On the eigenfunctions of Sturm--Liouville operators with potentials --- distributions

TL;DR

This paper investigates the eigenfunctions of Sturm--Liouville operators with distributional potentials, clarifying their asymptotic behavior and properties of associated biorthogonal systems, extending previous theoretical frameworks.

Contribution

It provides new asymptotic formulas for eigenfunctions and biorthogonal systems of Sturm--Liouville operators with distributional potentials, advancing the understanding of such operators.

Findings

Asymptotic formulas for eigenfunctions are refined.

Properties of biorthogonal systems are characterized.

Theoretical framework for distributional potentials is extended.

Abstract

In this paper we study a Sturm--Liouville operator $Ly=-y"+q(x)y$ in the space $L_2[0,\pi]$ with Direchlet boundary conditions. Here the potential $q$ is a first order distribution: $q\in W_2^{-1}[0,\pi]$. Such operators were defined in our previous papers. Here we clerify two leading terms in asymptotic formulae for eigenfunctions of such operators and for functions of biorthogonal system.