Examples of self-adjoint operators on $L^2(\mu)$

Question

I'd like to come up with a number of simple examples of (formally) self-adjoint operators on $L^2(\mu)$, where $L^2(\mu)$ denotes $L^2(\mathbb R)$ with respect to the Gaussian measure $d\mu$

$$d\mu(x) = \frac{1}{\sqrt{π}} e^{−x^2} dx.$$

So far I have these:

(1) $A$, where $A f(x) = f''(x) - x f'(x)$ $\hspace{5mm}$ (Ornstein-Uhlenbeck),

(2) $B$, where $B f(x)= x f(x)$ $\hspace{5mm}$ (position ),

(3) $C$, where $C f(x) = i f'(x)$ $\hspace{5mm}$ (momentum ).

but would like to add at least a few more.

Any related comments/corrections/references very welcome too.

Answer 1

If you're only interested in formally selfadjoint, then \begin{align} L & = e^{x^2}\left[-\frac{d}{dx}p(x)\frac{d}{dx}\right]+q(x) \\ & = -pe^{x^2}\frac{d^2}{dx^2}-p'e^{x^2}\frac{d}{dx}+q \end{align} for real functions $p$, $q$. For example, if $p=e^{-x^2}$, then $$ L = -\frac{d^{2}}{dx^2}+2x\frac{d}{dx}+q $$ I think your example (1) may have been generated with weight $e^{-x^2/2}$ instead of $e^{-x^2}$.

A first order operator that is formally selfadjoint is $$ M = e^{x^2}\left[i\frac{d}{dx}\right]+r $$