Expansion in the eigenfunctions of the schrödinger equation with a potential having a periodic asymptotic behavior

Abstract

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = −d ^z/dx ²+q(x)(−∞<x<∞)with a potential q(x), satisfying the condition

$$\int\limits_0^{ + \infty } {(1 + x^2 )|q(x) - q_ \pm (x)|dx< \infty ,} $$

where q±(x) are periodic functions.