Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis
| Autor: | Nikolaev, Nikita |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Nagoya Mathematical Journal 250 (2023) 434-469 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/nmj.2022.38 |
| Popis: | The singularly perturbed Riccati equation is the first-order nonlinear ODE $\hbar \partial_x f = af^2 + bf + c$ in the complex domain where $\hbar$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., they are Borel summations of the formal divergent $\hbar$-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schr\"odinger equation with a rational potential. Comment: Paper has been reorganised; notation, terminology, and theorem statements made clearer. Essential content unchanged |
| Databáze: | arXiv |
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