Séminaire Orléans
# Resonances near spectral thresholds for multichannel discrete Schrödinger operators

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Description

We study the distribution of resonances of discrete Hamiltonians of the form H_0 + ωV, ω ∈ C, near the thresholds of the spectrum of H_0. The unperturbed operator H_0 is a multichannel Laplace type operator on Z^2 ⊗ G whose spectrum admits a band structure in the complex plane, and V is a suitable compact perturbation. Here G is an abstract separable Hilbert space. We distinguish two cases. If G is of finite dimension, resonances exist and do not accumulate at the thresholds. We compute exactly their number and give a precise description on their location in clusters around some special points in the complex plane. If G is of infinite dimension, an accumulation phenomenon occurs at some thresholds. We describe it by means of an asymptotic analysis of the counting function of resonances. Consequences on the distribution of the complex eigenvalues are also given.

This is part of a joint work with Marouane Assal, Olivier Bourget and Diomba Sambou