# Convergence of expansions for eigenfunctions and asymptotics of the spectral data of the Sturm-Liouville problem

## TL;DR

This paper proves uniform convergence of eigenfunction expansions for a Sturm-Liouville problem with summable potential and uses this to derive more accurate asymptotic formulas for eigenvalues and norming constants.

## Contribution

It establishes uniform convergence of eigenfunction expansions for Sturm-Liouville problems with summable potentials, enabling refined asymptotic analysis of spectral data.

## Key findings

- Proved uniform convergence for eigenfunction expansions.
- Derived more precise asymptotic formulas for eigenvalues.
- Enhanced understanding of spectral data asymptotics.

## Abstract

Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $-y" + q \left( x \right) y = \mu y,$ $y \left(0\right)=0,$ $y\left( \pi \right)\cos \beta + y'\left( \pi \right)\sin \beta = 0,$ $\beta \in \left( 0, \pi \right)$ with summable potential $q \in L_{\mathbb{R}}^1 \left[0, \pi \right]$ is proved. This result is used to obtain more precise asymptotic formulae for eigenvalues and norming constants of this problem.