Spectrum of a multidimensional periodic lattice

Ondřej Turek

Published 2018-01-08Version 1

We examine spectral properties of a periodic $3$-dimensional cuboidal lattice graph and its $d$-dimensional generalization. Assuming $\delta$ couplings in the vertices, we demonstrate that the spectrum of the Hamiltonian can have any number of gaps, from zero through nonzero finite to infinite. In particular, we provide the first example to date of a quantum graph periodic in dimension $d\geq3$ with a nonzero finite number of spectral gaps.