Spectral density and Sobolev inequalities for pure and mixed states

TL;DR

This paper establishes universal Sobolev-type inequalities for positive operators with specific spectral decay, applicable to both pure and mixed quantum states, and relates spectral properties to topological invariants.

Contribution

It introduces general Sobolev inequalities for operators without submarkovian assumptions, extending their applicability to mixed states and spectral bounds on domains.

Findings

Derived Sobolev inequalities for positive operators with spectral decay.

Established bounds on Dirichlet spectrum distribution for domains.

Connected Novikov-Shubin invariants to cohomology torsion vanishing.

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.