A semiclassical Birkhoff normal form for constant-rank magnetic fields

IF 1.8 1区 数学 Q1 MATHEMATICS
Léo Morin
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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 B = dA has constant rank and admits a discrete well. On the classical part, we exhibit a harmonic oscillator for the Hamiltonian H = |p A(q)|2 near the zero-energy surface: the cyclotron motion. On the semiclassical part, we describe the semiexcited spectrum of the magnetic Laplacian = (id + A)(id + A). We construct a semiclassical Birkhoff normal form for and deduce new asymptotic expansions of the smallest eigenvalues in powers of 12 in the limit 0. In particular we see the influence of the kernel of B on the spectrum: it raises the energies at order 32.

恒级磁场的半经典伯克霍夫正则表达式
本文讨论任意维数的黎曼流形上的经典和半经典非消失磁场。我们假定磁场 B=dA 具有恒定秩,并且存在离散井。在经典部分,我们展示了零能面附近哈密顿 H=|p-A(q)|2 的谐振子:回旋运动。在半经典部分,我们描述了磁拉普拉斯的半激发光谱ℒℏ= (iℏd+A)∗(iℏd+A) 。我们为ℒℏ构建了一个半经典的伯克霍夫(Birkhoff)正态形式,并推导出在极限ℏ→ 0 时以ℏ1∕2 的幂为单位的最小特征值的新渐近展开。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Analysis & PDE
Analysis & PDE MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.80
自引率
0.00%
发文量
38
审稿时长
6 months
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.
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