EXACT SOLUTIONS FOR THE SINGULARLY PERTURBED RICCATI EQUATION AND EXACT WKB ANALYSIS release_2ne52ynm55hcfdti3sxi2gne3m

Abstract

<jats:title>Abstract</jats:title> The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763022000381_inline1.png" /> <jats:tex-math> $\hbar \partial _x f = af^2 + bf + c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in the complex domain where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763022000381_inline2.png" /> <jats:tex-math> $\hbar $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763022000381_inline3.png" /> <jats:tex-math> $\hbar \to 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0027763022000381_inline4.png" /> <jats:tex-math> $\hbar $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.
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