On the asymptotic distribution of eigenvalues

Abstract

It is well known that the asymptotic distribution of the eigenvalues of the one-dimensional Schrödinger equation is provided by the so-called W. K. B. formula. Most proofs of this depend on the approximate solution of the equation in two regions and the joining up of these solutions at the boundaries of the regions in a certain way. These methods are not easily generalized to the Schrödinger equation for dimensions greater than one. In the present paper the methods of Courant & Hilbert are applied to this problem and they lead very simply to a proof of the known result in one dimension and to analogous formulae for the eigenvalue distribution of the Schrödinger equation in two and three dimensions.