Resonances in asymptotic solutions of the Cauchy problem for the Schrödinger equation with rapidly oscillating finite-zone potential. (English. Russian original) Zbl 0667.35058

Dobrokhotov, S. Yu.

Math. Notes 44, No. 3-4, 656-668 (1988); translation from Mat. Zametki 44, No. 3, 319-340 (1988).

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.

Reviewer: E.D.Belokolos

MSC:

35P25	Scattering theory for PDEs
35J10	Schrödinger operator, Schrödinger equation

Keywords:

scattering of a wave; rapidly oscillating algebro-geometric potential; Cauchy problem; Schrödinger operator; theta-function; Maslov method; interactions; infinite set of resonances

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