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

Dobrokhotov, S. Yu.; Shafarevich, A. I.

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

Keywords:

Schrödinger equation; rapidly varying potential; semiclassical asymptotic behavior