Gaps in the Spectrum of the Laplacian in a Strip with Periodic Delta Interaction

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F19%3A50016100" target="_blank" >RIV/62690094:18470/19:50016100 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/content/pdf/10.1134%2FS0081543819040047.pdf" target="_blank" >https://link.springer.com/content/pdf/10.1134%2FS0081543819040047.pdf</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1134/S0081543819040047" target="_blank" >10.1134/S0081543819040047</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Gaps in the Spectrum of the Laplacian in a Strip with Periodic Delta Interaction

Popis výsledku v původním jazyce

We consider the Laplace operator in an infinite planar strip with a periodic delta interaction. The width of the strip is fixed and for simplicity is chosen equal to pi. The delta interaction is introduced on a periodic system of curves. Each curve consists of a finite number of segments, each having smoothness C-1. The curves are supposed to be strictly internal and do not intersect the boundaries of the strip. The period of their location is 2 epsilon pi, where epsilon is a sufficiently small number. The function describing the delta interaction is also periodic on the system of curves and is assumed to be bounded and measurable. The main result is the following fact. If epsilon &lt;= epsilon(0), where epsilon(0) is a certain explicitly calculated number and the norm of the function describing the delta interaction is smaller than some explicit constant, then the lower part of the spectrum of the operator has no internal gaps. The lower part is understood as the band of the spectrum until some point, which is explicitly calculated in terms of the parameter epsilon as a rather simple function. This result can be considered as a first step to the proof of the strong Bethe-Sommerfeld conjecture on the complete absence of gaps in the spectrum of an operator for a sufficiently small period of location of delta interactions.

Název v anglickém jazyce

Gaps in the Spectrum of the Laplacian in a Strip with Periodic Delta Interaction

Popis výsledku anglicky

We consider the Laplace operator in an infinite planar strip with a periodic delta interaction. The width of the strip is fixed and for simplicity is chosen equal to pi. The delta interaction is introduced on a periodic system of curves. Each curve consists of a finite number of segments, each having smoothness C-1. The curves are supposed to be strictly internal and do not intersect the boundaries of the strip. The period of their location is 2 epsilon pi, where epsilon is a sufficiently small number. The function describing the delta interaction is also periodic on the system of curves and is assumed to be bounded and measurable. The main result is the following fact. If epsilon &lt;= epsilon(0), where epsilon(0) is a certain explicitly calculated number and the norm of the function describing the delta interaction is smaller than some explicit constant, then the lower part of the spectrum of the operator has no internal gaps. The lower part is understood as the band of the spectrum until some point, which is explicitly calculated in terms of the parameter epsilon as a rather simple function. This result can be considered as a first step to the proof of the strong Bethe-Sommerfeld conjecture on the complete absence of gaps in the spectrum of an operator for a sufficiently small period of location of delta interactions.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název periodika

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

ISSN

0081-5438

e-ISSN

—

Svazek periodika

305

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

8

Strana od-do

"S16"-"S23"

Kód UT WoS článku

000491519000004

EID výsledku v databázi Scopus

—