Spectral asymptotics for the Schrödinger operator with bounded, unstructured potentials.

Abstract: High energy spectral asymptotics for Schrödinger operators on compact manifolds have been well studied since the early 1900s and it is now well known that they are intimately related to the structure of periodic geodesics. In this talk, we discuss analogous questions for Schrödinger operators on $R^d$ with potentials bounded together with all their derivatives. Since the geodesic flow on $R^d$ has no periodic trajectories (or indeed looping trajectories) one might guess that the spectral projector has a complete asymptotic expansion in powers of the spectral parameter. We show that when $d=1$, this is indeed the case. When $d=2$, we give a large class of potentials whose spectral projectors have complete asymptotics. Nevertheless, in $d\ge 2$, we construct examples where complete asymptotics fails. Based on joint work with J. Galkowski and R. Shterenberg

mathematical physicsspectral theory

Audience: researchers in the topic

Munich-Copenhagen-Santiago Mathematical Physics seminar

Series comments: The MAS-MP seminar series has now changed name to MCS-MP.

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