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Existence of solutions for Schrödinger evolution equations. (English) Zbl 0638.35036

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equation

iu/t=(-1/2)Δu+V(t,x)u,u(0)=u 0 ·

We provide sufficient conditions on V(t,x) such that the equation generates a unique unitary propagator U(t,s) and such that

U(t,s)u 0 C 1 (,L 2 )C 0 (,H 2 ( n ))

for u 0 H 2 ( n ). The conditions are general enough to accomodate moving singularities of type |x| -2+ϵ (n4) or |x| -n/2+ϵ (n3).

Reviewer: K.Yajima

MSC:
35K15Second order parabolic equations, initial value problems
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B65Smoothness and regularity of solutions of PDE
35B40Asymptotic behavior of solutions of PDE
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