Spectral Zeta function

[mhill]

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.

 

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

 

[denian]

The spectral zeta function is a powerful tool in studying the properties of a quantum mechanical system described by the Schrödinger operator. This function relates the eigenvalues of the operator to the trace of the operator, which is defined as the sum of the diagonal elements in any basis. In the case of the Schrödinger operator, the trace can be calculated using the spectral zeta function, which allows for a more efficient and elegant approach to solving problems.

However, as mentioned in the content, one of the challenges in using the spectral zeta function is the need to know the eigenvalues or the determinant of the Schrödinger operator. This information can be difficult to obtain, especially for complex systems. In such cases, it may be necessary to use approximations or numerical methods to estimate the eigenvalues.

Fortunately, there are techniques and algorithms available to help with this task, such as the WKB approximation or the variational method. Additionally, in some cases, the eigenvalues may be known analytically or can be obtained through symmetry considerations.

Overall, the spectral zeta function provides a powerful and elegant way to study the properties of quantum mechanical systems, but it may require some additional calculations or approximations to fully utilize its potential.