README.md

Computation of Klein-Gordon Eigenvalues

This repository contains a MATLAB routine that computes the eigenvalues of the Klein-Gordon equation in one space dimension. The algorithm is based on a recent research paper in the framework of the Solvability Complexity Hierarchy. The numerical method is based a Birman-Schwinger type decomposition of the Klein-Gordon operator, followed by a matrix approximation, which converges in operator norm. This process identifies the eigenvalues as the poles of a matrix-valued function $z\mapsto \|(I-K(z))^{-1}\|$ in the complex plane. The MATLAB routine returns a contour plot of $\|(I-K(z))^{-1}\|$, as well as the locations of its poles, computed via a gradient ascent method. The contents of this package are:

• main.m: Main script that computes and plots the eigenvalues for a potential given in potential.m;
• potential.m: Potential function;
• build_K.m: Computes the matrix approximation to the function $K(z)$;
• simpson_integral.m, potential_integrals.m: Technical functions that compute the matrix elements of the relevant operators via Simpson's integral method;
• GD.m: Computes the poles of $\|(I-K(z))^{-1}\|$ via standard gradient ascent.

Any comments or queries are welcome at https://frank-roesler.github.io/contact/