Gaps in the spectrum of a cuboidal periodic lattice graph

TL;DR

This paper investigates the spectral gaps of a Hamiltonian on a periodic cuboidal lattice graph, establishing conditions for finiteness of gaps and linking gap structure to continued fractions of edge length ratios, aiding spectral design.

Contribution

It introduces a novel connection between spectral gaps and continued fractions of edge ratios, enabling construction and partial control of spectral gaps in quantum graphs.

Findings

Finite number of spectral gaps under certain conditions

Connection between gap structure and continued fractions

Method for constructing graphs with desired spectral gaps

Abstract

We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As the main result, we find a connection between the arrangement of the gaps and the coefficients in a continued fraction associated with the ratio of edge lengths of the lattice. This knowledge enables a straightforward construction of a periodic quantum graph with any required number of spectral gaps and---to some degree---to control their positions; i.e., to partially solve the inverse spectral problem.