Eigenfunctions and matrix elements for a class of eigenvalue problems with staggered ladder spectra

Abstract

We present an alternate solution to an eigenvalue problem which arises in the study of the Fokker-Planck equation for generalized Ornstein-Uhlenbeck processes. We obtain the staggered ladder spectra found previously but in addition we obtain the normalized eigenfunctions in terms of associated Laguerre polynomials. The representation of the eigenfunctions in this form greatly simplifies the evaluation of matrix elements required in calculating ensemble averages and correlation coefficients for various observables. The solution to the eigenvalue problem is given in the generic and general cases.