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July  2000, 6(3): 705-722. doi: 10.3934/dcds.2000.6.705

The Schrödinger equation with singular time-dependent potentials

Holger Teismann 1,

1. 

Department of Mathematics and Statistics, University of Victoria, P.O. BOX 3045, Victoria, B.C., Canada

Received  August 1999 Revised  March 2000 Published  April 2000

The aim of this note is to extend the theory of (linear) Schrödinger equations with time-dependent potentials developed by K. Yajima [26, 27] to slightly more singular potentials. This is done by proving that the well-known Strichartz estimates for the Schrödinger group remain valid if the usual Lebesgue spaces$^1$ are replaced by the Lorentz spaces $L^{p,2}$. Moreover, the regularity of the solutions can be described more precisely by utilizing a generalized Leibniz rule for fractional derivatives.
Keywords: Strichartz estimates, well-posedeness, generalized Leibnitz rule., time-dependent potentials, Schrödinger equation, generalized Sobolev spaces, Lorentz spaces, evolution systems, fractional derivatives.
Mathematics Subject Classification: 35Q40, 47D06, 35J1.
Citation: Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
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