Resonances in asymptotic solutions of the Cauchy problem for the Schrödinger equation with rapidly oscillating finite-zone potential
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.
rapidly oscillating algebro-geometric potential
scientific article; zbMATH DE number 4092018