Inverse Scattering, the Coupling Constant Spectrum, and the Riemann Hypothesis

It is well known that the s-wave Jost function for a potential, V, is an entire function of with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the coupling constant, , will all be real and negative, _n (0)<0. By rescaling , such that _n <–1/4, and changing variables to s, with =s(s–1), it follows that as a function of s the Jost function has only zeros on the line s _n =1/2+i _n . Thus, finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search.In this paper we make a significant enlargement of the class of potentials needed for a generalization of the above idea. We also make this new class amenable to construction via inverse scattering methods. We show that all one needs is a one parameter class of potentials, U(s;x), which are analytic in the strip, 0Res1, Ims>T ₀, and in addition have an asymptotic expansion in powers of s(s–1)]^–1, i.e. U(s;x)=V ₀(x)+gV ₁(x)+g ² V ₂(x)++O(g ^N ), with g=s(s–1)]^–1. The potentials V _n (x) are real and summable. Under suitable conditions on the V _n s and the O(g ^N ) term we show that the condition, ₀ |f ₀(x)|² V ₁(x)dx0, where f ₀ is the zero energy and g=0 Jost function for U, is sufficient to guarantee that the zeros g _n are real and, hence, s _n =1/2+i _n , for _n T ₀.Starting with a judiciously chosen Jost function, M(s,k), which is constructed such that M(s,0) is Riemann's (s) function, we have used inverse scattering methods to actually construct a U(s;x) with the above properties. By necessity, we had to generalize inverse methods to deal with complex potentials and a nonunitary S-matrix. This we have done at least for the special cases under consideration.For our specific example, ₀ |f ₀(x)|² V ₁(x)dx=0 and, hence, we get no restriction on Img _n or Res _n . The reasons for the vanishing of the above integral are given, and they give us hints on what one needs to proceed further. The problem of dealing with small but nonzero energies is also discussed.