Spectral analysis of magnetic Laplacians on conformally cusp manifolds

Abstract

We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When $B$ is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When $B$ is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect. In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limit absorption principle. We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.

52 pages. Revised version: references added. To appear in Annales Henri Poincaré

References in corpus (5)

• Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds
• Decay Preserving Operators and Stability of the Essential Spectrum
• Discreteness of spectrum for the magnetic Schrödinger operators. I
• Scattering by a toroidal coil
• The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds

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• Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators
• Magnetic bottles on geometrically finite hyperbolic surfaces
• Limiting Absorption Principle for Schrödinger Operators with Oscillating Potential
• Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends
• Regularity of the eta function on manifolds with cusps
• Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area