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

- $\mu$ is a probability measure on some separable measure space $E$
- The spectrum of $L$ is discrete
- $L1=0$
- $-L$ is positive
- $L$ 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.