Dispersive Estimates for the Schrödinger Equation for C

Dispersive Estimates for Schrödinger Operators: a Survey

Dispersive Estimates for Schrödinger Operators with Measure-valued Potentials in R

We prove dispersive estimates for the linear Schrödinger evolution associated to an operator −∆+V in R3, where the potential is a signed measure with fractal dimension at least 3/2.

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...

Strichartz Estimates in Wiener Amalgam Spaces for the Schrödinger Equation

We study the dispersive properties of the Schrödinger equation. Precisely, we look for estimates which give a control of the local regularity and decay at infinity separately. The Banach spaces that allow such a treatment are the Wiener amalgam spaces, and Strichartz-type estimates are proved in this framework. These estimates improve some of the classical ones in the case of large time.

Zero Energy Scattering for One-dimensional Schrödinger Operators and Applications to Dispersive Estimates

We show that for a one-dimensional Schrödinger operator with a potential, whose (j + 1)-th moment is integrable, the j-th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case.