105 research outputs found
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
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
Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
Diagonalization of a certain operator in irreducible representations of the
positive discrete series of the quantum algebra U_q(su(1,1)) is studied.
Spectrum and eigenfunctions of this operator are found in an explicit form.
These eigenfunctions, when normalized, constitute an orthonormal basis in the
representation space. The initial U_q(su(1,1))-basis and the basis of
eigenfunctions are interrelated by a matrix with entries, expressed in terms of
big q-Laguerre polynomials. The unitarity of this connection matrix leads to an
orthogonal system of functions, which are dual with respect to big q-Laguerre
polynomials. This system of functions consists of two separate sets of
functions, which can be expressed in terms of q-Meixner polynomials
M_n(x;b,c;q) either with positive or negative values of the parameter b. The
orthogonality property of these two sets of functions follows directly from the
unitarity of the connection matrix. As a consequence, one obtains an
orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A
biorthogonal system of functions (with respect to the scalar product in the
representation space) is also derived.Comment: 15 pages, LaTe
Factorization method for difference equations of hypergeometric type on nonuniform lattices
We study the factorization of the hypergeometric-type difference equation of
Nikiforov and Uvarov on nonuniform lattices. An explicit form of the raising
and lowering operators is derived and some relevant examples are given.Comment: 21 page
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
On a q-extension of Mehta's eigenvectors of the finite Fourier transform for q a root of unity
It is shown that the continuous q-Hermite polynomials for q a root of unity
have simple transformation properties with respect to the classical Fourier
transform. This result is then used to construct q-extended eigenvectors of the
finite Fourier transform in terms of these polynomials.Comment: 12 pages, thoroughly rewritten, the q-extended eigenvectors now
N-periodic with q an M-th root of
Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System
We consider a quantum system consisting of a one-dimensional chain of M
identical harmonic oscillators with natural frequency , coupled by
means of springs. Such systems have been studied before, and appear in various
models. In this paper, we approach the system as a Wigner Quantum System, not
imposing the canonical commutation relations, but using instead weaker
relations following from the compatibility of Hamilton's equations and the
Heisenberg equations. In such a setting, the quantum system allows solutions in
a finite-dimensional Hilbert space, with a discrete spectrum for all physical
operators. We show that a class of solutions can be obtained using generators
of the Lie superalgebra gl(1|M). Then we study the properties and spectra of
the physical operators in a class of unitary representations of gl(1|M). These
properties are both interesting and intriguing. In particular, we can give a
complete analysis of the eigenvalues of the Hamiltonian and of the position and
momentum operators (including multiplicities). We also study probability
distributions of position operators when the quantum system is in a stationary
state, and the effect of the position of one oscillator on the positions of the
remaining oscillators in the chain
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