Searcharxiv⌕ Search

arXiv subjects

Charles F. Dunkl

Publications and source records attributed to Charles F. Dunkl.

At least 37 records · Page 2Linked to original sources
arXiv2017

Vector-valued Jack Polynomials and Wavefunctions on the Torus

The Hamiltonian of the quantum Calogero-Sutherland model of $N$ identical particles on the circle with $1/r^{2}$ interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials taking values in modules of the symmetric group and the matrix solution of a system of linear differential equations one constructs novel eigenfunctions of the Hamiltonian. Like the usual wavefunctions each eigenfunction determines a symmetric probability density on the $N$-torus. The construction applies to any irreducible representation of the symmetric group. The methods depend on the theory of generalized Jack polynomials due to Griffeth, and the Yang-Baxter graph approach of Luque and the author.

math-ph↗
arXiv2016

Planar harmonic and monogenic polynomials of type A

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on $\mathbb{R}^{N}$. The Dunkl operators are denoted by $T_{j}$ for $1\leq j\leq N$, and the Laplacian $Δ_κ=\sum_{j=1}^{N}T_{j}^{2}$. This paper finds the homogeneous harmonic polynomials annihilated by all $T_{j}$ for $j>2$. The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.

math.CA↗
arXiv2016

Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

For each irreducible module of the symmetric group on $N$ objects there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the $N$-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the Fourier-Stieltjes coefficients of the measure are established, and used to identify parameter values for which the construction fails. It is shown that the absolutely continuous part of the measure satisfies a first-order system of differential equations.

math.RT↗
arXiv2015

Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States

Previously, a formula, incorporating a $5F4$ hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments $\left\langle \left\vert ρ^{PT}\right\vert ^{n}\left\vert ρ\right\vert ^{k}\right\rangle /\left\langle \left\vert ρ\right\vert ^{k}\right\rangle$ of $4 \times 4$ density-matrices ($ρ$), and their partial transposes ($ρ^{PT}|$) was applied with $k=0$ to the generalized two-qubit separability-probability question. The formula can, further, be viewed we note here, as an averaging over "induced measures in the space of mixed quantum states". The associated induced-measure separability probabilities ($k =1, 2,\ldots$) are found--{\it via} a high-precision density approximation procedure--to assume interesting, relatively simple rational values in the two-re[al]bit ($α= \frac{1}{2}$), (standard) two-qubit ($α= 1$) and two-quater[nionic]bit ($α=2$) cases. We deduce rather simple companion (rebit, qubit, quaterbit,\ldots) formulas that successfully reproduce the rational values assumed for {\it general} $k$. These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

quant-ph↗
arXiv2015

Separability Probability Formulas and Their Proofs for Generalized Two-Qubit X-Matrices Endowed with Hilbert-Schmidt and Induced Measures

Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert-Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general 2-qubit states--deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter $α$ employed.

quant-ph↗
arXiv2015

Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities

Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($ρ$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|ρ|>|ρ^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The new determinantal $4F3$-hypergeometric moment formula is, then, adjusted (Appendices B and C) to the boundary case of minimally degenerate states ($|ρ|=0$), and its consistency manifested-also using density-approximation-with a theorem of Szarek, Bengtsson and {Ż}yczkowski (arXiv:quant-ph/0509008). This theorem states that the Hilbert-Schmidt separability probabilities of generic minimally degenerate two-qubit states are (again) one-half those of the corresponding generic nondegenerate states.

quant-ph↗
arXiv2014

Real numerical shadow and generalized B-splines

Restricted numerical shadow $P^X_A(z)$ of an operator $A$ of order $N$ is a probability distribution supported on the numerical range $W_X(A)$ restricted to a certain subset $X$ of the set of all pure states - normalized, one-dimensional vectors in ${\mathbb C}^N$. Its value at point $z \in {\mathbb C}$ equals to the probability that the inner product $< u |A| u >$ is equal to $z$, where $u$ stands for a random complex vector from the set $X$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. For a Hermitian operator $A$ of order $N$ we derive an explicit formula for its shadow restricted to real states, $P^{\mathbb R}_A(x)$, show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the $B$-spline. Furthermore, for operators acting on a space with tensor product structure, ${\cal H}_A \otimes {\cal H}_B$, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order N=4.

math.FA↗
arXiv2014

Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case for even dihedral groups). The matrix weight function for the Gaussian form is found explicitly by solving a boundary value problem, and then computing the normalizing constant. An orthogonal basis for the homogeneous harmonic polynomials is constructed. The coefficients of these polynomials are found to be balanced terminating $_4F_3$-series.

math.CA↗
arXiv2013

Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action. II

This is a sequel to [SIGMA 9 (2013), 007, 23 pages, arXiv:1210.1177], in which there is a construction of a $2\times2$ positive-definite matrix function $K (x)$ on $\mathbb{R}^{2}$. The entries of $K(x)$ are expressed in terms of hypergeometric functions. This matrix is used in the formula for a Gaussian inner product related to the standard module of the rational Cherednik algebra for the group $W(B_2)$ (symmetry group of the square) associated to the (2-dimensional) reflection representation. The algebra has two parameters: $k_{0}$, $k_{1}$. In the previous paper $K$ is determined up to a scalar, namely, the normalization constant. The conjecture stated there is proven in this note. An asymptotic formula for a sum of $_{3}F_{2}$-type is derived and used for the proof.

math.CA↗
arXiv2013

Clustering properties of rectangular Macdonald polynomials

The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.

math-ph↗
arXiv2013

Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action

The structure of orthogonal polynomials on $\mathbb{R}^{2}$ with the weight function $| x_{1}^{2}-x_{2}^{2}|^{2k_{0}}| x_{1}x_{2}|^{2k_{1}}e^{-(x_{1}^{2}+x_{2}^{2})/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0}+k_{1}>-\frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique 2-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $(k_{0},k_{1})$ satisfy $-\frac{1}{2}<k_{0}\pm k_{1}<\frac{1}{2}$. For vector polynomials $(f_{i})_{i=1}^{2}$, $(g_{i})_{i=1}^{2}$ the inner product has the form $\iint_{\mathbb{R}^{2}}f(x) K(x) g(x)^{T}e^{-(x_{1}^{2}+x_{2}^{2})/2}dx_{1}dx_{2}$ where the matrix function $K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.

math.CA↗
arXiv2012

Some Rarita-Schwinger Type Operators

In this paper we study a generalization of the classical Rarita-Schwinger type operators and construct their fundamental solutions. We give some basic integral formulas related to these operators. We also establish that the projection operators appearing in the Rarita-Schwinger operators and the Rarita-Schwinger equations are conformally invariant. We further obtain the intertwining operators for other operators related to the Rarita-Schwinger operators under actions of the conformal group.

math.CV↗
arXiv2012

Dunkl Operators and Related Special Functions

Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of functions like Gegenbauer, Jacobi and Jack symmetric polynomials and Bessel functions are labeled by parameters. These could be unspecified transcendental numbers or drawn from large sets of real numbers, for example the complement of {-1/2, -3/2, -5/2,...}. One aim of this chapter is to provide a harmonic analysis setting in which parameters play a natural role. The basic objects are finite reflection (Coxeter) groups and algebras of operators on polynomials which generalize the algebra of partial differential operators. These algebras have as many parameters as the number of conjugacy classes of reflections in the associated groups.

math.CA↗
arXiv2012

Restricted numerical shadow and geometry of quantum entanglement

The restricted numerical range $W_R(A)$ of an operator $A$ acting on a $D$-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset $R$ of the of set of pure quantum states of dimension $D$. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of $A$ -- a normalized probability distribution on the complex plane supported in $W_R(A)$. Its value at point $z \in {\mathbbm C}$ is equal to the probability that the expectation value $< ψ|A|ψ>$ is equal to $z$, where $|ψ>$ represents a random quantum state in subset $R$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini--Study measure. Studying restricted shadows of operators of a fixed size $D=N_A N_B$ we analyse the geometry of sets of separable and maximally entangled states of the $N_A \times N_B$ composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on $D=2^3$ dimensional Hilbert space allows us to investigate the structure of the orbits of $GHZ$ and $W$ quantum states of a three--qubit system.

quant-ph↗
arXiv2011

Numerical shadow and geometry of quantum states

The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.

quant-ph↗
arXiv2011

Vector-Valued Jack Polynomials from Scratch

Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak S}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak S}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups $G(r,p,N)$ and studied by one of the authors (C. Dunkl) in the specialization $r=p=1$ (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.

math.CO↗