Kronig-Penney electron in a homogeneous electric field. (English) Zbl 0935.34072

Buslaev, V. (ed.) et al., Differential operators and spectral theory. M. Sh. Birman’s 70th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 189(41), 45-57 (1999).
The one-dimensional Schrödinger equation on the real axis with a potential being the sum of a periodic singular potential of the form \(p(x)=V\sum_{-\infty}^\infty\delta(x-n)\) and a potential of the external homogeneous field \(v(x)=-Fx\) is considered. The aim of the paper is to reduce the spectral problem for this equation to a certain new discrete spectral problem. An asymptotic relation between these problems is described. The main result is the asymptotic behaviour of the solutions for large values of the argument.
For the entire collection see [Zbl 0911.00011].


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)