@article{2354,
abstract = {We investigate the ground state properties of a gas of interacting particles confined in an external potential in three dimensions and subject to rotation around an axis of symmetry. We consider the Gross-Pitaevskii (GP) limit of a dilute gas. Analysing both the absolute and the bosonic ground states of the system, we show, in particular, their different behaviour for a certain range of parameters. This parameter range is determined by the question whether the rotational symmetry in the minimizer of the GP functional is broken or not. For the absolute ground state, we prove that in the GP limit a modified GP functional depending on density matrices correctly describes the energy and reduced density matrices, independent of symmetry breaking. For the bosonic ground state this holds true if and only if the symmetry is unbroken.},
author = {Robert Seiringer},
journal = {Journal of Physics A: Mathematical and Theoretical},
number = {37},
pages = {9755 -- 9778},
publisher = {IOP Publishing Ltd.},
title = {{Ground state asymptotics of a dilute, rotating gas}},
doi = {10.1088/0305-4470/36/37/312},
volume = {36},
year = {2003},
}
@article{2357,
abstract = {The classic Poincaré inequality bounds the L q-norm of a function f in a bounded domain Ω ⊂ ℝ n in terms of some L p-norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Τ from Ω and concentrate our attention on Λ = Ω \ Τ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of Τ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that f has a finite L p gradient norm on the whole of Ω, not just on Λ. The important point is that the Poincaré inequality thus obtained bounds the L q-norm of f in terms of the L p gradient norm on Λ (not Ω) plus an additional term that goes to zero as the volume of Τ goes to zero. This error term depends on Τ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the 'nice' domain Ω. In the second generalization we are given a vector field A and replace ∇ by ∇ + iA(x) (geometrically, a connection on a U(1) bundle). Unlike the A = 0 case, the infimum of ∥(∇ + iA)f∥ p over all f with a given ∥f∥ q is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.},
author = {Lieb, Élliott H and Robert Seiringer and Yngvason, Jakob},
journal = {Annals of Mathematics},
number = {3},
pages = {1067 -- 1080},
publisher = {Princeton University Press},
title = {{Poincaré inequalities in punctured domains}},
doi = {10.4007/annals.2003.158.1067 },
volume = {158},
year = {2003},
}
@article{2358,
abstract = {A study was conducted on the one-dimensional (1D) bosons in three-dimensional (3D) traps. A rigorous analysis was carried out on the parameter regions in which various types of 1D or 3D behavior occurred in the ground state. The four parameter regions include density, transverse, longitudinal dimensions and scattering length.},
author = {Lieb, Élliott H and Robert Seiringer and Yngvason, Jakob},
journal = {Physical Review Letters},
number = {15},
pages = {1504011 -- 1504014},
publisher = {American Physical Society},
title = {{One-dimensional Bosons in three-dimensional traps}},
doi = {10.1103/PhysRevLett.91.150401},
volume = {91},
year = {2003},
}
@phdthesis{2414,
author = {Uli Wagner},
publisher = {ETH Zurich},
title = {{On k-Sets and Their Applications}},
doi = {10.3929/ethz-a-004708408},
year = {2003},
}
@inproceedings{2422,
abstract = {We prove a lower bound of 0.3288(4 n) for the rectilinear crossing number cr̄(Kn) of a complete graph on n vertices, or in other words, for the minimum number of convex quadrilaterals in any set of n points in general position in the Euclidean plane. As we see it, the main contribution of this paper is not so much the concrete numerical improvement over earlier bounds, as the novel method of proof, which is not based on bounding cr̄(Kn) for some small n.},
author = {Uli Wagner},
pages = {583 -- 588},
publisher = {SIAM},
title = {{On the rectilinear crossing number of complete graphs}},
year = {2003},
}