ON MANDELSTAM'S PROGRAM IN POTENTIAL SCATTERING
Mandelstam's program for constructing the scattering amplitude from its analytic properties and unitarity was analyzed in the case of nonrelativistic scattering by a cut-off potential or by a hard sphere. The asymptotic behavior of the scattering amplitude in the momentum transfer plane was obtained, leading to a double dispersion representation for the amplitude. The usefuienss of this representation is limited by an essential singularity at infinity in the momentum transfer plane. An infinite system of dispersion relations, connecting each partial wave with all succeeding ones, was derived from the dispersion relation for fixed momentum transfer. The partialwave amplitudes must be constructed from this system together with the unitarity conditions. Possible ambiguities in the solution of this problem were investigated. It is shown that ambiguities in the exact solution affecting only a finlte number of partial waves (Castillejo, Dalitz and Dyson ambiguities) do not exist. They would arise, however, in approximate soiutions and it would be very hard, in practice, to eliminate them from the exact solution. The ambiguities can be formulated in terms of the positions of the poles of the S-matrix. A series of sum rules which must be fulfilled by the poles is derived. The solution of the system was investigated in the particular case of scattering by a hard sphere. In this case, if one assumes that the exact solution is known for anguiar momenta larger than some (arbitrarily given) value, each partial-wave dispersion relation for smaller values of the angular momentum can be exactly solved, and it follows from the sum rules that the solution is unique. (auth)
- Research Organization:
- Rio de Janeiro. Centro Brasileiro de Pesquisas Fisicas
- NSA Number:
- NSA-17-003965
- OSTI ID:
- 4756963
- Report Number(s):
- NP-12262
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
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