Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
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: Subgroups of Lie groups and separation of variables
abstract: Separable systems of coordinates for the Helmholtz equation ΔdΨ =EΨ in pseudo‐Riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator Δd. They have also been characterized geometrically by the form that the metric ds2=gik(x)dxidxk can take. We complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of G, the symmetry group of Δd. For d=3 we study all separable coordinates that can be characterized in terms of the Lie algebra L of G and show that they are of eight types, seven of which are related to the subgroup structure of G. Our method clearly generalizes to the case d≳3. Although each separable system corresponds to a pair of commuting symmetry operators, there do exist pairs of commuting symmetries S1,S2 that are not associated with separable coordinates. For subgroup related operators we show in detail just...

title: Matrix operator symmetries of the Dirac equation and separation of variables
abstract: The set of all matrix‐valued first‐order differential operators that commute with the Dirac equation in n‐dimensional complex Euclidean space is computed. In four dimensions it is shown that all matrix‐valued second‐order differential operators that commute with the Dirac operator in four dimensions are obtained as products of first‐order operators that commute with the Dirac operator. Finally some additional coordinate systems for which the Dirac equation in Minkowski space can be solved by separation of variables are presented. These new systems are comparable to the separation in oblate spheroidal coordinates discussed by Chandrasekhar [S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford U.P., Oxford, 1983)].

title: Series solutions for the Dirac equation in Kerr-Newman space-time
abstract: The Dirac equation is solved for an electron in a Kerr–Newman geometry using an adaptation of the procedure of Chandrasekhar. The corresponding eigenfunctions obtained can be represented as series of Jacobi polynomials. The spectrum of eigenvalues can be calculated using continued fraction techniques. Representations for the eigenvalues and eigenfunctions are obtained for various ranges of the parameters appearing in the Kerr–Newman metric. Some comments concerning the bag model of nucleons are made.

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: Observation in the MINOS far detector of the shadowing of cosmic rays by the sun and moon
abstract: The shadowing of cosmic ray primaries by the moon and sun was observed by the MINOS far detector at a depth of 2070 mwe using 83.54 million cosmic ray muons accumulated over 1857.91 live-days. The shadow of the moon was detected at the 5.6 σ level and the shadow of the sun at the 3.8 σ level using a log-likelihood search in celestial coordinates. The moon shadow was used to quantify the absolute astrophysical pointing of the detector to be 0.17 ± 0.12°. Hints of interplanetary magnetic field effects were observed in both the sun and moon shadow.

title: Spontaneous light emission from fibers in MINOS
abstract: We report on the observation and measurements of unexpected background rates in the MINOS Far Detector. The noise level at the Far Detector is significantly greater than that expected from natural radioactivity and intrinsic photomultiplier dark current. We have conducted a series of additional tests which demonstrate that the excess rate is caused by spontaneous light emission in the wavelength-shifting fibers, which are used to read out signals from scintillator strips. This noise due to fibers exhibits an exponential fall off with time with a decay time constant of the order of 100 days.

title: ATLAS pixel detector electronics and sensors
abstract: The silicon pixel tracking system for the ATLAS experiment at the Large Hadron Collider is described and the performance requirements are summarized. Detailed descriptions of the pixel detector electronics and the silicon sensors are given. The design, fabrication, assembly and performance of the pixel detector modules are presented. Data obtained from test beams as well as studies using cosmic rays are also discussed.

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.

title: Improved search for Muon-neutrino to electron-neutrino oscillations in MINOS.
abstract: We report the results of a search for ν(e) appearance in a ν(μ) beam in the MINOS long-baseline neutrino experiment. With an improved analysis and an increased exposure of 8.2 × 10(20) protons on the NuMI target at Fermilab, we find that 2 sin(2) (θ(23))sin(2)(2θ(13))<0.12(0.20) at 90% confidence level for δ = 0 and the normal (inverted) neutrino mass hierarchy, with a best-fit of 2sin(2) (θ(23))sin(2)(2θ(13)) = 0.041(-0.031)(+0.047) (0.079(-0.053) (+0.071)). The θ(13) = 0 hypothesis is disfavored by the MINOS data at the 89% confidence level.