# On Completeness of Root Functions of Sturm-Liouville Problems with Discontinuous Boundary Operators

## TL;DR

This paper proves the completeness of root functions for a class of Sturm-Liouville problems with discontinuous boundary operators, using perturbation and spectral methods, in bounded domains with Robin boundary conditions.

## Contribution

It introduces a novel approach combining perturbation and spectral methods to establish completeness for Sturm-Liouville problems with discontinuous boundary operators.

## Key findings

- Root functions form a complete system in Lebesgue and Sobolev spaces.
- The method handles discontinuities in boundary operator coefficients.
- Completeness holds for various types of function spaces.

## Abstract

We consider a Sturm--Liouville boundary value problem in a boun\-ded domain $\cD$ of $\mathbb{R}^n$. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in $\cD$ and the boundary conditions are of Robin type on $\partial \cD$. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.