Willard Miller - Profile on Academia.edu (original) (raw)
Papers by Willard Miller
Symmetry and Separation of Variables: References
Journal of Mathematical Physics, 2007
Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties... more Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties. [Journal of Mathematical Physics 48, 113518 (2007)]. EG Kalnins, JM Kress, W. Miller, Jr. Abstract. A classical (or quantum ...
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symm... more We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ≡ (Δ_2 +V)Ψ=EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one ...
Journal of Mathematical Physics, 1977
Lie theory and the wave equation in spacetime. 3. Semisubgroup coordinates. [Journal of Mathemat... more Lie theory and the wave equation in spacetime. 3. Semisubgroup coordinates. [Journal of Mathematical Physics 18, 271 (1977)]. EG Kalnins, W. Miller, Jr. Abstract. We classify and study those coordinate systems which permit ...
Journal of Mathematical …, 2002
Complete sets of invariants for dynamical systems that admit a separation of variables. [Journal ... more Complete sets of invariants for dynamical systems that admit a separation of variables. [Journal of Mathematical Physics 43, 3592 (2002)]. EG Kalnins, JM Kress, W. Miller, Jr., GS Pogosyan. Abstract. Consider a classical Hamiltonian ...
Journal of mathematical physics, 2006
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean sp... more Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries. [Journal of Mathematical Physics 47, 033502 (2006)]. EG Kalnins, W. Miller, Jr., GS Pogosyan. Abstract. We ...
Superintegrability on the two-dimensional hyperboloid. II
Journal of Mathematical …, 1999
Superintegrability on the two-dimensional hyperboloid. II. [Journal of Mathematical Physics 40, 2... more Superintegrability on the two-dimensional hyperboloid. II. [Journal of Mathematical Physics 40, 2291 (1999)]. EG Kalnins, W. Miller, Jr., Ye. M. Hakobyan, GS Pogosyan. Abstract. This work is devoted to the investigation of the quantum ...
Quantum Theory and Symmetries
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symm... more We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (∆2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. Most of this paper is devoted to describing the method. Details will be provided elsewhere. As examples we revisit the Tremblay and Winternitz derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. The purpose of this project is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.
Journal of Physics A: Mathematical and Theoretical
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symm... more We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (∆2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and all 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. Our aim is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.
Journal of Physics A: Mathematical and Theoretical
Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intr... more Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators but their symmetry algebras don't close and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bôcher contractions of the conformal Lie algebra so(5, C) to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, and all from Bôcher contractions of a single "generic" system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.
Journal of …, 1991
Crucial to a knowledge of the perturbations of Robertson Walker cosmological models is a knowledg... more Crucial to a knowledge of the perturbations of Robertson Walker cosmological models is a knowledge of complete sets of functions with which to expand such perturbations. For the open Robertson Walker cosmology, this question will be completely answered. In addition, some observations will be made concerning explicit solution by separation of variables of wave equations for spin s in a Riemannan space having an infinitesmal line element of which the Robertson Walker models are a special case. I. VECTOR AND TENSOR HARMONICS ON THREE-DIMENSIONAL SPACES OF CONSTANT RIEMANNIAN CURVATURE The original investigations of Lifshitz' and Lifshitz and Khalatnikov' into the gravitational stability of the Robertson Walker (RW) isotropic cosmological models" demonstrated the utility of scalar, vector, and tensor harmonics in giving a complete description of small perturbations. In particular these authors"' showed that in the synchronous gauge all perturbations involving pressure, density, velocity, and metric fluctuations can be obtained once a complete set ofsuch functions is found for S, (three-dimensional sphere),
Journal of Physics A: Mathematical …, 2010
We refine a method for finding a canonical form of symmetry operators of arbitrary order for the ... more We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (∆ 2 + 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 = αr 2 + β/r 2 cos 2 kθ + γ/r 2 sin 2 kθ (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.
A Geometrical Perspective on the Coherent Multimode Optical Field and Mode Coupling Equations
IEEE Journal of Quantum Electronics, 2015
Journal of Differential Geometry
O a Class of Vector-Valued Functions Covariant Under the Classical Groups, with Applications to Physics
Invariant Tensor Fields in Physics and the Classical Groups
Siamam, 1971
Lie Theory and Separation of Variables. II: Parabolic Coordinates
Siam J Math Anal, 1974
SYMMETRY AND SEPARATION OF VARIABLES FOR LINEAR PARTIAL DIFFERENTIAL AND HAMILTON-JACOBI EQUATIONS**Research partially supported by NSF Grant MC S76-04838
Group Theoretical Methods in Physics, 1977
Symmetry, Integrability and Geometry: Methods and Applications, 2005
A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an int... more A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, the system is second order superintegrable. Such systems have remarkable properties. Typical properties are that 1) they are integrable in multiple ways and comparison of ways of integration leads to new facts about the systems, 2) they are multiseparable, 3) the second order symmetries generate a closed quadratic algebra and in the quantum case the representation theory of the quadratic algebra yields important facts about the spectral resolution of the Schrödinger operator and the other symmetry operators, and 4) there are deep connections with expansion formulas relating classes of special functions and with the theory of Exact and Quasi-exactly Solvable systems. For n = 2 the author, E.G. Kalnins and J. Kress, have worked out the structure of these systems and classified all of the possible spaces and potentials. Here I discuss our recent work and announce new results for the much more difficult case n = 3. We consider classical superintegrable systems with nondegenerate potentials in three dimensions and on a conformally flat real or complex space. We show that there exists a standard structure for such systems, based on the algebra of 3 × 3 symmetric matrices, and that the quadratic algebra always closes at order 6. We describe the Stäckel transformation, an invertible conformal mapping between superintegrable structures on distinct spaces, and give evidence indicating that all our superintegrable systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We also indicate how to extend the classical 2D and 3D superintegrability theory to include the operator (quantum) case.
Analysis and Applications, 2014
Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of sec... more Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quan...
Symmetry and Separation of Variables: References
Journal of Mathematical Physics, 2007
Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties... more Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties. [Journal of Mathematical Physics 48, 113518 (2007)]. EG Kalnins, JM Kress, W. Miller, Jr. Abstract. A classical (or quantum ...
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symm... more We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ≡ (Δ_2 +V)Ψ=EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one ...
Journal of Mathematical Physics, 1977
Lie theory and the wave equation in spacetime. 3. Semisubgroup coordinates. [Journal of Mathemat... more Lie theory and the wave equation in spacetime. 3. Semisubgroup coordinates. [Journal of Mathematical Physics 18, 271 (1977)]. EG Kalnins, W. Miller, Jr. Abstract. We classify and study those coordinate systems which permit ...
Journal of Mathematical …, 2002
Complete sets of invariants for dynamical systems that admit a separation of variables. [Journal ... more Complete sets of invariants for dynamical systems that admit a separation of variables. [Journal of Mathematical Physics 43, 3592 (2002)]. EG Kalnins, JM Kress, W. Miller, Jr., GS Pogosyan. Abstract. Consider a classical Hamiltonian ...
Journal of mathematical physics, 2006
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean sp... more Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries. [Journal of Mathematical Physics 47, 033502 (2006)]. EG Kalnins, W. Miller, Jr., GS Pogosyan. Abstract. We ...
Superintegrability on the two-dimensional hyperboloid. II
Journal of Mathematical …, 1999
Superintegrability on the two-dimensional hyperboloid. II. [Journal of Mathematical Physics 40, 2... more Superintegrability on the two-dimensional hyperboloid. II. [Journal of Mathematical Physics 40, 2291 (1999)]. EG Kalnins, W. Miller, Jr., Ye. M. Hakobyan, GS Pogosyan. Abstract. This work is devoted to the investigation of the quantum ...
Quantum Theory and Symmetries
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symm... more We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (∆2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. Most of this paper is devoted to describing the method. Details will be provided elsewhere. As examples we revisit the Tremblay and Winternitz derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. The purpose of this project is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.
Journal of Physics A: Mathematical and Theoretical
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symm... more We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (∆2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and all 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. Our aim is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.
Journal of Physics A: Mathematical and Theoretical
Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intr... more Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators but their symmetry algebras don't close and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bôcher contractions of the conformal Lie algebra so(5, C) to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, and all from Bôcher contractions of a single "generic" system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.
Journal of …, 1991
Crucial to a knowledge of the perturbations of Robertson Walker cosmological models is a knowledg... more Crucial to a knowledge of the perturbations of Robertson Walker cosmological models is a knowledge of complete sets of functions with which to expand such perturbations. For the open Robertson Walker cosmology, this question will be completely answered. In addition, some observations will be made concerning explicit solution by separation of variables of wave equations for spin s in a Riemannan space having an infinitesmal line element of which the Robertson Walker models are a special case. I. VECTOR AND TENSOR HARMONICS ON THREE-DIMENSIONAL SPACES OF CONSTANT RIEMANNIAN CURVATURE The original investigations of Lifshitz' and Lifshitz and Khalatnikov' into the gravitational stability of the Robertson Walker (RW) isotropic cosmological models" demonstrated the utility of scalar, vector, and tensor harmonics in giving a complete description of small perturbations. In particular these authors"' showed that in the synchronous gauge all perturbations involving pressure, density, velocity, and metric fluctuations can be obtained once a complete set ofsuch functions is found for S, (three-dimensional sphere),
Journal of Physics A: Mathematical …, 2010
We refine a method for finding a canonical form of symmetry operators of arbitrary order for the ... more We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (∆ 2 + 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 = αr 2 + β/r 2 cos 2 kθ + γ/r 2 sin 2 kθ (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.
A Geometrical Perspective on the Coherent Multimode Optical Field and Mode Coupling Equations
IEEE Journal of Quantum Electronics, 2015
Journal of Differential Geometry
O a Class of Vector-Valued Functions Covariant Under the Classical Groups, with Applications to Physics
Invariant Tensor Fields in Physics and the Classical Groups
Siamam, 1971
Lie Theory and Separation of Variables. II: Parabolic Coordinates
Siam J Math Anal, 1974
SYMMETRY AND SEPARATION OF VARIABLES FOR LINEAR PARTIAL DIFFERENTIAL AND HAMILTON-JACOBI EQUATIONS**Research partially supported by NSF Grant MC S76-04838
Group Theoretical Methods in Physics, 1977
Symmetry, Integrability and Geometry: Methods and Applications, 2005
A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an int... more A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, the system is second order superintegrable. Such systems have remarkable properties. Typical properties are that 1) they are integrable in multiple ways and comparison of ways of integration leads to new facts about the systems, 2) they are multiseparable, 3) the second order symmetries generate a closed quadratic algebra and in the quantum case the representation theory of the quadratic algebra yields important facts about the spectral resolution of the Schrödinger operator and the other symmetry operators, and 4) there are deep connections with expansion formulas relating classes of special functions and with the theory of Exact and Quasi-exactly Solvable systems. For n = 2 the author, E.G. Kalnins and J. Kress, have worked out the structure of these systems and classified all of the possible spaces and potentials. Here I discuss our recent work and announce new results for the much more difficult case n = 3. We consider classical superintegrable systems with nondegenerate potentials in three dimensions and on a conformally flat real or complex space. We show that there exists a standard structure for such systems, based on the algebra of 3 × 3 symmetric matrices, and that the quadratic algebra always closes at order 6. We describe the Stäckel transformation, an invertible conformal mapping between superintegrable structures on distinct spaces, and give evidence indicating that all our superintegrable systems are Stäckel transforms of systems on complex Euclidean space or the complex 3-sphere. We also indicate how to extend the classical 2D and 3D superintegrability theory to include the operator (quantum) case.
Analysis and Applications, 2014
Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of sec... more Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quan...