Potential scattering in the limit of large coupling

Abstract

We present a new method for the study of analytic properties of various quantities of interest occurring in potential scattering, such as the wave functions and the Jost functions. The method is based on a transformation of the Green's function of the reduced radial Schrodinger equation, and on some special properties of the Hankel functions, namely the location of their zeros and the behaviour of their moduli. Although the method seems to be adapted for studying other analytic properties, our main concern here are analytic and asymptotic properties with respect to the coupling constantg. We assume the potentialV(r) to i) be positive, and nonincreasing everywhere whenr increases, ii) satisfy the integrability conditions ((1), (2) and (3) below). In Sect.1, we give a brief survey of the result of Frank which are obtained by more complicated techniques and under the additional condition that the square root ofV should be a Laplace transform starting at a strictly positive value. Although our results are identical to those of Frank, our method, besides its simplicity, permits us to remove this last very restrictive condition. In Sect.2, we present our method, and use it to show essentially that not only the wave functions are analytic entire ing for each value of the momentumk in appropriate domains, but also that the order of these quantitics with respect tog is effectively 1/2, and their type effectively\(\int\limits_0^r {V^{\tfrac{1}{2}} } \) for the regular solution,\(\int\limits_r^\infty {V^{\tfrac{1}{2}} } \) for the Jost solution. In Sect.3 we study the Jost functions, and show that they are also of effective order 1/2 and effective type\(\int\limits_0^r {V^{\tfrac{1}{2}} } \). Concerning the type, we make a remark as to the inadequacy of the reasoning of Frank. We then study the implications of these results for the asymptotic behaviour of the number of bound states, and the phase shifts, wheng is large. There is an Appendix in which we give the proofs of some technical points. As we notice, our results should be of a more general character, and should hold ifV is positive, without being non-increasing.