On a General Condition of Heisenberg for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>Matrix

Abstract

It is shown that the $S$ matrix for an attractive exponential potential, which possesses redundant zeros, does not satisfy a general condition of Heisenberg. To insure the validity of Heisenberg's condition, we introduce the supplementary condition that the interaction potential should vanish for large distances from the scattering center. It is shown that the $S$ matrices for the attractive exponential and the Coulomb potential cut off at a large distance $R$ give correctly both the eigenvalues of energy and the asymptotic behavior of the wave functions for the $s$ states in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$.