PROOF OF MANDELSTAM REPRESENTATION FOR LOGARITHMICALLY SINGULAR POTENTIALS

The Mandelstam representation is proved in nonrelativistic potential-scattering theory for logarithmically singular potentials, that is potentials, analytic in Re r>0, having a repulsive core of the form V(r) |r|→0∼-r -2 ln r, and behaving at infinity like V(r) |r|→∞∼Cr -γ exp [-μr], μ>0, γ>5/4. The double spectral function is shown to be a continuous function for those γ. The proof rests on a detailed analysis, with a method developed by the author, of the partial-wave scattering amplitude for large energies and angular momenta. Using the thus obtained expressions for the partial-wave amplitude, the Martin proof of the Khuri dispersion relation, and the Bessis proof of the convergence of the double integral in the Mandelstam representation, are generalized to the case described above. © 1969 Società Italiana di Fisica.