Spectral analysis of magnetic Laplacians on conformally cusp manifolds

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.Comment: 52 pages. Revised version: references added. To appear in Annales Henri Poincar\'

Positive commutators, Fermi golden rule and the spectrum of zero temperature Pauli-Fierz Hamiltonians

We perform the spectral analysis of a zero temperature Pauli-Fierz system for small coupling constants. Under the hypothesis of Fermi golden rule, we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy. We rely on a positive commutator approach introduced by Skibsted and pursued by Georgescu-Gerard-Moller. We complete some results obtained so far by Derezinski-Jaksic on one side and by Bach-Froehlich-Segal-Soffer on the other side.Comment: 28 pages. References added and few typos correcte

A new look at Mourre's commutator theory

Mourre's commutator theory is a powerful tool to study the continuous spectrum of self-adjoint operators and to develop scattering theory. We propose a new approach of its main result, namely the derivation of the limiting absorption principle from a so called Mourre estimate. We provide a new interpretation of this result

Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies

We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime

Unboundedness of adjacency matrices of locally finite graphs

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium.Comment: Typos corrected. Examples added. Cute drawings. Simplification of the main condition. Case of the weight tending to zero more discussed

The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs

The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schr\"odinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schr\"odinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its the deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.Comment: Typos corrected. References and ToC added. Paper slightly reorganized. Section 3.2, about the diagonalization has been much improved. The older section about the stability of the deficiency indices in now in appendix. To appear in Journal of Mathematical Physic

Eigenvalue asymptotics for Schrödinger operators on sparse graphs

much better version.We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics