The Asymptotic Distribution of Eigenvalues for the Boundary Value Problem $y''(x) - \lambda ^2 p(x)y(x) = 0,y \in L_2 ( - \infty , + \infty )$

Abstract

In this paper we study the problem $y''(x) - \lambda ^2 p(x)y(x) = 0$, $y \in L_2 ( - \infty , + \infty )$. Sufficient conditions are given on $p(x)$ to insure the existence of eigenvalues and to enable one to compute the asymptotic distribution of all large positive eigenvalues. The equation considered is of a form of interest in wave mechanics and particle scattering.