Uniconvergence theorems for Sturm--Liouville operators with potentials from Sobolev space $W_2^{-1}[0,\pi]$

TL;DR

This paper proves a uniconvergence theorem for Sturm--Liouville operators with potentials in Sobolev space, showing that spectral projections converge uniformly to the function in the space of continuous functions.

Contribution

It establishes a uniconvergence result for Sturm--Liouville operators with potentials in Sobolev space, extending spectral convergence to uniform convergence.

Findings

Spectral projections converge uniformly to functions in $C[0,,)$.

Operators have a Riesz basis of eigenfunctions with Hilbert--Schmidt perturbation.

The uniconvergence theorem holds for potentials in $W_2^{-1}[0,,)$.

Abstract

We consider a Sturm--Liouville $Ly=-y''+q(x)y$ in space $L_2[0,\pi]$ with potential from Sobolev space $W_2^{-1}[0,\pi]$. Moreover, we assume, that $q=u'$, where $u\in L_2[0,\pi]$. We consider Direchlet boundary conditions $y(0)=y(\pi)=0$, although we can treat a boundary conditions of Sturm type. It is known, that operators of such class have a discrete spectr with only accumulation point $+\infty$ and the system $\{y_k\}_1^\infty$ of eigen and associated functions is a Riesz basis in $L_2[0,\pi]$. Moreover, this basis is a Hilbert--Schmidt perturbation of the basis $\{sin(kx)\}_1^\infty$. In this paper we prove the uniconvergence theorem: for any element $f\in L_2[0,\pi]$ the sequence $P_nf-S_nf\to0$ as $n\to\infty$ in $C[0,\pi]$ (here $P_n$ and $S_n$ are the Riesz projectors to $\{y_k\}_1^n$ and $\{\sin(kt)\}_1^n$ respectively).