Interpretation of Complex Roots of the S-Matrix

Abstract

IT has recently been pointed out by Kramers and Heisenberg¹ that from the knowledge of the collision cross-sections between free particles, the energy-levels of the discrete states can be deduced. For this purpose the asymptotic wave function in the continuous spectrum (for one particle under the influence of a fixed centre of force), is considered, where S_l is the diagonal form of the S-matrix and a function of the real momentum k. The stationary states follow then from the condition k = ix, and and, for relativistic particles, It is evidently necessary that x < m. The upper half of the k-plane (x < 0) gives only a repetition of the same physical facts if the roles of zeros and poles of S are interchanged.