Spectral density and Sobolev inequalities for pure and mixed states

Abstract

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of some l^{p,2}-cohomology.

References in corpus (1)

• Sobolev Inequalities for Differential Forms and $L_{q,p}$-cohomology

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