Exponentially convergent functional-discrete method for eigenvalue transmission problems with discontinuous flux and potential as a function in the space $L_1$

TL;DR

This paper introduces an exponentially convergent FD-method for solving eigenvalue transmission problems with discontinuous flux and $L_1$ potentials, applicable to linear and nonlinear cases, supported by numerical validation.

Contribution

It develops a new FD-method with superexponential convergence for eigenvalue problems involving discontinuous flux and $L_1$ potentials, including both linear and nonlinear cases.

Findings

Superexponential convergence rate established.

Numerical examples confirm theoretical results.

Insights into eigensolutions of nonself-adjoint operators gained.

Abstract

Based on the functional-discrete technique (FD-method), an algorithm for eigenvalue transmission problems with discontinuous flux and integrable potential is developed. The case of the potential as a function belonging to the functional space $L_1$ is studied for both linear and nonlinear eigenvalue problems. The sufficient conditions providing superexponential convergence rate of the method were obtained. Numerical examples are presented to support the theory. Based on the numerical examples and the convergence results, conclusion about analytical properties of eigensolutions for nonself-adjoint differential operators is made.