arXiv subjects

Pavel Winternitz

Publications and source records attributed to Pavel Winternitz.

At least 19 recordsLinked to original sources

arXiv2025

Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals

In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, $G_2$ rational, or $I_6$) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index $k$. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure $g^{(k)}$ with various $k$, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra $g^{(k)}$ for a certain $k$. In all presented cases the algebra of integrals is a 4-generated $(H, I_1, I_2, I_{12}\equiv[I_1, I_2])$ infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.

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arXiv2020

Second-order delay ordinary differential equations, their symmetries and application to a traffic problem

This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_-$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_-$. Two previous articles were devoted to {\it first}-order DODSs, here we concentrate on a large class of {\it second}-order ones. We show that within this class the symmetry algebra can be of dimension $n$ with $0 \leq n \leq 6$ for nonlinear DODSs and must be $n=\infty$ for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.

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arXiv2019

Cylindrical type integrable classical systems in a magnetic field

We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space $\mathbb{E}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H =\frac{1}{2}\left(\vec{p}+\vec{A}(\vec{x})\right)^2+W(\vec{x})$, $X_1=(p_ϕ^A)^2+s_1^r(r, ϕ, Z)p_r^A+s_1^ϕ(r, ϕ, Z)p_ϕ^A+s_1^Z(r, ϕ, Z)p_Z^A+m_1(r,ϕ,Z)$, $X_2=(p_Z^A)^2+s_2^r(r, ϕ, Z)p_r^A+s_2^ϕ(r, ϕ, Z)p_ϕ^A+s_2^Z(r, ϕ, Z)p_Z^A+m_2(r,ϕ,Z)$. Infinite families of such systems are found, in general depending on arbitrary functions or parameters. This leaves open the possibility of finding superintegrable systems among the integrable ones (i.e. systems with 1 or 2 additional independent integrals).

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arXiv2019

Higher Order Quantum Superintegrability: a new "Painlevé conjecture"

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N= 3$, 4 and 5 these equations have the Painlevé property. We conjecture that this is true for all $N\geq3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

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arXiv2018

Two-dimensional superintegrable systems from operator algebras in one dimension

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian $H$ and a polynomial of order $N$ in the momentum $p.$ We assume that their Poisson commutator $\{H,K\}$ vanishes, is a constant, a constant times $H$, or a constant times $K$. In the quantum case $H$ and $K$ are operators and their Lie commutator has one of the above properties. We use two copies of such $(H,K)$ pairs to generate two-dimensional superintegrable systems in the Euclidean space $E_2$, allowing the separation of variables in Cartesian coordinates. All known separable superintegrable systems in $E_2$ can be obtained in this manner and we obtain new ones for $N=4.$

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arXiv2018

Lie point symmetries and ODEs passing the Painlevé test

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlevé transcendents only $P_{III}$, $P_V$ and $P_{VI}$ have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.

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arXiv2017

Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries

A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional {sub}algebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone. We present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

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arXiv2017

Lie group classification of first-order delay ordinary differential equations

A group classification of first-order delay ordinary differential equation (DODE) accompanied by an equation for delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs) which consists of linear DODEs and solution independent delay relations have infinite-dimensional symmetry algebras, as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension $n$, $0 \leq n \leq 3$. It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

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arXiv2017

Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials

We consider a two dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e potentials V (x, y) = V_1(x)+V_2(y) where neither V_1(x) nor V_2(y) satisfy a linear ODE) allowing the existence of such an integral. All of these potentials are found to have the Painlevé property. Most of them are expressed in terms of known Painlevé transcendents or elliptic functions but some may represent new higher order Painlevé transcendents.

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arXiv2017

Conformally invariant elliptic Liouville equation and its symmetry preserving discretization

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra $o(3,1)$ as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane $E_2$. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a discretisation procedure developed earlier, we present a difference scheme that is invariant under the group $O(3,1)$ and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under $O(3,1)$ and is itself invariant under a subgroup of $O(3,1)$, namely the $O(2)$ rotations of the Euclidean plane.

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arXiv2017

Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials

A study is presented of two-dimensional superintegrable systems separating in Cartesian coordinates and allowing an integral of motion that is a fourth order polynomial in the momenta. All quantum mechanical potentials that do not satisfy any linear differential equation are found. They do however satisfy nonlinear ODEs. We show that these equations always have the Painlevé property and integrate them in terms of known Painlevé transcendents or elliptic functions.

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arXiv2015

Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.

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arXiv2015

Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations

Ordinary differential equations (ODEs) and ordinary difference systems (O$Δ$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective transformations of the independent variables $x$ and dependent variables $y$ are constructed. The ODEs are continuous limits of the O$Δ$Ss, or conversely, the O$Δ$Ss are invariant discretizations of the ODEs. The invariant O$Δ$Ss are used to calculate numerical solutions of the invariant ODEs of order up to five. The solutions of the invariant numerical schemes are compared to numerical solutions obtained by standard Runge-Kutta methods and to exact solutions, when available. The invariant method performs at least as well as standard ones and much better in the vicinity of singularities of solutions.

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arXiv2015

Three-dimensional superintegrable systems in a static electromagnetic field

We consider a charged particle moving in a static electromagnetic field described by the vector potential $\vec A(\vec x)$ and the electrostatic potential $V(\vec x)$. We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely $(P_1,P_2)$, $(L_3,P_3)$ and $(L_1,L_2,L_3)$. For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.

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arXiv2014

The Korteweg-de Vries equation and its symmetry-preserving discretization

The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the Eulerian form. We also propose invariant schemes that preserve the momentum. Numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes. We find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving Galilean transformations which is demonstrated numerically as well.

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arXiv2014

Superintegrable systems with spin and second-order integrals of motion

We investigate a quantum nonrelativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. We assume that the Hamiltonian is rotationally invariant and parity conserving and identify all such systems which allow additional integrals of motion that are second order matrix polynomials in the momenta. These integrals are assumed to be scalars, pseudoscalars, vectors or axial vectors. Among the superintegrable systems obtained, we mention a generalization of the Coulomb potential with scalar potential $V_0=\fracα{r}+\frac{3\hbar^2}{8r^2}$ and spin orbital one $V_1=\frac{\hbar}{2r^2}$.

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arXiv2014

Lie-point symmetries of the discrete Liouville equation

The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subalgebra $ SL_x \lf 2 , \mathbb{R} \rg \otimes SL_y \lf 2 , \mathbb{R} \rg $. The invariant scheme is an explicit one and provides a much better approximation of exact solutions than comparable standard (non invariant) schemes.

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arXiv2014

Lie group analysis of a generalized Krichever-Novikov differential-difference equation

The symmetry algebra of the differential--difference equation $$\dot u_n = [P(u_n)u_{n+1}u_{n-1} + Q(u_n)(u_{n+1}+u_{n-1})+ R(u_n)]/(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension $1 \le \mbox{dim}L \le 5$. When $P$, $Q$ and $R$ are specific second order polynomials in $u_n$ (depending on 6 constants) this is the integrable discretization of the Krichever--Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies $\mbox{dim}L=2$. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity $u_{n+N}=u_n$ and thus to reduce the differential--difference equation to a system of $N$ coupled ordinary three points difference equations.

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