78 results for Kalnins, Ernie G., Journal article

  • Lie theory and separation of variables. 3. The equation ftt−fss =γ2f

    Kalnins, Ernie G.; Miller, W., Jr. (1974-07)

    Journal article
    University of Waikato

    Kalnins has related the 11 coordinate systems in which variables separate in the equation ftt−fss = γ 2f to 11 symmetric quadratic operators L in the enveloping algebra of the Lie algebra of the pseudo-Euclidean group in the plane E(1,1). There are, up to equivalence, only 12 such operators and one of them, LE, is not associated with a separation of variables. Corresponding to each faithful unitary irreducible representation of E(1,1) we compute the spectral resolution and matrix elements in an L basis for seven cases of interest and also give overlap functions between different bases: Of the remaining five operators three are related to Mathieu functions and two are related to exponential solutions corresponding to Cartesian type coordinates. We then use these results to derive addition and expansion theorems for special solutions of ftt−fss = 2f obtained via separation of variables, e.g., products of Bessel, Macdonald and Bessel, Airy and parabolic cylinder functions. The exceptional operator LE is also treated in detail.

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  • Superintegrability and higher order integrals for quantum systems

    Kalnins, Ernie G.; Kress, Jonathan M.; Miller, W., Jr. (2010)

    Journal article
    University of Waikato

    We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger 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.

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  • Lie theory and separation of variables. 6. The equation iUt + ∆2U = 0

    Boyer, C.P.; Kalnins, Ernie G.; Miller, W., Jr. (1975-03)

    Journal article
    University of Waikato

    This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new.

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  • Lie theory and separation of variables. 4. The groups SO (2,1) and SO (3)

    Kalnins, Ernie G.; Miller, W., Jr. (1974-08)

    Journal article
    University of Waikato

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  • The relativistically invariant expansion of a scalar function on imaginary Lobachevski space

    Kalnins, Ernie G.; Miller, W., Jr. (1973-10)

    Journal article
    University of Waikato

    Using the previous analysis of Gel'fand and Graev a new relativistically invariant expansion of a scalar function on three-dimensional imaginary Lobachevski space L3(I) is given. The coordinate system used corresponds to the horospherical reduction SO(3,1) E2 SO(2) and covers all of L3(I).

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  • Mixed basis matrix elements for the subgroup reductions of SO(2,1)

    Kalnins, Ernie G. (1973-05)

    Journal article
    University of Waikato

    By using the irreducible decomposition on the two-dimensional light cone, the mixed basis matrix elements for the three subgroup reductions of SO(2,1) are calculated. These matrix elements are calculated for the principal series only and can be expressed in terms of well-known special functions. As a consequence of appearing in this context, some new properties of these special functions are given.

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  • Unitary Representations of the Homogeneous Lorentz Group in an O(1,1) O(2) Basis and Some Applications to Relativistic Equations

    Kalnins, Ernie G. (1972-09)

    Journal article
    University of Waikato

    Unitary irreducible representations of the homogeneous Lorentz group O(3, 1) belonging to the principal series are reduced with respect to the subgroup O(1,1) O(2). As an application we determine the mixed basis matrix elements between O(3) and O(1,1) O(2) bases and derive recurrence relations for them. This set of functions is then used to obtain invariant expansions of solutions of the Dirac and Proca free field equations. These expansions are shown to have the correct nonrelativistic limit.

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  • Lie theory and separation of variables. 11. The EPD equation

    Kalnins, Ernie G. (1976-03)

    Journal article
    University of Waikato

    We show that the Euler–Poisson–Darboux equation {∂tt -∂rr – [(2m+1)/r]∂r}Ө=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).

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  • Lie theory and separation of variables. 9. Orthogonal R-separable coordinate systems for the wave equation ψtt-∆2ψ=0

    Kalnins, Ernie G.; Miller, W., Jr. (1976-03)

    Journal article
    University of Waikato

    A list of orthogonal coordinate systems which permit R-separation of the wave equation ψtt-∆2ψ=0 is presented. All such coordinate systems whose coordinate curves are cyclides or their degenerate forms are given. In each case the coordinates and separation equations are computed. The two basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2).

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  • Erratum: Lie theory and separation of variables. 3. The equation ftt − fss = γ2f

    Kalnins, Ernie G.; Miller, W., Jr. (1975-07)

    Journal article
    University of Waikato

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  • Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials

    Boyer, C.P.; Kalnins, Ernie G.; Miller, W., Jr. (1975-03)

    Journal article
    University of Waikato

    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.

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  • Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxx −c/x2 U = 0

    Kalnins, Ernie G.; Miller, W., Jr. (1974-10)

    Journal article
    University of Waikato

    A detailed study of the group of symmetries of the time-dependent free particle Schrödinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well-defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x2 is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrödinger equation, several of which are new.

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  • Lie theory and separation of variables. 10. Nonorthogonal R-separable solutions of the wave equation ∂ttψ=∆2ψ

    Kalnins, Ernie G.; Miller, W., Jr. (1976-03)

    Journal article
    University of Waikato

    We classify and discuss the possible nonorthogonal coordinate systems which lead to R-separable solutions of the wave equation. Each system is associated with a pair of commuting operators in the symmetry algebra so(3,2) of this equation, one operator first order and the other second order. Several systems appear here for the first time.

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  • Lie theory and separation of variables. 8. Semisubgroup coordinates for Ψtt - ∆2Ψ = 0

    Kalnins, Ernie G.; Miller, W., Jr. (1975-12)

    Journal article
    University of Waikato

    We classify and study all coordinate systems which permit R-separation of variables for the wave equation in three space–time variables and such that at least one of the variables corresponds to a one-parameter symmetry group of the wave equation. We discuss 33 such systems and relate them to orbits of commuting operators in the enveloping algebra of the conformal group SO (3,2).

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  • Complete sets of functions for perturbations of Robertson–Walker cosmologies and spin 1 equations in Robertson–Walker-type space-times

    Kalnins, Ernie G.; Miller, W., Jr. (1991-03)

    Journal article
    University of Waikato

    Crucial to a knowledge of the perturbations of Robertson–Walker cosmological models are complete sets of functions with which to expand such perturbations. For the open Robertson–Walker cosmology an answer to this question is given. In addition some observations concerning explicit solution by separation of variables of wave equations for spin 1 in a Riemannian space having an infinitesimal line element of which the Robertson–Walker models are a special case are made.

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  • Series solutions for the Dirac equation in Kerr–Newman space-time

    Kalnins, Ernie G.; Miller, W., Jr. (1992-01)

    Journal article
    University of Waikato

    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.

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  • Separation of variables and contractions on two-dimensional hyperboloid

    Kalnins, Ernie G.; Pogosyan, G.S.; Yakhno, Alexander (2012)

    Journal article
    University of Waikato

    In this paper analytic contractions have been established in the R → ∞ con- traction limit for exactly solvable basis functions of the Helmholtz equation on the two- dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.

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  • Symmetry operators and separation of variables for spin-wave equations in oblate spheroidal coordinates

    Kalnins, Ernie G.; Williams, G.C. (1990-07)

    Journal article
    University of Waikato

    A family of second-order differential operators that characterize the solution of the massless spin s field equations, obtained via separation of variables in oblate spheroidal coordinates and using a null tetrad is found. The first two members of the family also characterize the separable solutions in the Kerr space-time. It is also shown that these operators are symmetry operators of the field equations in empty space-times whenever the space-time admits a second-order Killing–Yano tensor.

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  • Lie theory and the wave equation in space–time. 3. Semisubgroup coordinates

    Kalnins, Ernie G.; Miller, W., Jr. (1977-02)

    Journal article
    University of Waikato

    We classify and study those coordinate systems which permit R separation of variables for the wave equation in four-dimensional space–time and such that at least one of the variables corresponds to a one-parameter symmetry group of the wave equation. We discuss over 100 such systems and relate them to orbits of triplets of commuting operators in the enveloping algebra of the conformal group SO(4,2).

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  • Matrix operator symmetries of the Dirac equation and separation of variables

    Kalnins, Ernie G.; Miller, W., Jr.; Williams, G.C. (1986-07)

    Journal article
    University of Waikato

    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)].

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