Pr\"ufer Transformation and Spectral Analysis for a Sturm--Liouville-Type Equation

TL;DR

This paper introduces a generalized Pr"ufer transformation for a non-self-adjoint Sturm--Liouville-type problem, enabling spectral analysis through eigenvalue bounds and zero monotonicity proofs.

Contribution

It develops a novel generalized Pr"ufer transformation for complex Sturm--Liouville problems, facilitating spectral analysis and eigenvalue estimation.

Findings

Eigenfunction zeros are monotonic with respect to the spectral parameter.

Derived explicit upper and lower bounds for eigenvalues.

Established properties of solutions using the generalized Pr"ufer transformation.

Abstract

We study a second-order differential equation involving a quasi-derivative, leading to a non-self-adjoint Sturm--Liouville-type problem with four coefficient functions. To analyze this equation, we develop a generalized Pr\"ufer transformation that expresses solutions in terms of amplitude and phase variables. We further prove the monotonicity of eigenfunction zeros with respect to the spectral parameter and derive upper and lower bounds for the eigenvalues.