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 Lh = (ihd + A)∗(ihd + A). We construct a semiclassical Birkhoff normal form for Lh and deduce new asymptotic expansions of the smallest eigenvalues in powers of h1/2 in the limit h →0. In particular we see the influence of the kernel of B on the spectrum: it raises the energies at order h3/2.
| Original language | English |
|---|---|
| Journal | Analysis and PDE |
| Volume | 17 |
| Issue | 5 |
| Pages (from-to) | 1593-1532 |
| Number of pages | 62 |
| ISSN | 2157-5045 |
| DOIs | |
| Publication status | Published - 2024 |
Keywords
- magnetic Laplacian
- microlocal analysis
- normal form
- pseudodifferential operators
- semiclassical limit
- spectral theory
- symplectic geometry