Eigenvalue asymptotics for magnetic fields and degenerate potentials

Francoise Truc (1)

1 IF - Institut Fourier

We present various asymptotic estimates of the counting function of eigenvalues for Schrödinger operators in the case where the Weyl formula does not apply. The situations treated seem to establish a similarity between magnetic bottles (magnetic fields growing at infinity) and degenerate potentials, and this impression is reinforced by an explicit study in classical mechanics, where the classical Hamiltonian induced by an axially symmetric magnetic bottle can be seen as a perturbation of the Hamiltonian derived from an operator with a degenerate potential.

Mots clés: Spectrum; Min-max principle; Weyl asymptotics; Schrödinger operators; Magnetic bottles

Domaines: Physique mathématique [math-ph]

Francoise Truc. Eigenvalue asymptotics for magnetic fields and degenerate potentials. Conference on Operator Theory, Analysys and Mathematical Physics 2008, Jun 2008, Bedlewo, Poland. pp.135-170, ⟨10.1007/978-3-7643-9994-8⟩. ⟨hal-00579292⟩