Semiclassical analysis in the limit circle case

TL;DR

This paper develops a semiclassical approach to analyze second order differential equations in the limit circle case, providing explicit solutions, asymptotic formulas, and a comprehensive description of self-adjoint extensions and their resolvents.

Contribution

It introduces a common semiclassical Ansatz for all spectral parameters in the limit circle case, enabling detailed asymptotic analysis and characterization of self-adjoint realizations.

Findings

Constructed solutions with prescribed asymptotics

Derived asymptotic formulas for all solutions

Characterized self-adjoint boundary conditions and resolvents

Abstract

We consider second order differential equations with real coefficients that are in the limit circle case at infinity. Using the semiclassical Ansatz, we construct solutions (the Jost solutions) of such equations with a prescribed asymptotic behavior for $x\to\infty$. It turns out that in the limit circle case, this Ansatz can be chosen common for all values of the spectral parameter $z$. This leads to asymptotic formulas for all solutions of considered differential equations, both homogeneous and non-homogeneous. We also efficiently describe all self-adjoint realizations of the corresponding differential operators in terms of boundary conditions at infinity and find a representation for their resolvents.