## Semiclassical asymptotic behavior of the scattering of wave packets by the rapidly changing potential given by the equation $$-2| \nabla \Phi | ^ 2/\cosh ^ 2[\Phi (X)/h]+V_ 0(X)$$.(English. Russian original)Zbl 0696.35125

Sov. Phys., Dokl. 32, No. 8, 633-635 (1987); translation from Dokl. Akad. Nauk SSSR 295, 1347-1351 (1987).
Consider the Schrödinger equation $(1)\quad ih(\partial \psi /\partial t)=-h^ 2\Delta \psi +V\psi,\quad x\in {\mathbb{R}}^ n,\quad \psi (x,0)=\exp (iS_ 0(x)/h)\phi_ 0(x),$ describing the scattering of a rapidly oscillating wave packet by the rapidly varying potential $V=- 2| \nabla \phi |^ 2/\cosh^ 2[\phi (x)/h]+v_ 0(x).$ In this paper the semiclassical asymptotic behavior (h$$\to 0)$$ of the solution of the Cauchy problem (1) is investigated.
Reviewer: J.H.Tian

### MSC:

 35P25 Scattering theory for PDEs 35K15 Initial value problems for second-order parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs