15381990
Superintegrability and higher order integrals for quantum systems
We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrodinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. The flat space equations with potentials V = α(x + iy)k − 1/(x − iy)k + 1 in Cartesian coordinates, and V = αr2 + β/r2cos 2kθ + γ/r2sin 2kθ (the Tremblay, Turbiner and Winternitz system) in polar coordinates, have each been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.
120152328
Lie theory and separation of variables. 11. The EPD equation
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).
120822220
Separation of variables in Einstein spaces. I. Two ignorable and one null coordinate
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.
120256112
CLEBSCH--GORDAN COEFFICIENTS AND SPECIAL FUNCTION IDENTITIES. II. THE ROTATION AND LORENTZ GROUPS IN 3-SPACE.
It is shown that the construction of concrete models of Clebsch‐Gordan decompositions for tensor products of irreducible group representations leads to a wide variety of special function identities. In this paper the representation theory of the rotation and Lorentz groups in 3‐space is used to give elegant derivations of identities involving Languerre, Gegenbauer, hypergeometric, and generalized hypergeometric functions. Some of these identities may be new in this general form.
119430316
Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)
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.
122430224
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.
123121317
The CLEO III silicon tracker
Abstract The Cornell Electron Storage Ring is being upgraded to B-factory luminosities. The CLEO detector is also being upgraded with a new charged particle tracking system and with the addition of a ring imaging Cerenkov particle identification system. A major part of the tracking system upgrade is the construction of a new four-layer double-sided silicon tracker with 93% solid angle coverage and new readout electronics. The status of the silicon tracker including production tests and the expected performance of the final system are discussed.
118469809
Underground muons observed in the Soudan 2 detector from the directions of x-ray sources
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.
11232676
Search for sterile neutrino mixing in the MINOS long-baseline experiment
A search for depletion of the combined flux of active neutrino species over a 735 km baseline is reported using neutral-current interaction data recorded by the MINOS detectors in the NuMI neutrino beam. Such a depletion is not expected according to conventional interpretations of neutrino oscillation data involving the three known neutrino flavors. A depletion would be a signature of oscillations or decay to postulated noninteracting sterile neutrinos, scenarios not ruled out by existing data. From an exposure of 3.18×10^(20) protons on target in which neutrinos of energies between ∼500  MeV and 120 GeV are produced predominantly as ν_μ, the visible energy spectrum of candidate neutral-current reactions in the MINOS far detector is reconstructed. Comparison of this spectrum to that inferred from a similarly selected near-detector sample shows that of the portion of the ν_μ flux observed to disappear in charged-current interaction data, the fraction that could be converting to a sterile state is less than 52% at 90% confidence level (C.L.). The hypothesis that active neutrinos mix with a single sterile neutrino via oscillations is tested by fitting the data to various models. In the particular four-neutrino models considered, the mixing angles θ_(24) and θ_(34) are constrained to be less than 11° and 56° at 90% C.L., respectively. The possibility that active neutrinos may decay to sterile neutrinos is also investigated. Pure neutrino decay without oscillations is ruled out at 5.4 standard deviations. For the scenario in which active neutrinos decay into sterile states concurrently with neutrino oscillations, a lower limit is established for the neutrino decay lifetime τ_3/m_3>2.1×10^(-12) s/eV at 90% C.L.
123631242
The SOUDAN 2 detector: The design and construction of the tracking calorimeter modules
SOUDAN 2 is a 960-ton tracking calorimeter which has been constructed to search for nucleon decay and other phenomena. The full detector consists of 224 calorimeter modules each weighing 4.3 tons. The design and construction of the modules are described. The modules consist of finely segmented iron instrumented with 1 m long drift tubes of 15 mm internal diameter. The tubes enable three spatial coordinates and dE/dx to be recorded for charged particles traversing the tubes.
2264842
Measurement of the neutrino mass splitting and flavor mixing by MINOS.
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.