Numerical Computation of Non-Equilateral Quantum Graph Spectra

Abstract

In the broad range of studies related to quantum graphs, quantum graph spectra appear as a topic of special interest. They are important in the context of diffusion type problems posed on metric graphs. Theoretical findings suggest that quantum graph eigenvalues can be found as the solutions of a nonlinear eigenvalue problem, and in the special case of equilateral graphs, even as the solutions of a linear eigenvalue problem on the underlying combinatorial graph. The latter, remarkable relation to combinatorial graph spectra will be exploited to derive a solver for the general, non-equilateral case. Eigenvalue estimates from equilateral approximations will be applied as initial guesses in a Newton-trace iteration to solve the nonlinear eigenvalue problem.