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Full-Text Articles in Physical Sciences and Mathematics
The Schroedinger Equation With Potential In Rough Motion, Marius Beceanu, Avy Soffer
The Schroedinger Equation With Potential In Rough Motion, Marius Beceanu, Avy Soffer
Mathematics and Statistics Faculty Scholarship
This paper proves endpoint Strichartz estimates for the linear Schroedinger equation in 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.
Structure Of Wave Wperators In R^3, Marius Beceanu
Structure Of Wave Wperators In R^3, Marius Beceanu
Mathematics and Statistics Faculty Scholarship
We prove a structure formula for the wave operators in R^3 and their adjoints for a scaling-invariant class of scalar potentials V, under the assumption that zero is neither an eigenvalue, nor a resonance for -\Delta+V. The formula implies the boundedness of wave operators on L^p spaces, 1 \leq p \leq \infty, on weighted L^p spaces, and on Sobolev spaces, as well as multilinear estimates for e^{itH} P_c. When V decreases rapidly at infinity, we obtain an asymptotic expansion of the wave operators. The first term of the expansion is of order < y >^{-4}, commutes with the Laplacian, and exists when …