un 2 00 3 Dispersive estimates for Schrödinger operators in dimensions one and three
نویسندگان
چکیده
منابع مشابه
A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions
In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class C, α < n−3 2 , for which the solutions to the linear Schrödinger equation in R, −i∂tu = −∆u+ V u, u(0) = f, do not obey the usual L → L∞ dispersive estimate. This contrasts with known results in dimensions n ≤ 3, where a pointwise decay condition on V is generally sufficient to imply dispe...
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We investigate L → L∞ dispersive estimates for the Schrödinger equation iut − ∆u + V u = 0 in odd dimensions greater than three. We obtain dispersive estimates under the optimal smoothness condition for the potential, V ∈ C(n−3)/2(Rn), in dimensions five and seven. We also describe a method to extend this result to arbitrary odd dimensions.
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We show a family of virial-type identities for the Schrödinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension n ≥ 3, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trapping electromagnetic potential with almost Coulomb decay.
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We show a family of virial-type identities for the Schrödinger and wave equations with electromagnetic potentials. As a consequence, some weak dispersive inequalities in space dimension n ≥ 3, involving Morawetz and smoothing estimates, are proved; finally, we apply them to prove Strichartz inequalities for the wave equation with a non-trapping electromagnetic potential with almost Coulomb decay.
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