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A semiclassical Birkhoff normal form for constant-rank magnetic fields

Léo Morin

Vol. 17 (2024), No. 5, 1593–1632
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.

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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