The Experts below are selected from a list of 405 Experts worldwide ranked by ideXlab platform
Pablo Román - One of the best experts on this subject based on the ideXlab platform.
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Matrix Elements of Irreducible Representations of SU(n+1) x SU(n+1) and Multivariable Matrix-Valued Orthogonal Polynomials
Journal of Functional Analysis, 2020Co-Authors: Erik Koelink, Maarten Van Pruijssen, Pablo RománAbstract:Abstract In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as Simultaneous Eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case SU ( n + 1 ) meets all the conditions that we impose in Part 1. For any k ∈ N 0 we obtain families of orthogonal polynomials in n variables with values in the N × N -matrices, where N = ( n + k k ) . The case k = 0 leads to the classical Heckman-Opdam polynomials of type A n with geometric parameter. For k = 1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n ≥ 2 . We also give explicit expressions of the spherical functions that determine the matrix weight for k = 1 . These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n = 1 . The commuting family of differential operators that have the matrix-valued polynomials as Simultaneous Eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for ( n , k ) equal to ( 2 , 1 ) and ( 3 , 1 ) .
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Matrix elements of irreducible representations of $\mathrm{SU}(n+1)\times\mathrm{SU}(n+1)$ and multivariable matrix-valued orthogonal polynomials
arXiv: Representation Theory, 2017Co-Authors: Erik Koelink, Maarten Van Pruijssen, Pablo RománAbstract:In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as Simultaneous Eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case $\mathrm{SU}(n+1)$ meets all the conditions that we impose in Part 1. For any $k\in\mathbb{N}_{0}$ we obtain families of orthogonal polynomials in $n$ variables with values in the $N\times N$-matrices, where $N=\binom{n+k}{k}$. The case $k=0$ leads to the classical Heckman-Opdam polynomials of type $A_{n}$ with geometric parameter. For $k=1$ we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever $n\ge2$. We also give explicit expressions of the spherical functions that determine the matrix weight for $k=1$. These expressions are used to calculate the spherical functions that determine the matrix weight for general $k$ up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case $n=1$. The commuting family of differential operators that have the matrix-valued polynomials as Simultaneous Eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for $(n,k)$ equal to $(2,1)$ and $(3,1)$.
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Deformation of matrix-valued orthogonal polynomials related to Gelfand pairs
arXiv: Classical Analysis and ODEs, 2016Co-Authors: Maarten Van Pruijssen, Pablo RománAbstract:In this paper we present a method to obtain deformations of families of matrix-valued orthogonal polynomials that are associated to the representation theory of compact Gelfand pairs. These polynomials have the Sturm-Liouville property in the sense that they are Simultaneous Eigenfunctions of a symmetric second order differential operator and we deform this operator accordingly so that the deformed families also have the Sturm-Liouville property. Our strategy is to deform the system of spherical functions that is related to the matrix-valued orthogonal polynomials and then check that the polynomial structure is respected by the deformation. Crucial in these considerations is the full spherical function $\Psi_{0}$, which relates the spherical functions to the polynomials. We prove an explicit formula for $\Psi_{0}$ in terms of Krawtchouk polynomials for the Gelfand pair $(\mathrm{SU}(2)\times\mathrm{SU}(2),\mathrm{diag}(\mathrm{SU}(2)))$. For the matrix-valued orthogonal polynomials associated to this pair, a deformation was already available by different methods and we show that our method gives same results using explicit knowledge of $\Psi_{0}$. Furthermore we apply our method to some of the examples of size $2\times2$ for more general Gelfand pairs. We prove that the families related to the groups $\mathrm{SU}(n)$ are deformations of one another. On the other hand, the families associated to the symplectic groups $\mathrm{Sp}(n)$ give rise to a new family with an extra free parameter.
Miki Wadati - One of the best experts on this subject based on the ideXlab platform.
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The Calogero Model: Integrable Structure and Orthogonal Basis
Calogero—Moser— Sutherland Models, 2000Co-Authors: Miki Wadati, Hideaki UjinoAbstract:Integrability, algebraic structures, and orthogonal basis of the Calogero model are studied by the quantum Lax and Dunkl operator formulations. The commutator algebra among operators including conserved operators and creation-annihilation operators has the structure of the W-algebra. Through an algebraic construction of the Simultaneous Eigenfunctions of all the commuting conserved operators, we show that the Hi-Jack (hidden-Jack) polynomials, which are an multivariable generalization of the Her mit e polynomials, form the orthogonal basis.
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An algebraic approach to the non-symmetric Macdonald polynomial
Nuclear Physics B, 1999Co-Authors: Akinori Nishino, Hideaki Ujino, Miki WadatiAbstract:Abstract In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are Simultaneous Eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them.
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Fock Spaces for the Calogero Models with Distinguishable Particles
Journal of the Physical Society of Japan, 1998Co-Authors: Hideaki Ujino, Akinori Nishino, Miki WadatiAbstract:We present the creation, annihilation and number operators for the A N -1 - and B N -Calogero models with distinguishable particles. In a parallel way to the construction of the Fock space of the quantum harmonic oscillators, we construct the Fock spaces as the Simultaneous Eigenfunctions of the number operators for the Calogero models. Relationships between the Fock spaces and the known nonsymmetric orthogonal bases, which are spanned by the variants of the Jack polynomials, are presented.
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Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model
arXiv: Statistical Mechanics, 1998Co-Authors: Akinori Nishino, Hideaki Ujino, Miki WadatiAbstract:Using a similarity transformation that maps the Calogero model into $N$ decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their Simultaneous Eigenfunctions. The Simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the $B_{N}$-Calogero model.
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algebraic construction of a new symmetric orthogonal basis for the calogero model
Journal of the Physical Society of Japan, 1998Co-Authors: Hideaki Ujino, Miki WadatiAbstract:We demonstrate an algebraic construction of all the Simultaneous Eigenfunctions of the conserved operators for the Calogero model. The eigenfunction is a deformation of the symmetrized number state (bosonic state) of the N decoupled quantum harmonic oscillators. The eigenfunction does not coincide with the Hi-Jack polynomial, which shows that the conserved operators derived from the number operators of the decoupled quantum harmonic oscillators are algebraically different from those derived by the Dunkl operator formulation.
Harsh Mathur - One of the best experts on this subject based on the ideXlab platform.
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Particle in a box in PT -symmetric quantum mechanics and an electromagnetic analog
Physical Review A, 2013Co-Authors: Anirudh Dasarathy, Joshua P. Isaacson, Katherine Jones-smith, Jason Tabachnik, Harsh MathurAbstract:In PT quantum mechanics a fundamental principle of quantum mechanics, that the Hamiltonian must be hermitian, is replaced by another set of requirements, including notably symmetry under PT, where P denotes parity and T denotes time reversal. Here we study the role of boundary conditions in PT quantum mechanics by constructing a simple model that is the PT symmetric analog of a particle in a box. The model has the usual particle in a box Hamiltonian but boundary conditions that respect PT symmetry rather than hermiticity. We find that for a broad class of PT-symmetric boundary conditions the model respects the condition of unbroken PT-symmetry, namely that the Hamiltonian and the symmetry operator PT have Simultaneous Eigenfunctions, implying that the energy eigenvalues are real. We also find that the Hamiltonian is self-adjoint under the PT inner product. Thus we obtain a simple soluble model that fulfils all the requirements of PT quantum mechanics. In the second part of this paper we formulate a variational principle for PT quantum mechanics that is the analog of the textbook Rayleigh-Ritz principle. Finally we consider electromagnetic analogs of the PT-symmetric particle in a box. We show that the isolated particle in a box may be realized as a Fabry-Perot cavity between an absorbing medium and its conjugate gain medium. Coupling the cavity to an external continuum of incoming and outgoing states turns the energy levels of the box into sharp resonances. Remarkably we find that the resonances have a Breit-Wigner lineshape in transmission and a Fano lineshape in reflection; by contrast in the corresponding hermitian case the lineshapes always have a Breit-Wigner form in both transmission and reflection.
Fokko J. Van De Bult - One of the best experts on this subject based on the ideXlab platform.
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Ruijsenaars' hypergeometric function and the modular double of U-q(sl(2)(C))
Advances in Mathematics, 2006Co-Authors: Fokko J. Van De BultAbstract:Abstract Simultaneous Eigenfunctions of two Askey–Wilson second-order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra U q ( sl 2 ( C ) ) . These Eigenfunctions are shown to be equal to Ruijsenaars’ hypergeometric function under a proper parameter correspondence.
Hideaki Ujino - One of the best experts on this subject based on the ideXlab platform.
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The Calogero Model: Integrable Structure and Orthogonal Basis
Calogero—Moser— Sutherland Models, 2000Co-Authors: Miki Wadati, Hideaki UjinoAbstract:Integrability, algebraic structures, and orthogonal basis of the Calogero model are studied by the quantum Lax and Dunkl operator formulations. The commutator algebra among operators including conserved operators and creation-annihilation operators has the structure of the W-algebra. Through an algebraic construction of the Simultaneous Eigenfunctions of all the commuting conserved operators, we show that the Hi-Jack (hidden-Jack) polynomials, which are an multivariable generalization of the Her mit e polynomials, form the orthogonal basis.
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An algebraic approach to the non-symmetric Macdonald polynomial
Nuclear Physics B, 1999Co-Authors: Akinori Nishino, Hideaki Ujino, Miki WadatiAbstract:Abstract In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are Simultaneous Eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them.
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Fock Spaces for the Calogero Models with Distinguishable Particles
Journal of the Physical Society of Japan, 1998Co-Authors: Hideaki Ujino, Akinori Nishino, Miki WadatiAbstract:We present the creation, annihilation and number operators for the A N -1 - and B N -Calogero models with distinguishable particles. In a parallel way to the construction of the Fock space of the quantum harmonic oscillators, we construct the Fock spaces as the Simultaneous Eigenfunctions of the number operators for the Calogero models. Relationships between the Fock spaces and the known nonsymmetric orthogonal bases, which are spanned by the variants of the Jack polynomials, are presented.
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Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model
arXiv: Statistical Mechanics, 1998Co-Authors: Akinori Nishino, Hideaki Ujino, Miki WadatiAbstract:Using a similarity transformation that maps the Calogero model into $N$ decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their Simultaneous Eigenfunctions. The Simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that is a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformation and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent sets of mutually commuting conserved operators, as is the case with the hydrogen atom. We also confirm the same story for the $B_{N}$-Calogero model.
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algebraic construction of a new symmetric orthogonal basis for the calogero model
Journal of the Physical Society of Japan, 1998Co-Authors: Hideaki Ujino, Miki WadatiAbstract:We demonstrate an algebraic construction of all the Simultaneous Eigenfunctions of the conserved operators for the Calogero model. The eigenfunction is a deformation of the symmetrized number state (bosonic state) of the N decoupled quantum harmonic oscillators. The eigenfunction does not coincide with the Hi-Jack polynomial, which shows that the conserved operators derived from the number operators of the decoupled quantum harmonic oscillators are algebraically different from those derived by the Dunkl operator formulation.
