The asymptotic form of the continuum wavefunctions and redundant poles in the Heisenberg condition

Abstract

The derivation of the Heisenberg condition is re-examined to show why it is not an identity for potentials possessing redundant poles. Consideration of several such potentials for which exact solutions are known reveals that, in the process of taking an asymptotic limit, the usual derivation of the Heisenberg condition improperly neglects a set of terms. These terms are just those necessary to make the Heisenberg condition an identity; more importantly, it is demonstrated that these terms, providing information on the redundant poles, i.e., the sum of the residues of S(k) at the redundant poles, come from the asymptotic expansion of the continuum wavefunction. By this we are able to give the details of the nature of the asymptotic expansion of the continuum wavefunction and the information contained therein.