Gaps in the Spectrum of a Cuboidal Periodic Lattice Graph

Abstract

We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with δ 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.

One-sided Diophantine approximations

“…All examples of periodic quantum graphs studied in the literature until 2017 led to energy spectra with either infinitely many gaps, or no gaps at all. The first examples of quantum graphs that obey the conjecture in a nontrivial manner, i.e., that have a finite nonzero number of gaps in their energy spectra, appeared in [8] and [21]. In accord with [8] let us call a quantum graph having a finite nonzero number of gaps in its energy spectrum to be of the Bethe-Sommerfeld type.…”

“…Let be 1 or 2. Corollary 4.4 implies that every best lower or upper approximations of the -th kind to α has form (21). Conversely, each fraction (21) is a BLDA( ) or a BUDA( ) to α due to Corollary 4.2.…”