{"title":"A semiclassical Birkhoff normal form for constant-rank magnetic fields","authors":"Léo Morin","doi":"10.2140/apde.2024.17.1593","DOIUrl":null,"url":null,"abstract":"<p>This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi>\n<mo>=</mo>\n<mi>d</mi><mi>A</mi></math> has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>H</mi>\n<mo>=</mo>\n<mo>|</mo><mi>p</mi>\n<mo>−</mo>\n<mi>A</mi><mo stretchy=\"false\">(</mo><mi>q</mi><mo stretchy=\"false\">)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></math> near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the semiexcited spectrum of the magnetic Laplacian <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">ℒ</mi></mrow><mrow><mi>ℏ</mi></mrow></msub>\n<mo>=</mo> <msup><mrow><mo stretchy=\"false\">(</mo><mi>i</mi><mi>ℏ</mi><mi>d</mi>\n<mo>+</mo>\n<mi>A</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo>∗</mo></mrow></msup><mo stretchy=\"false\">(</mo><mi>i</mi><mi>ℏ</mi><mi>d</mi>\n<mo>+</mo>\n<mi>A</mi><mo stretchy=\"false\">)</mo></math>. We construct a semiclassical Birkhoff normal form for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">ℒ</mi></mrow><mrow><mi>ℏ</mi></mrow></msub></math> and deduce new asymptotic expansions of the smallest eigenvalues in powers of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℏ</mi></mrow><mrow><mn>1</mn><mo>∕</mo><mn>2</mn></mrow></msup></math> in the limit <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℏ</mi>\n<mo>→</mo> <mn>0</mn></math>. In particular we see the influence of the kernel of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>B</mi></math> on the spectrum: it raises the energies at order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℏ</mi></mrow><mrow><mn>3</mn><mo>∕</mo><mn>2</mn></mrow></msup></math>. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":"37 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis & PDE","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/apde.2024.17.1593","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
This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the semiexcited spectrum of the magnetic Laplacian . We construct a semiclassical Birkhoff normal form for and deduce new asymptotic expansions of the smallest eigenvalues in powers of in the limit . In particular we see the influence of the kernel of on the spectrum: it raises the energies at order .
期刊介绍:
APDE aims to be the leading specialized scholarly publication in mathematical analysis. The full editorial board votes on all articles, accounting for the journal’s exceptionally high standard and ensuring its broad profile.