Eigenvalues of Fokker-Planck operators

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Abstract
Fourier-transformed Fokker-Planck transport operators, T(k)=-(grad- nu . zeta nu .grad+ik. nu , have discrete spectra for most physically reasonable friction tensors zeta ( nu ). The set of eigenvalues ( lambda alpha (k)) represents a collection of single and/or multiple-valued functions analytic in k, with no other singularities than branch points, which are also branch points for the eigen projections Palpha (k). Corresponding analytic behaviour is found with the eigenfunctions phi alpha (k, nu ), chosen in such a way that ( phi alpha , phi beta *)= delta alpha beta and Palpha =(., phi alpha *) phi alpha . Eigen nilpotents and generalised eigenfunctions (which are not proper eigenfunctions) only appear at branch points and possibly at some other meetings points of eigenvalues. Each eigenvalue remains real for sufficiently small real k, until it meets its first branch point. Within this region, the eigenfunctions are conveniently classified by three labels analogous to quantum numbers alpha =(n,l,m). A necessary condition for the meeting point of two eigenvalues lambda nlm and lambda n'l'm' to be a branch point is m=m'.

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