Suppose I have a process
$$\eta = \frac{2-\tau}{\tau}$$
which is a Markov process taking values in $[1,\infty)$, which has infinitesimal generator
$$\mathcal{L} = \frac{1}{a}\frac{\partial}{\partial \eta}(\eta^2-1)\frac{\partial}{\partial \eta}$$
where $a$ is a constant. How do I show $\mathcal{L}$ is self-adjoint, ie
$$\int \phi(\eta)\mathcal{L}\psi(\eta) d\eta = \int \phi(\eta)\mathcal{L}^{\star}\psi(\eta) d\eta$$
where $\mathcal{L^{\star}}$ is the adjoint of $\mathcal{L}.$
Hint You need to show that $$\int \phi\mathcal{L}\psi dx = \int \phi\mathcal{L}^{\star}\psi dx$$
By definition $$\int \phi\mathcal{L}^{\star}\psi dx= \int \psi\mathcal{L}\phi dx$$
Therefore, to show that $\mathcal L$ is selfadjoint you need to show that $$\int \phi\mathcal{L}\psi d\eta =\int \psi\mathcal{L}\phi d\eta$$