In the following, I am referring to this paper, p. 9.
Consider the operator $L:=-\partial_v (\partial_v + v)$ which is self-adjoint on $$ \mathcal{M}:=\mu L^2(\mu\, dv)=\{g = \mu f: f\in L^2(\mu\, dv)\} $$ (with norm $\Vert g\Vert_\mathcal{M}=\Vert g/\mu\Vert_{L^2(\mu\, dv)}=\int\vert g\vert^2\mu^{-1}\, dv$ and inner product $\langle f,g\rangle_\mathcal{M}=\int fg\mu^{-1}\, dv$ where $\mu(v)=\frac{1}{\sqrt{2\pi}}e^{-v^2/2}$ and $\mu\, dv=\mu(v)\, dv$).
It is said without proof that $L$ has compact resolvent.
I would like to verify this claim. Unfortunately, I did not manage to do so; in fact, I do not even know how to determine the resolvent itself...