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A semiclassical Birkhoff normal form for constant-rank magnetic fields

Léo Morin

Abstract

We consider the semiclassical magnetic Laplacian $\mathcal{L}_h$ 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 $\mathcal{L}_h$ in the semiclassical limit $\hbar \rightarrow 0$. We deduce an expansion of all the eigenvalues under a threshold, in powers of $\hbar^{1/2}$.

A semiclassical Birkhoff normal form for constant-rank magnetic fields

Abstract

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

Paper Structure

This paper contains 28 sections, 288 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Related questions and perspectives
  5. Structure of the paper
  6. Reduction of the principal symbol $H$
  7. Notations
  8. Structure of $\Sigma$
  9. Construction of $\Phi_1$ and proof of Theorem \ref{['thm.H.rond.Phi']}
  10. Construction of the normal form $\mathcal{N}_{\hbar}$
  11. Formal series
  12. Formal Birkhoff normal form
  13. Quantizing the normal form
  14. Comparing the spectra of $\mathcal{L}_{\hbar}$ and $\mathcal{N}_{\hbar}$
  15. Spectrum of $\mathcal{N}_{\hbar}$
  16. ...and 13 more sections

Theorems & Definitions (21)

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  • ...and 11 more