Sturmian eigenvalue equations with a Chebyshev polynomial basis

Abstract

A Chebyshev polynomial basis is proposed for the solution of Sturmian eigenvalue equations of the form Aν = f which are encountered in Quantum Scattering theory. A is a non-self-adjoint second order differential operator and the solution is regular at the origin and has an outgoing wave boundary condition asymptotically. Introduction of the boundary conditions of the problem transforms the polynomial expansion into a set of linearly independent basis functions, or Chebyshev set, where each member of the set satisfies the boundary conditions. Substitution of this set into the eigenvalue equation leads to a finite, complex general matrix problem which is solved by conventional techniques. Detailed computation of eigenvalues and eigenfunctions for five cases including analytical and physically realistic examples confirms the inherent polynomial stability of the method characteristic of the minimax norm.