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. (197407)
Journal article
University of WaikatoKalnins 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 pseudoEuclidean 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 WaikatoWe 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. (197503)
Journal article
University of WaikatoThis 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. (197408)
Journal article
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University of Waikato 
The relativistically invariant expansion of a scalar function on imaginary Lobachevski space
Kalnins, Ernie G.; Miller, W., Jr. (197310)
Journal article
University of WaikatoUsing the previous analysis of Gel'fand and Graev a new relativistically invariant expansion of a scalar function on threedimensional 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 Rseparable coordinate systems for the wave equation ψtt∆2ψ=0
Kalnins, Ernie G.; Miller, W., Jr. (197603)
Journal article
University of WaikatoA list of orthogonal coordinate systems which permit Rseparation 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. (197507)
Journal article
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University of Waikato 
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. (197503)
Journal article
University of WaikatoAs 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. (197410)
Journal article
University of WaikatoA detailed study of the group of symmetries of the timedependent free particle Schrödinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a welldefined 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 Rseparable solutions of the wave equation ∂ttψ=∆2ψ
Kalnins, Ernie G.; Miller, W., Jr. (197603)
Journal article
University of WaikatoWe classify and discuss the possible nonorthogonal coordinate systems which lead to Rseparable 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. (197512)
Journal article
University of WaikatoWe classify and study all coordinate systems which permit Rseparation of variables for the wave equation in three space–time variables and such that at least one of the variables corresponds to a oneparameter 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–Walkertype spacetimes
Kalnins, Ernie G.; Miller, W., Jr. (199103)
Journal article
University of WaikatoCrucial 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 spacetime
Kalnins, Ernie G.; Miller, W., Jr. (199201)
Journal article
University of WaikatoThe 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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Lie theory and the wave equation in space–time. 3. Semisubgroup coordinates
Kalnins, Ernie G.; Miller, W., Jr. (197702)
Journal article
University of WaikatoWe classify and study those coordinate systems which permit R separation of variables for the wave equation in fourdimensional space–time and such that at least one of the variables corresponds to a oneparameter 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. (198607)
Journal article
University of WaikatoThe set of all matrixvalued firstorder differential operators that commute with the Dirac equation in ndimensional complex Euclidean space is computed. In four dimensions it is shown that all matrixvalued secondorder differential operators that commute with the Dirac operator in four dimensions are obtained as products of firstorder 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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Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0
Boyer, C.P.; Kalnins, Ernie G.; Miller, W., Jr. (197801)
Journal article
University of WaikatoWe present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the wellknown Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each Rseparable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra o (3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split off, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known twobody problems, including the harmonic oscillator and Kepler problems, as special cases.
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Lie theory and the wave equation in spacetime. 5. Rseparable solutions of the wave equation ψttΔ3ψ = 0
Kalnins, Ernie G.; Miller, W., Jr. (1977)
Journal article
University of WaikatoA detailed classification is made of orthogonal coordinate systems for which the wave equation ψtt  Δ3ψ = 0 admits an R separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have been found in previous articles. We find 106 new coordinates to give a total of 367 conformally inequivalent orthogonal coordinates for which the wave equation admits an R separation of variables.
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Superintegrability on the twodimensional hyperboloid
Kalnins, Ernie G.; Miller, W., Jr.; Pogosyan, G.S. (199710)
Journal article
University of WaikatoIn this work we examine the basis functions for classical and quantum mechanical systems on the twodimensional 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.
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Quadrics on complex Riemannian spaces of constant curvature, separation of variables, and the Gaudin magnet
Kalnins, Ernie G.; Kuznetsov, V.B.; Miller, W., Jr. (199404)
Journal article
University of WaikatoIntegrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature are considered herein. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of the authors, extends to coordinates of this type. The complete classification of these separable coordinate systems is provided by means of the corresponding L matrices for the Gaudin magnet. The limiting procedures (or calculus) which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems.
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Models of qalgebra representations: Matrix elements of the qoscillator algebra
Kalnins, Ernie G.; Miller, W., Jr.; Mukherjee, Sanchita (199311)
Journal article
University of WaikatoThis article continues a study of function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by qhypergeometric functions. Here a q analog of the oscillator algebra (not a quantum algebra) is considered. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the ``group operators'' on these representation spaces are computed. This ``local'' approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. It is shown that the matrix elements themselves transform irreducibly under the action of the algebra. q analogs of a formula are found for the product of two hypergeometric functions 1F1 and the product of a 1F1 and a Bessel function. They are interpreted here as expansions of the matrix elements of a ``group operator'' (via the exponential mapping) in a tensor product basis (for the tensor product of two irreducible oscillator algebra representations) in terms of the matrix elements in a reduced basis. As a byproduct of this analysis an interesting new orthonormal basis was found for a q analog of the Bargmann–Segal Hilbert space of entire functions.
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