Related papers: The quantum $H_3$ integrable system
The quantum $H_4$ integrable system is a 4D system with rational potential related to the non-crystallographic root system $H_4$ with 600-cell symmetry. It is shown that the gauge-rotated $H_4$ Hamiltonian as well as one of the integrals,…
It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate…
It is shown that all 3-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional…
We study aspects of the quantum and classical dynamics of a $3$-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual…
We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives…
We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general cubic algebra and we present specific…
We study the most general form of a three dimensional classical integrable system with axial symmetry and invariant under the axis reflection. We assume that the three constants of motion are the Hamiltonian, $H$, with the standard form of…
In recent paper Fakkousy et al. show that the 3D H\'{e}non-Heiles system with Hamiltonian $ H = \frac{1}{2} (p_1 ^2 + p_2 ^2 + p_3 ^2) +\frac{1}{2} (A q_1 ^2 + C q_2 ^2 + B q_3 ^2) + (\alpha q_1 ^2 + \gamma q_2 ^2)q_3 + \frac{\beta}{3}q_3…
A method for deriving superintegrable Hamiltonians with a spin orbital interaction is presented. The method is applied to obtain a new superintegrable system in Euclidean space $\mathbb{E}_3$ with the following properties. It describes a…
A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the…
Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form…
We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it…
Integrability is a cornerstone of classical mechanics, where it has a precise meaning. Extending this notion to quantum systems, however, remains subtle and unresolved. In particular, deciding whether a quantum Hamiltonian - viewed simply…
We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific…
A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant…
The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that…
We prove nonintegrability of a model Hamiltonian system defined on the Lie algebra $\mathfrak{su}_3$ suitable for investigation of connections between classical and quantum characteristics of chaos.
It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system, introduced by Sokolov-Turbiner in [arXiv:1409.7439], is equivalent to the $\mathfrak{gl}(3)$ quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian as…
A family of classical integrable systems defined on a deformation of the two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed through Hamiltonians defined on the non-standard quantum deformation of a sl(2) Poisson…
The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable…