Spectral Zeta function

[mhill]

Let be H an Schrodinguer operator so $$H \phi =E_n \phi$$

then we have the identity

$$\sum_ {n} E_{n}^{-s} = \frac{1}{\Gamma (s)} \int_{0}^{\infty} dt t^{s-1} Tr[e^{-tH}]$$

the problem is , that to define the Trace of an operator i should know the Eigenvalues or the Determinant of the Schrodinguer operator, anyone can help ? thanks.

 

[lbrits]

The assumption is that you can diagonalize it. Being Hermitian, it has real eigenvalues. The similarity transformation that makes it diagonal gets killed by the trace, so you can just pretend $$H = \operatorname{diag}(E_1, E_2, ..., E_n)$$