# Is this second order differential operator Hermitian (2 variables)?

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 $T = T^{+}$)?

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What's the Hilbert space here? $L^2$ on $\mathbb R^2$ or on some domain? Any boundary conditions? –  user31373 Sep 23 '12 at 23:44
$\large {\rm i}/2$ at the end ?. –  Felix Marin Nov 23 '13 at 1:17

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.