimits_0^{\xi} \phi \ast \phi (z + t, \xi - t) {\mathrm{d}}{t}, \end{align} $$

where $\phi (x, 0) = \phi _0^{\pm } (z) {\mathrel {\mathop :}=} a_0 (z) = \pm \tfrac {5}{18} z^{-2}$ . This is equation (50) from §5.1.

Step 3: Method of successive approximations.

The integral equation (103) is solved by the method of successive approximations. This method yields a sequence ${\left \{ \phi _n^{\pm } \right \}}_{n=0}^{\infty }$ of holomorphic functions given by $\phi _0^{\pm } = a_0 = \pm \tfrac {5}{18} z^{-2}$ , $\phi _1^{\pm } = 0$ , and for $n \geq 2$ ,

$$ \begin{align*} \phi_n^{\pm} &= \frac{1}{2} \sum_{i + j = n-2} \int\nolimits_0^{\xi} \phi_i^{\pm} \ast \phi_j^{\pm} (z + t, \xi - t) {\mathrm{d}}{t} \\ &= \frac{1}{2} \sum_{i + j = n-2} \int\nolimits_0^{\xi} \int\nolimits_0^{\xi - t} \phi_i^{\pm} (z + t, \xi - t - y) \phi_j^{\pm} (z + t, y) {\mathrm{d}}{y} {\mathrm{d}}{t}. \end{align*} $$

This is equation (53) from §5.1. It is easy to see that $\phi _n = 0$ for all n odd because $\phi _1 = 0$ . The first few even terms of this sequence are

$$ \begin{align*} \phi_0^{\pm} &= \pm \tfrac{5}{18} z^{-2}; \\ \phi_2^{\pm} &= \tfrac{1}{12} \big(\pm \! \tfrac{5}{18}\big)^2 \tfrac{3z + 2\xi}{z^3 (z + \xi)^2} \xi^2 \sim \tfrac{1}{6} \big(\pm \! \tfrac{5}{18}\big)^2 z^{-3} \xi \qquad\text{as } \xi \to + \infty; \\ \phi_4^{\pm} &= \tfrac{1}{48} \big(\pm \! \tfrac{5}{18}\big)^3 \tfrac{\xi^4}{z^4 (z + \xi)^2} \sim \tfrac{1}{48} \big(\pm \! \tfrac{5}{18}\big)^3 z^{-4} \xi^2 \qquad\text{as } \xi \to + \infty; \\ \phi_6^{\pm} &= \tfrac{1}{2} \big(\pm \! \tfrac{5}{18}\big)^4 \left( \tfrac{\xi \left(16 \xi^5+810 z^5+1,650 \xi z^4+915 \xi ^2 z^3+70 \xi ^3 z^2-8 \xi ^4 z\right)}{4,320 z^5 (z + \xi)^2 (2z + \xi)} + \log \left(\tfrac{z}{z + \xi}\right) \tfrac{7 \xi^2 + 27 z^2 + 28 \xi z }{72 z^2 (2 z + \xi)^2}\right)\\ &\sim \tfrac{1}{270} \big(\pm \! \tfrac{5}{18}\big)^4 z^{-5} \xi^3 \qquad\text{as } \xi \to + \infty; \\ \phi_8^{\pm} &\sim \tfrac{7}{35,640} \big(\pm \! \tfrac{5}{18}\big)^5 z^{-6} \xi^4 \qquad\text{as } \xi \to + \infty. \end{align*} $$

An exact expression for $\phi _8^{\pm }$ involves logarithms and dilogarithms; it is very long and not very useful, occupying almost half of this page. However, the pattern is clear:

$$ \begin{align*} \phi_{2n}^{\pm} \in {O} \left( \tfrac{{L}^n}{n!} z^{-2} \big( \xi / z^2 \big)^n \right) \qquad\text{as } \xi \to + \infty, \end{align*} $$

for some constant $ {L}> 0$ independent of n and z. It follows that the solution to the integral equation (103) satisfies

$$ \begin{align*} \phi_{\pm} (z, \xi) = \sum_{n=0}^{\infty} \phi_n (z, \xi)\quad\preccurlyeq\quad\sum_{n=0}^{\infty} \tfrac{1}{n!} z^{-2} \big( {L} \xi / z^2 \big)^n = z^{-2} e^{{L} \xi / z^2}\qquad\quad \text{as } \xi \to + \infty. \end{align*} $$

This asymptotic inequality yields the exponential estimate (55) from §5.1 with ${ {A}} = 4$ and $ {K} = 0$ .

Step 4: Laplace transform.

Applying the Laplace transform to $\phi _{\pm }$ , we obtain exact solutions ${ {F}}_{\pm }$ of the two Riccati equations (101):

$$ \begin{align*} {{F}}_{\pm} (z, \xi) {\mathrel{\mathop:}=} \int\nolimits_0^{+\infty} e^{-\xi/\hbar} \phi_{\pm} (z, \xi) {\mathrm{d}}{\xi}. \end{align*} $$

It is evident from the asymptotic behavior of $\phi _{\pm }$ as $\xi \to +\infty $ that this Laplace integral is uniformly convergent for all $z \in \mathsf{H}_0$ and all $\hbar \in \mathsf{S}_0 {\mathrel {\mathop :}=} {\left \{ {\operatorname {Re}} (1/\hbar )> {L} \right \}}$ . Note that it is not uniformly convergent for $z \in \mathsf{H}$ , because the constant ${ {A}}$ in the estimate for $\phi _{\pm }$ grows like $|z|^{-2}$ .

Finally, using the inverse Liouville transformation $\Phi ^{-1}_0: z \mapsto \big ( \frac {3}{4} z \big )^{2/3}$ to go back to the x-variable, we obtain two exact solutions of the Riccati equation (98):

Transforming back to the original Riccati equation (91) via the identities (99) and

, we obtain two exact solutions of the original Riccati equation (91):

These are the two canonical exact solutions on $\mathsf{U}_0$ .