# Trace of a differential operator

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.

• In general, this Wikipedia entry may help; in the specific case, Jon's answer gives you how to compute. – Willie Wong May 24 '13 at 15:04

## 1 Answer

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.

• In general, the trace of an operator is the sum over sandwiching it between eigenstates. Thus $\text{tr} f(A)=\sum f(\lambda_n)$. – Sharkos May 24 '13 at 15:06
• @Sharkos: Correct. – Jon May 24 '13 at 15:08
• Thanks, though I wasn't seeking confirmation, but merely emphasizing the underlying idea for the OP's benefit... – Sharkos May 24 '13 at 15:10