NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2017, 8 (3), P. 305–309
Quantum graphs with the Bethe–Sommerfeld property
P. Exner – Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519 Prague; Department of Theoretical Physics, Nuclear Physics Institute CAS, 25068 Řež near Prague, Czech Republic; email@example.com
O. Turek – Department of Theoretical Physics, Nuclear Physics Institute CAS, 25068 Řež near Prague, Czech Republic; Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia; Laboratory for Unified Quantum Devices, Kochi University of Technology, Kochi 782-8502, Japan; firstname.lastname@example.org
In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned δ-coupling at the vertices.
Keywords: periodic quantum graphs, gap number, δ-coupling, rectangular lattice graph, scaleinvariant coupling, Bethe-Sommerfeld conjecture, golden mean.
PACS 03.65.w, 02.30.Tb, 02.10.Db, 73.63.Nm