The Feshbach-Schur map and perturbation theory

TL;DR

This paper introduces a perturbation theory for discrete spectra of self-adjoint operators using the Feshbach-Schur map, providing explicit estimates, simplicity, and a fixed point approach, with applications to helium-like ions.

Contribution

It develops a novel perturbation framework based on the Feshbach-Schur map that offers explicit bounds and a fixed point formulation for eigenvalues and eigenfunctions.

Findings

Provides rigorous eigenvalue and eigenfunction estimates with explicit constants.

Simplifies perturbation analysis through a compact, elementary approach.

Applies theory to obtain bounds on ground states of helium-like ions.

Abstract

This paper deals with perturbation theory for discrete spectra of linear operators. To simplify exposition we consider here self-adjoint operators. This theory is based on the Feshbach-Schur map and it has advantages with respect to the standard perturbation theory in three aspects: (a) it readily produces rigorous estimates on eigenvalues and eigenfunctions with explicit constants; (b) it is compact and elementary (it uses properties of norms and the fundamental theorem of algebra about solutions of polynomial equations); and (c) it is based on a self-contained formulation of a fixed point problem for the eigenvalues and eigenfunctions, allowing for easy iterations. We apply our abstract results to obtain rigorous bounds on the ground states of Helium-type ions.