Complex analytic theory of Sturm-Liouville operators with Schatten $p$-class resolvents

TL;DR

This paper develops a complex analytic framework to analyze the spectral properties of Sturm-Liouville operators with Schatten $p$-class resolvents, revealing universal asymptotics and constructing minimal order characteristic functions.

Contribution

It introduces a universal spectral dependence based on Schatten class properties, constructs minimal order characteristic functions, and derives contour integral representations of spectral zeta-functions.

Findings

Spectral blowup/decay rates depend on the largest Schatten class failure.

Constructed minimal order characteristic functions for Schatten $p$-class resolvent problems.

Derived contour integral formulas for spectral zeta-functions.

Abstract

We use the theory of entire functions of finite order to prove a universal spectral dependence of the blowup/decay rate of solutions of the Sturm-Liouville eigenvalue equation for problems with Schatten $p$-class resolvents. The general form of the asymptotics turns out to depend exclusively on the largest integer $\mathfrak{p}$ such that the underlying resolvents fail to be in the Schatten $\mathfrak{p}$-class. We then use the above result to construct a characteristic function of minimal order for Sturm-Liouville problems with Schatten $p$-class resolvents. This immediately yields contour integral representations of spectral $\zeta$-functions that were previously only known for quasi-regular problems (except for a few examples). We also demonstrate how our methods lead to new results in connection to important classic topics of Liouville-Green (or WKB) asymptotics and the approximation of the spectrum of singular problems via underlying truncated regular problems. All our applications are accompanied by illustrative examples, including the Airy differential equation, harmonic oscillator (and general power potentials), and the Laguerre differential equation.