Lie theory and separation of variables. 4. The groups SO(2,1) and SO(3)
Winternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid x02−x12−x22=1 separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one‐variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well‐known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lame functions. We show that there is virtually a perfect correspondence between the known theory of the Lame functions and the representation theory of SO (2,1) and SO (3).

title: Superintegrability on the two-dimensional hyperboloid. II
abstract: 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.

title: Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)
abstract: We consider the ``missing label'' problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3)⊃O(3)⊃O(2). We prove that only two independent O (3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L2 and the two SU(3) Casimir operators C(2) and C(3). Any one of these two operators (of third and fourth order in the generators) can be added to C(2), C(3), L2, and L3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X(3) and X(4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

title: Separation of variables in Einstein spaces. I. Two ignorable and one null coordinate
abstract: Computes and classifies all Einstein spaces and their corresponding coordinate systems such that the Hamilton-Jacobi equation admits a separation of variables with two ignorable and one essential null coordinates. The authors find that all such spaces correspond to plane-fronted gravitational waves of Petrov type N. They also give necessary conditions which a general space-time must have in order to admit coordinates of the type studied. Furthermore, the integrability conditions for Helmholtz (Schrodinger) separability are completely solved.

title: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
abstract: As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

title: Lie theory and separation of variables. 11. The EPD equation
abstract: We show that the Euler–Poisson–Darboux equation {∂tt−∂rr−[(2m+1)/r]∂r}Θ=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second‐order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second‐order operators in the enveloping algebra of SL(2,C).

title: Measurement of the neutrino mass splitting and flavor mixing by MINOS.
abstract: Measurements of neutrino oscillations using the disappearance of muon neutrinos from the Fermilab NuMI neutrino beam as observed by the two MINOS detectors are reported. New analysis methods have been applied to an enlarged data sample from an exposure of 7.25×10(20) protons on target. A fit to neutrino oscillations yields values of |Δm(2)|=(2.32(-0.08)(+0.12))×10(-3) eV(2) for the atmospheric mass splitting and sin(2)(2θ)>0.90 (90% C.L.) for the mixing angle. Pure neutrino decay and quantum decoherence hypotheses are excluded at 7 and 9 standard deviations, respectively.

title: First observations of separated atmospheric nu(mu) and anti-nu(mu) events in the MINOS detector.
abstract: The complete 5.4 kton MINOS far detector has been taking data since the beginning of August 2003 at a depth of 2070 meters water-equivalent in the Soudan mine, Minnesota. This paper presents the first MINOS observations of muon neutrino and muon anti-neutrino charged-current atmospheric neutrino interactions based on an exposure of 418 days. The ratio of upward to downward-going events in the data is compared to the Monte Carlo expectation in the absence of neutrino oscillations giving: R_data(up/down)/R_MC(up/down) = 0.62^{+0.19}_{-0.14} (stat.) +- 0.02 (sys.). An extended maximum likelihood analysis of the observed L/E distributions excludes the null hypothesis of no neutrino oscillations at the 98 % confidence level. Using the curvature of the observed muons in the 1.3 T MINOS magnetic field muon neutrino and muon anti-neutrino interactions are separated. The ratio of muon neutrino to muon anti-neutrino events in the data is compared to the Monte Carlo expectation assuming neutrinos and anti-neutrinos oscillate in same manner giving: R_data(numubar/numu)/R_MC(numubar/numu) = 0.96^{+0.38}_{-0.27} (stat.) +- 0.15 (sys.), where the errors are the statistical and systematic uncertainties. Although the statistics are limited, this is the first direct observation of atmospheric neutrino interactions separately for muon neutrinos and muon anti-neutrinos.

title: Underground muons observed in the Soudan 2 detector from the directions of x-ray sources
abstract: We report on observations of underground muons made with the Soudan 2 proton decay detector at a depth of 2090 mwe from the directions of the binaries Hercules X-1 and 1E2259+586 and the Crab pulsar.

title: The 2010 Interim Report of the Long-Baseline Neutrino Experiment Collaboration Physics Working Groups
abstract: In early 2010, the Long-Baseline Neutrino Experiment (LBNE) science collaboration initiated a study to investigate the physics potential of the experiment with a broad set of different beam, nearand far-detector configurations. Nine initial topics were identified as scientific areas that motivate construction of a long-baseline neutrino experiment with a very large far detector. We summarize the scientific justification for each topic and the estimated performance for a set of far detector reference configurations. We report also on a study of optimized beam parameters and the physics capability of proposed Near Detector configurations. This document was presented to the collaboration in fall 2010 and updated with minor modifications in early 2011. PACS numbers: 14.60.Lm, 14.60.Pq, 95.85.Ry, 13.15+g, 13.30.Ce, 11.30.Fs, 14.20.Dh, 26.65.+t, 25.30.Pt, 29.40.Ka, 29.40.Gx ∗

title: Search for Lorentz Invariance and CPT Violation with Muon Antineutrinos in the MINOS Near Detector
abstract: We have searched for sidereal variations in the rate of antineutrino interactions in the MINOS Near Detector. Using antineutrinos produced by the NuMI beam, we find no statistically significant sidereal modulation in the rate. When this result is placed in the context of the Standard Model Extension theory we are able to place upper limits on the coefficients defining the theory. These limits are used in combination with the results from an earlier analysis of MINOS neutrino data to further constrain the coefficients.