123571163
A classification of second‐order raising operators for Hamiltonians in two variables
We develop a group theoretic method based on results of Winternitz et al. to compute and classify all first‐ and second‐order raising and lowering operators admitted by Hamiltonians of the form H¯=−(1/2)Δ2+V(x,y). The key to our results, which generalize to higher dimensions, is a proof that H¯ admits a second‐order raising operator only if the Schrodinger equation separates in Cartesian, polar, or elliptic coordinates.
121892369
Lie theory and the wave equation in space–time. I. The Lorentz group
In this article we begin a study of the relationship between separation of variables and the conformal symmetry group of the wave equation ψtt −Δ3ψ=0 in space–time. In this first article we make a detailed study of separation of variables for the Laplace operator on the one and two sheeted hyperboloids in Minkowski space. We then restrict ourselves to homogeneous solutions of the wave equation and the Lorentz subgroup SO(3,1) of the conformal group SO(4,2). We study the various separable bases by using the methods of integral geometry as developed by Gel’fand and Graev. In most cases we give the spectral analysis for these bases, and a number of new bases are developed in detail. Many of the special function identities derived appear to be new. This preliminary study is of importance when we subsequently study models of the Hilbert space structure for solutions of the wave equation and the Klein–Gordon equation ψtt−Δ3ψ=λψ.
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.
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.
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).
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.
1036381
Measurement of neutrino oscillations with the MINOS detectors in the NuMI beam.
This Letter reports new results from the MINOS experiment based on a two-year exposure to muon neutrinos from the Fermilab NuMI beam. Our data are consistent with quantum-mechanical oscillations of neutrino flavor with mass splitting |Deltam2| = (2.43+/-0.13) x 10(-3) eV2 (68% C.L.) and mixing angle sin2(2theta) > 0.90 (90% C.L.). Our data disfavor two alternative explanations for the disappearance of neutrinos in flight: namely, neutrino decays into lighter particles and quantum decoherence of neutrinos, at the 3.7 and 5.7 standard-deviation levels, respectively.
110729769
ATLAS pixel detector electronics and sensors
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.
16015306
Observation of muon intensity variations by season with the MINOS far detector
The temperature of the upper atmosphere affects the height of primary cosmic ray interactions and the production of high-energy cosmic ray muons which can be detected deep underground. The MINOS far detector at Soudan, MN, has collected over 67 × 10^6 cosmic ray induced muons. The underground muon rate measured over a period of five years exhibits a 4% peak-to-peak seasonal variation which is highly correlated with the temperature in the upper atmosphere. The coefficient, α_T, relating changes in the muon rate to changes in atmospheric temperature was found to be α_T = 0.873 ± 0.009(stat) ± 0.010(syst). Pions and kaons in the primary hadronic interactions of cosmic rays in the atmosphere contribute differently to αT due to the different masses and lifetimes. This allows the measured value of α_T to be interpreted as a measurement of the K/π ratio for Ep ≳ 7  TeV of 0.12^(+0.07)_(-0.05), consistent with the expectation from collider experiments.
13850851
Observation of muon neutrino disappearance with the MINOS detectors in the NuMI neutrino beam.
This Letter reports results from the MINOS experiment based on its initial exposure to neutrinos from the Fermilab NuMI beam. The rates and energy spectra of charged current nu(mu) interactions are compared in two detectors located along the beam axis at distances of 1 and 735 km. With 1.27 x 10(20) 120 GeV protons incident on the NuMI target, 215 events with energies below 30 GeV are observed at the Far Detector, compared to an expectation of 336+/-14 events. The data are consistent with nu(mu) disappearance via oscillations with |Delta(m)2/32|=2.74 +0.44/-0.26 x10(-3)eV(2) and sin(2)(2theta(23))>0.87 (68% C.L.).
124171013
Status of the Soudan 2 detector experiment
The status of the Soudan 2 experiment is discussed. The detector parameters, running schedule, and physics program are presented. Limits on proton decay, the flux of highly ionizing magnetic monopoles and neutrinos from active galactic nuclei are given.