68 results for Miller, W., Jr., 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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  • 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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  • Jacobi elliptic coordinates, functions of Heun and Lame type and the Niven transform

    Kalnins, Ernie G.; Miller, W., Jr. (2005)

    Journal article
    University of Waikato

    Lame and Heun functions arise via separation of the Laplace equation in general Jacobi ellipsoidal or conical coordinates. In contrast to hypergeometric functions that also arise via variable separation in the Laplace equation, Lame and Heun functions have received relatively little attention, since they are rather intractable. Nonetheless functions of Heun type do have remarkable properties, as was pointed out in the classical book \Modern Analysis" by Whittaker and Watson who devoted an entire chapter to the subject. Unfortunately the beautiful identities appearing in this chapter have received little notice, probably because the methods of proof seemed obscure. In this paper we apply the modern operator characterization of variable separation and exploit the conformal symmetry of the Laplace equation to obtain product identities for Heun type functions. We interpret the Niven transform as an intertwining operator under the action of the conformal group. We give simple operator derivations of some of the basic formulas presented by Whittaker and Watson and then show how to generalize their results to more complicated situations and to higher dimensions.

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  • Separable coordinates for three-dimensional complex riemannian spaces

    Kalnins, Ernie G.; Miller, W., Jr. (1979)

    Journal article
    University of Waikato

    In this paper we study the problem of separation of variables for the equations: Helmholtz equation & Hamilton-Jacobi equation.

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  • Superintegrable systems in Darboux spaces

    Kalnins, Ernie G.; Kress, Jonathan M.; Miller, W., Jr. (2003-12)

    Journal article
    University of Waikato

    Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via "coupling constant metamorphosis" (or equivalently, via Stäckel multiplier transformations). We present a table of the results.

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  • Superintegrability in a non-conformally-flat space

    Kalnins, Ernie G.; Kress, Jonathan M.; Miller, W., Jr. (2013)

    Journal article
    University of Waikato

    Superintegrable systems in two- and three-dimensional spaces of constant curvature have been extensively studied. From these, superintegrable systems in conformally flat spaces can be constructed by Stackel transform. In this paper a method developed to establish the superintegrability of the Tremblay-Turbiner-Winternitz system in two dimensions is extended to higher dimensions and a superintegrable system on a non-conformally-flat four-dimensional space is found. In doing so, curvature corrections to the corresponding classical potential are found to be necessary. It is found that some subalgebras of the symmetry algebra close polynomially.

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  • Superintegrability and higher-order constants for classical and quantum systems

    Kalnins, Ernie G.; Miller, W., Jr.; Pogosyan, G.S. (2011)

    Journal article
    University of Waikato

    We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classicalmechanics, there are higher-order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.

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  • Examples of complete solvability of 2D classical superintegrable systems

    Chen, Yuxan; Kalnins, Ernie G.; Li, Qiushi; Miller, W., Jr. (2015)

    Journal article
    University of Waikato

    Classical (maximal) superintegrable systems in n dimensions are Hamiltonian systems with 2n - 1 independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton’s equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler’s laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of “metronome orbits”, trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

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  • Superintegrability on the two-dimensional hyperboloid. II

    Kalnins, Ernie G.; Miller, W., Jr.; Hakobyan, Ye M.; Pogosyan, G.S. (1999-05)

    Journal article
    University of Waikato

    This work is devoted to the investigation of the quantum mechanical systems on the two-dimensional hyperboloid which admits separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C. P. Boyer, E. G. Kalnins, and P. Winternitz [J. Math. Phys. 24, 2022 (1983)], which have not yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.

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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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  • Models for the 3D Singular Isotropic Oscillator Quadratic Algebra

    Kalnins, Ernie G.; Miller, W., Jr.; Post, Sarah (2010-02-01)

    Journal article
    University of Waikato

    We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.

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