Related papers: Harmonic Analysis on Quantum Complex Hyperbolic Sp…
The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are…
For certain situations we give a geometrical background for quasiclassical KP calculations based on an explicit connection to quantum mechanics and the collapse of coherent states to coadjoint orbits for classical operators.
In this paper, we show Hardy-Rellich identities for polyharmonic operators $\Delta^m$ and radial Laplacian $\Delta_r^m$ in $\mathbb{R}^n$ with Hardy-H\'enon weight $|x|^\alpha$ for all $m, n\in \mathbb{N}, \alpha\in \mathbb{R}$. Moreover,…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…
We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from $L^\infty(\mathbb B)$ to $\mathcal B$, and from $C_0(\mathbb B)$ to $\mathcal B_0$ are onto. We…
The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing $h_0$.…
A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian…
We develop the quantum harmonic analysis framework in the reducible setting and apply our findings to polyanalytic Fock spaces. In particular, we explain some phenomena observed in arXiv:2201.10230 and answer a few related open questions.…
We improve a specific method to obtain the dimension of the eigenspaces of the Laplace-Beltrami operator on lens spaces and establish some applications related to the explicit description of the dimension of the smallest positive eigenvalue…
We describe various aspects of the Al-Salam-Carlitz $q$-Charlier polynomials. These include combinatorial descriptions of the moments, the orthogonality relation, and the linearization coefficients.
Our concern in this paper is to study the qualitative properties for harmonic functions related to the fractional Laplacian. Firstly we classify the polynomials in the whole space and in the half space for the fractional Laplacian defined…
We construct explicit complex-valued $p$-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the…
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…
Lie-algebraic and quantum-algebraic techniques are used in the analysis of thermodynamic properties of molecules and solids. The local anharmonic effects are described by a Morse-like potential associated with the $su(2)$ algebra. A…
In this paper we use a bicharacter construction to define an $H_D$-quantum vertex algebra structure corresponding to the quantum vertex operators describing certain classes of symmetric polynomials.
Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description as advocated by Loop Quantum Gravity. Here we extend previous results on the semiclassical properties of quantum…
When several inequivalent supercharges form a closed superalgebra in Quantum Mechanics it entails the appearance of hidden symmetries of a Super-Hamiltonian. We examine this problem in one-dimensional QM for the case of periodic potentials…
We study the eigenvalues of a Laplace-Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. By assigning partitions with the restricted uniform measure,…