A simple modification of the definition of the S-matrix is proposed. It is expected that the divergences related to nonzero self-energies are considerably milder with the modified definition than with the usual one. This conjecture is verified in a few examples using perturbation theory. The proposed formula is written in terms of the total Hamiltonian operator and a free Hamiltonian operator and is therefore applicable in any case when these Hamiltonian operators are known.
The analytic structure of the non-relativistic unitary and non-unitary S-matrix is investigated for the cases of the unknown interactions with the unknown motion equations inside a sphere of radius a, surrounded by the centrifugal and rapidly decreasing (exponentially or by the Yukawian law or by the more rapidly decreasing) potentials. The one-channel case and special examples of many-channel cases are considered. Some kinds of symmetry conditions are imposed. The Schroedinger equation for r > a for the particle motion and the condition of the completeness of the correspondent wave functions are assumed. The connection of the obtained results with the usual (temporal) causality is examined. Finally a scientific program is presented as a clear continuation and extension of the obtained results.
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