TITLE: Trace of a differential operator
QUESTION [8 upvotes]: Given the differential operator:
$$A=\exp(-\beta H)$$
where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$
and $\beta\gt 0$
How can I get the trace of this operator?
Thanks in advance.

REPLY [7 votes]: The trace of this operator is easily obtained in the following way:
$$
   Z={\rm Tr}\exp(-\beta H).
$$
that is equivalent to
$$
 Z=\sum_n \langle n|\exp(-\beta H)|n\rangle.
$$
Assuming $H|n\rangle=E_n|n\rangle$, this is just
$$
  Z=\sum_n\exp(-\beta E_n).
$$
Your case is the harmonic oscillator $E_n=n+\frac{1}{2}$ and the sum is just a geometric series easy to perform.