On the oscillation properties of eigenfunctions of Sturm--Liouville problem with singular coefficients

TL;DR

This paper investigates the oscillation properties of eigenfunctions in a singular Sturm--Liouville problem with generalized coefficients, extending classical results to more general, less regular cases.

Contribution

It proves that classical properties of eigenfunction zeros hold even when coefficients are singular or less regular, broadening the understanding of Sturm--Liouville problems.

Findings

Eigenfunctions have zeros with properties similar to the smooth case.

Distribution of zeros follows classical oscillation theorems.

Results apply to problems with singular coefficients and boundary conditions.

Abstract

In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized functions $q,r\in W_2^{-1}[0,1]$ are real-valued and unitary matrix $U\in\mathbb C^{2\times 2}$ is diagonal. The main goal is to prove that well-known (for smooth case) facts about number and distribution of zeros of eigenfunctions hold in general case.