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Reconstruction of singularities for solutions of Schrödinger’s equation. (English) Zbl 0554.35031

This paper is concerned with the time-behaviour of singularities of solutions to some Schrödinger equations. There are three types of Hamiltonians considered: \(H=H_ 0+V\), where the perturbation V is a symbol of order 0 (i.e. \(| D_ x^{\alpha}V(x)| \leq c_{\alpha}(1+| x|)^{-| \alpha |})\) and \(H_ 0\) is the Laplacian (the free particle case), the isotropic oscillator or the anisotropic oscillator respectively. In the (perturbed) free particle case it is shown that the kernel \(k_ v(t,x,y)\) of the unitary time evolution operator \(\exp (-itH)\) is smooth in (x,y) for all t. For the isotropic oscillator case, i.e. if \(H_ 0=-1/2\Delta +1/2| x|^ 2\), the singular support of the kernel \(k_ v(t,x,y)\) is determined and it is shown that it decays rapidly in x if \(t=m\pi\), \(m\in {\mathbb{N}}.\)
In the general case of the anisotropic oscillator i.e. if \(H_ 0=- 1/2\Delta +1/2\sum^{n}_{k=1}\omega^ 2_ kx^ 2_ k,\) it is shown that the wave front sets are stable under perturbations V of order 0 in the sense that \(WF(k_ v(t,\cdot,y))\subseteq WF(k(t,\cdot,y))\) where k denotes the kernel of the unperturbed propagator \(\exp (-itH_ 0).\) This is the main result of the paper.
Reviewer: H.Cycon

MSC:

35J10 Schrödinger operator, Schrödinger equation
47Gxx Integral, integro-differential, and pseudodifferential operators
35A20 Analyticity in context of PDEs
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