Bloch electron in an external field. (Russian) Zbl 0714.34128

The scattering problem is studied for the equation \(H\psi =E\psi\) where \(H=-\partial^ 2_ x+p(x)-\epsilon x,\) p: \({\mathbb{R}}\to {\mathbb{R}}\) is a smooth periodic function, \(\epsilon >0\) is a parameter, E is the spectral parameter. Basic objects of investigation are the analogue of the Jost function M(E) and the reflection coefficient \(r(E)=\overline{M(E)}/M(E)\). Under additional assumptions on the analytical properties of the periodic potential p, the analytical properties of the solutions of the equation, of the function M and of the reflection coefficient r are studied, and the distribution of resonances (zeros and poles of the reflection coefficient) in the complex E-plane is investigated. Based on the study of the asymptotic behaviour of the solutions of the equation \(H\psi =E\psi\) as \(\epsilon\to 0\), an asymptotic formula for the function M is obtained and the relationship between the resonance chains and the well- known Wannier-Stark ladders is clarified.
Reviewer: W.Müller


34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U30 Dispersion theory, dispersion relations arising in quantum theory