OFF-ENERGY-SHELL PARTIAL-WAVE AMPLITUDES

We present dispersion relations which give the full off-energy-shell T-matrix elements Tll′(p,p′;s) for all values of the parametric energy s in terms of bound-state form factors, a subtraction constant, and the half-off-shell T-matrix elements Tll′(p,s12;s) in the scattering (s>0) region. Study of the half-off-shell T matrix for s>0 shows that it can be written as the product of a real matrix Hll′(p,s) and the on-shell T matrix. We combine these results to obtain a representation of the full off-energy-shell T-matrix elements in terms of experimental on-shell T-matrix elements, the real half-off-shell factors Hll′(p,s), a subtraction constant and bound-state form factors. Our results are based only on assumptions of time-reversal invariance, off-energy-shell unitarity, analyticity, and asymptotic behavior. The results are independent of any specific dynamical assumptions. We conclude with a discussion of the special case of uncoupled partial waves and the advantages of a separable representation of the half-off-shell factors. © 1969 The American Physical Society.