Approximation of eigenvalues of Schr\"odinger operators

TL;DR

This paper derives sharp convergence rate estimates for eigenvalues of Schrödinger operators, presents a numerical algorithm for eigenvalue computation in $L^2( eal)$, and discusses extensions to metric graphs.

Contribution

It provides new sharp estimates for the convergence rates of eigenvalues of Schrödinger operators and introduces an effective numerical algorithm with error bounds.

Findings

Derived sharp convergence rate estimates for eigenvalues.

Developed a numerical algorithm for eigenvalue computation.

Extended results to Schrödinger operators on metric graphs.

Abstract

It is known that convergence of l.s.b. closed symmetric sesquilinear forms implies norm resolvent convergence of the associated self-adjoint operators and this in turn convergence of discrete spectra. In this paper in both cases sharp estimates for the rate of convergence are derived. An algorithm for the numerical computation of eigenvalues of generalized Schr\"{o}dinger operators in $L^2(\mathbb{R})$ is presented and illustrated by explicit examples; the mentioned general results on the rate of convergence are applied in order to obtain error estimates for these computations. An extension of the results to Schr\"{o}dinger operators on metric graphs is sketched.