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Analytic properties of the scattering amplitude for interaction via nonlocal potentials Davis, Ronald Stuart

Abstract

The derivation of a partial-wave amplitude for scattering by a separable, nonlocal potential given by MCMillan in Nuovo Cimento 29, 4153 (1963) is reviewed. Using his results, an exact expression for the amplitude is derived for a potential of the form -g V(r)V(r¹), where V(r) = ra e-μr , and its analytic properties are studied. The asymptotic behaviour of the amplitude as |ℓ| → ∞ (where ℓ is the usual angular-momentum parameter) is derived, and is shown to permit a Sommerfeld-Watson transformation to be performed on the series expression for the total scattering amplitude in terms of the partial-wave amplitudes. By means of this transformation, a double-dispersion relation is derived for the total amplitude in both the complex-energy and complex-cos θ planes. Explicit forms are derived for the weight functions, and the convergence of the integrals involved is studied. In addition to the usual branch cuts along the positive, real energy and cos θ axes, an extra cut along the negative , real energy axis is found which is not present for the local case. Its origin is traced to the fact that the Wronskian of two solutions of the nonlocal radical Schroedinger equation is not necessarily a constant, as it is in the purely local case; and to the conditions necessary to ensure convergence of the extra integral in the nonlocal Schroedinger equation.

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