A semiclassical Birkhoff normal form for constant-rank magnetic fields

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 $\hslash \to 0$. We deduce an expansion of all the eigenvalues under a threshold, in powers of ${\hslash }^{1/2}$.

References in corpus (2)

• Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells
• Magnetic WKB constructions on surfaces