Monotonic Local Decay Estimates

Details Monotonic Local Decay Estimates Monotonic Local Decay Estimates

2011-10-29

arxiv.org/abs/1110.6549v1

Mathematics

Analysis of PDEs

Submitted, 44 pages

Scientific paper

For the Hamiltonian operator H = -{\Delta}+V(x) of the Schr\"odinger Equation with a repulsive potential, the problem of local decay is considered. It is analyzed by a direct method, based on a new, L^2 bounded, propagation observable. The resulting decay estimate, is in certain cases monotonic in time, with no "Quantum Corrections". This method is then applied to some examples in one and higher dimensions. In particular the case of the Wave Equation on a Schwarzschild manifold is redone: Local decay, stronger than the known ones are proved (minimal loss of angular derivatives and lower order of radial derivatives of initial data). The method developed here can be an alternative in some cases to the Morawetz type estimates, with L^2-multipliers replacing the first order operators. It provides an alternative to Mourre's method, by including thresholds and high energies.

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