Spectral density and Sobolev inequalities for pure and mixed states

Details Spectral density and Sobolev inequalities for pure and mixed states Spectral density and Sobolev inequalities for pure and mixed states

2009-06-16

arxiv.org/abs/0906.2902v1

Geometric and Functional Analysis, vol 20, no3 (2010), p.817-844

Mathematics

Functional Analysis

Scientific paper

10.1007/s00039-010-0075-6

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.

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