The energy of dilute Bose gases

For a dilute system of non-relativistic bosons interacting through a positive $L^1$ potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(\rho) \geq 4\pi a \rho^2 (1+ \frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} +o(\sqrt{\rho a^3}\,))$, thereby proving the Lee-Huang-Yang formula for the energy density.Comment: 64 pages, minor correction

Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics

In one and two spatial dimensions there is a logical possibility for identical quantum particles different from bosons and fermions, obeying intermediate or fractional (anyon) statistics. We consider applications of a recent Lieb-Thirring inequality for anyons in two dimensions, and derive new Lieb-Thirring inequalities for intermediate statistics in one dimension with implications for models of Lieb-Liniger and Calogero-Sutherland type. These inequalities follow from a local form of the exclusion principle valid for such generalized exchange statistics.Comment: Revised and accepted version. 49 pages, 2 figure

Excess charge for pseudo-relativistic atoms in Hartree-Fock theory

We prove within the Hartree-Fock theory of pseudo-relativistic atoms that the maximal negative ionization charge and the ionization energy of an atom remain bounded independently of the nuclear charge Z and the fine structure constant \alpha as long as Z\alpha is bounded.Comment: 48 Page

Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb-Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality \cite{ES-IV} and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.Comment: latex file, 31 pages. Dedicated to Elliott H. Lieb on his 70-th birthda

Uniform Lieb-Thirring inequality for the three dimensional Pauli operator with a strong non-homogeneous magnetic field

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.Comment: latex file. Revised final version: typos corrected, the definition of the lengthscale simplified, references added/update