21,877 research outputs found
Spectra of observables in the q-oscillator and q-analogue of the Fourier transform
Spectra of the position and momentum operators of the Biedenharn-Macfarlane
q-oscillator (with the main relation aa^+-qa^+a=1) are studied when q>1. These
operators are symmetric but not self-adjoint. They have a one-parameter family
of self-adjoint extensions. These extensions are derived explicitly. Their
spectra and eigenfunctions are given. Spectra of different extensions do not
intersect. The results show that the creation and annihilation operators a^+
and a of the q-oscillator for q>1 cannot determine a physical system without
further more precise definition. In order to determine a physical system we
have to choose appropriate self-adjoint extensions of the position and momentum
operators.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Minimal representations of unitary operators and orthogonal polynomials on the unit circle
In this paper we prove that the simplest band representations of unitary
operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the
unit circle play an essential role in the development of this result, and also
provide a parametrization of such five-diagonal representations which shows
specially simple and interesting decomposition and factorization properties. As
an application we get the reduction of the spectral problem of any unitary
Hessenberg matrix to the spectral problem of a five-diagonal one. Two
applications of these results to the study of orthogonal polynomials on the
unit circle are presented: the first one concerns Krein's Theorem; the second
one deals with the movement of mass points of the orthogonality measure under
monoparametric perturbations of the Schur parameters.Comment: 31 page
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
Orthogonal polynomials for the weakly equilibrium Cantor sets
Let be the weakly equilibrium Cantor type set introduced in [10].
It is proven that the monic orthogonal polynomials with respect to
the equilibrium measure of coincide with the Chebyshev polynomials
of the set. Procedures are suggested to find of all degrees and the
corresponding Jacobi parameters. It is shown that the sequence of the Widom
factors is bounded below
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