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The Stark effect and long range scattering in two Hilbert spaces. (English) Zbl 0695.35144

Quantum mechanical scattering is considered in the presence of a constant electric field. The “free” Hamiltonian is \(-\Delta -x_ 1\) and the full Hamiltonian is a perturbation by a long range potential V: \[ D^{\alpha}V(x)=O(1+| x|)^{-| \alpha | /2+\epsilon}\quad for\quad all\quad multi-indices\quad \alpha, \] for some \(\epsilon >0\). (A short range potential can also be included.) It is shown that the two Hilbert space wave operators exist and are complete. The approach is by Enss’s method using Isozaki and Kitada’s “time independent modifiers”. Central to their approach is the construction of the “modifier” which requires approximating a solution of a certain partial differential equation. This approximation is done here by an iteration scheme which is unlike the approximation method of earlier works.
Reviewer: D.A.W.White

MSC:

35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
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