We show that all bounded trajectories in the two dimensional classical system with the potential $V(r,ϕ)=ω^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k ϕ}}+ \frac{βk^2}{r^2 \sin^2 {k ϕ}}$ are closed for all integer and rational values of $k$. The period is $T=\fracπ{2ω}$ and does not depend on $k$. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.
We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.
A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of el…
Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
We construct a class of interacting spin Calogero-Moser type systems. They can be regarded as a many particle system with spin degrees of freedom and as an integrable spin chain of Gaudin type. We prove that these Hamiltonian systems are superintegrable.
We discuss a family of Hamiltonians given by particular rational extensions of the singular oscillator in two-dimensions. The wave functions of these Hamiltonians can be expressed in terms of products of Laguerre and exceptional Jacobi polynomials. We show that these systems are superintegrable and admit an integral of motion that is of fourth-order. As such systems have been classified, we see that these potential satisfy a non-linear equation related to Painlevé VI. We begin by demonstrating the process with the simpler example of rational extensions of the harmonic oscillator and use the cl…
Structure results for higher order symmetry algebras of 2D classical superintegrable systems
最近，作者和J. M. Kress提出了一种特殊的函数递推关系方法来证明一个可积二维系统的量子超可积性，该系统包含了高阶对称的显式构造以及由这些对称生成的闭代数的结构关系.我们应用该方法的5个家庭的系统，每个依赖于一个合理的参数k，包括最显着的笼各向异性振荡器，Tremblay，Turbiner和Winternitz系统和变形开普勒-库仑系统。在这里，我们为所有相关的经典哈密顿系统，以及为一个家庭，包括一般的潜在的2-球上的这些结构的类似物。我们并没有在每种情况下证明生成对称是最低可能的顺序，但我们相信这是通过我们的方法的扩展。
We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N γ_n(q_1 p_1 + q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $γ_n$ are arbitrary coefficients, and $N\in \mathbb N$. For $N=2$, being both $γ_1,γ_2$ different from zero, this reduces to the classical Zernike system. We prove that $\mathcal H_N$ always provides a superintegrable system (for any value of $γ_n$ and $N$) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta…
This paper has studied the three-dimensional Dunkl oscillator models in a generalization of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. Their symmetries are obtained by the Jordan-Schwinger representations in the family of the Cayley-Klein orthogonal algebras using the creation and annihilation operators of the dynamical $sl_{-1}(2)$ algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of $so_{κ…
Quantum Calogero-Moser spin system is a superintegable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of corresponding simple Lie group. In this paper the underlying Lie group G is a compact connected, simply connected simple Lie group. It has a natural generalization known as quantum Calogero-Moser spin chain. In the first part of the paper we show that quantum Calogero-Moser spin chain is a quantum superintegrable systems. Then we show that the Euclidean multi-time propagator for this model can be written as a partition fun…
We extend the old formalism of cut-and-join operators in the theory of Hurwitz $τ$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $τ$-functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the $β$ deformation of the matrix model as well. The general interpolating WLZZ model is generated by a $W$-represe…
We consider a family of periodic tight-binding models (combinatorial graphs) that have the minimal number of links between copies of the fundamental domain. For this family we establish a local condition of second derivative type under which the critical points of the dispersion relation can be recognized as global maxima or minima. Under the additional assumption of time-reversal symmetry, we show that any local extremum of a dispersion band is in fact its global extremum if the dimension of the periodicity group is three or less, or (in any dimension) if the critical point in question is a s…
