An ODE with Gaussians

Question

I'm tackling an equation of this type: $$\left( a - b e^{-x^2} - e^{-x^2/2} \frac{1}{x} \frac{\partial}{\partial x} x \frac{\partial}{\partial x} e^{- x^2 /2} \right) f(x) = g(x) $$ where $f(x)$ is my unknown and $g(x)$ is given, both are defined for $x\geq 0$, both smooth and square integrable, while $a$ and $b$ are positive parameters. Alas, the differential operator on the LHS isn't easy to invert. My attempts to compute the resolvent have consisted mostly of projecting $f(x)$ on various bases. How would you proceed?