Computing Klein-Gordon Spectra

TL;DR

This paper develops a constructive numerical method to compute eigenvalues of the Klein-Gordon equation with guaranteed error bounds, and applies it to physically relevant potentials, advancing computational spectral theory.

Contribution

It introduces a new algorithm for eigenvalue computation of the Klein-Gordon equation with error guarantees, and provides abstract enclosures for the spectrum.

Findings

Algorithm successfully computes eigenvalues with guaranteed bounds.

Abstract enclosures accurately predict the spectrum.

Numerical results match theoretical enclosures for various potentials.

Abstract

We study the computational complexity of the eigenvalue problem for the Klein-Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein-Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein-Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.