# Examples for Markov generators with pure point spectrum

I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ is a probability measure. In functional analytic terms, $L$ is a self-adjoint operator on $L^2(\mu)$ sending the constants to zero, whose discrete spectrum is contained in the interval $(-\infty,0]$.

When looking at $L^2(\mathbb{R},\mu)$, where $\mu$ admits exponential moments, and one assumes that the eigenfunctions of $L$ are orthogonal polynomials, the only three generators that can arise are the Ornstein-Uhlenbeck, Laguerre and Beta generator with the Hermite, Laguerre and Jacobi polynomials as eigenfunctions and spectrum $-\mathbb{N}_0 = \{ -n \mid n \geq 0 \}$ (OU- and Laguerre) or $\{ -n^2 \mid n \geq 0 \}$ (Jacobi). This has been shown in "Classification of diffusion semigroups in R associated with orthogonal polynomials [French]" by O. Mazet, see MathSciNet review

I'm looking for other examples of such generators (with or without explicit semigroup) and any other connected results which describe the structure of the spectrum. I'm also thankful for references where I might find such examples or results.

To clarify, I'm looking for a self adjoint linear operator $L$ on $L^2(E,\mu)$ where

1. $\mu$ is a probability measure on some separable measure space $E$
2. The spectrum of $L$ is discrete
3. $L1=0$
4. $-L$ is positive
5. $L$ is diffusive, i.e. for all smooth functions $\varphi$, one has that $$L \varphi(f) = \varphi'(f) Lf + \varphi''(f) \Gamma(f,f),$$ where $\Gamma(f,g) = \frac{1}{2} \left( L(fg) - f Lg - g Lf \right)$ is the carre du champ operator associated to $L$. This condition describes that $L$ generates a diffusion semigroup.
• I am not sure if this is what you want but Grigorchuk and Zuk showed that the Markov operator for a simple random walk on the lamplighter group with appropriate generating set has pure point spectrum. – Benjamin Steinberg Sep 24 '14 at 14:29
• I don't quite understand what you mean by saying that "the only three generators that can arise are the Ornstein-Uhlenbeck, Laguerre and Beta generator". If you consider the Laplacian as a (very) special case of the Ornstein-Uhlenbeck operator, then I agree. In any case, $\Delta$ on $L^2(\Omega)$, $|\Omega|=1$, with Neumann boundary conditions seems to be what you need: It generates a Markov semigroup under very mild regularity assumption on $\partial\Omega$ and its spectrum can be almost arbitrarily complicated, the main constraint being the (dimension dependent) Weyl asyptotic formula. – Delio Mugnolo Sep 24 '14 at 18:52
• How does $L$ act on non-constant functions? Or is an abstract definition of $L$ contained in what you wrote? – limanac Sep 27 '14 at 7:01
• @DelioMugnolo I've clarified my question and added a reference. Could you please give me a hint where I can find the results you quoted? I'm not an expert in Markov semigroups and not very familiar with the literature. Thank you very much! – herrsimon Sep 30 '14 at 10:57