Bruno Colbois , Corentin Léna , Luigi Provenzano , Alessandro Savo
{"title":"Geometric bounds for the magnetic Neumann eigenvalues in the plane","authors":"Bruno Colbois , Corentin Léna , Luigi Provenzano , Alessandro Savo","doi":"10.1016/j.matpur.2023.09.014","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with uniform magnetic field <span><math><mi>β</mi><mo>></mo><mn>0</mn></math></span> and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields <span><math><mi>β</mi><mo>=</mo><mi>β</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> on a simply connected domain in a Riemannian surface.</p><p>In particular: we prove the upper bound <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mi>β</mi></math></span> for a general plane domain for a constant magnetic field, and the upper bound <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover></mrow></msub><mo></mo><mrow><mo>|</mo><mi>β</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></mrow></math></span> for a variable magnetic field when Ω is simply connected.</p><p>For smooth domains, we prove a lower bound of <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> depending only on the intensity of the magnetic field <em>β</em> and the rolling radius of the domain.</p><p>The estimates on the Riesz mean imply an upper bound for the averages of the first <em>k</em> eigenvalues which is sharp when <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span> and consists of the semiclassical limit <span><math><mfrac><mrow><mn>2</mn><mi>π</mi><mi>k</mi></mrow><mrow><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mfrac></math></span> plus an oscillating term.</p><p>We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is always small.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"179 ","pages":"Pages 454-497"},"PeriodicalIF":2.1000,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001356","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of with uniform magnetic field and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields on a simply connected domain in a Riemannian surface.
In particular: we prove the upper bound for a general plane domain for a constant magnetic field, and the upper bound for a variable magnetic field when Ω is simply connected.
For smooth domains, we prove a lower bound of depending only on the intensity of the magnetic field β and the rolling radius of the domain.
The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when and consists of the semiclassical limit plus an oscillating term.
We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which is always small.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.