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A semi-classical picture of quantum scattering. (English) Zbl 0858.35106

The author considers the asymptotics as \(h\to0\) for Schrödinger equations in the form \[ ih\partial_tu^h=\Bigl[- {\textstyle {h^2\over2}\Delta+ U\Bigl({x\over h}}\Bigr)+ V(x)\Bigr]u^h, \] in \(\mathbb{R}^t\times \mathbb{R}^d\). The case \(U\equiv 0\) corresponds to the “standard” semi-classical problem. In the general case the author shows how the limiting evolution keeps trace of quantum effects and provides a picture of quantum scattering. The essential point is to realize the matching between the asymptotics \(h\to0\) for \(-(h^2/2)+V(x)\), describing the evolution on a macroscopic scale and the asymptotics \(|y|\to+\infty\) for the operator \(-{1\over2}\Delta_y+ U(y)\) associated with the microscopic scale.
Reviewer: B.Helffer (Orsay)

MSC:

35Q40 PDEs in connection with quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
35P25 Scattering theory for PDEs
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References:

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