268 research outputs found

    Planar Harmonic Polynomials of Type B

    Full text link
    The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to partial derivative operators. This paper finds all the polynomials in N variables which are annihilated by the sum of the squares (T_1)^2+(T_2)^2 and by all T_i for i>2 (harmonic). They are given explicitly in terms of a novel basis of polynomials, defined by generating functions. The harmonic polynomials can be used to find wave functions for the quantum many-body spin Calogero model.Comment: 17 pages, LaTe

    Vector valued Macdonald polynomials

    Full text link
    This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type AN−1A_{N-1}. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several objects and properties are analyzed, such as the canonical bilinear form which pairs polynomials with those arising from reciprocals of the original parameters, and the symmetrization of the Macdonald polynomials. The main tool of the study is the Yang-Baxter graph. We show that these Macdonald polynomials can be easily computed following this graph. We give also an interpretation of the symmetrization and the bilinear forms applied to the Macdonald polynomials in terms of the Yang-Baxter graph.Comment: 85 pages, 5 figure

    Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model

    Full text link
    We study the hole propagator of the Calogero-Sutherland model with SU(2) internal symmetry. We obtain the exact expression for arbitrary non-negative integer coupling parameter β\beta and prove the conjecture proposed by one of the authors. Our method is based on the theory of the Jack polynomials with a prescribed symmetry.Comment: 12 pages, REVTEX, 1 eps figur

    On Some One-Parameter Families of Three-Body Problems in One Dimension: Exchange Operator Formalism in Polar Coordinates and Scattering Properties

    Get PDF
    We apply the exchange operator formalism in polar coordinates to a one-parameter family of three-body problems in one dimension and prove the integrability of the model both with and without the oscillator potential. We also present exact scattering solution of a new family of three-body problems in one dimension.Comment: 10 pages, LaTeX, no figur

    Supertraces on the algebra of observables of the rational Calogero model based on the classical root system

    Get PDF
    A complete set of supertraces on the algebras of observables of the rational Calogero models with harmonic interaction based on the classical root systems of B_N, C_N and D_N types is found. These results extend the results known for the case A_N. It is shown that there exist Q independent supertraces where Q(B_N)=Q(C_N) is a number of partitions of N into a sum of positive integers and Q(D_N) is a number of partitions of N into a sum of positive integers with even number of even integers.Comment: 10 pages, LATE

    Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics

    Full text link
    We study self-adjoint operators defined by factorizing second order differential operators in first order ones. We discuss examples where such factorizations introduce singular interactions into simple quantum mechanical models like the harmonic oscillator or the free particle on the circle. The generalization of these examples to the many-body case yields quantum models of distinguishable and interacting particles in one dimensions which can be solved explicitly and by simple means. Our considerations lead us to a simple method to construct exactly solvable quantum many-body systems of Calogero-Sutherland type.Comment: 17 pages, LaTe

    A Spectral Method for Elliptic Equations: The Neumann Problem

    Full text link
    Let Ω\Omega be an open, simply connected, and bounded region in Rd\mathbb{R}^{d}, d≥2d\geq2, and assume its boundary ∂Ω\partial\Omega is smooth. Consider solving an elliptic partial differential equation −Δu+γu=f-\Delta u+\gamma u=f over Ω\Omega with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball BB, and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials unu_{n} of degree ≤n\leq n that is convergent to uu. The transformation from Ω\Omega to BB requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For u∈C∞(Ω‾)u\in C^{\infty}(\overline{\Omega}) and assuming ∂Ω\partial\Omega is a C∞C^{\infty} boundary, the convergence of ∥u−un∥H1\Vert u-u_{n}\Vert_{H^{1}} to zero is faster than any power of 1/n1/n. Numerical examples in R2\mathbb{R}^{2} and R3\mathbb{R}^{3} show experimentally an exponential rate of convergence.Comment: 23 pages, 11 figure
    • …
    corecore