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where and how is found the scattering quatniztion condition

$ det (1-zS)= 0 $ so the energies or bounded states of scattering come out from this

here $ S=S(k)=e^{2i\delta (k)} $ where could i find some info about it ??

wouldn't be easier to impose the condtion on the functional determinant of the Schroedinguer operator $ det(E-H)=0 $

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