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Existence of solutions for Schrödinger evolution equations

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Abstract

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationi∂u/∂t=(−1/2)Δu+V(t,x)u,u(0)=u 0. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u 0εC 1(ℝ,L 2) ∩C 0(ℝH 2(ℝn)) foru 0εH 2(ℝn). The conditions are general enough to accommodate moving singularities of type ∣x∣−2+ɛ(n≧4) or ∣x∣−n/2+ɛ(n≧3).

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  1. Department of Pure and Applied Sciences, University of Tokyo, 3-8-1 Komaba, Meguroku, 153, Tokyo, Japan

    Kenji Yajima

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  1. Kenji Yajima
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Communicated by B. Simon

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Yajima, K. Existence of solutions for Schrödinger evolution equations. Commun.Math. Phys. 110, 415–426 (1987). https://doi.org/10.1007/BF01212420

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  • DOI: https://doi.org/10.1007/BF01212420

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