Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices
Résumé
Spaces S(omega), S{(omega) over bar}, S((omega) over bar) of ultradecreasing ultradifferentiable (or for short, ultra-S) functions, depending on a weight e(omega(x)), are introduced in the context of quantum statistics. The corresponding coefficient spaces in the Fock basis are identified, and it is shown that the Hermite expansion is a tame isomorphism between these spaces. These results are used to link decay properties of density matrices to corresponding properties of the Wigner distribution.