quantization of this hamiltonian?

let be the Hamiltonian $ H=f(xp) $ if we consider canonical quantization so

$ f( -ix \frac{d}{dx}-\frac{i}{2})\phi(x)= E_{n} \phi(x)$

here 'f' is a real valued function so i believe that $ f(xp) $ is an Hermitian operator, however

How could i solve for a generic 'f' the eqution above ? , in case 'f' is a POlynomial i have no problem to solve it the same in case f is an exponential but for a generic 'f' how could i use operator theory to get the eigenvalue spectrum ?¿

Answer

$xp$ is not Hermitian, but $(xp+px)/2$ is. Thus you need to use the latter as an argument.

To solve the time-independent Schroedinger equatione, first solve it for $H=(xp+px)/2$, and note that the eigenvectors don't change when taking a function of this operator.