TY - GEN AB - We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state. AU - Lieb, Élliott H AU - Robert Seiringer ID - 2363 IS - 2 T2 - Communications in Mathematical Physics TI - Derivation of the Gross-Pitaevskii equation for rotating Bose gases VL - 264 ER - TY - JOUR AB - We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first-order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose-Einstein condensation. AU - Robert Seiringer ID - 2364 IS - 3 JF - Reviews in Mathematical Physics TI - A correlation estimate for quantum many-body systems at positive temperature VL - 18 ER - TY - JOUR AB - We consider a gas of fermions with non-zero spin at temperature T and chemical potential μ. We show that if the range of the interparticle interaction is small compared to the mean particle distance, the thermodynamic pressure differs to leading order from the corresponding expression for non-interacting particles by a term proportional to the scattering length of the interparticle interaction. This is true for any repulsive interaction, including hard cores. The result is uniform in the temperature as long as T is of the same order as the Fermi temperature, or smaller. AU - Robert Seiringer ID - 2365 IS - 3 JF - Communications in Mathematical Physics TI - The thermodynamic pressure of a dilute fermi gas VL - 261 ER - TY - JOUR AB - Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential. AU - Frank, Rupert L AU - Laptev, Ari AU - Lieb, Élliott H AU - Robert Seiringer ID - 2366 IS - 3 JF - Letters in Mathematical Physics TI - Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials VL - 77 ER - TY - CHAP AB - The recent experimental success in creating Bose-Einstein condensates of alkali atoms, honored by the Nobel prize awards in 2001 [1,5], led to renewed interest in the mathematical description of interacting Bose gases. AU - Robert Seiringer ED - Dereziński, Jan ED - Siedentop, Heinz ID - 2368 T2 - Large Coulomb Systems TI - Dilute, trapped Bose gases and Bose-Einstein condensation VL - 695 ER - TY - CHAP AB - One of the most remarkable recent developments in the study of ultracold Bose gases is the observation of a reversible transition from a Bose Einstein condensate to a state composed of localized atoms as the strength of a periodic, optical trapping potential is varied. In [1] a model of this phenomenon has been analyzed rigorously. The gas is a hard core lattice gas and the optical lattice is modeled by a periodic potential of strength λ. For small λ and temperature Bose- Einstein condensation (BEC) is proved to occur, while at large λ BEC disappears, even in the ground state, which is a Mott-insulator state with a characteristic gap. The inter-particle interaction is essential for this effect. This contribution gives a pedagogical survey of these results. AU - Aizenman, Michael AU - Lieb, Élliott H AU - Robert Seiringer AU - Solovej, Jan P AU - Yngvason, Jakob ED - Asch, Joachim ED - Joye, Alain ID - 2369 T2 - Mathematical Physics of Quantum Mechanics TI - Bose-Einstein condensation as a quantum phase transition in an optical lattice VL - 690 ER - TY - CHAP AU - Bang-Jensen, Jørgen AU - Reed, Bruce AU - Schacht, Bruce AU - Šámal, Robert AU - Toft, Bjarne AU - Uli Wagner ID - 2416 T2 - Topics in Discrete Mathematics TI - On six problems posed by Jarik Nešetřil VL - 26 ER - TY - JOUR AB - We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√̃) colors in the worst case. We then derive two efficient variants of this simple algorithm. The first is deterministic and uses O(log 2 n) colors, and the second is randomized and uses O(log n) colors with high probability. We also show that the O(log 2 n) bound on the number of colors used by our deterministic algorithm is tight on the worst case. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case. AU - Chent, Ke AU - Fiat, Amos AU - Kaplan, Haim AU - Levy, Meital B AU - Matoušek, Jiří AU - Mossel, Elchanan AU - Pach, János AU - Sharir, Micha AU - Smorodinsky, Shakhar AU - Uli Wagner AU - Welzl, Emo ID - 2430 IS - 5 JF - SIAM Journal on Computing TI - Online conflict-free coloring for intervals VL - 36 ER - TY - CONF AB - We prove an upper bound, tight up to a factor of 2, for the number of vertices of level at most t in an arrangement of n halfspaces in R , for arbitrary n and d (in particular, the dimension d is not considered constant). This partially settles a conjecture of Eckhoff, Linhart, and Welzl. Up to the factor of 2, the result generalizes McMullen's Upper Bound Theorem for convex polytopes (the case ℓ = O) and extends a theorem of Linhart for the case d ≤ 4. Moreover, the bound sharpens asymptotic estimates obtained by Clarkson and Shor. The proof is based on the h-matrix of the arrangement (a generalization, introduced by Mulmuley, of the h-vector of a convex polytope). We show that bounding appropriate sums of entries of this matrix reduces to a lemma about quadrupels of sets with certain intersection properties, and we prove this lemma, up to a factor of 2, using tools from multilinear algebra. This extends an approach of Alon and Kalai, who used linear algebra methods for an alternative proof of the classical Upper Bound Theorem. The bounds for the entries of the h-matrix also imply bounds for the number of i-dimensional faces, i > 0, at level at most ℓ. Furthermore, we discuss a connection with crossing numbers of graphs that was one of the main motivations for investigating exact bounds that are valid for arbitrary dimensions. AU - Uli Wagner ID - 2431 TI - On a geometric generalization of the Upper Bound Theorem ER - TY - JOUR AB - We show, with an elementary proof, that the number of halving simplices in a set of n points in 4 in general position is O(n4-2/45). This improves the previous bound of O(n4-1/134). Our main new ingredient is a bound on the maximum number of halving simplices intersecting a fixed 2-plane. AU - Matoušek, Jiří AU - Sharir, Micha AU - Smorodinsky, Shakhar AU - Uli Wagner ID - 2429 IS - 2 JF - Discrete & Computational Geometry TI - K-sets in four dimensions VL - 35 ER -