Let
$ T= \frac{ \partial^{2}}{\partial _{x} \partial _{y}}+iay\frac{ \partial}{\partial _{y}}+i(1-a)x \frac{ \partial}{\partial _{x}}+ \frac{i}{2}$
be the second order differential operator, where $ i =\sqrt{-1} $ and $ a $ is a real parameter.
Can we prove that this Hamiltonian is Hermitian (so $ T = T^{+} $)?
It is symmetric on $L^2({\mathbb R}^2)$, because $D_x = i \partial/\partial x$ and $D_y = i \partial/\partial y$ are symmetric, and this is $(S + S^+)/2$ where $S = - D_x D_y + a y D_y + (1-a) x D_x$. Self-adjointness might be trickier.