Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
Nikita Nikolaev
Abstract
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.
Topics & Concepts
OdeWKB approximationUniquenessMathematicsOrder (exchange)Complex systemMathematical analysisApplied mathematicsNonlinear systemPhysicsComputer scienceQuantum mechanicsArtificial intelligenceEconomicsFinanceDifferential Equations and Numerical MethodsNonlinear Waves and SolitonsNumerical methods for differential equations