Two charges on plane in a magnetic field: special trajectories

A classical mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and $(e_2, m_2)$ subject to a constant magnetic field perpendicular to a plane is considered. Special "superintegrable" trajectories (circular and linear) for which the distance between charges remains unchanged are indicated as well as their respectful constants of motion. The number of the independent constants of motion for special trajectories is larger for generic ones. A classification of pairs of charges for which special trajectories occur is given. The special trajectories for three particular cases of two electrons, (electron - positron), (electron - $\alpha$-particle) are described explicitly.Comment: 22 pages, 5 figure

Inelastic and elastic collision rates for triplet states of ultracold strontium

We report measurement of the inelastic and elastic collision rates for ^{88}Sr atoms in the (5s5p)^3P_0 state in a crossed-beam optical dipole trap. This is the first measurement of ultracold collision properties of a ^3P_0 level in an alkaline-earth atom or atom with similar electronic structure. Since the (5s5p)^3P_0 state is the lowest level of the triplet manifold, large loss rates indicate the importance of principle-quantum-number-changing collisions at short range. We also provide an estimate of the collisional loss rates for the (5s5p){^3P_2} state.Comment: 4 pages 5 figure

Degenerate Fermi Gas of 87^{87}Sr

We report quantum degeneracy in a gas of ultra-cold fermionic $^{87}$Sr atoms. By evaporatively cooling a mixture of spin states in an optical dipole trap for 10.5\,s, we obtain samples well into the degenerate regime with $T/T_F=0.26^{+.05}_{-.06}$. The main signature of degeneracy is a change in the momentum distribution as measured by time-of-flight imaging, and we also observe a decrease in evaporation efficiency below $T/T_F \sim 0.5$.Comment: 4 pages, 3 figure

Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials

Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are considered. We present all real standard potentials that allow separation of variables in polar coordinates and admit an independent fourth-order integral of motion. The general form of the potentials satisfies a linear ODE. In the classical case, the standard potentials coincide with the Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models. In the quantum case new superintegrable systems are obtained, in addition to the TTW and PW ones. Their classical limit is free motion.Comment: 30 page

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.Comment: 49 pages. Final version, to appear in Calc. Var. Partial Differential Equation