The ground state properties of interacting Bose gases in external potentials, as considered in recent experiments, are usually described by means of the Gross-Pitaevskii energy functional. We present here the first proof of the asymptotic exactness of this approximation for the ground state energy and particle density of a dilute Bose gas with a positive interaction.

The ground state properties of interacting Bose gases in external potentials, as considered in recent experiments, are usually described by means of the Gross-Pitaevskii energy functional. We present here the first proof of the asymptotic exactness of this approximation for the ground state energy and particle density of a dilute Bose gas with a positive interaction.

Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. A unique account of the subject, this book provides a complete, self-contained description of research on the stability of matter problem. It introduces the necessary quantum mechanics to mathematicians, and aspects of functional analysis to physicists. The topics covered include electrodynamics of classical and quantized fields, Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, gravitational stability of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit. The book is an up-to-date account for researchers, and its pedagogical style makes it suitable for advanced undergraduate and graduate courses in mathematical physics.

We consider two-dimensional attractive Bose-Einstein condensates in a trap V (x) rotating at the velocity Ω, which can be described by the complex-valued Gross-Pitaevskii energy functional. We prove that there exists a critical rotational velocity 0 < Ω * := Ω * (V ) ≤ ∞, depending on the general potential V (x), such that for any small rotational velocity 0 ≤ Ω < Ω * , minimizers exist if and only if the attractive interaction strength a satisfies a < a * = w 2 2 , where w is the unique positive radial solution of ∆w − w + w 3 = 0 in R 2 . For any 0 < Ω < Ω * , we also analyze the limit behavior of the minimizers as a ր a * , where all the mass concentrates at a global minimum point of VΩ(x) := V (x) − Ω 2 4 |x| 2 . Specially, if V (x) = |x| 2 and 0 < Ω < Ω * (= 2), we further prove that, up to the phase rotation, all minimizers are real-valued, unique and free of vortices as a ր a * .

We show that the Lieb-Thirring inequalities on moments of negative eigenvalues of Schrödinger-like operators remain true, with possibly different constants, when the critical Hardy-weight C | x | − 2 C |x|^{-2} is subtracted from the Laplace operator. We do so by first establishing a Sobolev inequality for such operators. Similar results are true for fractional powers of the Laplacian and the Hardy-weight and, in particular, for relativistic Schrödinger operators. We also allow for the inclusion of magnetic vector potentials. As an application, we extend, for the first time, the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Z α = 2 / π Z\alpha =2/\pi , for α \alpha less than some critical value.