Periodic quantum graphs from the Bethe–Sommerfeld perspective

Abstract

Abstract. The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is finite. To the date its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the 'Bethe-Sommerfeld property', that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with the vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a δ coupling at the vertices and a suitable irrational ratio of the edges. Our result allows to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to the date.

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.…”

“…The same is true for BUDA(2). (Figure (a)), take grid points [q, p] ∈ N 2 that lie immediately below the graph of f (x) = αx, i.e., [1,2], [2,4], [3,6], [4,8] and [5,11]. Their vertical distances to the graph are approximately 0.24, 0.47, 0.71, 0.94 and 0.18, respectively.…”

“…(25) § We thank the referee for pointing our attention to that result. The argument given in [8,Prop. 3.5] relies on Lemma 3.4 ibidem.…”

Gaps in the Spectrum of a Cuboidal Periodic Lattice Graph

“…Formula (23) was derived in [11], where function υ(γ) ("upsilon") was originally introduced. Ibidem the following basic properties were derived:…”

“…Indeed, every periodic quantum graph studied in the literature proved to have either infinitely many gaps, or no gaps at all. The very existence of a periodic quantum graph featuring a nonzero finite number of gaps was an open problem until 2017, when an example of a graph with this property was explicitly constructed [11,12]. The graph had a form of a planar rectangular lattice supporting δ couplings in the vertices, with the edge lengths and coupling strength carefully adjusted.…”