Weyl Calculus and Arithmetic

Abstract

In this chapter, we consider the even and odd parts of the metaplectic representation of $\tilde G$ (the twofold cover of G = SL(2,ℝ)) in L²(ℝ). Two families of coherent states, parametrized by the hyperbolic half-plane ?, and denoted as (u _z ) and (u ¹ _z ), are built with the help of the eigenstates with lowest energy levels of the harmonic oscillator. Equation (1.3) will be considered in Section 2: this will provide an opportunity to recall some of the main properties of the Weyl calculus and related concepts; the Weyl calculus is also a model for other symbolic calculi to be introduced later. Families of coherent states of an arithmetic nature will be constructed in Section 3: they are related to the Dedekind eta-function, a 24th root of the Ramanujan Δ-function. A succession of two intertwining operators, one a quadratic change of variable from the line to the half-line, the other a version of the Laplace transformation, will provide the link: the latter one will also be used later, in a τ-dependent context. Section 4 is devoted to calculations regarding the Wigner function of a pair of arithmetic coherent states. The spectral resolution of the automorphic function obtained from the diagonal matrix elements of an operator with a radial Weyl symbol against a family of coherent states of an arithmetic nature is finally obtained in Section 5.