Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{\"o}dinger operators

TL;DR

This paper introduces a novel numerical approach combining Feshbach-Schur perturbation theory with Fourier spectral discretization to solve eigenvalue problems for Schrödinger operators, supported by theoretical analysis and numerical validation.

Contribution

It develops an abstract Feshbach-Schur framework with minimal regularity assumptions and applies it to Fourier spectral discretizations for Schrödinger eigenvalue problems.

Findings

The method effectively approximates eigenvalues of Schrödinger operators.

Theoretical analysis confirms the method's convergence under minimal regularity.

Numerical results validate the theoretical predictions.

Abstract

In this article, we propose a new numerical method and its analysis to solve eigenvalue problems for self-adjoint Schr{\"o}dinger operators, by combining the Feshbach-Schur perturbation theory with the spectral Fourier discretization. In order to analyze the method, we establish an abstract framework of Feshbach-Schur perturbation theory with minimal regularity assumptions on the potential that is then applied to the setting of the new spectral Fourier discretization method. Finally, we present some numerical results that underline the theoretical findings.