478,203 research outputs found
Doubling (Dual) Hahn Polynomials: Classification and Applications
We classify all pairs of recurrence relations in which two Hahn or dual Hahn
polynomials with different parameters appear. Such couples are referred to as
(dual) Hahn doubles. The idea and interest comes from an example appearing in a
finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J.
Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our
classification shows there exist three dual Hahn doubles and four Hahn doubles.
The same technique is then applied to Racah polynomials, yielding also four
doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set
of symmetric orthogonal polynomials related to the Christoffel and Geronimus
transformations. For each case, we also have an interesting class of
two-diagonal matrices with closed form expressions for the eigenvalues. This
extends the class of Sylvester-Kac matrices by remarkable new test matrices. We
examine also the algebraic relations underlying the dual Hahn doubles, and
discuss their usefulness for the construction of new finite oscillator models
The Hahn Quantum System
Using a formulation of quantum mechanics based on the theory of orthogonal
polynomials, we introduce a four-parameter system associated with the Hahn and
continuous Hahn polynomials. The continuum energy scattering states are written
in terms of the continuous Hahn polynomial whose asymptotics give the
scattering amplitude and phase shift. On the other hand, the finite number of
discrete bound states are associated with the Hahn polynomial.Comment: 18 pages, 7 figure
Hahn echo and criticality in spin-chain systems
We establish a relation between Hahn spin-echo of a spin-
particle and quantum phase transition in a spin-chain, which couples to the
particle. The Hahn echo is calculated and discussed at zero as well as at
finite temperatures. On the example of XY model, we show that the critical
points of the chain are marked by the extremal values in the Hahn echo, and
influence the Hahn echo in surprising high temperature. An explanation for the
relation between the echo and criticality is also presented.Comment: 5 pages, 6 figure
The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise
in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are
the bispectral polynomials of a discrete argument obtained from a q-> -1 limit
of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators
including an involution, it is also a q-> -1 limit of the quantum algebra
sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator.
The algebra H, a two-parameter generalization of u(2) with an involution as
additional generator, is first derived from the recurrence relation of the -1
Hahn polynomials. It is then shown that H can be realized in terms of the
generators of two added sl_{-1}(2) algebras, so that the Clebsch-Gordan
coefficients of sl_{-1}(2) are dual -1 Hahn polynomials. An irreducible
representation of H involving five-diagonal matrices and connected to the
difference equation of the dual -1 Hahn polynomials is constructed.Comment: 15 pages, Some minor changes from version #
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