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The eigenfunctions and eigenvalues of the Fokker-Planck operator with lineardrift and constant diffusion are required for expanding time-dependentsolutions and for evaluating our recent perturbation expansion for probabilitydensities governed by a nonlinear master equation. Although well-known in onedimension, for multiple dimensions the eigenfunctions are not explicitly givenin the literature. We develop raising and lowering operators for theFokker-Planck (FP) operator and its adjoint, and use them to obtain expressionsfor the corresponding eigenvalues and eigenfunctions. We show that theeigenfunctions for the forward and adjoint FP operators form a bi-orthogonalset, and that the eigenfunctions reduce to sums of products of Hermitefunctions in a particular coordinate system.

Eigenfunctions of the Multidimensional Linear Noise Fokker-Planck Operator via Ladder Operators