Estimates for negative eigenvalues of a random Schrödinger operator

Abstract

where γ ≥ 0 for d ≥ 3, γ > 0 for d = 2 and γ ≥ 1/2 for d = 1. The estimate (1.1) is called the classical Lieb-Thirring inequality. One needs to remark that although for any V ∈ L the eigenvalue sum ∑ j |λj | converges for both V and −V , it follows from our results that the converse need not be true. The sum ∑ j |λj | can converge even for potentials that are not functions of the class L . In the present paper we study the question: how typical is the situation when the right-hand side of (1.1) is infinite, but nevertheless the series in the left-hand side converges? For that purpose, we introduce a certain class of potentials that either decay slower than L-functions or do not decay at all. Potentials in this class will depend on a parameter ω, which runs over a space with a probability measure, so that one can distinguish between typical and not typical ω (typical ω’s run over a set of measure one). Instead of a decay of the potential, our theorems require random oscillations of V = Vω, which ensure that E[Vω(x)] = 0 for all x. First, we establish the estimate