Balanced distribution-energy inequalities and related entropy bounds

Abstract

Abstract. Let A be a self-adjoint operator acting over a space X endowed with a partition. We give lower bounds on the energy of a mixed state ρ from its distribution in the partition and the spectral density of A. These bounds improve with the refinement of the partition, and generalize inequalities by Li-Yau and Lieb-Thirring for the Laplacian in R n . They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of ρ, as seen from X, and some spectral entropy, with respect to its energy distribution. On R n , this yields lower bounds on the sum of the entropy of the densities of ρ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on A.

Cwikel's theorem and the CLR inequality

“…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).…”

“…Roughly speaking, the only assumption is the existence of a density of states, and the energy dependence of this density of states determines the way in which γ(x, x) enters the right side of (1.2). This generality of [Ru1,Ru2] was of crucial importance for the results in [FrOl, FrLeLiSe]. In this paper we do not aim at highest possible generality, but we do include a new theorem about operators T on arbitrary measure spaces X.…”

Energy Contribution of a Point-Interacting Impurity in a Fermi Gas

“…We shall follow the method introduced by Rumin in [26]. With the aid of the spectral projections P e 1 1(|∆ L + µ| ≥ e) we have the layer cake representation…”

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.…”