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

Abstract

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

Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis

“…From the assumption that D 0 ≡ 0, it follows that at each order in , we can uniquely solve for f ± k . This establishes the formula (12), from which the other statements readily follow.…”

“…Our main result can be used to give a positive answer to this question in a large class of problems (generalizing in particular the recent results of Nemes). This is briefly described in a special case in §6.3, and a full description is given in [12].…”

“…x ψ(x, ) = xψ(x, ) (see [12]). In this case, X = C x and the sectorial domain S is the open right half-plane {Re( ) > 0}.…”

A Panorama Of Physical Mathematics c. 2022

“…See [279] for a recent discussion. Very recently the Borel summability of the WKB series in many quantum problems has been rigorously established [280].…”

Gevrey Asymptotic Implicit Function Theorem

“…This type of condition is exactly adapted to the Borel-Laplace method, see §A.2. Similar methods are also used in the construction of exact WKB solutions for singularly perturbed ODEs such as the Schrödinger equation [Nik21].…”