Spectral properties and bound states of the Dirac equation on periodic quantum graphs

TL;DR

This paper studies the spectral properties and bound states of the Dirac equation on periodic quantum graphs, introducing a variational approach to establish existence and multiplicity results for bound states.

Contribution

It develops a variational framework for nonlinear Dirac equations on periodic quantum graphs, proving existence and multiplicity of bound states under certain nonlinear conditions.

Findings

Existence of at least one bound state.

Infinitely many bound states when nonlinearity is even.

Spectral decomposition of the Dirac operator on periodic graphs.

Abstract

We investigate nonlinear Dirac equations on a periodic quantum graph $G$ and develop a variational approach to the existence and multiplicity of bound states. After introducing the Dirac operator on $G$ with a $\mathbb Z^{d}$-periodic potential, we describe its spectral decomposition and work in the natural energy space. Under asymptotically linear or superquadratic assumptions on the nonlinearity, we establish the required linking geometry and a Cerami-type compactness property modulo $\mathbb Z^{d}$-translations. As a consequence, we prove the existence of at least one bound state and, when the nonlinearity is even, infinitely many geometrically distinct bound states.