Sylvain Golenia

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 limiting 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.

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 studying 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 the Appendix. We also discuss some applications to the dispersive Helmholtz 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.

THE MAGNETIC LAPLACIAN ACTING ON DISCRETE CUSPS

We study several toy-models of cups-like weighted graphs. We prove that the form-domain of the magnetic Laplacian and that of the non-magnetic Laplacian can be different. We establish the emptiness of the essential spectrum and compute the asymptotic of eigenvalues for the magnetic Laplacian.

Comment on “The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs” [J. Math. Phys. 52, 063512 (2011)]

In this note we answer negatively to our conjecture concerning the deficiency indices. More precisely, given any non-negative integer

Un nouveau regard sur l’estimation de Mourre

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A New Look at Mourre’s Commutator Theory

, there is locally finite graph on which the adjacency matrix has deficiency indices

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

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The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

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Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

Resume. La theorie de Mourre est un outil puissant pour etudier le spectre continu d’operateurs auto-adjoints et pour developper une theorie de la diffusion. Dans cet expose nous proposons d’un nouveau regard sur la theorie de Mourre en donnant une nouvelle approche du resultat principal de la theorie : le principe d’aborption limite (PAL) obtenu a partir de l’estimation de Mourre. Nous donnons alors une nouvelle interpretation de ce resultat. Cet expose a aussi pour but d’etre une introduction a la theorie.