Evolution Equations on Non-Flat Waveguides |
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Authors: | Piero D’Ancona Reinhard Racke |
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Institution: | (2) Dipartimento di Matematica, Unversit? di Roma “La Sapienza”, Roma, Italy; |
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Abstract: | We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator $$H=-\Delta_{x}-\Delta_{y}+V(x,y)$$ with Dirichlet boundary conditions on an unbounded domain ??, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If ?? is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu???u?=?f. As consequences, we prove smoothing estimates for the Schr?dinger and wave equations associated to H, and Strichartz estimates for the Schr?dinger equation. Additionally, we deduce that the operator H does not admit eigenvalues. |
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