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By Stanislaw Penczek, Przemyslaw Kubisa, Krzysztof Matyjaszewski

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146) if V is hermitean. 151) between Sqq' and T qq , tells us that the poles of S are the same as for T. The latter ones oeeur, as we saw, just at the bound state energies. What will happen if we change the strength A of the interaction, V == A v? Clearly the binding energies E b will move and therefore the poles. Let us weaken A such that one Eb-value goes to zero. For even smaller A'S it has to disappear from the energy plane cut along 0 ~ E < 00. The reason is, that as an eigenvalue of H it has always to be real, and for loeal potentials it eannot be embedded in the eontinuous speetrum E ~ 0 of H.

65) We see that only the normal component of Pi leads to an additional nonzero contribution to the cross section. Again parity conservation rules out a dependence of I on components of Pi in the scattering plane. Assuming that P j is transversal to the beam direction we encounter a situation as shown in Fig. 1. The mo menta q and q I define the scattering plane. If q I leaves to the left of the beam direction at the angle & N points upwards, N(L)' whereas if q I leaves to the right at the same angle & N points downwards, N(R)' Now having the direction P j at our disposal, we can introduce an azimuthai angle ({J which for spin-dependent forces is a dynamically relevant quantity.

It connects the half-shell T-matrix elements with each other. Once it is solved the on-shell matrix element is also known, and one can calculate the cross section. + ie by an arbitrary complex value z. 127) is just the momentum representation of the operator equation T(z) = V + VGo(z) T(z) . 128) defines the offshell T-matrix. This extension will be needed in the description of systems with more than two particles, as we shall see in Chaps. 3 and 4. Once the new independent energy variable z has been introduced, one can investigate the analytic properties of T(z).

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