Approximation of eigenvalues of Schr\"odinger operators

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.