The Hardy inequality and the heat equation with magnetic field in any dimension

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61389005%3A_____%2F16%3A00462436" target="_blank" >RIV/61389005:_____/16:00462436 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1080/03605302.2016.1179317" target="_blank" >http://dx.doi.org/10.1080/03605302.2016.1179317</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1080/03605302.2016.1179317" target="_blank" >10.1080/03605302.2016.1179317</a>

Jazyk výsledku

angličtina

Název v původním jazyce

The Hardy inequality and the heat equation with magnetic field in any dimension

Popis výsledku v původním jazyce

n the Euclidean space of any dimension d, we consider the heat semi group generated by the magnetic Schrodinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrodinger operator on the (d-1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrodinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.

Název v anglickém jazyce

The Hardy inequality and the heat equation with magnetic field in any dimension

Popis výsledku anglicky

n the Euclidean space of any dimension d, we consider the heat semi group generated by the magnetic Schrodinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrodinger operator on the (d-1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrodinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BE - Teoretická fyzika

OECD FORD obor

—

Projekt

<a href="/cs/project/GA14-06818S" target="_blank" >GA14-06818S: Rigorózní metody v kvantové dynamice: geometrie a magnetická pole</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

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

Název periodika

Communications in Partial Differential Equations

ISSN

0360-5302

e-ISSN

—

Svazek periodika

41

Číslo periodika v rámci svazku

7

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

33

Strana od-do

1056-1088

Kód UT WoS článku

000380142200003

EID výsledku v databázi Scopus

2-s2.0-84975282623