Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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^{+} $)?

share|improve this question
    
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
add comment

1 Answer 1

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.