A semiclassical Birkhoff normal form for constant-rank magnetic fields

Abstract

We consider the semiclassical magnetic Laplacian L on a Riemannian manifold, with a constant-rank and non-vanishing magnetic field B. Under the localization assumption that B admits a unique and non-degenerate well, we construct three successive Birkhoff normal forms to describe the spectrum of L in the semiclassical limit → 0. We deduce an expansion of all the eigenvalues under a threshold, in powers of 1/2 .

Trace formula for the magnetic Laplacian

“…The proof of this formula, based on the use of the heat equation, was given in [11]. We refer the reader to the papers [4,11,30,54,58] and the bibliography given there for the study of the heat kernel associated 1 N H N , as well as to the papers [64,21], where the asymptotic Weyl formula was studied in the case when the curvature R L is non-degenerate. There is also an extensive literature devoted to studies of the asymptotic behavior of the low-lying eigenvalues of H N .…”

“…There is also an extensive literature devoted to studies of the asymptotic behavior of the low-lying eigenvalues of H N . See, for example, books and review papers [34,39,40,73], as well as recent papers [59,64,65] (and the references therein).…”

Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle with constant rank curvature

“…Applying the results of [10] to get spectral asymptotics for L (j) h , we deduce from Theorem 1 the following corollary.…”

“…In [11], similar results were obtained in the special case of the magnetic Laplacian, using a Birkhoff normal form, but also giving a description of semi-excited states, and a Weyl law. In [10], these result are generalized to constant-rank magnetic fields.…”

“…In this paper, we use Agmon-like estimates to reduce the spectral theory of the Bochner Laplacian with magnetic wells to magnetic Laplacians. Then, we deduce spectral asymptotics for the Bochner Laplacian using the results of [11,10].…”