On the Chebyshev properties of system of eigenfunctions for Sturm--Liouville problem with singular coefficients

TL;DR

This paper investigates whether classical Chebyshev properties of eigenfunctions in Sturm--Liouville problems extend to cases with singular coefficients, broadening understanding of spectral theory in less regular settings.

Contribution

The paper proves that Chebyshev properties of eigenfunctions hold for Sturm--Liouville problems with singular coefficients, generalizing known results from smooth to singular cases.

Findings

Chebyshev property extends to singular coefficient cases

Eigenfunctions maintain orthogonality and oscillation properties

Results applicable to problems with generalized functions and matrices

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 function $q\in W_2^{-1}[0,1]$ is real-valued, generalized weight function $r\in W_2^{-1}[0,1]$ is positive and unitary matrix $U\in\mathbb C^{2\times 2}$ is diagonal. The goal is to prove that well-known (for smooth case) facts about Chebyshev property of eigenfunctions hold in general case.