240 research outputs found
On discrete q-ultraspherical polynomials and their duals
We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q),
which corresponds to a=b=-c, leads to a discrete orthogonality relation for
imaginary values of the parameter a (outside of its commonly known domain 0<a<
q^{-1}). Since P_n(x;q^\alpha, q^\alpha, -q^\alpha; q) tend to Gegenbauer (or
ultraspherical) polynomials in the limit as q->1, this family represents yet
another q-extension of these classical polynomials, different from the
continuous q-ultraspherical polynomials of Rogers. The dual family with respect
to the polynomials P_n(x;a,a,-a;q) (i.e., the dual discrete q-ultraspherical
polynomials) corresponds to the indeterminate moment problem, that is, these
polynomials have infinitely many orthogonality relations. We find orthogonality
relations for these polynomials, which have not been considered before. In
particular, extremal orthogonality measures for these polynomials are derived.Comment: 14 page
Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom
The Kravchuk and Meixner polynomials of discrete variable are introduced for
the discrete models of the harmonic oscillator and hydrogen atom. Starting from
Rodrigues formula we construct raising and lowering operators, commutation and
anticommutation relations. The physical properties of discrete models are
figured out through the equivalence with the continuous models obtained by
limit process.Comment: LaTeX, 14 pages (late submission
A finite oscillator model related to sl(2|1)
We investigate a new model for the finite one-dimensional quantum oscillator
based upon the Lie superalgebra sl(2|1). In this setting, it is natural to
present the position and momentum operators of the oscillator as odd elements
of the Lie superalgebra. The model involves a parameter p (0<p<1) and an
integer representation label j. In the (2j+1)-dimensional representations W_j
of sl(2|1), the Hamiltonian has the usual equidistant spectrum. The spectrum of
the position operator is discrete and turns out to be of the form
, where k=0,1,...,j. We construct the discrete position wave
functions, which are given in terms of certain Krawtchouk polynomials. These
wave functions have appealing properties, as can already be seen from their
plots. The model is sufficiently simple, in the sense that the corresponding
discrete Fourier transform (relating position wave functions to momentum wave
functions) can be constructed explicitly
Bulk spectral function sum rule in QCD-like theories with a holographic dual
We derive the sum rule for the spectral function of the stress-energy tensor
in the bulk (uniform dilatation) channel in a general class of strongly coupled
field theories. This class includes theories holographically dual to a theory
of gravity coupled to a single scalar field, representing the operator of the
scale anomaly. In the limit when the operator becomes marginal, the sum rule
coincides with that in QCD. Using the holographic model, we verify explicitly
the cancellation between large and small frequency contributions to the
spectral integral required to satisfy the sum rule in such QCD-like theories.Comment: 16 pages, 2 figure
On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials
This paper studies properties of q-Jacobi polynomials and their duals by
means of operators of the discrete series representations for the quantum
algebra U_q(su_{1,1}). Spectrum and eigenfunctions of these operators are found
explicitly. These eigenfunctions, when normalized, form an orthogonal basis in
the representation space. The initial U_q(su_{1,1})-basis and the bases of
these eigenfunctions are interconnected by matrices, whose entries are
expressed in terms of little and big q-Jacobi polynomials. The orthogonality by
rows in these unitary connection matrices leads to the orthogonality relations
for little and big q-Jacobi polynomials. The orthogonality by columns in the
connection matrices leads to an explicit form of orthogonality relations on the
countable set of points for {}_3\phi_2 and {}_3\phi_1 polynomials, which are
dual to big and little q-Jacobi polynomials, respectively. The orthogonality
measure for the dual little q-Jacobi polynomials proves to be extremal, whereas
the measure for the dual big q-Jacobi polynomials is not extremal.Comment: 26 pages, LaTeX, the exposition is slightly improved and some
additional references have been adde
A superintegrable finite oscillator in two dimensions with SU(2) symmetry
A superintegrable finite model of the quantum isotropic oscillator in two
dimensions is introduced. It is defined on a uniform lattice of triangular
shape. The constants of the motion for the model form an SU(2) symmetry
algebra. It is found that the dynamical difference eigenvalue equation can be
written in terms of creation and annihilation operators. The wavefunctions of
the Hamiltonian are expressed in terms of two known families of bivariate
Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials
form bases for SU(2) irreducible representations. It is further shown that the
pair of eigenvalue equations for each of these families are related to each
other by an SU(2) automorphism. A finite model of the anisotropic oscillator
that has wavefunctions expressed in terms of the same Rahman polynomials is
also introduced. In the continuum limit, when the number of grid points goes to
infinity, standard two-dimensional harmonic oscillators are obtained. The
analysis provides the limit of the bivariate Krawtchouk
polynomials as a product of one-variable Hermite polynomials
Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials
By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established.3a допомогою двох операторів, зображуваних матрицями Якобі, введено сім'ю q-ортогональних многочленів, що є дуальними по відношенню до альтернативних q-многочленів Шарльє. Для цих многочленів отримано дискретне співвідношення ортогональності та властивість повноти
More on the q-oscillator algebra and q-orthogonal polynomials
Properties of certain -orthogonal polynomials are connected to the
-oscillator algebra. The Wall and -Laguerre polynomials are shown to
arise as matrix elements of -exponentials of the generators in a
representation of this algebra. A realization is presented where the continuous
-Hermite polynomials form a basis of the representation space. Various
identities are interpreted within this model. In particular, the connection
formula between the continuous big -Hermite polynomials and the continuous
-Hermite polynomials is thus obtained, and two generating functions for
these last polynomials are algebraically derived
- …