Harmonic Oscillator on the SO(2,2) Hyperboloid

In the present work the classical problem of harmonic oscillator in the hyperbolic space H²₂: z²₀+z²₁−z²₂−z²₃=R² has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on H²₂, as in the other spaces with constant curvature, is exactly solvable...

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Дата:	2015
Автори:	Petrosyan, D.R., Pogosyan, G.S.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2015
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147158
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Harmonic Oscillator on the SO(2,2) Hyperboloid / D.R. Petrosyan, G.S. Pogosyan // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 51 назв. — англ.

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spelling	irk-123456789-1471582019-02-14T01:24:48Z Harmonic Oscillator on the SO(2,2) Hyperboloid Petrosyan, D.R. Pogosyan, G.S. In the present work the classical problem of harmonic oscillator in the hyperbolic space H²₂: z²₀+z²₁−z²₂−z²₃=R² has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on H²₂, as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones. 2015 Article Harmonic Oscillator on the SO(2,2) Hyperboloid / D.R. Petrosyan, G.S. Pogosyan // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 51 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E60; 37J15; 37J50; 70H20 DOI:10.3842/SIGMA.2015.096 http://dspace.nbuv.gov.ua/handle/123456789/147158 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In the present work the classical problem of harmonic oscillator in the hyperbolic space H²₂: z²₀+z²₁−z²₂−z²₃=R² has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on H²₂, as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones.
format	Article
author	Petrosyan, D.R. Pogosyan, G.S.
spellingShingle	Petrosyan, D.R. Pogosyan, G.S. Harmonic Oscillator on the SO(2,2) Hyperboloid Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Petrosyan, D.R. Pogosyan, G.S.
author_sort	Petrosyan, D.R.
title	Harmonic Oscillator on the SO(2,2) Hyperboloid
title_short	Harmonic Oscillator on the SO(2,2) Hyperboloid
title_full	Harmonic Oscillator on the SO(2,2) Hyperboloid
title_fullStr	Harmonic Oscillator on the SO(2,2) Hyperboloid
title_full_unstemmed	Harmonic Oscillator on the SO(2,2) Hyperboloid
title_sort	harmonic oscillator on the so(2,2) hyperboloid
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147158
citation_txt	Harmonic Oscillator on the SO(2,2) Hyperboloid / D.R. Petrosyan, G.S. Pogosyan // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 51 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT petrosyandr harmonicoscillatorontheso22hyperboloid AT pogosyangs harmonicoscillatorontheso22hyperboloid
first_indexed	2025-07-11T01:29:32Z
last_indexed	2025-07-11T01:29:32Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 096, 23 pages Harmonic Oscillator on the SO(2, 2) Hyperboloid? Davit R. PETROSYAN † and George S. POGOSYAN ‡§ † Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980, Russia E-mail: petrosyan@theor.jinr.ru ‡ Departamento de Matematicas, CUCEI, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico E-mail: george.pogosyan@cucei.udg.mx § International Center for Advanced Studies, Yerevan State University, A. Manoogian 1, Yerevan, 0025, Armenia E-mail: pogosyan@ysu.am Received April 24, 2015, in final form November 20, 2015; Published online November 25, 2015 http://dx.doi.org/10.3842/SIGMA.2015.096 Abstract. In the present work the classical problem of harmonic oscillator in the hyperbolic space H2 2 : z20+z21−z22−z23 = R2 has been completely solved in framework of Hamilton–Jacobi equation. We have shown that the harmonic oscillator on H2 2 , as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones. Key words: superintegrable systems; harmonic oscillator; hyperbolic space; Hamilton–Jacobi equation 2010 Mathematics Subject Classification: 22E60; 37J15; 37J50; 70H20 1 Introduction The harmonic oscillator as a distinguished dynamical system plays the fundamental role in theoretical and mathematical physics due to many special properties outgoing from its hidden symmetry. Together with the Kepler–Coulomb problem they are only one among the central potentials for which all classical trajectories are closed (Bertrand theorem) and in quantum me- chanics all energy state are multiply degenerate (accidental degeneracy). The other consequence of hidden symmetry is the existence of additional functionally (in quantum mechanics linearly) independent integrals of motion and the phenomena of multiseparability, that is separability of variables in Hamilton–Jacobi or Schrödinger equation in more than one orthogonal systems of coordinate. It has long been known [9, 12, 26] that the harmonic oscillator problem possesses five functionally independent integrals of motion, which generate the separation of variable into eight systems of coordinates [11, 17]. In most of them harmonic oscillator admits the exact so- lution, the fact which makes it attractive to use as a model of molecular, atomic and nuclear physics and other branches of theoretical physics. The generalization of Kepler–Coulomb system and oscillator problem on the spaces of con- stant curvature start from the work of Lobachevsky, who first identified the Kepler potential in hyperbolic space H3 (two-sheeted hyperboloid) and found the trajectories of classical motion [38] (see also the articles [7, 10, 34, 47]). The extension of the harmonic oscillator problem on the ?This paper is a contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html mailto:petrosyan@theor.jinr.ru mailto:george.pogosyan@cucei.udg.mx mailto:pogosyan@ysu.am http://dx.doi.org/10.3842/SIGMA.2015.096 http://www.emis.de/journals/SIGMA/Benenti.html 2 D.R. Petrosyan and G.S. Pogosyan spherical and hyperbolic geometries has already been done in the book of Liebmann [37], who also discussed the geometric character of the conics in noneuclidean geometry. The investigation of Kepler–Coulomb problem in quantum mechanics was motivated to compare the properties of the Coulomb potential in the “open hyperbolic” or “closed” universe to that of an “open but flat” universe. Schrödinger [46] was the first who discussed this problem and discovered that for “hydrogen atom” on three-dimensional sphere only discrete spectrum exists. Virtually at the same time, Infeld and Shild [25] found that in an open hyperbolic universe there is only a finite (but very large) number of bound states. The motion in Coulomb field on imaginary Lobachevsky space (one-sheet hyperboloid), as shown by Grosche [16], has some peculiarities. It is not singularfor any value of variable and their discrete spectra infinite degenerate. The essential advance in the theory of systems with hiddensymmetry in the spaces with constant curvature was made by Higgs [24], Leemon [36] and Belorussian authors in [2]. They have shown that the complete degeneracy of the spectrum of the Coulomb and oscillator problems on the three- dimensional sphere and hyperboloid is caused by an additional integrals of motion: “curved” Runge–Lenz’s vector (for the Coulomb potential) and Demkov–Fradkin tensor (for the oscillator). However, in contrast to the flat space, commutation relations between the compo- nents of Runge–Lenz’s vectorand Demkov–Fradkin tensor on the sphere and hyperboloid form the quadratic or cubic algebra. Later it was proven that these properties are inherent in all class of maximally second-order superintegrable systems, which also belong to the Kepler–Coulomb and oscillator potentials(see for instance recent review [40] and references therein). We recall that in general, in an N -dimensional space, maximal superintegrability means that the classical Hamiltonian allows (2N − 1) functionally independent integrals of motion (including the Hamiltonian) that are well defined functions on phase space. The first searchof superintegrable systems in two- and three-dimensional flat space was done in the pioneering works of Winternitz and Smorodinsky with co-authors in [39, 50], later the notion of super- integrability in the spaces of constant curvature has been introduced in theseries of papers [17, 18, 19, 20]. The complete classification of superintegrable systems on the two-dimensional complex sphere,which include to real spaces, sphere and hyperboloid, as particular cases have done in the work [28]. Some of the superintegrable systems have been constructed on SN and HN spaces in [23]. We can also mention some articles devoted to the investigationof various aspects of both classical and quantum superintegrable systems in the spaces of constant curvature, for instance [2, 21, 27, 30, 31, 32]. The classical and quantum mechanical systems on the spaces of constant curvature (positive and negative) have always drawn a great attention due to their connection with the relativistic physics and gravity. The 2D and 3D one-sheeted and SO(2, 2) hyperboloids are the models of the relativistic spacetime with a constant curvature, namely de Sitter and anti de Sitter spaces, which is a crucial point for its wide application in the field theories [42, 49], quantum gravity and cosmology [1, 14, 48], integrable Yang–Mills–Higgs equation (or Bogomolny equation) [33, 51]. Among other applications we can mention also quantum Hall effect [3] and coherent statequan- tization [13]. However, as far as we know, the superintegrable systems on imaginary Lobachevski space H1 2 : SO(3, 1)/SO(2, 1), (de Sitter space time dS2+1) on hyperboloid H2 2 = SO(2, 2)/SO(2, 1), (Anti de Sitter space time AdS2+1), have not been studied with the same degree of detail and need to be further investigated.It appears that the first work in this direction (if we do not take into account the paper [16]) was the article [8] (see also more general case in [4]) where the authors, using the reduction procedure to the free Hamiltonian on the homogeneous space SU(2, 2)/U(2, 1), obtain the eleven different types of maximally superintegrable systems on the hyperboloid H2 2 . Later, in paper [22], the superintegrable generalization of harmonic oscillator and Kepler–Coulomb potentials covering the six three-dimensional spaces of constant curvature (including de Sitter and anti de Sitter spaces) in unified way, parametrized by two Harmonic Oscillator on the SO(2, 2) Hyperboloid 3 contraction parameters defining themetric in each space, have been constructed. In these papers the classical superintegrable systems are only identified but have not been solved. Recently, also the main properties of two-dimensional harmonic oscillator problem have been investigated in [6], using again two parameters approach, in nine standard two-dimensional Cayley–Klein spaces, including the de Sitter dS1+1 and anti de Sitter AdS1+1 spaces. The present work in a sense can be considered as a continuation of our previous articles [43, 44, 45], devoted to the investigation of classical and quantum Kepler–Coulomb problem and quantum harmonic oscillator problem on the configuration hyperbolic space with constant cur- vature H2 2 . The given paper aims to investigate the harmonic oscillator problem on the whole hy- perbolic space H2 2 from the point of view of classical mechanics, which, to our knowledge, has not been elucidated in literature so far. This task seems more complicated but also more interesting than the analogous problem in the other three-dimensional hyperbolic spaces. It mainly derive from the complexity of the space H2 2 which includes such subspaces as the one- and two-sheeted hyperboloids. This study will hopefully also help us to better understand the quantum case. 2 The hyperbolic space H2 2 and constants of motion A three-dimensional hyperboloid H2 2⊂R2,2 is described by the equation z20 + z21 − z22 − z23 = R2 (2.1) To be more specific we parametrize the hyperboloid (2.1) using the geodesic pseudo-spherical coordinate (r, τ, ϕ) [29, 43], namely z0 = ±R cosh r, z1 = R sinh r sinh τ, z2 = R sinh r cosh τ cosϕ, z3 = R sinh r cosh τ sinϕ, (2.2) where r ≥ 0 is the “geodesic radial angle”, τ ∈ (−∞,∞), and ϕ ∈ [0, 2π). The connection between two sets of coordinates z0 → −z0 corresponds to the complex transformation of radial angle r → iπ − r. The system of coordinate (2.2) is valid only for \|z0\| ≥ R and the missing part of the surface for \|z0\| < R may also be taken into account if we use another form of the pseudo-spherical coordinate z0 = ±R cosχ, z1 = R sinχ coshµ, z2 = R sinχ sinhµ cosϕ, z3 = R sinχ sinhµ sinϕ, (2.3) where now χ ∈ (−π 2 , π 2 ), µ ∈ (−∞,∞) and ϕ ∈ [0, 2π). It is also easy to see that the two pseudo-spherical system of coordinate (2.2) and (2.3) are connected by r → iχ, τ → µ− iπ/2. (2.4) Here we shall make use of the pseudo-spherical system of coordinate in form (2.2). To investigate the motion in the region \|z0\| ≤ R, everywhere below, we will use the transformation (2.4). The restriction of the pseudo-euclidean metric ds2 = Gµνdz µdzν , Gµν = diag(−1,−1, 1, 1), (µ, ν = 0, 1, 2, 3) on R2,2 to H2 2 leads to the following formula ds2 R2 = dr2 − sinh2 rdτ2 + sinh2 r cosh2 τdϕ2. Then the kinetic energy is given by T = R2 2 ( ṙ2 − sinh2 r ( τ̇2 − cosh2 τϕ̇2 )) 4 D.R. Petrosyan and G.S. Pogosyan and the canonical momenta can be obtained in a usual way pr = ∂T ∂ṙ = R2ṙ, pτ = ∂T ∂τ̇ = −R2 sinh2 rτ̇ , pϕ = ∂T ∂ϕ̇ = R2 sinh2 r cosh2 τϕ̇. Thus the free Hamiltonian in the pseudo-spherical phase space (r, τ, ϕ; pr, pτ , pϕ) with respect to the canonical Lie–Poisson brackets {f, g} = 3∑ i=1 ( ∂f ∂qi ∂g ∂pi − ∂g ∂qi ∂f ∂pi ) , (2.5) has the form Hfree = 1 2R2 { p2r − 1 sinh2 r ( p2τ − p2ϕ cosh2 τ )} . (2.6) It is clear that isometry group of H2 2 hyperboloid is given by SO(2, 2) group. The correspon- ding Lie algebra is six dimensional. The generators of so(2, 2) algebra can be written in terms of the ambient space R2,2 coordinates zµ and momenta pµ as L1 = −(z2p3 − z3p2), L2 = −(z1p3 + z3p1), L3 = (z1p2 + z2p1), N1 = (z0p1 − z1p0), N2 = −(z0p2 + z2p0), N3 = −(z0p3 + z3p0), (2.7) and the Lie–Poisson brackets (2.5) with the help of three-dimensional metric ḡik=diag{1,−1,−1} reads {Li,Lj} = ḡimḡjnεmnkLk, {Ni,Nj} = ḡimḡjnεmnkLk, {Ni,Lj} = ḡimḡjnεmnkNk, where i, j, k = 1, 2, 3. There are two Casimir invariants, the first of which vanishes in realiza- tion (2.7): C1 = L ·N = N · L = ḡikNiLk = N1L1 −N2L2 −N3L3 = 0, (2.8) and the second one is C2 = N2 + L2, (2.9) where N2 = N ·N = ḡikNiNk = N 2 1 −N 2 2 −N 2 3 , L2 = L · L = ḡikLiLk = L21 − L22 − L23. (2.10) The next step is computing the relationship between the ambient momenta and the geodesic polar one. Taking into account that four-dimensional canonical momentum pµ (µ = 0, 1, 2, 3) pµ = ∂L ∂żµ = Gµν ż ν , L = 1 2 Gµν ż µżν , where L is a kinetic energy in the ambient space R2,2, we obtain that R · p0 = −R · ∂z0 ∂t = − sinh r pr, R · p1 = −R · ∂z1 ∂t = − cosh r sinh τ pr + cosh τ sinh r pτ , Harmonic Oscillator on the SO(2, 2) Hyperboloid 5 R · p2 = R · ∂z2 ∂t = cosh r cosh τ cosϕpr − sinh τ cosϕ sinh r pτ − sinϕ sinh r cosh τ pϕ, R · p3 = R · ∂z3 ∂t = cosh r cosh τ sinϕpr − sinh τ sinϕ sinh r pτ + cosϕ sinh r cosh τ pϕ. Then the generators (2.7) in geodesic pseudo-spherical coordinates and momenta are given by the formulas N1 = − sinh τ pr + cosh τ coth r pτ , N2 = − cosh τ cosϕpr + coth r sinh τ cosϕpτ + coth r sinϕ cosh τ pϕ, N3 = − cosh τ sinϕpr + coth r sinh τ sinϕpτ − coth r cosϕ cosh τ pϕ, L3 = − cosϕpτ + sinϕ coth τ pϕ, L2 = − sinϕpτ − cosϕ coth τ pϕ, L1 = pϕ. (2.11) Using now equations (2.9), (2.10) and (2.11) it is easy to see the second Casimir operator C2 is related with the free Hamiltonian (2.6) by C1 = −2R2Hfree. Thus all the quantities (2.11) Poisson commute with free Hamiltonian (2.6) and are constants of the motion. From the seven integrals of the motion {Hfree,Ni,Li} only five are functionally independent, because of the relation (2.9) and constraint (2.8). Hence the geodesic motion with the Hamiltonian (2.6) turns out to be a maximally superintegrable system. Let us now consider the spherically symmetric model, namely the Hamiltonian H = Hfree + V(r), where Hfree is given by equation (2.6) and V(r) is a potential function. It is obvious that the Hamilton–Jacobi equation H = E for any central potential admit separation of variables in the pseudo-spherical system of coordinates (2.2) (and (2.3))1. The pseudo-spherical system of coordinates corresponds to the subgroup chains SO(2, 2) ⊃ SO(2, 1) ⊃ SO(2). Thus, the central symmetry of Hamiltonian H implies the conservation low of the vector L = (L1,L2,L3) with the scalar product (2.8), which we can interpreted as Lorenzian “angular momentum”. In particular the first component of angular momentum L1 = pϕ and Casimir invariant of algebra so(2, 1): L2 = L21 − L22 − L23 = − ( p2τ − p2ϕ cosh2 τ ) , (2.12) together with the Hamiltonian H: H = 1 2R2 { p2r + L2 sinh2 r } + V(r), form the mutually Poisson-involutive system of constants of motion. As it follows from the equation (2.12): p2ϕ/cosh2 τ − L2 ≥ 0, the quantity L2, in contrast to the motion in Euclidean space (or spheres and two-sheeted hyperboloids), can take not only the positive or zero but also the negative value. Another difference is that at the fixed values of L2: p2ϕ ≥ L2. The existence of an additional independent constant of motion L2 (L3 then not independent) means that the problem is at least once degenerate and the trajectories placed on the two-dimensional surface. For the case of positive L2 putting τ = 0, or L2 = p2ϕ, we obtain that the motion takes place on the two-dimensional subspace, namely two-sheeted hyperboloid z20 − z22 − z23 = R2, while for negative L2, we may put ϕ = 0 or p2ϕ = 0, and restricted to the one-sheeted hyperboloid z20 + z21 − z22 = R2. 1Beside of the pseudo-spherical system of coordinates (2.2) the Hamilton–Jacobi equation Hfree = E and free Schrödinger equation on H2 2 hyperboloid allow the separation of variables additionally in 70th orthogonal systems of coordinates (see for details [29]). 6 D.R. Petrosyan and G.S. Pogosyan In the case of \|z0\| < R the formulas for so(2, 2) generators (2.11) are changed accordingly to the transformation (2.4). We have L2 = − ( p2µ + p2ϕ sinh2 µ ) . Hence by virtue of above relation, the L2 takes only negative value. Without the loss of generality we can put ϕ = 0 or p2ϕ = 0 and the motion on H2 2 again restricted to the one-sheeted hyperboloid z20 + z21 − z22 = R2. 3 Harmonic oscillator potential Let us now concentrate on the spherically symmetric model, namely harmonic oscillator system. In the article [45] we have extended the Euclidean isotropic harmonic oscillator potential with the frequency ω to our space H2 2 , which is given by V osc = ω2R2 2 ( z22 + z23 − z21 z20 ) =  ω2R2 2 tanh2 r, \|z0\| ≥ R, −ω 2R2 2 tan2 χ, \|z0\| ≤ R. Respectively the Hamiltonian may be expressed as follow Hosc = 1 2R2 ( p2r + L2 sinh2 r ) + ω2R2 2 tanh2 r (3.1) for \|z0\| ≥ R, and Hosc = − 1 2R2 ( p2χ + L2 sin2 χ ) − ω2R2 2 tan2 χ (3.2) for \|z0\| ≤ R. The Hamiltonian of the harmonic oscillator system, besides the angular momentum L has ad- ditional integrals of motion quadratic in the momenta, which are associated with the generators (N1,N2,N3), the so called Demkov–Fradkin tensor [9, 12]: Dik = 1 R2 NiNk + ω2R2 zizk z20 , Dik = Dki, i, k = 1, 2, 3. The components of Dik tensor Poisson commute with Hamiltonian of harmonic oscillator (3.1) and (3.2), but not necessarily with each other. In the pseudo-spherical coordinates the diagonal components of this tensor has the form D11 = N 2 1 R2 + ω2R2 sinh2 τ tanh2 r, D22 = N 2 2 R2 + ω2R2 cosh2 τ cos2 ϕ tanh2 r, D33 = N 2 2 R2 + ω2R2 cosh2 τ sin2 ϕ tanh2 r, so the harmonic oscillator Hamiltonian is given by Hosc = −D11 +D22 +D33 − L2 2R2 . (3.3) Harmonic Oscillator on the SO(2, 2) Hyperboloid 7 In addition to this, the Demkov–Fradkin tensor has the algebraic properties∑ i LiDik = ∑ i DkiLi = 0, k = 1, 2, 3. (3.4) It is clear that the ten integrals of motion {H,Li,Dik} cannot be functionally independent because of the relations (3.3) and (3.4), and that {L1D11} = {L2D22} = {L3D33} = 0. Only five integrals of motion, which we can choose as {H, L2,L1,L2,D33}, are functionally in- dependent. Thus Hosc is a maximally superintegrable Hamiltonian. The components of angular momentum and Demkov–Fradkin tensor forms the quadratic algebra. The nonvanishing Poisson brackets have been presented in Appendix A. In the contraction limit R→∞ theH2 2 hyperbolic space turns into the Minkowski space M2+1. Let us pass to Beltrami coordinates xi = R zi z0 = R zi√ R2 + z22 + z23 − z21 , i = 1, 2, 3. (3.5) Then, at the limit R→∞ we have that lim R→∞ V osc(r) = ω2 2 ( −x21 + x22 + x23 ) , which can be interpreted as a harmonic oscillator potential on the M2+1 Minkowski space (x1, x2, x3). 4 Integration of the Hamilton–Jacobi equation The Hamilton–Jacobi equation, associated with the Hamiltonian (3.1), is obtained after the substitution pµi → ∂S/∂µi, where µi = (r, τ, ϕ). Therefore we get H = 1 2R2 {( ∂S ∂r )2 − 1 sinh2 r ( ∂S ∂τ )2 + 1 sinh2 r cosh2 τ ( ∂S ∂ϕ )2 } + ω2R2 2 tanh2 r = E. This equation is completely separable, and the coordinate ϕ is cyclic. We look the solution for the classical action S(r, τ, ϕ) in form S(r, τ, ϕ) = pϕϕ+ S1(r) + S2(τ)− Et, and obtain( ∂S2 ∂τ )2 − p2ϕ cosh2 τ = −L2, (4.1) 1 2R2 ( ∂S1 ∂r )2 + ω2R2 2 tanh2 r + L2 2R2 sinh2 r = E. (4.2) The “quasi-radial” equation (4.2) describes the motion in field of effective potential Ueff(r) = ω2R2 2 tanh2 r + L2 2R2 sinh2 r . (4.3) At the large r ∼ ∞ the effective potential Ueff(r) tends to a constant value equal to ω2R2/2, whereas the behavior at the point r = 0 is determined by the angular momentum L2. 8 D.R. Petrosyan and G.S. Pogosyan Figure 1. Effective potential Ueff(r) in case of 0 ≤ L2 < ω2R4 for value of L2 = 0, 1/16, 1/8, 1/4; ω = R = 1. Figure 2. Effective potential Ueff(r) in case of L2 ≥ ω2R4 for value of L2 = 2, 3, 4; ω = R = 1. In case 0 ≤ L2 < ω2R4 potential (4.3) has a minimum at r0 = tanh−1 4 √ L2/ω2R4 (see Fig. 1), and at this point 0 ≤ Ueff(r0) = ω √ L2 − L2 2R2 < ω2R2 2 , (4.4) where equality is possible only in case of L2 = 0. For L2 ≥ ω2R4 the potential Ueff(r) is repulsive on the whole semi-axis r ∈ [0,∞) (see Fig. 2). In the case of negative L2 the effective potential (4.4) is attractive and has a singularity for a small r as ∼ r−2 (see Fig. 3). For the region \|z0\| < R the differential equations (4.1) and (4.2) are transformed to the following ones( ∂S2 ∂µ )2 + p2ϕ sinh2 µ = −L2, 1 2R2 ( ∂S1 ∂χ )2 + ω2R2 2 tan2 χ+ L2 2R2 sin2 χ = −E. The first equation admits only negative value of L2. Therefore we take into account the motion inside the region \|z0\| < R when investigate we the case of negative value of L2. Harmonic Oscillator on the SO(2, 2) Hyperboloid 9 Figure 3. Effective potential Ueff(r) in case of L2 < 0 for value of L2 = −1,−2,−3; ω = R = 1. Integrating now equations (4.1) and (4.2) we get S1(r) = ∫ √ 2R2E − ω2R4 tanh2 r − L2 sinh2 r dr, (4.5) S2(τ) = ∫ √ −L2 + p2ϕ cosh2 τ dτ. (4.6) Since we are interested only the trajectories we will follow the usual procedures [35] and consider the equations ∂S ∂E = ∂S1 ∂E − t = −t0, ∂S ∂L2 = ∂S1 ∂L2 + ∂S2 ∂L2 = β, ∂S ∂pϕ = ϕ+ ∂S2 ∂pϕ = ϕ0, (4.7) where t0, ϕ0 and β are the constants. 4.1 Integration of quasi-radial part From equations (4.5) and (4.7) we get that t− t0 = 1 ω ∫ tanh rdr√ − tanh4 r + 2 (E/ω2R2 + L2/2ω2R4) tanh2 r − L2/ω2R4 . (4.8) Below we consider separately all four cases: 0 < L2 < ω2R4, L2 ≥ ω2R4, L2 < 0 and L2 = 0. 1. The case 0 < L2 < ω2R4. For the roots in the radical expression of denominator in (4.8) we have X1,2 = (2R2E + L2)± √ (2R2E + L2)2 − 4L2ω2R4 2ω2R4 , (4.9) where X = tanh2 r ∈ [0, 1]. It’s obvious that the radicand in equation (4.9) is positive for any values of energy E > Emin = Ueff(r0) and equal zero for E = Emin. Thus the roots X1,2 (X1 ≤ X2) are positive. It is easy to see that for Emin ≤ E < ω2R2/2 both roots satisfy the inequality condition 0 < X1 < X2 < 1. At E ≥ ω2R2/2: 0 < X1 < 1 ≤ X2 and equality X2 = 1 is possible only for E = ω2R2/2. The bounded motion exists exclusively for Emin ≤ E < ω2R2/2. Below we will consider separately all possible cases, namely: Emin < E < ω2R2/2, E = Emin, E > ω2R2/2 and E = ω2R2/2. 10 D.R. Petrosyan and G.S. Pogosyan A. Performing the integration in formula (4.8) we get for Emin < E < ω2R2/2 2ω2R2 sinh2 r = (1− 2E/ω2R2)−1 {( 2E − L2/R2 ) + √( 2E + L2/R2 )2 − 4L2ω2 sin [ 2ω √ 1− 2E/ω2R2(t− t0) ]} . Thus the motion is bounded and periodic. The period is given by T (R) = π ω 1√ 1− 2E/ω2R2 . (4.10) The total frequency ω0 = ω √ 1− 2E/ω2R2 and unlike the motion in Euclidean space, depends on the energy of particle E and curvature of the space κ = −1/R2 as a parameter, but it is constant for each of the orbits at a fixed value of the energy2. This property is common to all closed orbits of superintegrable systems on the spaces with constant curvature. The contraction limit R→∞ give us the correct Euclidean period: T (R)R→∞ = π ω . The period of motion on H2 2 always larger than in Euclidean space by the factor: 1/ √ 1− 2E/ω2R2 and tends to infinity at the limit E → ω2R2/2, that is the closed orbits changes to the infinite open ones. B. In the case of minimum energy: E = Emin = Ueff(r0) or Emin = ω √ L2 − L2/2R2 the integral in (4.8) is not defined and we must solve directly the equation (4.2). From equation (4.2) we obtain( ∂S1 ∂r )2 = − (√ L2 coth r − √ ω2R4 tanh r )2 ≥ 0, or ∂S1/∂r = 0 and tanh2 r = √ L2/ω2R4. Therefore r = tanh−1 √1− √ 1− 2E ω2R2  , (4.11) i.e., the trajectories are circles. Here from two values of √ L2 allowed by equation E = Ueff(r0), we choose the smaller one √ L2 = ωR2 ( 1− √ 1− 2E/ω2R2 ) because it satisfies the condition 0 < L2 < ω2R4. In case of contraction limit R → ∞ we obtain E = Emin = ω √ L2 and r = √ E/ω. C. In case of E > ω2R2/2 after integration in (4.8) we have 2ω2R2 sinh2 r = ( 2E/ω2R2 − 1 )−1{( L2/R2 − 2E ) + √( 2E + L2/R2 )2 − 4L2ω2 cosh [ 2ω √ 2E/ω2R2 − 1(t0 − t) ]} , (4.12) i.e., the motion is not bounded. D. For the limiting case of E = ω2R2/2 the roots of denominator are X1 = L2/ω2R4, X2 = 1, thus L2/ω2R4 < tanh2 r < 1 and motion is not bounded because of tanh−1(L2/ω2R4) < r <∞. The integration in (4.8) yield cosh2 r = ( 1− L2/ω2R4 )−1 + ω2 ( 1− L2/ω2R4 ) (t− t0)2. (4.13) 2. Let us consider now the case of L2 ≥ ω2R4 (see Fig. 2). From equation (4.8) we get that the only possible value for energy is E > ω2R2/2 and the roots satisfy the inequality 2The Euclidean harmonic oscillator is a classical example of an isochronous system [5]. The period of motion of Euclidean oscillator depends only from frequency and is the same for all orbit. Harmonic Oscillator on the SO(2, 2) Hyperboloid 11 0 < X1 < 1 < X2. Thereby, the equation of motion is determined by the formula (4.12). The motion of particle is limited only by the point rmin = tanh−1 √ X1, i.e., it has the ability to go to infinity. 3. Let us consider finally the case of L2 ≤ 0. From the equation (4.8) we have that the roots of denominator are X1,2 = ( 2ER2 − \|L2\| ) ± √ (2ER2 − \|L2\|)2 + 4\|L2\|ω2R4 2ω2R4 , where again X = tanh2 r ∈ [0, 1]. It can be seen that X1 < 0 < X2 is independent of the value of A and energy E. For the region E ≥ ω2R2/2 one of the roots is X2 > 1, so the radicand is positive for any values of variable r, including the point r = 0: r ∈ [0,∞). The same situation develops for region E < ω2R2/2, where r ∈ [0, tanh−1 √ X2]. Therefore in case of negative A the particle can penetrate from the region z0 ≥ R to 0 ≤ z0 ≤ R. Performing the integration in formula (4.8), we have for E < ω2R2/2 sinh2 r = 2R2E + \|L2\| 2R2(ω2R2 − 2E) + √ (2R2E − \|L2\|)2 + 4\|L2\|ω2R4 2R2(ω2R2 − 2E) sin [ 2ω √ 1− 2E/ω2R2(t− t0) ] , (4.14) while for E > ω2R2/2 sinh2 r = 2R2E + \|L2\| 2R2(ω2R2 − 2E) + √ (2R2E − \|L2\|)2 + 4\|L2\|ω2R4 2R2(2E − ω2R2) cosh [ 2ω √ 2E/ω2R2 − 1(t− t0) ] . (4.15) From the formula (4.14) it follows that the motion at E < ω2R2/2 is bounded and periodic with period (4.10). Below we will construct the bounded trajectories lying on the whole hyperboloid, namely not only in the region \|z0\| ≥ R, but also \|z0\| ≤ R. In case when E = ω2R2 the integration in (4.8) leads, up to a transformation L2 → −\|L2\|, to a result similar to the formula (4.13). In the limiting case of L2 = 0 the formulas (4.14), (4.15) and (4.13) are simplified. For 0 < E < ω2R2/2 we get sinh2 r = 2E/ω2R2 1− 2E/ω2R2 cos2 ( ω √ 1− 2E/ω2R2(t− t0)− π 4 ) , while in case of E > ω2R2/2 sinh r = √ 2E/ω2R2 2E/ω2R2 − 1 sinh ( ω √ 2E/ω2R2 − 1(t0 − t) ) . Finally for E = ω2R2/2 we obtain sinh r = ω(t− t0). 4.2 Integration of the angular parts 1. Let us first consider the case when L2 > 0. From (4.5) and (4.6) we obtain ∂S1 ∂L2 = −1 2 ∫ dr sinh2 r √ 2R2E − ω2R4 tanh2 r − L2/ sinh2 r , (4.16) 12 D.R. Petrosyan and G.S. Pogosyan ∂S2 ∂L2 = −1 2 ∫ dτ√ −L2 + p2ϕ/ cosh2 τ . (4.17) The integrals can be easily calculated to give [15] ∂S2 ∂L2 = − 1√ 4L2 arcsin  sinh τ√ p2ϕ/L 2 − 1  , ∂S1 ∂L2 = 1 4 √ A arcsin [ 2L2 coth2 r − (2ER2 + L2)√ (2ER2 + L2)2 − 4L2ω2R4 ] . Here we require − √ p2ϕ/L 2 − 1 < sinh τ < √ p2ϕ/L 2 − 1, and ∣∣2L2 coth2 r − ( 2ER2 + L2 )∣∣ <√(2ER2 + L2 )2 − 4L2ω2R4. (4.18) The condition (4.18) is equivalent to z1 < coth r < z2, where z1,2 are the roots of denominator in integral (4.16): z1,2 = ( 2ER2 + L2 ) ± √( 2ER2 + L2 )2 − 4L2ω2R4 2L2 , E ≥ Emin = ω √ L2 − L2/2R2. The final condition z2 > 1 implies that L2 > ω2R4 and E > ω2R2/2 or 0 < L2 < ω2R4 and E > Emin. Therefore for ∂S/∂L2 we have ∂S ∂L2 = 1 4 √ L2 arcsin  2L2 coth2 r − ( 2ER2 + L2 )√( 2ER2 + L2 )2 − 4L2ω2R4  − 2 arcsin  sinh τ√ p2ϕ/L 2 − 1  = β. (4.19) Next, from (4.6) and (4.7) we obtain ∂S ∂pϕ = ϕ+ ∫ pϕdτ cosh2 τ √ −L2 + p2ϕ/ cosh2 τ = ϕ+ arcsin tanh τ√ 1− L2/p2ϕ = ϕ0, (4.20) and hence tanh τ = √ 1− L2/p2ϕ sin(ϕ0 − ϕ). (4.21) 2. Let us consider the integration in formulas (4.16), (4.17) and (4.20) in the case L2 ≤ 0. Instead of equation (4.19) we obtain [15] ∂S ∂L2 = 1 4 √ \|L2\| arccosh  2\|L2\| coth2 r + ( 2ER2 − \|A\| )√( 2ER2 − \|L2\| )2 + 4\|L2\|ω2R4  Harmonic Oscillator on the SO(2, 2) Hyperboloid 13 − 2 arcsinh  sinh τ√ 1 + p2ϕ/\|L2\|  = β, (4.22) and sin(ϕ0 − ϕ) = pϕ√ p2ϕ + \|L2\| tanh τ, (4.23) with the restriction for r: coth2 r ≥ ( 1 2 − ER2 \|L2\| ) + √( 1 2 − ER2 \|L2\| )2 + ω2R4 \|L2\| . The limiting case of L2 = 0 could be easily calculated directly from equations (4.22) and (4.23). So, we get ∂S ∂L2 ∣∣∣∣ L2=0 = √ 2E coth2 r − ω2R2 4ER − sinh τ 2pϕ = β, sinh τ = tan(ϕ0 − ϕ) (4.24) with the obvious restriction coth2 r ≥ ω2R2/2E. 5 The trajectories for L2 > 0 From (4.19) and (4.21) we have coth2 r = ( ER2 L2 + 1 2 ) + √( ER2 L2 + 1 2 )2 − ω2R4 L2 sin ( 2ψ + 4 √ L2β ) , (5.1) where ψ = arcsin  sinh τ√ p2ϕ/L 2 − 1  = arcsin  1√ 1 + L2/p2ϕ cot2(ϕ0 − ϕ) . (5.2) Now we can rewrite the equation (5.1) in form tanh2 r = 1( ER2 L2 + 1 2 ) + √( ER2 L2 + 1 2 )2 − ω2R4 L2 sin ( 2ψ + 4 √ L2β ) . (5.3) Thus we see from (5.2) that the dependence of angle τ in the equation of trajectories (5.3) can be eliminated. On the other hand from the formula (4.21) it follows that the motion of particle on the hyperboloid is restricted to the additional condition z1 z3 = tanh τ sinϕ = √ 1− L2/p2ϕ. Therefore, without the loss of generality we can choose τ = 0 or L2 = p2ϕ. Taking into account that the formula (5.3) is invariant about transformation r → iπ − r we can conclude that all trajectories of motion, given by this formula, lie on the upper (z0 ≥ R) or lower (z0 ≤ −R) sheets of the two-sheeted hyperboloid: z20 − z22 − z23 = R2. Obviously they are symmetric with respect to transformation z0 → −z0. 14 D.R. Petrosyan and G.S. Pogosyan Putting now L2 = p2ϕ in (4.21) we obtain that ψ = (ϕ0 − ϕ) and the formula (5.3) gain the following form (equation of orbits) tanh2 r = p 1 + ε(R) cos 2ϕ , (5.4) where we use the notations p(R) = ( ER2 L2 + 1 2 )−1 > 0, ε(R) = √ 1− 4ω2R4L2( 2ER2 + L2 )2 < 1, (5.5) and choose ϕ0 = −2β √ A+ π 4 that the points ϕ = 0 will be the nearest to the center. It is clear that radicand is always positive because of E > Ueff(r0) for 0 < A < ω2R4 and E > ω2R2/2 for A ≥ ω2R4. It is well-known that as in the Euclidean plane it is possible to introduce the conic (section) on the two-dimensional spaces of constant curvature [7, 10, 34] (see also the definition of curves on the two dimensional hyperboloid in [41]). The conics on the spaces with constant curvatures are the curves of the intersection between two-sheeted hyperboloid (or sphere) and second order quadric cone with the origin in the center of hyperboloid (sphere). Geometrically the conic on the spaces of constant curvature possesses many properties characteristic of conic section in Euclidean plane, particularly we can speak about the focuses F1 and F2 and can determine the conic as the point set, from which the sum (ellipses) or difference (hyperbolas) 2a of distances r1 and r2 to two given points (focuses F1 and F2) are constant. Let us now analysis of the oscillator orbit (5.4). The formula of trajectories (5.4) may be written in more convenient form 1 tanh2 r = cos2 ϕ B2 + sin2 ϕ A2 , (5.6) or in term of the Beltrami coordinate (3.5): x22 B2 + x23 A2 = R2, (5.7) where the constant A and B are B2 = p(R) 1 + ε(R) , A2 = p(R) 1− ε(R) , 0 < B2 ≤ A2. (5.8) The orbit equation of the type (5.6) has been studied in detail in the paper [6] (see also [10]) at the investigation of two-dimensional harmonic oscillator in the space of constant curvature in polar coordinates. The curves (5.6) are always conic on the hyperbolic plane, but its type depends on the value of A and B. It is obvious that if the value A2 > 1 and B2 > 1, then for any polar angle ϕ it follows that tanh r > 1, and this case cannot produce any oscillator orbit. In the case of B2 < A2 < 1 the conic (5.6) takes the form of hyperbolic ellipses. The quantities A and B are related to the lengths of the large and small semiaxes a and b, running the interval [0,∞), defined as the values of r at ϕ = π/2 and ϕ = 0. Then the values A, B can be written in term of hyperbolic tangent of a, b: A2 = tanh2 a and B2 = tanh2 b and the equation of orbit (5.6) is 1 tanh2 r = cos2 ϕ tanh2 b + sin2 ϕ tanh2 a . (5.9) Harmonic Oscillator on the SO(2, 2) Hyperboloid 15 In the contraction limit R→∞ we have r → r̃/R where r̃ = √ x22 + x23 is the radial variable in the Euclidean plane. Taking into account the limit ε(R)→ ε̃ = √ 1− ω2L2 E2 , R2p(R)→ p̃ ≡ L2 E , we get from (5.7) that the equation of trajectories transforms into the oscillator one x22 B̃2 + x23 Ã2 = 1, B̃2 = p̃ 1 + ε̃ , Ã2 = p̃ 1− ε̃ The next interesting case is when B2 < 1 < A2. This conic (5.6) is neither the ellipse nor the hyperbola. Following the paper [6] we will call this conic as the ultraellipse. Only one semi- axis b belongs to the hyperbolic plane and the next one formally is not on the real distance. It is possible to introduce a new “semiaxis” ã (situated on the complex plane on the line ã = a+iπ/2) which related with the quantity A by A2 = coth ã. Thus, instead of (5.9) we have the conic 1 tanh2 r = cos2 ϕ tanh2 b + tanh2 ãsin2 ϕ. (5.10) There is a joint point of two conics (5.9) and (5.10), namely A2 = 1 (a→∞ or ã→∞). In this case the conic is given by 1 tanh2 r = cos2 ϕ tanh2 b + sin2 ϕ. This conic is an equidistant curve with equidistance b from the axis z2 [6]. Let us now consider all the possible trajectories of motion depending on the energy and angular momentum L2. A. First we consider the case when Ueff(r0) < E < ω2R2/2 and 0 < L2 < ω2R4. It is clear that B2 ≤ A2 =  ( ER2 L2 + 1 2 ) − √( ER2 L2 + 1 2 )2 − ω2R4 L2  −1 < 1 and the oscillator orbits are described by the equation (5.9). Denote the minimum b = rmin, (ϕ = 0) and maximum a = rmax, (ϕ = π/2) points on the orbit as a distance from the center of field. From (5.8) and (5.9) we have tanh2 rmin = p 1 + ε(R) , tanh2 rmax = p 1− ε(R) , and correspondingly rmin = coth−1  √√√√(ER2 L2 + 1 2 ) + √( ER2 L2 + 1 2 )2 − ω2R4 L2  , rmax = coth−1  √√√√(ER2 L2 + 1 2 ) − √( ER2 L2 + 1 2 )2 − ω2R4 L2  . Thus we find that the trajectories of motion are ellipses lying symmetrically to the point z0 = R, z1 = z2 = z3 = 0 on the upper sheet of the two-sheeted hyperboloid (see Fig. 4). 16 D.R. Petrosyan and G.S. Pogosyan Figure 4. The figure shows the elliptic trajectories lying on the upper sheet of the two-sheeted hyper- boloid z20 − z22 − z23 = R2, z0 > R for the value ε = 0.3 and p = 0.3, 0.4, 0.5. Figure 5. The cyclic orbits: ε = 0 and p = 0.2, 0.5, 0.8. B. In case of minimum energy E = Emin = Ueff(r0) we have from (5.5) that ε = 0 and p = ωR2/ √ L2 and consequently tanh2 r = B2 = A2 = ωR2/ √ L2. Thus the orbits are circles with the radius given by the formula (4.11) (see Fig. 5). C. For the case of energy values E = ω2R2/2 we get that p(R) = 2A ω2R4 + L2 , ε(R) = \|ω2R4 − L2\| ω2R4 + L2 , therefore for 0 < L2 < ω2R4 we get B2 = L2/ω2R4 < 1 and A2 = 1. The conic is 1 tanh2 r = ω2R4 L2 cos2 ϕ+ sin2 ϕ, which represents the equidistant curves (see Fig. 6). The minimal distance rmin from the center Harmonic Oscillator on the SO(2, 2) Hyperboloid 17 Figure 6. The figure shows the equidistant orbits lying on the upper sheet of the two-sheeted hyperboloid z20 − z22 − z23 = R2, z0 > R with the value of pairs (p, ε) = (1/3, 2/3); (2/3, 1/3); (8/9, 1/9). is given by the formula rmin = coth−1 ( ωR2 √ L2 ) . Let L2 = ω2R4. Then B2 = A2 = 1 and the conic is a “largest” circle with radius r = ∞. For the case L2 > ω2R4 we obtain that B2 = 1, A2 = L2/ω2R4 > 1. Then from the formula (5.6) it follows that tanh r > 1 and no any oscillator orbits exist. D. For the energy E > ω2R2/2 it is easy to see that for any positive L2 > 0 A2 =  ( ER2 L2 + 1 2 ) − √( ER2 L2 + 1 2 )2 − ω2R4 L2  −1 > 1, B2 < 1. The motion of a particle is determined by the equation (5.10) where tanh2 ã = 1/A2. The trajectories are ultraellipses and describe the motion of a particle from the minimum point rmin: rmin = coth−1  √√√√(ER2 L2 + 1 2 ) + √( ER2 L2 + 1 2 )2 − ω2R4 L2  , to infinity (see Fig. 7). On the other hand side B2 · A2 = ω2R4/L2, so that for L2 < ω2R4 we get 1/A2 < B2 < 1, whereas for L2 > ω2R4 follows that B2 < 1/A2 < 1 and the value of L2 = ω2R4 or B2 = 1/A2 separates two set of ultraellipses. Let us also note that the in contraction limit R→∞ these orbits corresponds to the Euclidean oscillator orbits with the large values of energy (the straight line x22 = B̃2). 6 The trajectories for L2 ≤ 0 To simplify further formulas we set first pϕ = 0. Then, from equation (4.23) it follows that the motion occurs at a constant value of the azimutal angle ϕ = ϕ0 that is limited by the condition z3/z2 = tanϕ0. To further simplify it is enough to choose ϕ0 = 0 or ϕ0 = π. Thus we 18 D.R. Petrosyan and G.S. Pogosyan Figure 7. The figure shows the ultraellipses lying on the upper sheet of the two-sheeted hyperboloid z20 − z22 − z23 = R2, z0 > R for the value ε = 0.8 and p = 0.2, 0.5, 0.8. get that trajectory of the motion lies on the one-sheeted hyperboloid z20 + z21 − z22 = R2. The formula (4.22) gives us the equation of the trajectory in the region z0 > R: coth2 r = ( 1 2 − ER2 \|L2\| ) + √( 1 2 − ER2 \|L2\| )2 + ω2R4 \|L2\| cosh ( 2τ + 4 √ \|L2\|β ) . (6.1) Performing the further transformation r → iχ and τ → µ− iπ/2 in formula (6.1), we obtain the equation of the trajectory in the region 0 < z0 < R: cot2 χ = − ( 1 2 − ER2 \|L2\| ) + √( 1 2 − ER2 \|L2\| )2 + ω2R4 \|L2\| cosh ( 2µ+ 4 √ \|L2\|β ) . (6.2) In the formula of trajectory (6.1) we must distinguish two cases, namely for the value of energy E < ω2R2/2 and E ≥ ω2R2/2. In the first case E < ω2R2/2 from equation (6.1) it follows that for any value of the variable τ ∈ (−∞,∞) we have that coth r > 1. Therefore, the trajectory of the motion extends from the point r = 0 at the τ → −∞ (z0 = R, z1 < 0, z2 > 0) to its maximum rmax = coth−1 √√√√(1 2 − ER2 \|L2\| ) + √( 1 2 − ER2 \|L2\| )2 + ω2R4 \|L2\| , at the point τ = −2 √ \|L2\|β and then goes back to the point r = 0 when τ →∞ (z0 = R, z1 > 0, z2 > 0). Further on, the particle penetrates through the point z0 = R from the region z0 > R to the region 0 < z0 < R, which, as it follows from the equation (6.2), corresponds to the value of angles µ→∞ and χ→ 0, (z0 < R, z1 > 0, z2 > 0). Further trajectory extends to the maximal value χmax: χmax = cot−1 √√√√−(1 2 − ER2 \|L2\| ) + √( 1 2 − ER2 \|L2\| )2 + ω2R4 \|L2\| ≤ π 2 , Harmonic Oscillator on the SO(2, 2) Hyperboloid 19 Figure 8. The trajectories of motion in the case of \|L2\| = 1; E = −3/2,−1/2, 1/4, 1/2, 3/2; ω = R = 1. at the point µ = −2 √ \|L2\|β, and then continue to µ → −∞, χ → 0 (z0 < R, z1 > 0, z2 < 0). After, the particle again passes the point z0 = R and penetrates to the region z0 ≥ R. Further using similar reasoning it can be shown that the trajectories in case of E < ω2R2/2, are a closed curve lying on the one-sheeted hyperboloid z20+z21−z22 = R2, z0 > 0, so the motions are bounded and periodic. The same situation takes place for the case of z0 < 0. In the case of E ≥ ω2R2/2 it is easy to see that the inequality√( 1 2 − ER2 \|A\| )2 + ω2R4 \|L2\| ≤ 1 2 + ER2 \|L2\| is valid. Thus the trajectory of the motion, depending on the sign of variable τ is split into two paths. One of the paths begins from the large r at the minimal point τmin = −2 √ \|L2\|β + 1 2 cosh−1 ( 1 2 + ER2 \|L2\| ) √( 1 2 − ER2 \|L2\| )2 + ω2R4 \|L2\| . and continues to the point r = 0 at τ → ∞ (z0 = R, z2 > 0). Then the trajectory passing the part of 0 < z0 < R goes back from (z0 = R, z2 < 0) at the point r = 0, τ ∼ ∞ to r ∈ ∞ at τmin. The second path is symmetric with respect to axis z1. Thus the trajectories of motion in the case of E ≥ ω2R2/2 are not bounded. Some examples of trajectories for the fixed negative L2 and various values of energy E, are presented on the Fig. 8. In the case L2 = 0 it is easy to get from (4.24) coth2 r = ω2R2 2E +R √ E (2β − tanϕ/pϕ)2 , with ϕ0 = 0. In the case of E < ω2R2/2 the bounded motion takes place rmin = 0 (ϕ = π/2) and rmax = coth−1 √ ω2R2 2E (ϕ = arctan 2βpϕ), whereas for the E ≥ ω2R2/2 the orbits are infinite: r ∈ [0,∞). The trajectories of the motion can be presented on the hyperbolic cylinder z20 − z22 = R2, z21 = z23 , z0 ≥ R (see Fig. 9). 20 D.R. Petrosyan and G.S. Pogosyan Figure 9. The bounded and infinite trajectories of the motion for L2 = 0 lying on the hyperbolic cylinder z20 − z22 = R2, z21 = z23 , and z0 ≥ R. The figure shows the cases E = 0.2, 0.5, 0.8; ω = R = pϕ = 1. 7 Conclusion We have shown that the notion of harmonic oscillator problem can be extended not only to the sphere and two-sheeted hyperboloid but also to the hyperbolic space H2 2 . It was proved that the harmonic oscillator problem on H2 2 is exactly solvable and also belongs to the class of superintegrable systems. We have constructed the dynamical algebra of symmetry for this system, which is nonlinear and quadratic (so-called Higgs algebra). We completely solved the Hamilton–Jacobi equation for harmonic oscillator problem in the geodesic pseudo-spherical sys- tems of coordinates. It was shown that for positive value of the Lorentzian momentum L2 > 0 all trajectories of motion lie on the upper (or lower) sheets of two dimensional two-sheeted hyper- boloid z20 − z22 − z23 = R2. These trajectories are always conics centered in the origin of potential r = 0. For the special values of energy Emin < E < ω2R2/2 and momentum L2 < ω2R4 all the orbits are ellipses (or circles for E = Emin). In case when E > ω2R2/2 independently of the value of L2, the oscillator orbits are ultraellipses or equidistant curves for E = ω2R2/2. We have seen that in case of negative values of Loreinzian momentum L2 ≤ 0 the oscillator orbits lie on the one-sheeted hyperboloid z20 + z21 − z22 = R2 and are bounded and periodic for E < ω2R2/2 and infinite for E ≥ ω2R2/2. The similar situation is valid for L2 = 0, but in this case the orbits lie on the hyperbolic cylinder z20 − z22 = R2, z21 = z23 . Let us make short comments concerning the connection of the classical and quantum case. The quantum-mechanical counterpart of the angular momentum operator (2.7) comes through the replacement pµ → −i∂/∂zµ and is given by L̂1 = −i(z2∂3 − z3∂2), L̂2 = −i(z1∂3 + z3∂1), L̂3 = i(z1∂2 + z2∂1). Then in the pseudo-spherical coordinates (2.2) the operator L̂2 takes the form L̂2 = L̂2 1 − L̂2 2 − L̂2 3 = ( 1 cosh τ ∂ ∂τ cosh τ ∂ ∂τ − 1 cosh2 τ ∂2 ∂ϕ2 ) , and coincide with the Casimir operator of SO(2, 1) group. Thus the Schrödinger equation for the harmonic oscillator potential can be written as 1 sinh2 r ∂ ∂r sinh2 r ∂Ψ ∂r + [ 2R2E − L̂2 sinh2 r − ω2R4 tanh2 r ] Ψ = 0, (7.1) Harmonic Oscillator on the SO(2, 2) Hyperboloid 21 and solved by separation of variables via the ansatz Ψ(r, τ, ϕ) = R(r)Y(τ, ϕ). The pseudo- spherical function Y is a eigenfunction of operator L̂2Y = `(`+1)Y which describes the quantum geodesic motion on the two-dimensional one-sheeted hyperboloid. The spectrum of ` can take as well as the real values: ` = 0, 1, . . . (discrete series of representation of SO(2, 1) group) and complex value ` = −1/2 + iρ, ρ > 0 (continuous principal series). In the first case the eigenvalue of L̂2 operator is positive and in the second one negative. The exact solution of the Schrödinger equation (7.1) for the positive eigenvalues of operator L̂2 has been constructed in the previous paper [45]. It was shown that as in the caseof two-sheeted hyperboloid, the energy spectrum contains the scattering states and a finite number of degenerate bound states. This fact coincides with the existence of closed and infinite orbits for positive L2 in classical case. We have not considered in the article [45] the quantum motion in the case of negative eigenvalue of L̂2 because of the strong singularity at the center of harmonic oscillator potential, although it is clear that the system has a discrete spectrum. This work is in progress. Finally, we wish to emphasize that the Kepler–Coulomb and harmonic oscillator potentials are the “building block” upon which most of superintegrable potentials can be constructed. Thus the investigation of these systems is important for the further study and understanding of more complicated superintegrable systems in the hyperbolic space H2 2 . A Symmetry algebra The nonvanishing Poisson brackets between the components of Demkov–Fradkin tensor Dij and Li: {D12,L1} = −D13, {D12,L2} = −D23, {D12,L3} = −D11 −D22, {D13,L1} = D12, {D13,L2} = −D11 −D33, {D13,L3}] = −D23, {D23,L1} = D22 −D33, {D23,L2} = −D12, {D23,L3} = −D13, {D11,L2} = −2D13, {D11,L3} = −2D12, {D22,L1} = −2D23, {D22,L3] = −2D12, {D33,L1] = 2D23, {D33,L2} = −2D13, The same between Dik: {D11,D12} = 2ω2L3 + 2 R2 L3D11, {D11,D13} = 2ω2L2 + 2 R2 L2D11, {D11,D23} = 2 R2 (L2D12 + L3D13), {D11,D22} = 4 R2 L3D12, {D22,D12} = 2ω2L3 − 2 R2 L3D22, {D22,D13} = − 2 R2 (L3D23 + L1D12), {D22,D23} = 2ω2L1 − 2 R2 L1D22, {D22,D33} = − 4 R2 L1D23, {D33,D12} = − 2 R2 (L2D23 − L1D13), {D33,D13} = 2ω2L2 − 2 R2 L2D33, {D33,D23} = −2ω2L1 + 2 R2 L1D33, {D33,D11} = − 4 R2 L2D13, {D12,D13} = − ( 2ω2 − 1 4R4 ) L1 + 1 R2 (L1D11 + L2D12 + L3D13) , {D12,D23} = ( 2ω2 − 1 4R4 ) L2 + 1 R2 (L1D12 + L2D22 − L3D23) , {D13,D23} = − ( 2ω2 − 1 4R4 ) L3 + 1 R2 (−L1D13 + L2D23 − L3D33) . 22 D.R. 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Harmonic Oscillator on the SO(2,2) Hyperboloid

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