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PS

I work in this way.

I calculate the eigenvalues and eigenvectors from the Green operator.

Specifically, I have the hamiltonian operator

H = H0 + V

H0 has only continuous spectrum. Using a perturbative expansion, I find the Green operator

G(z) = 1/(z-H)

in terms of V and of the Green operator of H0

G0(z) = 1/(z-H0)

So I find that G(z) has a branch cut and one simple pole. This is consistent with various works and books. Then I calculate the eigenvalues and the corrispondent eigenvectors. So I express ψ and χ in terms of a common basis, and using the Fourier coefficients I can calculate the inner product.