Half-off-shell phase equations for non-local interactions

The Sobel equations for half-shell phase functions are generalized to cover non-local interactions via explicit expressions of the half-shell phase shift in terms of the on-shell phase functions. All partial waves on equal footing are treated. It can be shown that the two types of on-shell phase functions are asymptotically equivalent for non-local interactions too. Therefore, the two types of the half-shell phase functions will be found to coincide asymptotically, providing thus an expression of the half-shell phase shift itself in terms of the on-shell phase function. (D.P.)