Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices

Abstract

Spaces $S_{\omega },S_{\{\omega \}},S_{(\omega )}$ 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 decrease properties of density matrices to corresponding properties of the Wigner distribution.

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