Realization of the Riemann Hypothesis via Coupling Constant Spectrum

We present a Non-relativistic Quantum mechanical model, which exhibits the realization of Riemann Conjecture. The technique depends on exposing the S-wave Jost function at zero energy and in identifying it with the Riemann ξ(s) function following a seminal paper of N. N. Khuri. ∗I dedicate this note to my teacher, George Sudarshan of the University of Texas at Austin We begin by recalling the all-too-familiar lore that the Riemann hypothesis has been the Holy Grail of mathematics and physics for more than a century [1]. It asserts that all the zeros of ξ(s) have σ = 1 2 , where s = σ±itn, n = 1, 2, 3 . . .∞. It is believed all zeros of ξ(s) are simple. The function ζ(s) is related to the Riemann ξ(s) function via the defining relation [1], ξ(s) = 1 2 s(s− 1)π S 2 Γ (s 2 ) ζ(s) (1) so that ξ(s) is an entire function, where