DIFFERENTIAL GEOMETRY/PDE SEMINAR

Tuesday, June 29, 2004 Padelford C-36 1:30 pm

On a spectral properties of a family of differential operators

Michael Solomyak (Weizmann Institute, Israel)

We study the spectrum of a family A_{α} of self-adjoint partial differential operators, depending on a real parameter α. The differential expression which defines the action of the operator does not involve α, it appears only in the boundary conditions. From the point of view of the Perturbation Theory, we are dealing with the operators, defined via their quadratic forms, and the perturbation is only form-bounded, but not form-compact with respect to the unperturbed operator. This situation is rather unusual for this class of problems. This is reflected in the character of results. There exists a ”borderline” value α_{0}, such that the spectral properties of A_{α} for α < α_{0} and for α > α_{0} are quite different. To a large extent, these properties are determined by an auxiliary Jacobi matrix, which also depends on α.

The family studied was suggested by U. Smilansky as a model of an irreversible quantum system. A large part of the results was obtained in cooperation with S.N. Naboko.

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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