Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs-Reference-Cited by-同舟云学术

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

Published:2023-01-17 Issue:1 Volume:400 Page:463-517
ISSN:0010-3616
Container-title:Communications in Mathematical Physics
language:en
Short-container-title:Commun. Math. Phys.

Author:

Nikolaev Nikita^ORCID

Abstract

AbstractWe 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.

Funder

HORIZON EUROPE Marie Sklodowska-Curie Actions

Publisher

Springer Science and Business Media LLC

Subject

Mathematical Physics,Statistical and Nonlinear Physics

Link

https://link.springer.com/content/pdf/10.1007/s00220-022-04603-7.pdf

Reference99 articles.

1. Nikolaev, N.: Exact solutions for the singularly perturbed Riccati equation and exact wkb analysis. Nagoya Math. J. 1–36. https://doi.org/10.1017/nmj.2022.38; arXiv:2008.06492 [math.CA] (to appear)

2. Jeffreys, H.: On certain approximate solutions of linear differential equations of the second order. Proc. Lond. Math. Soc. 2(23), 428–436 (1924)

3. Wentzel, G.: Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik. Z. Phys. 38, 518–529 (1926). https://doi.org/10.1007/BF01397171

4. Kramers, H.A.: Wellenmechanik und halbzahlige Quantisierung. Z. Phys. 39, 828–840 (1926). https://doi.org/10.1007/BF01451751

5. Brillouin, L.: La mécanique ondulatoire de Schrödinger; une méthode générale de résolution par approximations successives. Compt. Rend. Hebd. Seances Acad. Sci. 183, 24–26 (1926)

Cited by 4 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Quantization of Classical Spectral Curves via Topological Recursion;Communications in Mathematical Physics;2024-04-25

2. Resurgent Structure of the Topological String and the First Painlevé Equation;Symmetry, Integrability and Geometry: Methods and Applications;2024-04-02

3. Gevrey asymptotic implicit function theorem;L’Enseignement Mathématique;2024-01-10

4. Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices;Letters in Mathematical Physics;2023-07-04