A Fermi golden rule for quantum graphs

TL;DR

This paper develops a Fermi golden rule to estimate decay rates of states in quantum graphs and similar systems, supported by numerical experiments and a resonance existence result.

Contribution

It introduces a new Fermi golden rule applicable to quantum graphs and boundary problems on surfaces with cusps, including uniform resonance existence results.

Findings

Derived a Fermi golden rule for quantum graphs

Established a resonance existence result uniform over energies

Validated results through numerical experiments

Abstract

We present a Fermi golden rule giving rates of decay of states obtained by perturbing embedded eigenvalues of a quantum graph. To illustrate the procedure in a notationally simpler setting we also present a Fermi Golden Rule for boundary value problems on surfaces with constant curvature cusps. We also provide a resonance existence result which is uniform on compact sets of energies and metric graphs. The results are illustrated by numerical experiments.