Coulomb-oscillator duality in spaces of constant curvature

In this paper we construct generalizations to spheres of the well known Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and quantum mechanical analogues of the Kepler-Coulomb problem on these spheres are discussed.Comment: 33 pages, LaTeX fil

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two-dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n-dimensional isotropic quantum oscillator

Superintegrability on the two-dimensional hyperboloid

In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras

Superintegrability on the two dimensional hyperboloid II

This work is devoted to the investigation of the quantum mechanical systems on the two dimensional hyperboloid which admit separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C.P.Boyer, E.G.Kalnins and P.Winternitz, which haven't yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.Comment: 18 pages, LaTex; 1 figure (eps

Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd kk

In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of $k$, some novel exactly solvable and integrable quantum Hamiltonian $H_k$ on a plane is superintegrable and that the additional integral of motion is a $2k$th-order differential operator $Y_{2k}$. Here we demonstrate the conjecture for the infinite family of Hamiltonians $H_k$ with odd $k \ge 3$, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some $D_{2k}$-extended and invariant Hamiltonian \chh_k, which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a $D_{2k}$-invariant integral of motion \cyy_{2k}, from which $Y_{2k}$ can be obtained by projection in the $D_{2k}$ identity representation space.Comment: 14 pages, no figure; change of title + important addition to sect. 4 + 2 more references + minor modifications; accepted by JPA as an FT

Cosmic microwave background snapshots: pre-WMAP and post-WMAP

Abbreviated: We highlight the remarkable evolution in the CMB power spectrum over the past few years, and in the cosmological parameters for minimal inflation models derived from it. Grand unified spectra (GUS) show pre-WMAP optimal bandpowers are in good agreement with each other and with the one-year WMAP results, which now dominate the L < 600 bands. GUS are used to determine calibrations, peak/dip locations and heights, and damping parameters. These CMB experiments significantly increased the case for accelerated expansion in the early universe (the inflationary paradigm) and at the current epoch (dark energy dominance) when they were combined with `prior' probabilities on the parameters. A minimal inflation parameter set is applied in the same way to the evolving data. Grid-based and and Monte Carlo Markov Chain methods are shown to give similar values, highly stable over time and for different prior choices, with the increasing precision best characterized by decreasing errors on uncorrelated parameter eigenmodes. After marginalizing over the other cosmic and experimental variables for a weak+LSS prior, the pre-WMAP data of Jan03 cf. the post-WMAP data of Mar03 give Omega_{tot} =1.03^{+0.05}_{-0.04} cf. 1.02^{+0.04}_{-0.03}. Adding the flat prior, n_s =0.95^{+0.07}_{-0.04} cf. 0.97^{+0.02}_{-0.02}, with < 2\sigma evidence for a log variation of n_s. The densities have concordance values. The dark energy pressure-to-density ratio is not well constrained by our weak+LSS prior, but adding SN1 gives w_Q < -0.7. We find \sigma_8 = 0.89^{+0.06}_{-0.07} cf. 0.86^{+0.04}_{-0.04}, implying a sizable SZ effect; the high L power suggest \sigma_8 \sim 0.94^{+0.08}_{-0.16} is needed to be SZ-compatible.Comment: 36 pages, 5 figures, 5 tables, Jan 2003 Roy Soc Discussion Meeting on `The search for dark matter and dark energy in the Universe', published PDF (Oct 15 2003) is http://www.cita.utoronto.ca/~bond/roysoc03/03TA2435.pd

Magnetization plateau in the S=1/2 spin ladder with alternating rung exchange

We have studied the ground state phase diagram of a spin ladder with alternating rung exchange $J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} \delta ]$ in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an $XXZ$ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is $\sim \delta$. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as $\delta^{\nu}$, where $\nu =4/5$ in the case of ladder with isotropic antiferromagnetic legs and $\nu =2$ in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields ($H^{\pm}_{c1}$ and $H^{\pm}_{c2}$) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.Comment: 6 pages, 2 figure

Complete sets of invariants for dynamical systems that admit a separation of variables

Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} = ij. The 2n–1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion