Related papers: The big q-Jacobi function transform
By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of…
In this paper, we shall consider the notion of bicomplex inner product and define bicomplex Hilbert space. We shall define $L^{2}[a,b]$ where the functions take bicomplex values. We shall prove the Theorem for a bounded self adjoint…
For two positive integers $m$ and $n$, we let ${\mathbb H}_n$ be the Siegel upper half plane of degree $n$ and let ${\mathbb C}^{(m,n)}$ be the set of all $m\times n$ complex matrices. In this article, we study differential operators on the…
This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various…
We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order…
We study the set ${\cal C}$ consisting of pairs of orthogonal projections $P,Q$ acting in a Hilbert space ${\cal H}$ such that $PQ$ is a compact operator. These pairs have a rich geometric structure which we describe here. They are parted…
We introduce families of rational functions that are biorthogonal with respect to the $q$-hypergeometric distribution. A triplet of $q$-difference operators $X$, $Y$, $Z$ is shown to play a role analogous to the pair of bispectral operators…
A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on…
Following the pioneering work of Duistermaat and Gr\"unbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one…
For a (classically) integrable quantum mechanical system with two degrees of freedom, the functional dependence $\hat{H}=H_Q(\hat{J}_1,\hat{J}_2)$ of the Hamiltonian operator on the action operators is analyzed and compared with the…
In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of…
The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are…
All the hermitian representations of the ``symmetric" $q$-oscillator are obtained by means of expansions. The same technique is applied to characterize in a systematic way the $k$-order boson realizations of the $q$-oscillator and…
A simple connection between the universal $R$ matrix of $U_q(sl(2))$ (for spins $\demi$ and $J$) and the required form of the co-product action of the Hilbert space generators of the quantum group symmetry is put forward. This gives an…
The q-special functions appear naturally in q-deformed quantum mechanics and both sides profit from this fact. Here we study the relation between the q-deformed harmonic oscillator and the q-Hermite polynomials. We discuss: recursion…
The observable algebra O of SO_q(3)-symmetric quantum mechanics is generated by the coordinates of momentum and position spaces (which are both isomorphic to the SO_q(3)-covariant real quantum space R_q^3). Their interrelations are…
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…
The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra…
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…
Formally self-adjoint, conformally covariant, polydifferential operators provide a general framework for studying variational problems, such as prescribing the scalar, $Q$-, or $\sigma_2$-curvatures, within a conformal class. We describe…