Periodic quantum graphs from the Bethe-Sommerfeld perspective

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F17%3A00486254" target="_blank" >RIV/61389005:_____/17:00486254 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21340/17:00319053

Výsledek na webu

<a href="http://dx.doi.org/10.1088/1751-8121/aa8d8d" target="_blank" >http://dx.doi.org/10.1088/1751-8121/aa8d8d</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1088/1751-8121/aa8d8d" target="_blank" >10.1088/1751-8121/aa8d8d</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Periodic quantum graphs from the Bethe-Sommerfeld perspective

Popis výsledku v původním jazyce

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 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 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 delta coupling at the vertices and a suitable irrational ratio of the edges. Our result allows one to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to date.

Název v anglickém jazyce

Periodic quantum graphs from the Bethe-Sommerfeld perspective

Popis výsledku anglicky

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 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 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 delta coupling at the vertices and a suitable irrational ratio of the edges. Our result allows one to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to date.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10301 - Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect)

Projekt

<a href="/cs/project/GA17-01706S" target="_blank" >GA17-01706S: Matematicko-fyzikální modely nových materiálů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Physics A-Mathematical and Theoretical

ISSN

1751-8113

e-ISSN

—

Svazek periodika

50

Číslo periodika v rámci svazku

45

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

32

Strana od-do

—

Kód UT WoS článku

000423284300001

EID výsledku v databázi Scopus

2-s2.0-85032215939