Prüfer Transformation and Spectral Analysis for a Sturm--Liouville-Type Equation

TL;DR

This work develops a generalized Prüfer transformation for a non-self-adjoint Sturm--Liouville-type equation with a quasi-derivative, decoupling solutions into amplitude and phase via $p(y'+sy)=-r\cos\theta$ and $y=r\sin\theta$. The authors prove the equivalence between the original second-order equation and a first-order Prüfer system, establish existence and uniqueness of the phase dynamics, and show that zeros are governed by the phase $\theta$, enabling monotonicity results for eigenfunction zeros with respect to the spectral parameter. They derive sharp eigenvalue bounds by converting the problem to a self-adjoint form through $v=e^{\bar s}y$ and applying Rayleigh quotient and Liouville substitutions, yielding explicit lower and upper bounds. Finally, they present a phase-based eigenvalue criterion that reduces spectral questions to root-finding for $\theta(b;\lambda)$, with potential benefits for numerical computation and further generalizations, including time-scale extensions.

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 Prüfer 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.

Theorems & Definitions (15)

• Theorem 2.1: Equivalence of Prüfer system and original system
• proof
• Lemma 2.2: Lipschitz Continuity
• proof
• Corollary 2.3: Global existence and uniqueness
• Remark 3.1
• Theorem 3.2: Monotonicity of zeros with respect to $\lambda$
• proof
• Remark 3.3
• Theorem 3.4: Lower bound on the $n^{\text{th}}$ eigenvalue