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The Schrödinger equation with a potential in rough motion. (English) Zbl 1245.35099

Summary: This paper proves endpoint Strichartz estimates for the linear Schrödinger equation in \(\mathbb R^3\), with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large in norm. The potential is also subjected to a time-dependent rescaling, with a non-differentiable dilation parameter. We use the Strichartz estimates to prove the non-dispersion of bound states, when the path is small in norm, as well as boundedness of energy. We also include a sample nonlinear application of the linear results.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
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