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 \[ i \partial u/\partial t=(-1/2)\Delta u+V(t,x)u,\quad u(0)=u_ 0. \] We provide sufficient conditions on V(t,x) such that the equation generates a unique unitary propagator \(U(t,s)\) and such that \[ U(t,s)u_ 0\in C^ 1({\mathbb{R}},L^ 2)\cap C^ 0({\mathbb{R}},H^ 2({\mathbb{R}}^ n)) \] for \(u_ 0\in H^ 2({\mathbb{R}}^ n)\). The conditions are general enough to accomodate moving singularities of type \(| x|^{-2+\epsilon}\) \((n\geq 4)\) or \(| x|^{-n/2+\epsilon}\quad (n\leq 3)\).
Reviewer: K.Yajima


35K15 Initial value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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