Quantum graphs with the Bethe-Sommerfeld property

0480621 - ÚJF 2018 RIV RU eng J - Článek v odborném periodiku
Exner, Pavel - Turek, Ondřej
Quantum graphs with the Bethe-Sommerfeld property.
Nanosystems: Physics, Chemistry, Mathematics. Roč. 8, č. 3 (2017), s. 305-309. ISSN 2220-8054
Grant CEP: GA ČR GA17-01706S
Institucionální podpora: RVO:61389005
Klíčová slova: periodic quantum graphs * gap number * delta-coupling * rectangular lattice graph * scale-invariant coupling * Bethe-Sommerfeld conjecture * golden mean
Obor OECD: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)

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 delta-coupling at the vertices.
Trvalý link: http://hdl.handle.net/11104/0276356