Compactness of operator related to harmonic oscillator hamiltonian

Question

Let define the operator $$A = -\partial_x^2-\partial_y^2+2iy\partial_x+y^2$$ with domain in the Schwartz space of the rapidly decreasing functions. Using partial Fourier transform in $x$, the operator can be written as $$ \tilde{A}=k^2 -\partial^2_y-2yk+y^2=(k-y)^2-\partial_y^2$$ I have to prove that the resolvent operator of $A$ is compact, that is $(A-z)^{-1}$ is compact for some $z\in \rho(A)$. There is an hint to relate $\tilde{A}$ to the hamiltonian of harmonic oscillator and then work on the explicit resolvent (Mehler formula), but I actually don't get it.