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Dispersive estimates for the Schr\"{o}dinger equation with finite rank perturbations
In this paper, we investigate dispersive estimates for the time evolution of
Hamiltonians where each
satisfies certain smoothness and decay conditions. We show that,
under a spectral assumption, there exists a constant such that
As far as we are aware, this seems to provide the first study of
estimates for finite rank perturbations of the Laplacian in
any dimension.
We first deal with rank one perturbations (). Then we turn to the
general case. The new idea in our approach is to establish the Aronszajn-Krein
type formula for finite rank perturbations. This allows us to reduce the
analysis to the rank one case and solve the problem in a unified manner.
Moreover, we show that in some specific situations, the constant grows polynomially in . Finally, as an
application, we are able to extend the results to and deal with some
trace class perturbations.Comment: 78 page
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