Eigenvalue asymptotics for magnetic fields and degenerate potentials Franc¸oise Truc Abstract. We present various asymptotic estimates of the counting function of eigenvalues for Schrodinger operators in the case where the Weyl formula does not apply. The situations treated seem to establish a similarity between mag- netic 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. Table of contents 1. Introduction 2 2. Degenerate potentials 5 2.1. The Tauberian approach 5 2.2. The min-max approach 7 3. Magnetic bottles 9 3.1. General setting 9 3.2. The Euclidean case 10 3.3. The hyperbolic half-plane 13 3.4. Geometrically finite hyperbolic surfaces 19 4. A Neumann problem with magnetic field 21 4.1. A problem arising from super-conductivity 21 4.2. The spectrum in the case of the half-space, for a constant field and for h=1 22 4.3. Non-Weyl-type asymptotics when the field is nearly tangent to the boundary 23 5. A problem of magnetic bottle in classical mechanics 24 5.1. The Lorentz equation 24 5.2. Adiabatic invariants 25 Received by the editors April 2O, 2009. 2000 Mathematics Subject Classification.
- ginzburg-landau func- tional associated
- field
- symmetric magnetic
- counting func
- constant field
- neumann realization
- homogeneous potential
- universal constant