Existence and mapping properties of the wave operator for the Schrödinger equation with singular potential

Georgiev, Vladimir; Ivanov, Angel

doi:10.1090/S0002-9939-05-07854-8

Existence and mapping properties of the wave operator for the Schrödinger equation with singular potential
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by Vladimir Georgiev and Angel Ivanov

Proc. Amer. Math. Soc. 133 (2005), 1993-2003

DOI: https://doi.org/10.1090/S0002-9939-05-07854-8

Published electronically: February 15, 2005

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Abstract:

We consider the Schrödinger equation in three-dimensional space with small potential in the Lorentz space $L^{3/2,\infty }$ and we prove Strichartz-type estimates for the solution to this equation. Moreover, using Cook’s method, we prove the existence of the wave operator. In the last section we prove the equivalence between the homogeneous Sobolev spaces $\dot {H}^s$ and $\dot {H}^s_V$ in the case $0 \leq s < \frac {3}{2}$.

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