Analysis of Regge poles for the Schrödinger equation

Abstract

We study the question addressed by Barut and Dilley ( Barut & Dilley 1963 J. Math. Phys . 4 , 1401–1408) of counting the number of Regge poles for a radial Schrödinger equation. Using the asymptotics of Rudolph Langer, we acquire estimates for the free solutions at infinity for large generalized complex angular momentum | λ |. These estimates allow us to calculate the Wronskian of two particular solutions, which is the function whose zeros are the Regge poles, for large | λ | in the right-half λ -plane. These angular momentum asymptotics are rigorously related to the large-radius asymptotics by generalizing Marianna Shubova’s idea of formulating an integral equation for the solution at infinity. This leads to the proof that for integrable potentials there are only finitely many Regge poles. This should be compared with the ideas of Barut and Dilley, who require that the potential be analytic in the right-half plane with r 2 V ( r ) remaining bounded.