Lax-Phillips Scattering for Atomorphic Functions Based on the Eisenstein Transform

Abstract

We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all functions with finite energy-form for the nonEuclidean wave equation. The use of the Eisenstein transform along with some properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.

The Schrödinger operator with Morse potential on the right half-line

“…A slightly different version of this interpretation is given by Lax and Phillips [44], see in particular [44, Appendix 2 to Section 7]. Recently, an operator formulation of the Lax-Phillips approach was given by Uetake [72]. There are also interpretations of the zeta zeros in terms of eigenvalues of operators on various function spaces, given by Connes [18].…”

Concerning Riemann Hypothesis

“…where ξ(s) is Riemann's ξ function [1], Eq. ( 1) and we have shifted the variable 's' by 1 2 , following the convention of Uetake [7]. Since all zeros of Jost function F + (s) lie on the critical line, Rs = − 1 4 , we conclude that the Riemann hypothesis is valid.…”

“…We now follow Uetake's analysis closely [7].The Eisenstein series of two variables E(z, s) on a fundamental domain is by definition, is real analytic for z = x + iy ∈ |H| and E(z, s) is meromorphic in s in the complex plane C. It is regular for each z ∈ |H|, with respect to 's' in Rs 0, except at s = 1 2 . In fact, E(z, 1 2 + s) is an automorphic function on |H|:…”

“…We invoke the well-known results of Lax-Phillips [3] and Faddeev-Pavlov [4] scattering theory of automorphic functions in the Poincare' upper-half plane, z = x + iy, y > 0, −∞ < x < ∞, which was motivated by Gelfand's [5] observation of the analogy between the Eisenstein functions [6] and the scattering matrix, S(λ). Recently, Yoichi Uetake [7] has undertaken a detailed study of Lax-Phillips and Faddeev-Pavlov analysis, by resorting to the technique of Eisenstein transform.…”

“…where the scattering matrix S(s) has the following form, for orthogonal incoming and outgoing subspaces D − and D + [7]:…”