Spectral Zeta function

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The discussion focuses on the Spectral Zeta function related to the Schrödinger operator, denoted as H, where the identity involving the eigenvalues E_n is established. The identity states that the sum of the eigenvalues raised to the power of -s is equal to a specific integral involving the trace of the operator e^{-tH}. The challenge presented is the need to determine the eigenvalues or the determinant of the Schrödinger operator to define the trace. It is confirmed that the operator can be diagonalized due to its Hermitian nature, ensuring real eigenvalues.

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Let be H an Schrodinguer operator so [tex]H \phi =E_n \phi[/tex]

then we have the identity

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

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.
 
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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 [tex]H = \operatorname{diag}(E_1, E_2, ..., E_n)[/tex]
 

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