Uniform asyptotic formulae for eigenfunctions of Sturm--Liouville operators with singular potentials

TL;DR

This paper derives uniform asymptotic formulas for eigenfunctions of Sturm--Liouville operators with singular potentials, including distributional and Sobolev space cases, providing precise estimates of the eigenfunctions' behavior.

Contribution

It extends asymptotic analysis of eigenfunctions to Sturm--Liouville operators with singular potentials in Sobolev and distributional spaces, with uniform estimates.

Findings

Derived uniform asymptotic formulas for eigenfunctions.

Extended results to potentials in Sobolev spaces $W_2^{ heta-1}$.

Provided estimates for eigenfunctions with singular potentials.

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 fitst order distribution $q\in W_2^{-1}[0,\pi]$. Such operators were defined in our previous papers. Here we study an asymptotic behaviour of eigenfunctions with uniform estimates of rests. We obtain this estimates also for potentials from Sobolev spaces $q\in W_2^{\theta-1}$, where $\theta\in[0,1/2)$.