A semiclassical Birkhoff normal form for constant-rank magnetic fields

Morin, Léo

doi:10.2140/apde.2024.17.1593

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A semiclassical Birkhoff normal form for constant-rank magnetic fields

Léo Morin

Vol. 17 (2024), No. 5, 1593–1632

DOI: 10.2140/apde.2024.17.1593

Abstract

This paper deals with classical and semiclassical nonvanishing magnetic fields on a Riemannian manifold of arbitrary dimension. We assume that the magnetic field $B = d A$ 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 $ℒ_{ℏ} = {(i ℏ d + A)}^{*} (i ℏ d + A)$ . We construct a semiclassical Birkhoff normal form for $ℒ_{ℏ}$ and deduce new asymptotic expansions of the smallest eigenvalues in powers of $ℏ^{1 ∕ 2}$ in the limit $ℏ \to 0$ . In particular we see the influence of the kernel of $B$ on the spectrum: it raises the energies at order $ℏ^{3 ∕ 2}$ .

Keywords

magnetic Laplacian, normal form, spectral theory, semiclassical limit, pseudodifferential operators, microlocal analysis, symplectic geometry

Mathematical Subject Classification

Primary: 35P15, 81Q20

Secondary: 35S05, 37J40, 70H05

Milestones

Received: 22 July 2021

Revised: 24 August 2022

Accepted: 8 November 2022

Published: 20 June 2024

Authors

Léo Morin
Department of Mathematics Aarhus University Aarhus Denmark

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Vol. 17, No. 5, 2024