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Gevrey smoothing properties of the Schrödinger evolution group in weighted Sobolev spaces. (English) Zbl 0837.35128

The paper studies the Schrödinger evolution group \(U(t)= \exp(-it H)\), where \(H= -{1\over 2} \Delta+ V\) and \(V\) is a potential satisfying various conditions. In particular, Jensen’s approach is used to derive estimates for the derivatives of the solution of the Schrödinger equation \(i {\partial u\over \partial t}= -{1\over 2} \Delta u+ Vu\) with initial data in suitably \(L^2\) spaces.
Estimates for the time derivatives still hold when \(V= V_1+ V_2\), where \(V_1\) is a smooth potential and \(V_2\) is a rough potential that decays sufficiently rapidly at infinity. The methods and results of the paper are applied also to scattering theory to obtain the location of the resonance poles of the scattering matrix for the equation \(\Delta u+ k^2 u- V(x) u= 0\).
Reviewer: S.Totaro (Firenze)

MSC:

35Q40 PDEs in connection with quantum mechanics
35J10 Schrödinger operator, Schrödinger equation
47A40 Scattering theory of linear operators
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