Abstract
We study the ⋆-exponential function U(t;X) of any element X in the affine symplectic Lie algebra of the Moyal ⋆-product on the symplectic manifold (ℝ × ℝ;ω). When X is a compact element, a natural specific candidate for U (t;X) to be the exponential function is suggested by the study we make in the non-compact case. U (t;X) has singularities in the t variable. The analytic continuation U(z;X),z = t + iy, defines two boundary values δ+ U (t;X) = limy↓0 U(z;X) and δ-(t;X) = limy↑0 U(z; X). δ+ U (t;X) is a distribution while δ- U (t;X) is a Beurling-type, Gevrey-class s — 2 ultradistribution. We compute the Fourier transforms in t of δ± U (t;X). Both Fourier spectra are discrete but different (e.g. opposite in sign for the harmonic oscillator). The Fourier spectrum of δ+ U(t;X) coincides with the spectrum of the self adjoint operator in the Hilbert space L 2(ℝ) whose Weyl symbol is X. Only the boundary value δ+ U(t;X) should be considered as the ⋆-exponential function for the element X, since δ- U(t;X) has no interpretation in the Hilbert space L 2(ℝ).
Programa Praxis XXI 2 /2.1/MAT/458/94
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Moreno, C., da Silva, J.A.P. (2000). Star-products, spectral analysis, and hyperfunctions. In: Dito, G., Sternheimer, D. (eds) Conférence Moshé Flato 1999. Mathematical Physics Studies, vol 21/22. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1276-3_16
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DOI: https://doi.org/10.1007/978-94-015-1276-3_16
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