Resonances in asymptotic solutions of the Cauchy problem for the Schrödinger equation with rapidly oscillating finite-zone potential
type: Item
instance of: scholarly article
title: Resonances in asymptotic solutions of the Cauchy problem for the Schrödinger equation with rapidly oscillating finite-zone potential
zbMATH Open document ID: 0667.35058
DOI: 10.1007/BF01159126
author: Sergei Yu. Dobrokhotov
published in: Mathematical Notes
publication date: 1988-01-01T00:00:00Z
review text: The scattering of a wave on a rapidly oscillating algebro-geometric potential is investigated, i.e. the Cauchy problem is solved for the Schrödinger operator \[ ih\psi_ t=-h^ 2\psi_{xx}+v\psi \] with algebrogeometric potential \[ v=v(\Phi (x)/n,E(x)),\quad v(\tau,E)=C(E)- 2(\Phi_ x(E)\partial_ t)^ 2 \ln \theta (\tau,E) \] and initial data \(\psi |_{t=0}=A(x)\exp (iS(x)/h).\) Here \(\theta\) is the Riemann g-dimensional theta-function, E(x) is a slowly moving \((2g+I)\)-dimensional vector of spectral boundaries, \(\Phi\) (x) is a slowly changing g-dimension phase etc. A generalization of the well known Maslov method is used for the solution of the problem. The main difficulty of the problem consists in the necessity to take in account interactions with an infinite set of resonances. According to the paper these interactions nevertheless give results only of the order \(\sqrt{h}\) in comparison with the leading term.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 4092018
zbMATH Keywords: scattering of a wave
zbMATH Keywords: rapidly oscillating algebro-geometric potential
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: theta-function
zbMATH Keywords: Maslov method
zbMATH Keywords: interactions
zbMATH Keywords: infinite set of resonances
reviewed by: E. D. Belokolos
MaRDI profile type: Publication
cites work: Q4090743
cites work: Q4183612
cites work: Q3344582
cites work: Q4189539
cites work: Q3317424
full work available at URL: https://doi.org/10.1007/bf01159126
OpenAlex ID: W2130664253
description: scientific article; zbMATH DE number 4092018