Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell-Schrödinger equations.(English)Zbl 1251.35095

Summary: We consider Schrödinger equations in $$\mathbf R^{1+2}$$ with electro-magnetic potentials. The potentials belong to $$H^{1}$$, and typically they are time-independent or determined as solutions to inhomogeneous wave equations. We prove Kato type smoothing estimates for solutions. We also apply this result to the Maxwell-Schrödinger equations in the Lorentz gauge and prove unique solvability of this system in the energy space.

MSC:

 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35Q61 Maxwell equations
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References:

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