Guowu Meng
The Kepler problem is a physical problem about two bodies which attract each other by a force proportional to the inverse square of the distance. The MICZ-Kepler problems are its natural cousins and have been previously generalized from dimension three to dimension five. In this paper, we construct and analyze the (quantum) MICZ-Kepler problems in all dimensions higher than two.
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We present some generalization of 16D oscillator by anisotropic and nonlinear inharmonic terms and its dual analog for 9D related MICZ-Kepler systems by generalized version of the Kustaanheimo-Stiefel transformation. The solvability of the Schrödinger equation of the these problems by the variables separation method are discussed in different coordinates.
We present the classical solutions of the two-center MICZ-Kepler and MICZ-Kepler-Stark systems. Then we suggest the model of multi-center MICZ-Kepler system on the curved spaces equipped with $so(3)$-invariant conformal flat metrics.
Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid
提出三维球面和双曲面上广义MICZ-Kepler系统的类似物，构造其能谱和归一化波函数，发现系统是极小超可积的。
We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of a simple mechanical system on the cone over the rotation group SO(n). As a corollary we derive an elementary formula for its general solution. The punch-line of our computation is that the additional MICZ-Kepler $|ϕ|^2/r^2$ type potential term is the rotational part of the cone's kinetic energy.
For each integer $n\ge 2$, we demonstrate that a 2n-dimensional generalized MICZ-Kepler problem has an $\widetilde{\mr{Spin}}(2, 2n+1)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\widetilde{\mr{Spin}}(2, 2n+1)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\widetilde{\mr{Spin}}(2, 2n+1)$-module.
The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz vector $\mathbf A$, with the parameter space consisting of the pairs of 3D vectors $(\mathbf A, \mathbf L)$ with ${\mathbf L}\cdot {\mathbf L} > (\mathbf L\cdot \mathbf A)^2$. The recent 4D perspective of the Kepler problem yields a new parametrization, with the parameter space consisting of the pairs of Minkowski vectors $(a,l)$ with $l\cdot l =-1$, $a\cdot…
We discuss the generalizations of the MICZ-Kepler system (the system describing the motion of the charged particle in the field of Dirac dyon), to the curved spaces, arbitrary potentials and to the multi-dyon background.
We propose the integrable (pseudo)spherical generalization of the four-dimensional anisotropic oscillator with additional nonlinear potential. Performing its Kustaanheimo-Stiefel transformation we then obtain the pseudospherical generalization of the MICZ-Kepler system with linear and $\cosθ$ potential terms. We also present the generalization of the parabolic coordinates, in which this system admits the separation of variables. Finally, we get the spherical analog of the presented MICZ-Kepler-like system.
We study the radial part of the MICZ-Kepler problem in an algebraic way by using the $su(1,1)$ Lie algebra. We obtain the energy spectrum and the eigenfunctions of this problem from the $su(1,1)$ theory of unitary representations and the tilting transformation to the stationary Schrödinger equation. We construct the physical Perelomov number coherent states for this problem and compute some expectation values. Also, we obtain the time evolution of these coherent states.
For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an $\mr{Spin}(2, 2n+2)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n+1)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\mr{Spin}(2, 2n+2)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\mr{Spin}(2, 2n+2)$-module.
We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere …
The nonrelativistic motion of a charged particle around a dyon in (9+1) spacetime is known as the nine-dimensional MICZ-Kepler problem. This problem has been solved exactly by the variables-separation method in three different coordinate systems, spherical, parabolic, and prolate spheroidal. In the present study, we establish a relationship between the variable separation and the algebraic structure of SO(10) symmetry. Each of the spherical, parabolic, or prolate spheroidal bases is proved to be a set of eigenfunctions of a corresponding nonuplet of algebraically-independent integrals of motio…
We propose the general scheme of incorporation of the Dirac monopoles into mechanical systems on the three-dimensional conformal flat space. We found that any system (without monopoles) admitting the separation of variables in the elliptic or parabolic coordinates can be extended to the integrable system with the Dirac monopoles located at the foci of the corresponding coordinate systems. Particular cases of this class of system are the two-center MICZ-Kepler system in the Euclidean space, the limiting case when one of the background dyons is located at the infinity as well as the model of par…
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that N-dimensionall singular oscillator and (n+1)-dimensional generalized MICZ-Kepler system are dual to each other and the duality transformation is the generalized version of the KS transformation. The solvability of the Schrodinger equation of these problems by the variables separation method is given in a double, spherical and parabolic coordinates. The quad…
Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension five. The dynamical symmetry group of this system is $\widetilde {\mathrm O}^*(4n)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm {O}^*}(4n)$ with highest weight $$(\underbrace{-1, ..., -1}_{2n-1}, -(1+\barσ)),$$ which oc…
Let $n\ge 2$ be an integer. To each irreducible representation $σ$ of $\mathrm O(1)$, an $\mathrm {O}(1)$-Kepler problem in dimension $n$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical symmetry group of this system is $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with the Hilbert space of bound states ${\mathscr H}(σ)$ being the unitary highest weight representation of $\widetilde {\mathrm{Sp}}_{2n}(\mathbb R)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_{n-1}, -(1/2+|σ|)),$$…
Let $n\ge 2$ be a positive integer. To each irreducible representation $σ$ of $\mr U(1)$, a $\mr U(1)$-Kepler problem in dimension $(2n-1)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is equivalent to a MICZ-Kepler problem. The dynamical symmetry group of this system is $\widetilde {\mr U}(n, n)$, and the Hilbert space of bound states ${\ms H}(σ)$ is the unitary highest weight representation of $\widetilde {\mr U}(n, n)$ with highest weight $$(\underbrace{-1/2, ..., -1/2}_n, 1/2+\bar σ, \underbrace{1/2, ..., 1/2}_{n-1})$$ when $\bar σ\ge 0$ or $$(\underbrace{…
We define the Landau problem on two-dimensional surfaces of revolution of the second order: ellipsoid, hyperboloid and paraboloid. We start form the two-center MICZ-Kepler system Hamiltonian and then making the reduction into the various two-dimensional surfaces listed above we obtain the Hamiltonians of the charged particle moving on the corresponding surface of revolution with the magnetic filed conserving the symmetry of the two-dimensional surface(Landau problem). For each case we figure out at which values of parameters the qualitative character of the moving coincides with that of a free…
It is shown that Kepler problem in deformed (quantum) four-dimensional space in non relativistic limit is integrable in quadratures. In non relativistic limit group of motion of quantum space coincide with Galilei one.
We study the regularized $(2n-1)$-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of $U(n,n)$. The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed and their equivalence is shown. Some integrable generalization (perturbation) of $(2n-1)$-Kepler problem is proposed.
It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the "external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.
Quantized Kepler-Coulomb dynamical models on two-dimensional constant curvature spaces
本文作为前期研究的延续，针对二维常曲率空间中的开普勒-库仑势模型，采用Sommerfeld多项式方法开展量子化研究，其成果可关联低维碳材料动力学及刚体动力学受限问题。
The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and in the second part a new quasi-bi-Hamiltonian structure is obtained by making use of the separability…
The quantum Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order supe…
We construct and analyze a generalization of the Kepler problem. These generalized Kepler problems are parameterized by a triple $(D, κ, μ)$ where the dimension $D\ge 3$ is an integer, the curvature $κ$ is a real number, the magnetic charge $μ$ is a half integer if $D$ is odd and is 0 or 1/2 if $D$ is even. The key to construct these generalized Kepler problems is the observation that the Young powers of the fundamental spinors on a punctured space with cylindrical metric are the right analogues of the Dirac monopoles.
The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature $\k$ as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when particularized for $\k>0$, or $\k<0$, respectively; in addition, the Euclidean case arises as the particular case $\k=0$. In the second part we stud…
Bound and scattering states of a non-central extension of the three-dimensional Kepler-Coulomb Hamiltonian are worked out analytically within the framework of the potential groups of the problem, SO(7) for bound states and SO(6,1) for scattering states. In the latter case, the S matrix is calculated by the method of intertwining operators.
The existence of quasi-bi-Hamiltonian structures for a two-dimensional superintegrable $(k_1,k_2,k_3)$-dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, the…
We introduce a new superintegrable Kepler-Coulomb system with non-central terms in $N$-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give a algebraic derivation of spectrum of the superintegrable system. We show how the $so(N+1)$ symmetry algebra of the $N$-dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure constants involving a Casimir operator of $so(N-1)$ Lie algebra. We…
Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards
本文研究Kustaanheimo-Stiefel变换与Birkhoff-Waldvogel变换在三维开普勒台球中的应用，揭示可积反射壁的旋转不变性，并建立Levi-Civita平面理论证明双中心台球的可积性。
We decouple the Dirac's radial equations in $D+1$ dimensions with Coulomb-type scalar and vector potentials through appropriate transformations. We study each of these uncoupled second-order equations in an algebraic way by using an $su(1,1)$ algebra realization. Based on the theory of irreducible representations, we find the energy spectrum and the radial eigenfunctions. We construct the Perelomov coherent states for the Sturmian basis, which is the basis for the unitary irreducible representation of the $su(1,1)$ Lie algebra. The physical radial coherent states for our problem are obtained b…
An explicit integrator uniform in the true anomaly and exactly preserving all integrals of motion in the three-dimensional Kepler problem
本文提出一种显式积分器，能够在三维开普勒问题中精确保持所有运动积分，从而保持轨道轨迹的精确形状和方向，避免伪预cession，具有高精度和长期稳定性。
It is shown that by an appropriate canonical transformation Kepler dynamics can be put in the form which allows to exhibit the structure of the symmetry transformations related to the superintegrability. They appear to fit nicely into general scheme of nonlinear realizations. In new coordiantes the Kepler dynamics results from dimensional reduction of that describing low energy mesons with spontaneously broken chiral symmetry.
The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…
A Schroedinger type equation on the superspace R^{D|2n} is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant "distance" away from the origin. An osp(2,D+1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respect…
The Gutzwiller's trace formula for the anisotropic Kepler problem is Fourier transformed with a convenient variable $u=1/\sqrt{-2E}$ which takes care of the scaling property of the AKP action $S(E)$. Proper symmetrization procedure (Gutzwiller's prescription) is taken by the introduction of half-orbits that close under symmetry transformations so that the two dimensional semi-classical formulas match correctly the quantum subsectors $m^π=0^+$ and $m^π=0^-$. Response functions constructed from half orbits in the POT side are explicitly given. In particular the response function $g_X$ from $X$-s…
Harmonic oscillator and the Kepler problem are superintegrable systems which admit more integrals of motion than degrees of freedom and all these integrals are polynomials in momenta. We present superintegrable deformations of the oscillator and the Kepler problem with algebraic and rational first integrals. Also, we discuss a family of superintegrable metrics on the two-dimensional sphere, which have similar first integrals.
The central force problem is considered in a three dimensional space in which the Poisson bracket among the spatial coordinates is the one by the SU(2) Lie algebra. It is shown that among attractive power-law potentials it is only the Kepler one that all of its bound-states make closed orbits. The analytic solution of the path equation under the Kepler potential is presented. It is shown that except the Kepler potential, in contrast to ordinary space, all of the potentials for which all of the almost circular orbits are closed are non-power-law ones. For the non-power-law potentials examples o…
Self-similarity in the Kepler-Heisenberg problem
本文研究Kepler–Heisenberg问题的零能量轨道，证明在温和条件下其具有自相似性，并据此将轨道分为未来碰撞、过去碰撞和无碰撞拟周期三类，且碰撞必在有限时间发生。
Numerical Methods and Closed Orbits in the Kepler-Heisenberg Problem
本文研究了凯普勒-海森堡问题中闭合轨道的数值方法，通过蒙特卡罗优化和射击法结合辛积分器发现了一类具有新对称类型的花形周期轨道，并证明这些轨道在由角动量参数化的一维初始条件集上密集分布。
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