arXiv subjects

Danilo Latini

Publications and source records attributed to Danilo Latini.

24 records · Page 2Linked to original sources

arXiv2022

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the $\mathfrak{gl}(n)$ Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

math-ph↗

arXiv2021

Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures

The recent interest in the study of higher-rank polynomial algebras related to $n$-dimensional classical and quantum superintegrable systems with coalgebra symmetry and their connection with the generalised Racah algebra $R(n)$, a higher-rank generalisation of the rank one Racah algebra $R(3)$, raises the problem of understanding the role played by the $n - 2$ quadratic subalgebras generated by the left and right Casimir invariants (sometimes referred as universal quadratic substructures) from this new perspective. Such subalgebra structures play a signficant role in the algebraic derivation of spectrum of quantum superintegrable systems. In this work, we tackle this problem and show that the above quadratic subalgebra structures can be understood, at a fixed $n > 3$, as the images of $n - 2$ injective morphisms of $R(3)$ into $R(n)$. We show that each of the $n-2$ quadratic subalgebras is isomorphic to the rank one Racah algebra $R(3)$. As a byproduct, we also obtain an equivalent presentation for the universal quadratic subtructures generated by the partial Casimir invariants of the coalgebra. The construction, which relies on explicit (symplectic or differential) realisations of the generators, is performed in both the classical and the quantum cases.

math-ph↗

arXiv2020

Embedding of the Racah Algebra R($\boldsymbol{n}$) and Superintegrability

The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable models in $2$D can be obtained by means of suitable limits and contractions. A higher rank generalization of $R(3)$, the so-called rank $n-2$ Racah algebra $R(n)$, has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterizing $n$D superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of $n$-dimensional maximally superintegrable models, the Smorodinsky-Winternitz system and the generalized Kepler-Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra $R(n)$ as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

math-ph↗

arXiv2015

The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.

math-ph↗

arXiv2015

The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion

The formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way to be suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System, the so-called Taub-Nut system, associated with the Hamiltonian: $$ \mathcal{H}_η({\mathbf{q}}, {\mathbf{p}}) = \mathcal{T}_η({\mathbf{q}}, {\mathbf{p}}) + \mathcal{U}_η({\mathbf{q}}) = \frac{|{\mathbf{q}}| {\mathbf{p}}^2}{2m(η+ |{\mathbf{q}}|)} - \frac{k}{η+ |{\mathbf{q}}|} \quad (k>0, η>0) \, .$$ In full agreement with the results recently derived by A. Ballesteros et al. for the quantum case, we show that the classical Taub-Nut system shares a number of essential features with the Kepler system, that is just its Euclidean version arising in the limit $η\to 0$, and for which a SUSYQM approach has been recently introduced by S. Kuru and J. Negro. In particular, for positive $η$ and negative energy the motion is always periodic; it turns out that the period depends upon $ η$ and goes to the Euclidean value as $η\to 0$. Moreover, the maximal superintegrability is preserved by the $η$-deformation, due to the existence of a larger symmetry group related to an $η$-deformed Runge-Lenz vector, which ensures that in $\mathbb{R}^3$ closed orbits are again ellipses. In this context, a deformed version of the third Kepler's law is also recovered. The closing section is devoted to a discussion of the $η<0$ case, where new and partly unexpected features arise.

math-ph↗

arXiv2014

Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum space describing point particles coupled to Einstein gravity in three space-time dimensions. We introduce different sets of coordinates on the group manifold and discuss their properties under Lorentz transformations. In particular we show how a certain set of coordinates exhibits an upper bound on the energy under deformed Lorentz boosts which saturate at the Planck energy. We discuss how this deformed symmetry framework is generally described by a quantum deformation of the Poincaré group: the quantum double of ${\rm SL}(2,\mathbb{R})$. We then illustrate how the space of functions on the group manifold momentum space has a dual representation on a non-commutative space of coordinates via a (quantum) group Fourier transform. In this context we explore the connection between Weyl maps and different notions of (quantum) group Fourier transform appeared in the literature in the past years and establish relations between them.

gr-qc↗