Resonances of invariant differential operators

Abstract: Given a self-adjoint differential operator with continuous spectrum acting on a Hilbert space H, its resonances are the poles of a meromorphic extension across the spectrum of its resolvent acting on a dense subspace of H in which the operator is no longer self-adjoint. They can be thought of as replacements of eigenvalues for problems on noncompact domains. In this talk we will first explore two well-understood cases: the Laplacian on Euclidean spaces and the Laplace-Beltrami operator on rank one Riemannian symmetric spaces of the noncompact type, two settings where a notion of Fourier analysis is available. In both cases, the Laplacian comes from the action of the Casimir operator through the left regular representation of the underlying group, and the Plancherel formula provides a direct integral decomposition of such representation. Elaborating from this point of view, in collaboration with A. Pasquale and T. Przebinda we started to develop methods to study resonances in more general settings. As an example of such methods, in the talk we will consider some instances of Capelli operators and see how one can exploit Howe’s theory for reductive dual pairs. We will also consider the problem of identifying the representations that are naturally attached to the resonances in these settings.

classical analysis and ODEsfunctional analysisrepresentation theoryspectral theory

Audience: researchers in the topic

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