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 F (λ) = χ A (]0, λ]) 1,∞ , without assuming e −tA is sub-Markovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum-mechanical sense. As an illustration, one can relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of the p,2 -cohomology for some p ≥ 2.

Cwikel's theorem and the CLR inequality

“…We now prove the dual form of Theorem 2.1. As we mentioned in the introduction, the proof follows closely some ideas of Rumin [Ru1,Ru2].…”

“…It also yields a rather good value for the constant. In this paper we shall derive the CLR inequality (1.1) from (1.2) and we shall extend (1.2) to L 2 (R d ) ⊗ G with constants independent of the dimension of the auxiliary Hilbert space G. Both results are new and go beyond [Ru1,Ru2]. Our results in the operator-valued case improve upon previous results of [Hu1] (who follows [Cw] and has larger constants) and [FrLiSe1] (who can only deal with (−∆) s for 0 < s ≤ 1).…”

“…One of our goals here is to provide a new and simple proof of Cwikel's theorem and the CLR inequality. Our starting point is the remarkable paper [Ru1] by Rumin which contains, among others, the inequality Tr γ 1/2 (−∆)γ 1/2 ≥ const…”

“…In addition to deriving the CLR inequality (1.1) from (1.2) we are able to answer the following conceptual question about (1.2). Namely, besides the new inequality (1.2) Rumin's papers [Ru1,Ru2] contain a new proof of the inequality…”

Book Review: The stability of matter in quantum mechanics

“…Of particular interest to readers of the book is a recent fairly elementary proof of the Lieb-Thirring inequality which appeared [14,15] after the publication of the book.…”