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Weyl Calculus and Arithmetic

Part of the Pseudo-Differential Operators book series (PDO, volume 1)

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 z 1 ), 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.

Keywords

Coherent State Modular Form Wigner Function Discrete Series Projective Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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