6533b851fe1ef96bd12a971a

RESEARCH PRODUCT

Star-products, spectral analysis, and hyperfunctions

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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).