SINGULARITIES OF DOUBLE SPECTRAL FUNCTION IN POTENTIAL SCATTERING THEORY

The double spectral function of the Mandelstam representation in nonrelativistic potential-scattering theory is analysed with the method sketched by Bessis. It is shown that it may have singularities, connected in the following way to the behaviour of the potential at infinity. For a potential satisfying {Mathematical expression}, μ>0, γ>0, the double spectral function is continuous for γ>5/4, but has for γ≤5/4 singularities of the type ( {Mathematical expression}, where 3/4<γ≤5/4, and δ(t-4 μ 2-μ 4/s), when γ=3/4. These singularities are integrable, and the Mandelstam representation is thus valid for γ≥3/4. For γ<3/4 the singularities are nonintegrable, and the Mandelstam representation in its familiar form is not valid. As follows from the considerations of the preceding paper, the results of this paper are not changed by the introduction of a logarithmically singular repulsive core. © 1969 Società Italiana di Fisica.