Star-products, spectral analysis, and hyperfunctions

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 x R;omega). 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 delta U+(t;X) = lim(y down arrow0)U(z;X) and delta _U(t;X) = lim(y up arrow0)U(t;X). delta U+(t;X) is a distribution while delta _U(t;X) is a Beurling-type, Gevrey-class s = 2 ultradistribution. We compute the Fourier transforms in I of delta U+/-(t;x). Both Fourier spectra are discrete but different (e.g. opposite in sign for the harmonic oscillator). The Fourier spectrum of delta U+(t;X) coincides With the spectrum of the selfadjoint operator in the Hilbert space L-2(R) whose Weyl symbol is X. Only the boundary value delta U+(t;X) should be considered as the *-exponential function for the element X, since delta _U(t;X) has no interpretation in the Hilbert space L-2(R).