A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the cur...

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Date:	2007
Main Authors:	Cariñena, J.F., Rañada, M.F., Santander, M.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2007
Series:	Symmetry, Integrability and Geometry: Methods and Applications
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/147830
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Cite this:	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog / J.F. Cariñena, M.F. Rañada, M. Santander // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 44 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1478302025-02-09T15:33:38Z A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog Cariñena, J.F. Rañada, M.F. Santander, M. Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom. This paper is a contribution to the Proceedings of the Workshop on Geometric Aspects of Integrable Systems (July 17–19, 2006, University of Coimbra, Portugal). Partial financial support of research projects BFM-2003-02532, FPA-2003-02948, MTM-2005-09183, DGA E24/1 and VA013C05 is acknowledged. 2007 Article A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog / J.F. Cariñena, M.F. Rañada, M. Santander // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 44 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37J35; 34A34; 34C15; 70H06 https://nasplib.isofts.kiev.ua/handle/123456789/147830 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom.
format	Article
author	Cariñena, J.F. Rañada, M.F. Santander, M.
spellingShingle	Cariñena, J.F. Rañada, M.F. Santander, M. A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Cariñena, J.F. Rañada, M.F. Santander, M.
author_sort	Cariñena, J.F.
title	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
title_short	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
title_full	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
title_fullStr	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
title_full_unstemmed	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
title_sort	super-integrable two-dimensional non-linear oscillator with an exactly solvable quantum analog
publisher	Інститут математики НАН України
publishDate	2007
url	https://nasplib.isofts.kiev.ua/handle/123456789/147830
citation_txt	A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog / J.F. Cariñena, M.F. Rañada, M. Santander // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 44 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 030, 23 pages A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog? José F. CARIÑENA †, Manuel F. RAÑADA † and Mariano SANTANDER ‡ † Departamento de F́ısica Teórica, Facultad de Ciencias Universidad de Zaragoza, 50009 Zaragoza, Spain E-mail: jfc@unizar.es, mfran@unizar.es ‡ Departamento de F́ısica Teórica, Facultad de Ciencias Universidad de Valladolid, 47011 Valladolid, Spain E-mail: msn@fta.uva.es Received October 31, 2006, in final form January 24, 2007; Published online February 24, 2007 Original article is available at http://www.emis.de/journals/SIGMA/2007/030/ Abstract. Two super-integrable and super-separable classical systems which can be con- sidered as deformations of the harmonic oscillator and the Smorodinsky–Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky–Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equa- tion reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom. Key words: deformed oscillator; integrability, super-integrability; Hamilton–Jacobi separa- bility; Hamilton–Jacobi super-separability; quantum solvable systems 2000 Mathematics Subject Classification: 37J35; 34A34; 34C15; 70H06 1 Super-integrable systems There are few integrable systems in the Arnold–Liouville sense: Hamiltonian systems in a 2n- dimensional symplectic manifold for which there exist n functionally independent constants of motion fi in involution (including the Hamiltonian H itself), i.e. {fi, fj} = 0, ∀ i, j = 1, . . . , n, df1 ∧ · · · ∧ dfn 6= 0. The system is said to be super-integrable when it is integrable and there exists a set of m > n functionally independent constants of motion, i.e. it possesses more independent first integrals than degrees of freedom. The existence of these additional first integrals gives rise to a higher degree of regularity in the phase space (e.g. there exist periodic orbits) since the trajectories are restricted to submanifolds of dimension lower than n. In particular, a system with n degrees of freedom possessing 2n− 1 independent first integrals is said to be maximally super-integrable. It is also well-known [1] that the only central potentials in which all bounded orbits are closed (periodic) are given by V = (1/2)ω2 0r 2 and V = −k/r. From a modern point of view the existence of closed trajectories is considered as a consequence of the existence of the maximal number of functionally independent integrals of motion; thus, the result obtained by Bertrand ?This paper is a contribution to the Proceedings of the Workshop on Geometric Aspects of Integ- rable Systems (July 17–19, 2006, University of Coimbra, Portugal). The full collection is available at http://www.emis.de/journals/SIGMA/Coimbra2006.html mailto:jfc@unizar.es mailto:mfran@unizar.es mailto:msn@fta.uva.es http://www.emis.de/journals/SIGMA/2007/030/ http://www.emis.de/journals/SIGMA/Coimbra2006.html 2 J.F. Cariñena, M.F. Rañada and M. Santander is a proof of the super-integrability of these two systems. In fact, the harmonic oscillator shares with the Kepler problem a very distinguished property not only in Classical but also in Quantum Mechanics, where super-integrability leads to energy levels depending on a single quantum number. Fris et al. studied in 1965 the Euclidean n = 2 systems which admit separability in two different coordinate systems [2], and obtained four families Vr, r = a, b, c, d, of super-integrable potentials with constants of motion linear or quadratic in the velocities (momenta). The two first families Va = 1 2 ω2 0(x 2 + y2) + k2 x2 + k3 y2 , Vb = 1 2 ω2 0(4x 2 + y2) + k2x+ k3 y2 , can be considered as the more general Euclidean deformations (with strengths k2, k3) of the 1 : 1 and 2 : 1 harmonic oscillators preserving quadratic super-integrability (the other two families, Vc and Vd, were related with the Kepler problem). The super-integrability of Va, which is known as the ‘Smorodinsky–Winternitz’ (S-W) potential, has been studied by Evans [3, 4] for the general case of n degrees of freedom. A large number of papers have been published on super-integrability in these last years, most of them related with quadratic superintegrabilty (see [5] for the proceedings of a workshop on super-integrability and Refs. [6, 7, 8, 9] for some very recent studies on super-integrable systems with integrals quadratic in momenta). The idea is that if we call super-separable a system that admits Hamilton–Jacobi separation of variables (Schrödinger in the quantum case) in more than one coordinate system, then quadratic super-integrability (i.e., super-integrability with linear or quadratic constants of motion) can be considered as a property arising from super-separability. We note that these studies also include non-Euclidean Hamiltonian systems [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] and that in both cases, Euclidean and non- Euclidean, many of these systems are closely related with the harmonic oscillator. The rareness of integrable systems leads to the following question. Is it possible to deform a super-integrable system but preserve super-integrability of the system? We report here some results of previous works where such question is affirmatively answered. We shall show that this is possible for two important examples: the isotropic bi-dimensional harmonic oscillator and the Smorodinsky–Winternitz system. A very interesting example of a one-dimensional nonlinear oscillator depending of a parameter λ was studied in 1974 by Mathews and Lakshmanan [23, 24] and it has recently been proved [25] that this particular nonlinear system can be generalized to the two-dimensional case, and even to the n-dimensional case: these higher dimensional systems are super-integrable deformations admitting 2n − 1 quadratic constants of motion. It is also proved that there is a related λ-de- pendent two-dimensional oscillator isotonic oscillator that is a super-integrable deformation, i.e. a λ-dependent version of the Smorodinski–Winternitz system [26, 27]. Actually, the deformation introduced by the parameter λ modifies the Hamilton–Jacobi equation but preserves the exis- tence of a multiple separability. Moreover, we point out that a geometric interpretation of the higher-dimensional systems was proposed in relation with the dynamics on spaces of constant curvature. This paper must be considered as a survey summarizing recent works by the authors on properties related with the integrability and super-integrability of certain two-dimensional λ-de- pendent systems related with the harmonic oscillator and it is neither a review of the whole field nor a comparison with other approaches of different authors. It is mainly focused on the study of deformations of super-integrable systems that do not alter the super-integrability structure. It starts with a brief review of the basic properties of the above mentioned isotonic oscillator, S-W system and Mathews–Lakshmanan oscillator and some two-dimensional λ-dependent oscillator- like systems, first in in the classical approach and afterwards in their quantum counterparts. A Super-Integrable Two-Dimensional Non-Linear Oscillator 3 Roughly speaking this survey is divided into three main parts: first, the analysis of the classical λ-dependent oscillator including the existence of Hamilton–Jacobi multiple separability (existence of alternative coordinate systems in which the corresponding λ-dependent Hamilton– Jacobi equation separates). Second, the relation of this λ-dependent nonlinear model with the harmonic oscillator on the three spaces of constant curvature (S2,E2,H2). The final part is devoted to the analysis of these systems from the quantum viewpoint. The second part presents a geometrical approach and proves that these systems can be considered in two different ways: either as a nonlinear deformation of a linear system, or simply as a model of the oscillator on spaces of constant curvature. In the first case the parameter λ represents the strength of the deformation and in the second one the curvature of the space. Finally, in the third part, devoted to the quantum version of this λ-dependent oscillator, it is proved that the system is exactly solvable: the Hamiltonian can be factorized and the wave functions and energies can be explicitly obtained. It includes some points such as: (i) Analysis of the transition from the classical λ-dependent system to the quantum one, (ii) Exact resolution of the λ-dependent Schrödinger equation, factorization method and existence of operators A and A+, and shape-invariance property, (iii) Schrödinger multiple separability and quantum super-integrability. In more detail the structure of the paper is as follows: In next section we describe shortly two one-dimensional classical systems which can be considered as generalizations of the harmonic oscillator and the isotonic oscillator: the λ-deformed nonlinear oscillator and the deformed isotonic oscillator. Section 3 deals with two-dimensional generalizations of these classical systems and their deformations and Section 4 studies the separability properties of their Hamilton Jacobi equations. A geometric interpretation is given in Section 5 and a quantization of the nonlinear oscillator, is carried out in Section 6, and the quantum spectrum s computed by using the traditional power series expansion. The system is shown in Section 7 to admit a shape invariant factorization which allows us to explicitly compute the spectrum in an algebraic alternative way. Finally, in Section 8 we sketch the method to be used for the corresponding two-dimensional system and prove that he Hamiltonian can be written as a sum of three different terms such that each one commutes with the sum of the other two, what provides us alternative complete sets of compatible observables. 2 Some one-dimensional classical systems We start this section by reviewing the basic properties of two simple classical systems. 2.1 The harmonic oscillator The dynamics of the classical harmonic oscillator in one dimension is given by dx dt = v, dv dt = −ω2x and is described by a Lagrangian L = (1/2) ( v2 − ω2x2 ) . A complex variable z = ωx+ iv may be introduced and the equations of the motion become dz dt = −iωz, whose general solution is z = z0e −iωt, i.e. x = x0 cosωt− v0 ω sinωt = A cos(ωt+ ϕ), and therefore the solutions are periodic with angular frequency ω, while A and ϕ are arbitrary. 4 J.F. Cariñena, M.F. Rañada and M. Santander 2.2 The isotonic harmonic oscillator The isotonic oscillator is described by the Lagrangian [28, 29] L = 1 2 v2 x − 1 2 α2x2 − k x2 , k > 0, (like a harmonic oscillator plus a centripetal barrier). This is an important example of an isochronous system and actually they are the only two rational potentials giving rise to isochro- nous systems [30, 31]. The Euler–Lagrange equation ẍ+ α2x+ c x3 = 0, c = −2k, is a particular case of the so-called Pinney–Ermakov equation [32, 33] whose general solution can be written as x = (1/(αA) √ (α2A4 + c) sin2(αt+ φ)− c. The corresponding quantum system admits a shape-invariant factorization [34] and therefore is solvable by means of algebraic methods. 2.3 A 1-dimensional nonlinear oscillator In 1974 Mathews and Lakshmanan [23] studied the equation of motion( 1 + λx2 ) ẍ− λxẋ2 + α2x = 0, λ > 0. (1) The general solution takes the form x = A sin(ωt+ φ), with the following additional restriction linking angular frequency ω and amplitude A: ω2 = α2 1 + λA2 . The equation (1) is the Euler–Lagrange equation for the Lagrangian: Lλ(x, ẋ) = 1 2 1 1 + λx2 ( ẋ2 − α2x2 ) . It describes a system with nonlinear oscillations with an amplitude dependent frequency (or period). We can also allow negative values for λ [25], but when λ < 0 the values of x are limited by the condition \|x\| < 1/ √ \|λ\|. In the limit λ → 0 we recover the equation of motion and both the Lagrangian of the harmonic oscillator and the frequency become independent of the amplitude. Therefore, the system can be seen as a deformation of the harmonic oscillator. It can also be seen as an oscillator with a position-dependent effective mass which depends on λ: mλ = 1 1 + λx2 . The Hamiltonian for such a system is: Hλ(x, p) = 1 2 ( 1 + λx2 ) p2 + 1 2 α2x2 1 + λx2 . The important fact is that there is an interesting generalization to n = 2 or even arbitrary n [25]. Note that the Lagrangian for the one-dimensional free-particle (i.e. for α = 0) Lλ(x, vx) = T1(λ) = 1 2 v2 x 1 + λx2 A Super-Integrable Two-Dimensional Non-Linear Oscillator 5 is invariant under the vector field Xt x(λ) = √ 1 + λx2 ∂ ∂x + λxvx√ 1 + λx2 ∂ ∂vx . which is the natural lift to the phase space R× R of the vector field in R Xx(λ) = √ 1 + λx2 ∂ ∂x , (2) i.e. Xt x(λ) ( T1(λ) ) = 0. 2.4 A deformed isotonic oscillator We study next a deformed isotonic oscillator in one dimension described by the Lagrangian [26]: Lλ(x, vx, k) = 1 2 v2 x 1 + λx2 − 1 2 α2x2 1 + λx2 − k x2 . Here λ can be any real number, but when λ < 0 the possible values of x are such that \|x\| < 1/ √ \|λ\|. The Euler–Lagrange equation is d2x dt2 − λx 1 + λx2 ( dx dt )2 + α2x 1 + λx2 − 2k 1 + λx2 x3 = 0, and one can see that the general solution for bounded motions is [26]: x = 1 ωA √ (ω2A4 − 2k) sin2(ωt+ φ) + 2k, where ωR1 [ 2k + (ω2A4 − 2k) sin2(ωt+ φ) ]2 = 0 with R1 = λω2A4 − ( α2 − ω2 − 2kλ2 ) A2 + 2kλ. It can be done similarly for unbounded motions, the functions sin(ωt+φ) being then replaced by functions sinh(Ωt+ φ). There also exist limit unbounded motions of the form x = √ (At+B)2 + C. It will be shown that as in the harmonic oscillator case, there is an interesting generalization to the n = 2 case. 3 Some two-dimensional classical systems 3.1 The harmonic oscillator The Hamiltonian of a two-dimensional classical harmonic oscillator is (for simplicity m = 1): H(x, y, px, py) = 1 2 ( p2 x + p2 y ) + 1 2 ( ω2 1x 2 + ω2 2y 2 ) , 6 J.F. Cariñena, M.F. Rañada and M. Santander and we can easily check that the energy functions for each degree of freedom are constants of motion: I1 = Ex = 1 2 ( p2 x + ω2 1x 2 ) , I2 = Ey = 1 2 ( p2 v + ω2 2y 2 ) . The rational case, for which ω1 = n1ω0, ω2 = n2ω0, with n1, n2 ∈ N, is super-integrable [35]. In fact, let Kx and Ky be defined by Kx = px + in1ω0x, Ky = py + in2ω0y. The Hamiltonian H and the canonical symplectic form Ω0 can be expressed in terms of such functions as follows: H = 1 2 ( KxK ∗ x +KyK ∗ y ) , Ω0 = i 2n1ω0 dKx ∧ dK∗ x + i 2n2ω0 dKy ∧ dK∗ y , and therefore, as the fundamental Poisson brackets are {Kx,K ∗ x} = 2in1ω0 and {Ky,K ∗ y} = 2in2ω0, the evolution equations are d dt Kx = in1ω0Kx, d dt K∗ y = −in2ω0K ∗ y . Hence, the complex function J defined as Jn1,n2 = Kn2 x (K∗ y )n1 is a (complex) constant of motion which determines two different real first integrals: I3 = Im(Jn1,n2) and I4 = Re(Jn1,n2). They are polynomials in the momenta of degrees n1 + n2 − 1 and n1 + n2, respectively. Two particularly important examples are the isotropic harmonic oscillator (for n1 = n2 = 1) and the case n1 = 1, n2 = 2. In these cases the four constants are not independent but only three of them are independent. For instance, in the isotropic case the constants of motion take the form I3 = pxpy + ω2xy and I4 = xpy − ypx. 3.2 The Smorodinsky–Winternitz system A 2-dimensional generalization of the isotonic oscillator with rotational symmetry would be Vri = 1 2 ω2 0 ( x2 + y2 ) + k1 x2 + y2 . There is however another super-integrable generalization given by the Smorodinsky–Winternitz potential [2] VSW = 1 2 ω2 0 ( x2 + y2 ) + k2 x2 + k3 y2 , which is a set of two non-interacting isotonic systems with the same frequency and in general different constants k2 6= k3. Of course, the case k2 = k3 = 0 reduces to the usual isotropic harmonic oscillator. The energies of each degree of freedom are constants of the motion and there exists a third constant of motion given by C = J2 + k2 y2 x2 − k3 x2 y2 , with J = xvy − yvx. 3.3 A nonlinear oscillator in two dimensions It has recently been proved [25] that there exists a 1-parameter dependent generalization of the nonlinear oscillator for the 2-dimensional case with the following requirements: 1. The kinetic term T2(λ) is a quadratic function of the velocities invariant under rotations. 2. T2(λ) is invariant under (the tangent lifts of) the vector fields X1(λ) and X2(λ) given by X1(λ) = √ 1 + λr2 ∂ ∂x , X2(λ) = √ 1 + λr2 ∂ ∂y , which are extensions to R2 of the vector field Xx(λ) in the n = 1 case given by (2). A Super-Integrable Two-Dimensional Non-Linear Oscillator 7 These conditions lead to the following form for the kinetic energy: T2(λ) = 1 2 1 1 + λr2 [ v2 x + v2 y + λ(xvy − yvx)2 ] , r2 = x2 + y2. The term λ(xvy − yvx)2 represents a two-dimensional contribution that cannot appear in the one-dimensional case. When λ < 0 this function will have a singularity at 1− \|λ\|r2 = 0 and we should restrict our dynamics to the interior of the circle x2 + y2 < 1/\|λ\| where T2(λ) is positive definite. The kinetic energy T2(λ) is determined by the λ-dependent metric ds2(λ) = 1 1 + λr2 [( 1 + λy2 ) dx2 + ( 1 + λx2 ) dy2 − 2λxydxdy ] . T2(λ) remains invariant under the actions of the lifts of the vector fields X1(λ), X2(λ), and XJ , given by X1(λ) = √ 1 + λr2 ∂ ∂x , X2(λ) = √ 1 + λr2 ∂ ∂y , XJ = x ∂ ∂y − y ∂ ∂x , namely Xt 1(λ) = √ 1 + λr2 ∂ ∂x + λ xvx + yvy√ 1 + λr2 ∂ ∂vx , Xt 2(λ) = √ 1 + λr2 ∂ ∂y + λ xvx + yvy√ 1 + λr2 ∂ ∂vy , Xt J = x ∂ ∂y − y ∂ ∂x + vx ∂ ∂vy − vy ∂ ∂vx , These vector fields close on a Lie algebra: [X1(λ), X2(λ)] = λXJ , [X1(λ), XJ ] = X2(λ), [X2(λ), XJ ] = −X1(λ), which is either isomorphic to so(3,R) when λ > 0, to so(2, 1) when λ < 0, or to the Lie algebra of the Euclidean group for λ = 0. The appropriate generalization for the potential of the nonlinear two-dimensional λ-dependent oscillator is given by Vλ(x, y) = α2 2 x2 + y2 1 + λ(x2 + y2) . This bi-dimensional nonlinear oscillator is completely integrable [25], because one can show that, if K1 and K2 are the functions K1 = P1(λ) + iα x√ 1 + λr2 , K2 = P2(λ) + iα y√ 1 + λr2 , with P1(λ) = vx − λJy√ 1 + λr2 , P2(λ) = vy + λJx√ 1 + λr2 , J = xvy − yvx, then the complex functions Kij defined as Kij = KiK ∗ j , i, j = 1, 2, are constants of motion. In fact, the time-evolution of the functions K1 and K2 is d dt K1 = iα 1 + λr2 K1, d dt K2 = iα 1 + λr2 K2, from which we see that the complex functions Kij are constants of the motion. Therefore the system is super-integrable with the following first integrals of motion: I1(λ) = \|K1\|2, I2(λ) = \|K2\|2, I3 = Im(K12) = α(xvy − yvx). 8 J.F. Cariñena, M.F. Rañada and M. Santander 3.4 Nonlinear Smorodinsky–Winternitz system We can consider a deformed Smorodinsky–Winternitz system with Lagrangian [26] L = 1 2(1 + λr2) [ v2 x + v2 y + λ(xvy − yvx)2 ] − α2 2 x2 + y2 1 + λ(x2 + y2) + k2 x2 + k3 y2 . The E-L equations cannot be directly solved in a simple way, but one can check by direct computation the existence of three independent constants of motion: I1 = P 2 1 + α2x2 1 + λ(x2 + y2) + 2k2 1 + λy2 x2 , I2 = P 2 2 + α2y2 1 + λ(x2 + y2) + 2k3 1 + λx2 y2 , I3 = J2 + 2k2 y2 x2 + 2k3 x2 y2 . This proves that for any value of λ this system is completely integrable. Note that for λ = 0 the previous constants of motion reduce to the three first integrals of the S-W potential. Consequently, this system is a deformation of the S-W potential which preserves the completely integrability. 4 Hamilton–Jacobi super-separability of the previous systems The Legendre transformation for a Lagrangian Lλ = T2(λ)− V (x) is given by [25] px = (1 + λy2)vx − λxyvy 1 + λr2 , py = (1 + λx2)vy − λxyvx 1 + λr2 . Note that xpy − ypx = xvy − yvx. The general expression for a λ-dependent Hamiltonian is H(λ) = 1 2 [ p2 x + p2 y + λ(xpx + ypy)2 ] + 1 2 α2V (x, y), and hence the associated Hamilton–Jacobi equation is( ∂S ∂x )2 + ( ∂S ∂y )2 + λ ( x ∂S ∂x + y ∂S ∂y )2 + α2V (x, y) = 2E. This equation is not separable in (x, y) coordinates but there exist three particular orthogo- nal coordinate systems, and three particular families of associated potentials, for which such a Hamiltonian admits Hamilton–Jacobi separability: (i) In terms of the new coordinates (zx, y), zx = x/ √ 1 + λy2, the Hamilton–Jacobi equation becomes:( 1 + λz2 x )( ∂S ∂zx )2 + ( 1 + λy2 )2(∂S ∂y )2 + α2 ( 1 + λy2 ) V = 2 ( 1 + λy2 ) E so if the potential V (x, y) can be written in the form V = W1(zx) 1 + λy2 +W2(y) then the equation becomes separable. The potential is therefore integrable with the following two quadratic integrals of motion I1(λ) = ( 1 + λr2 ) p2 x + α2W1(zx), A Super-Integrable Two-Dimensional Non-Linear Oscillator 9 I2(λ) = ( 1 + λr2 ) p2 y − λJ2 + α2 ( W2(y)− λy2 1 + λy2 W1(zx) ) . and Hλ can be written as: Hλ = (1/2)(I1(λ) + I2(λ)). (ii) Similarly, using coordinates (x, zy), zy = y/ √ 1 + λx2, the Hamilton–Jacobi equation becomes: ( 1 + λx2 )2(∂S ∂x )2 + ( 1 + λz2 y )( ∂S ∂zy )2 + α2 ( 1 + λx2 ) V = 2 ( 1 + λx2 ) E, therefore, if the potential V (x, y) can be written on the form V = W1(x) + W2(zy) 1 + λx2 then the equation becomes separable. The potential is therefore integrable with the following two quadratic integrals of motion I1(λ) = ( 1 + λr2 ) p2 x − λJ2 + α2 ( W1(x)− λx2 1 + λx2 W1(zy) ) , I2(λ) = ( 1 + λr2 ) p2 y + α2W2(zy) and Hλ can be written as: Hλ = (1/2)(I1(λ) + I2(λ)). (iii) In polar coordinates (r, φ) the Hamiltonian H(λ) is H(λ) = 1 2 [( 1 + λr2 ) p2 r + p2 φ r2 ] + α2 2 V (r, φ) so that the Hamilton–Jacobi equation is given by ( 1 + λr2 )(∂S ∂r )2 + 1 r2 ( ∂S ∂φ )2 + α2V (r, φ) = 2E. If the potential V is of the form V = F (r) + G(φ) r2 , then the equation admits separability r2 ( 1 + λr2 )(∂S ∂r )2 + r2 ( α2F (r)− 2E ) + ( ∂S ∂φ )2 + α2G(φ) = 0. The potential V is integrable with the following two quadratic integrals of motion: I1(λ) = ( 1 + λr2 ) p2 r + 1− r2 r2 p2 φ + α2 [ F (r) + 1− r2 r2 G(φ) ] , I2(λ) = p2 φ + α2G(φ) and Hλ can be written as: Hλ = (1/2)(I1(λ) + I2(λ)). 10 J.F. Cariñena, M.F. Rañada and M. Santander 4.1 Super-separability of the nonlinear oscillator Using the previous results one can see that the potential Vλ = α2 2 x2 + y2 1 + λ(x2 + y2) is actually super-separable (separable in more than one system of coordinates) [25]. In fact, the potential for the deformed oscillator we are considering can be alternatively written as follows: Vλ = α2 2 1 1 + λy2 [ z2 x 1 + λz2 x + y2 ] = α2 2 1 1 + λx2 [ x2 + z2 y 1 + λz2 y ] = α2 2 r2 1 + λr2 . Consequently, the Hamiltonian H(λ) = 1 2 [ p2 x + p2 y + λ(xpx + ypy)2 ] + α2 2 x2 + y2 1 + λ(x2 + y2) admits the following decomposition H(λ) = H1(λ) +H2(λ)− λH3 where the three partial functions H1, H2, and H3 are: H1(λ) = 1 2 [( 1 + λr2 ) p2 x + α2 x2 1 + λr2 ] , H2(λ) = 1 2 [( 1 + λr2 ) p2 y + α2 y2 1 + λr2 ] , H3 = 1 2 (xpy − ypx)2, and each one has a vanishing Poisson bracket with H,{ H,H1(λ) } = 0, { H,H2(λ) } = 0, { H,H3(λ) } = 0. 4.2 Super-separability of the deformed S-W system The important point is that, in a similar manner, the λ-deformed Smorodinsky–Winternitz potential Vλ,k = α2 2 ( x2 + y2 1 + λ(x2 + y2) ) + k2 x2 + k3 y2 can be alternatively written in the following three different ways [26] Vλ,k = α2 2 ( 1 1 + λy2 )[ z2 x 1 + λz2 x + y2 ] + ( k2 1 + λy2 ) 1 z2 x + k3 y2 = α2 2 ( 1 1 + λx2 )[ x2 + z2 y 1 + λz2 y ] + k2 x2 + ( k3 1 + λx2 ) 1 z2 y = α2 2 ( r2 1 + λr2 ) + k2 r2 cos2 φ + k3 r2 sin2 φ . Therefore, it is super-separable since it is separable in three different systems of coordinates, (zx, y), (x, zy), and (r, φ). This remarkable property means that the Hamiltonian Hλ,k = 1 2 [ p2 x + p2 y + λ(xpx + ypy)2 ] + Vλ,k A Super-Integrable Two-Dimensional Non-Linear Oscillator 11 admits the following decomposition Hλ,k = Hpx + Hpy − λHJ , where the three partial func- tions H1, H2, and H3, are given by Hpx = 1 2 [( 1 + λr2 ) p2 x + α2 ( x2 1 + λr2 )] + k2 ( 1 + λy2 x2 ) , Hpy = 1 2 [( 1 + λr2 ) p2 y + α2 ( y2 1 + λr2 )] + k3 ( 1 + λx2 y2 ) , HJ = 1 2 ( xpy − ypx )2 + k2 y2 x2 + k3 x2 y2 . Each one of these three terms has a vanishing Poisson bracket with H for any value of the parameters λ, k2 and k3{ Hλ,k,Hpx } = 0, { Hλ,k,Hpy } = 0, { Hλ,k,HJ } = 0. Consequently, the Hamiltonian can be written as a sum, not of two, but of three integrals of motion. The third one represents the contribution of the angular momentum J to H, with the parameter λ as a coefficient, therefore vanishing in the limit λ→ 0. 5 A geometric interpretation The existence of additional constants of motion for the harmonic oscillator in a spherical ge- ometry was studied by Higgs in 1979 [36]. The Higgs approach considers the motion on Sn, embedded in the Euclidean space En+1, by means of a central (also known as gnomonic) projec- tion on a plane Πn tangent to Sn at a chosen point. In this way he could study the properties of the spherical version of the Fradkin tensor. A different alternative approach is discussed in [17, 20] by using curvature-dependent trigonometric and hyperbolic functions, where use is made of the curvature κ as a parameter so that the dynamics can be studied at the same time in the sphere S2 and in the hyperbolic plane H2. In differential geometric terms, the three spaces with constant curvature, sphere S2, Euclidean plane E2, and hyperbolic plane H2, can be considered as three different situations inside a family of Riemannian manifolds M2 κ = (S2 κ,E2,H2 κ) with the curvature κ as a parameter κ ∈ R. In order to obtain mathematical expressions valid for all the values of κ, it is convenient to make use of the following κ-trigonometric functions Cκ(x) =  cos √ κx if κ > 0, 1 if κ = 0, cosh √ −κx if κ < 0, Sκ(x) =  1√ κ sin √ κx if κ > 0, x if κ = 0, 1√ −κ sinh √ −κx if κ < 0, and the κ-dependent tangent function Tκ(x) defined in the natural way, Tκ(x) = Sκ(x)/Cκ(x). The fundamental properties of these curvature-dependent trigonometric functions are C2 κ(x) + κS2 κ(x) = 1, and Cκ(2x) = C2 κ(x)− κ S2 κ(x), d dx Sκ(x) = Cκ(x), Sκ(2x) = 2 Sκ(x) Cκ(x), d dx Cκ(x) = −κSκ(x). 12 J.F. Cariñena, M.F. Rañada and M. Santander If in the Lagrangian of the nonlinear oscillator in one dimension Lλ(x, ẋ) = 1 2 1 1 + λx2 ( ẋ2 − α2x2 ) , we consider the change of variable x = Sκ(u), where λ = −κ, we find that 1 + λx2 = 1− κx2 = 1− κS2 κ(u) = C2 κ(u) and as dx/du = Cκ(u), we see that ẋ = dx du u̇ = Cκ(u)u̇ and therefore the Lagrangian becomes Lκ(u, u̇) = 1 2 C2 κ(u)u̇2 C2 κ(u) − 1 2 α2 S2 κ(u) C2 κ(u) = 1 2 u̇2 − α2 2 T2 κ(u), the vector field Xx turns out to be X = Cκ(u) du dx ∂ ∂u = ∂ ∂u , and the metric giving rise to the Lagrangian g = du ⊗ du. We note that the expression T2 κ(u) of the above one-dimensional potential appears as directly related with the potential of the two-dimensional harmonic oscillator on a space with curvature κ. The expression of the differential element of distance in geodesic polar coordinates (ρ, φ) on the family M2 κ = (S2 κ,E2,H2 κ), can be written as follows ds2κ = dρ2 + S2 κ(ρ)dφ2, so that it reduces to ds21 = dρ2 + ( sin2 ρ ) dφ2, ds20 = dρ2 + ρ2dφ2, ds2−1 = dρ2 + ( sinh2 ρ ) dφ2, in the three particular cases κ = 1, 0,−1 of the unit sphere, Euclidean plane, and ‘unit’ Lobachewski plane, respectively. Note that ρ denotes the distance along a geodesic on the manifold M2 κ ; for example, in the spherical κ > 0 case, ρ is the distance of the point to the origin (e.g., the North pole) along a maximum circle. Therefore, the Lagrangian for the geodesic (free) motion on the spaces (S2 κ,E2,H2 κ) is Lκ(ρ, φ, vρ, vφ) = Tκ(ρ, φ, vρ, vφ) = 1 2 ( v2 ρ + S2 κ(ρ)v2 φ ) , and the Lagrangian for a general mechanical system (Riemannian metric minus a potential) is Lκ(ρ, φ, vρ, vφ) = 1 2 ( v2 ρ + S2 κ(ρ)v2 φ ) − U(ρ, φ, κ). The spherical and hyperbolic harmonic oscillators are characterized by the following Lagrangians with curvature κ [17, 20] Lκ(ρ, φ, vρ, vφ) = 1 2 ( v2 ρ + S2 κ(ρ)v2 φ ) − 1 2 ω2 0 T2 κ(ρ), i.e. the harmonic oscillators on the unit sphere (Higgs oscillator), on the Euclidean plane, or on the unit Lobachewski plane, are: U1(ρ) = 1 2 ω2 0 tan2 ρ, U0(ρ) = 1 2 ω2 0ρ 2, U−1(ρ) = 1 2 ω2 0 tanh2 ρ. A Super-Integrable Two-Dimensional Non-Linear Oscillator 13 Next we study the behavior of Lκ under two different changes of variables. 1. If we consider the κ-dependent change (ρ, φ) → (r′, φ) given by r′ = Tκ(ρ), then the Lagrangian Lκ becomes LHκ(r′, φ, vr′ , vφ) = 1 2 ( vr′ 2 (1 + κr′2)2 + r′2v2 φ (1 + κr′2) ) − 1 2 α2r′2. This function coincides, in the spherical κ > 0 case, with the Lagrangian studied by Higgs in [36]. In Cartesian coordinates (x, y) it reduces to LHκ(x, y, vx, vy) = 1 2 1 (1 + κr′2) [ v2 x + v2 y + κ(xvy − yvx)2 ] − 1 2 α2r′2, r′2 = x2 + y2. 2. Let us consider the κ-dependent change (ρ, φ) → (r, φ) given by r = Sκ(ρ), λ = −κ. Then the Lagrangian Lκ becomes Lλ(r, φ, vr, vφ) = 1 2 ( v2 r 1 + λr2 + r2v2 φ ) − α2 2 ( r2 1 + λr2 ) . Therefore, if we change to Cartesian coordinates (x, y) we arrive to Lλ(x, y, vx, vy) = 1 2 ( 1 1 + λr2 )[ v2 x + v2 y + λ(xvy − yvx)2 ] − α2 2 ( r2 1 + λr2 ) , r2 = x2 + y2. This function is just the Lagrangian obtained in [25] as the natural generalization of the one- dimensional Lagrangian Lλ(x, vx) for the nonlinear equation (1) of Mathews and Lakshmanan. We thus have three different and alternative ways of describing the harmonic oscillator on spaces of constant curvature: the original κ-dependent trigonometric (hyperbolic) Lagrangian Lκ and the two other approaches, LHκ and Lλ, obtained from it. The Higgs approach [36] has been studied by many authors (see e.g. [37, 38] and refe- rences therein) mainly in relation with the theory of dynamical symmetries. Concerning the λ-dependent Lagrangian Lλ, it has similarities with LHκ but it does not coincide with it. In the model of Higgs κ (or λ) is present in the kinetic term T in a different way and the potential V appears as κ-independent; this affects to the Hamiltonian formalism. On the other side each one of these three formalisms can be used for study of the κ-dependent version of the S-W system. In the language of Lκ the potential, that was studied in [13, 20], is given by Ua(r, φ, κ) = k1U 1 a + k2U 2 a + k3U 3 a + k0, k1 = 1 2 ω2 0, U1 a = T2 κ(ρ), U2 a = 1 (Sκ(ρ) cosφ)2 , U3 a = 1 (Sκ(ρ) sinφ)2 , with integrals of motion given by I1(κ) = P 2 1 (κ) + ω2 0 ( Tκ(ρ) cosφ )2 + 2k2( Tκ(ρ) cosφ )2 , I2(κ) = P 2 2 (κ) + ω2 0 ( Tκ(ρ) sinφ )2 + 2k3( Tκ(ρ) sinφ )2 , I3(κ) = J2(κ) + 2k2 cos2 φ + 2k3 sin2 φ , 14 J.F. Cariñena, M.F. Rañada and M. Santander with P1(κ), P2(κ) and J(κ) given by P1(κ) = (cosφ)vρ − (Cκ(ρ) Sκ(ρ) sinφ)vφ, P2(κ) = (sinφ)vρ + (Cκ(ρ) Sκ(ρ) cosφ)vφ, J(κ) = S2 κ(ρ)vφ. This S-W system has been studied, using a different approach, in the two- and three- dimensional sphere in [10], in the two-dimensional hyperboloid in [12] and in the complex two-sphere in [16]. One of the advantages of the Lλ approach is that the Euler–Lagrange equations can be directly solved and the general solution has a rather simple form that can be interpreted as “quasi- harmonic” nonlinear oscillations [25]; other important advantage is that it is very appropriate for the study of the quantum oscillator. In what follows we will focus our attention on the quantum Hamiltonian dynamics determined by the λ-dependent Lagrangian Lλ. 6 The one-dimensional quantum nonlinear oscillator Let us consider the quantum case for n = 1. The problem is to define the quantum operator defining the Hamiltonian of this position-dependent mass system, because the mass and the momentum P do not commute and this fact gives rise to an ambiguity in the ordering of factors. Instead of using traditional procedures as Weyl ordering we shall develop an alternative method of quantization [39]. We first remark that the vector field Xx(λ) = √ 1 + λx2 ∂ ∂x , which was a Killing vector for the metric corresponding to the kinetic energy g = ( 1 + λx2 )−1 dx⊗ dx and generates the translations in this Riemann space, does not leave invariant the natural measure in the real line but the only invariant measures are the multiples of dµ = ( 1 + λx2 )−1/2 dx. This suggests us to consider the Hilbert space L2(R, dµ) and the remarkable fact is that the adjoint of the differential operator √ 1 + λx2∂/∂x in such space is the opposite of such operator. The Legendre transformation corresponding to this deformed kinetic energy is defined by p = vx 1 + λx2 and then the Hamiltonian function for the free particle is given by H = ( 1 + λx2 )p2 2 = 1 2 (√ 1 + λx2p )2 . The usual prescription of canonical quantization does not present any ambiguity because the linear operator (we put ~ = 1) P̂ = −i √ 1 + λx2 ∂ ∂x is self-adjoint in the space L2(R, dµ). Note that with the above mentioned change of coordinates, x = Sκ(u), dµ = du and ∂ ∂u = Cκ(u) ∂ ∂x = √ 1− κS2 κ(u) ∂ ∂x = √ 1 + λx2 ∂ ∂x is self-adjoint in the space L2(R, du), and then P̂ generates translations in these coordinates u. A Super-Integrable Two-Dimensional Non-Linear Oscillator 15 The quantum Hamiltonian operator of the free particle is Ĥ = −1 2 (√ 1 + λx2 ∂ ∂x )2 = −1 2 ( 1 + λx2 ) ∂2 ∂x2 − 1 2 λx ∂ ∂x and in presence of an interaction V1(x) the Hamiltonian will be Ĥ1 = −1 2 ( 1 + λx2 ) d2 dx2 − 1 2 λx d dx + V1(x). (3) We are interested in the case of the nonlinear oscillator for which Ĥ1 = −1 2 ( 1 + λx2 ) d2 dx2 − 1 2 λx d dx + 1 2 α2x2 1 + λx2 . (4) If m and ~ are taken into account, and with the change of parameter, to be more clear later, given by α2 = β ( β + ~ m λ ) it is enough to define the dimensionless variables y = √ mβ ~ , Λ = ~ mβ λ and then Ĥ1 is written as Ĥ1 = β~ [ −1 2 ( 1 + Λy2 ) d2 dy2 − 1 2 Λy d dy + 1 2 (1 + Λ) y2 1 + λy2 ] . The time-independent Schrödinger equation Ĥ1Ψ = EΨ, E = ~βE becomes[ −1 2 ( 1 + Λy2 ) d2 dy2 − 1 2 Λy d dy + 1 2 (1 + Λ) y2 1 + λy2 ] Ψ = EΨ. With the change of variable [40] Ψ(y,Λ) = ϕ(y,Λ) ( 1 + Λy2 )−1/(2Λ) the eigenvalue equation becomes( 1 + Λy2 ) ϕ′′ + (Λ− 2)yϕ′ + (2E − 1)ϕ = 0 and assuming a power series development ϕ(y) = ∑ n≥0 any n the following recursion relation for the coefficients is obtained: an+2 = (−1)n an (n+ 2)(n+ 1) [n(Λn− 2 + (2E − 1)]. 16 J.F. Cariñena, M.F. Rañada and M. Santander Therefore the general solution is determined by the values of the coefficients a0 and a1. In particular we can write the solutions y0 and y1 determined by a0 = 1, a1 = 0 and a0 = 1, a1 = 0, respectively. The solution y0 is an even function while the y1 function is odd. The general solution will be written as y = a0y0 + a1y1. The convergence radius of such power series is R = 1√ \|Λ\| because lim n→∞ ∣∣∣∣an+2 an ∣∣∣∣ = \|Λ\|. This series y = a0y0 + a1y1 reduces to a polynomial of degree p when one of the two coefficients vanish and that means that there exists a positive integer number p such that 2E −1 = 2p−Λp2, and then Ep = p ( 1− Λ p 2 ) + 1 2 . The polynomial solutions can be easily found and they have a form quite similar to the corre- sponding Hermite polynomials. 7 Factorization method and shape-invariance The spectrum of the harmonic oscillator can be found using the factorization method we shall describe in this section [39]. Actually the Hamiltonian can be factorized and we arrive to a shape- invariant Hamiltonian for which the full point-spectrum can be found by algebraic methods. We take ~ = ω = 1. The eigenvalue problem is (up to a factor 1/2) Hψn = ( − d2 dx2 + x2 ) ψn = (2n+ 1)ψn, where the Hamiltonian H is such that H − 1 = a†a, with a = (d/dx+ x), a† = (−d/dx+ x), and then the ground state is found from aψ0 = ( d dx + x ) ψ0 = 0, i.e. ψ0 ∝ e−x2/2. The other eigenstates can be found by applying an iterative way the creation operator a† to the ground state: ψn = 1√ 2nn! a†nψ0 ∝ Hn(x)e−x2/2, with Hn(x) being the Hermite polynomials. A similar procedure can be used for the quantum nonlinear oscillator case. We should look for a function W (x), to be called super-potential, in such a way that the operator A and its adjoint operator A+, given by A = 1√ 2 (√ 1 + λx2 d dx +W (x) ) , A+ = 1√ 2 ( − √ 1 + λx2 d dx +W (x) ) , are such that Ĥ1 = A+A, i.e. Ĥ1 = A+A = 1 2 [ − √ 1 + λx2 d dx +W (x) ] [√ 1 + λx2 d dx +W (x) ] . A Super-Integrable Two-Dimensional Non-Linear Oscillator 17 Therefore, in order to the Hamiltonian so found be that of the deformed quadratic energy term together with a potential V1, the super-potential function W must satisfy√ 1 + λx2W ′ −W 2 + 2V1 = 0. We can define a new quantum Hamiltonian operator Ĥ2 = AA+ = [√ 1 + λx2 d dx +W (x) ] [ − √ 1 + λx2 d dx +W (x) ] which is called the partner Hamiltonian. The new potential V2 is given in terms of W by V2 = 1 2 (√ 1 + λx2W ′ + gW 2 ) . The important fact is that AĤ1 = Ĥ2A, A+Ĥ2 = Ĥ1A +. When Ĥ1\|Ψ〉 = E\|Ψ〉, then, Ĥ2A\|Ψ〉 = AĤ1\|Ψ〉 = EA\|Ψ〉. If A\|Ψ〉 6= 0, A\|Ψ〉 is an eigenvector of Ĥ2 corresponding to the same eigenvalue E, and similarly, if \|Φ〉 is an eigenvector of Ĥ2 with eigenvalue E and such that A+\|Φ〉 6= 0, then A+\|Φ〉 is an eigenvector of Ĥ1 corresponding to the same eigenvalue E. The spectra of Ĥ1 and Ĥ2 are then almost identical, the only differences are when either \|Ψ〉 is an eigenvector of Ĥ1 but A\|Ψ〉 6= 0, or \|Φ〉 is an eigenvector of Ĥ2 for which A+\|Φ〉 = 0. Some parameters may appear in the expression of V1, and the super-potential function W will also depend on them. The most important case is when the explicit forms of the potential and its partner are quite similar and only differ in the values of the parameters. In this case we say that the problem has shape invariance (see e.g. [41]). Suppose that a quantum Hamiltonian Ĥ1(α) admits a factorization Ĥ1(α) = A+(α)A(α) in such a way that the partner Hamiltonian Ĥ2(α) is of the same form as Ĥ1(α) but for a different value of the parameter α. More specifically, there exists a function f such that Ĥ2(α) = Ĥ1(α1) +R(α1), where α1 = f(α) and R(α) is a constant depending on the parameter α. In this case Genden- shtëın developed a method for exact computing of all the spectrum of Ĥ1 [41, 42, 43]. First, the bound state \|Ψ0〉 is found by solving A(α)\|Ψ0(α)〉 = 0, and has a zero energy. Then, \|Ψ0(α1)〉 is an eigenstate of Ĥ2(α) with E1 = R(α1), because Ĥ2(α)\|Ψ0(α1)〉 = (Ĥ1(α1) +R(α1))\|Ψ0(α1)〉 = R(α1)\|Ψ0(α1)〉, and A†(α)\|Ψ0(α1)〉 is the first excited state of Ĥ1(α), with energy E1 = R(α1), because: Ĥ1(α)A†(α)\|Ψ0(α1)〉 = A†(α)(Ĥ1(α1) +R(α1))\|Ψ0(α1)〉 = R(α1)A†(α)\|Ψ0(α1)〉. Iterating the process we find the sequence of energies for Ĥ1(α) Ek = k∑ j=1 R(αj), E0 = 0, the corresponding eigenfunctions being \|Ψn(x, α0)〉 = A†(α0)A†(α1) · · ·A†(αn−1)\|Ψ0(x, αn)〉, where α0 = α and αj+1 = f(αj), namely, αk = fk(α0) = fk(α). 18 J.F. Cariñena, M.F. Rañada and M. Santander Coming back to the nonlinear oscillator case, if β ∈ R, we define the linear operator in L2(R, dµ) A = 1√ 2 (√ 1 + λx2 d dx + βx√ 1 + λx2 ) , for which its adjoint operator is A+ = 1√ 2 ( − √ 1 + λx2 d dx + βx√ 1 + λx2 ) . Then, we find that if Ĥ ′ 1 = Ĥ1 − (1/2)β, Ĥ ′ 1 = A+A = −1 2 ( 1 + λx2 ) d2 dx2 − 1 2 λx d dx + 1 2 β(β + λ) x2 1 + λx2 − 1 2 β, Ĥ ′ 2 = AA+ = −1 2 ( 1 + λx2 ) d2 dx2 − 1 2 λx d dx + 1 2 β(β − λ) x2 1 + λx2 + 1 2 β. This shows that there is one positive number β for which the Hamiltonian Ĥ ′ 1 of the quantum non-linear oscillator admits a factorization with the parameters α and β related by α2 = β(β+λ). In our preceding case, the parameter being β, when comparing Ĥ ′ 1 with its partner, as Ĥ ′ 1(β − λ) = −1 2 [( 1 + λx2 ) d2 dx2 + λx d dx ] + 1 2 (β − λ)β ( x2 1 + λx2 ) − 1 2 (β − λ), then we see that Ĥ ′ 1(β − λ) = ( Ĥ ′ 2(β)− 1 2 β ) − 1 2 (β − λ) and therefore Ĥ ′ 2(β) = Ĥ ′ 1(f(β)) + β − 1 2 λ where f is the function f(β) = β − λ. If R is defined by R(β) = β + (1/2), then Ĥ ′ 2(β) = Ĥ ′ 1(β1) +R(β1). This shows that, as the quantum non-linear oscillator is shape invariant, we can develop the me- thod sketched before: First, the eigenvector \|Ψ0〉 is determined by the condition A(β0)\|Ψ0〉 = 0. More specifically, we should solve the differential equation d dx Ψ0 + β x 1 + λx2 Ψ0 = 0 and therefore the wave function of the fundamental state must be proportional to Ψ0 = 1 (1 + λx2)r0 , r0 = β 2λ . The energies of the first excited states will be E′ 1 = R(β1) = β − λ+ λ 2 A Super-Integrable Two-Dimensional Non-Linear Oscillator 19 and iterating the process we get E′ n = n∑ k=1 R(βk) = n∑ k=1 ( βk + λ 2 ) = n∑ k=1 ( β − λk + λ 2 ) , and therefore, E′ n = nβ + λ [ n 2 − n∑ k=1 k ] = nβ − n2 2 λ. The energy of the eigenstates of Ĥ1 = Ĥ ′ 1 + (1/2)β will be given by En = nβ − n2 2 λ+ 1 2 β. The method also provides us the corresponding eigenfunctions as \|Ψ1(β)〉 = A+(β)\|Ψ0(β1)〉, . . . . . . . . . . . . . . . . . . . . . . . . . . . , \|Ψn(β)〉 = A+(β)A+(β1) · · ·A+(βn−1)\|Ψ0(βn)〉. There is a clear difference between the case λ > 0 and the case λ < 0. Note that the lowest value for E′ n is E′ 0 = 0. Therefore: If λ > 0 only values such that β − λ n 2 ≥ 0 =⇒ n ≤ 2β λ are allowed. The eigenvalues are not equally spaced. On the contrary, when λ < 0 all natural numbers are allowed for n 8 Quantization of the 2-dimensional nonlinear harmonic oscillator There exist relatively few examples of quantum Hamiltonians in two dimensions whose spectrum can be fully determined by algebraic methods. We shall show that the deformed nonlinear oscillator is one of such examples. Furthermore, this can be done in different ways [44]. First, one can check that all the measures dµ = ρ(x, y)dx ∧ dy invariant under the vector fields X1 = √ 1 + λr2 ∂ ∂x , X2 = √ 1 + λr2 ∂ ∂y , are proportional to dµ = 1√ 1 + λr2 dx ∧ dy. Therefore we shall consider the Hilbert space L2(R2, dµ). The form of Killing vectors X1 and X2 generating ‘translations’ suggests us to take the momenta operators P̂x = −i~ √ 1 + λr2 ∂ ∂x , P̂y = −i~ √ 1 + λr2 ∂ ∂y , 20 J.F. Cariñena, M.F. Rañada and M. Santander and then the quantum Hamiltonian is Ĥ = − ~2 2m (( 1 + λr2 ) ∂2 ∂x2 + λx ∂ ∂x ) − ~2 2m (( 1 + λr2 ) ∂2 ∂y2 + λy ∂ ∂y ) + λ ~2 2m ( x2 ∂ 2 ∂y2 + y2 ∂ 2 ∂x2 − 2xy ∂2 ∂x∂y − x ∂ ∂x − y ∂ ∂y ) + 1 2 g r2 1 + λr2 , which can be written as Ĥ = Ĥ1 + Ĥ2 − λĴ2, with Ĥ1 = − ~2 2m (( 1 + λr2 ) ∂2 ∂x2 + λx ∂ ∂x ) + 1 2 g x2 1 + λr2 , Ĥ2 = − ~2 2m (( 1 + λr2 ) ∂2 ∂y2 + λy ∂ ∂y ) + 1 2 g y2 1 + λr2 , Ĵ2 = − ~2 2m ( x2 ∂ 2 ∂y2 + y2 ∂ 2 ∂x2 − 2xy ∂2 ∂x∂y − x ∂ ∂x − y ∂ ∂y ) + 1 2 g r2 1 + λr2 . Note that each term commutes with the sum of the other two and therefore with Ĥ. Conse- quently, we can consider three different (complete) systems of compatible observables: {Ĥ1, Ĥ2 − λJ2}, {Ĥ1 − λJ2, Ĥ2}, {Ĥ1 + Ĥ2, J}. Therefore we should solve one of these spectral problems: A) Ĥ1Ψ(E1, E2j) = E1Ψ(E1, E2j), (Ĥ2 − λJ2)Ψ(E1, E2j) = E2jΨ(E1, E2j); B) (Ĥ1 − λJ2)Ψ(E1j , E2) = E1jΨ(E1j , E2), Ĥ2Ψ(E1j , E2) = E2Ψ(E1j , E2); C) Ĥ1Ψ(E1, E2j) = E1Ψ(E1, E2j), (Ĥ2 − λJ2)Ψ(E1, E2j) = E2jΨ(E1, E2j). As in the one-dimensional case, it is convenient to use g = mα2 + λ~α and dimensionless variables x = √ ~ mα x̃, y = √ ~ mα ỹ, λ = mα ~ Λ, E = ~αe for which 1 + λr2 = 1 + Λr̃2 and the Schrödinger equation becomes − (( 1 + Λr2 ) ∂2 ∂x2 + Λx ∂ ∂x ) Ψ− ~2 2m (( 1 + Λr2 ) ∂2 ∂y2 + Λy ∂ ∂y ) Ψ + Λ ~2 2m ( x2 ∂ 2 ∂y2 + y2 ∂ 2 ∂x2 − 2xy ∂2 ∂x∂y − x ∂ ∂x − y ∂ ∂y ) Ψ + 1 2 (1 + Λ) r2 1 + Λr2 Ψ = eΨ. A) As the Hamilton–Jacobi equation separates in coordinates (zx, y) we shall use such coor- dinates to write the Schrödinger equation (z is used instead of zx) −1 2 ( 1 + Λz2 1 + Λy2 ∂2 ∂z2 + Λz 1 + Λy2 ∂ ∂z ) Ψ − 1 2 ( (1 + Λy2) ∂2 ∂y2 + 2Λy ∂ ∂z ) Ψ + 1 2 1 + Λ 1 + Λy2 ( z2 1 + Λz2 + y2 ) Ψ = eΨ, A Super-Integrable Two-Dimensional Non-Linear Oscillator 21 and assuming a factorization for Ψ(z, y) of the form Ψ(z, y) = Z(z)Y (y) we find the following equations: −1 2 (( 1 + Λz2 ) Z ′′ + ΛzZ ′ ) + 1 2 (1 + Λ) z2 1 + Λz2 Z = µZ, − 1 2 (( 1 + Λy2 )2 Y ′′ + 2Λy ( 1 + Λy2 ) Y ′ ) + ( 1 2 (1 + Λ)y2 − ( 1 + Λy2 ) e ) Y = −µY. In this way the two-dimensional problem has been decoupled in two 1-dimensional equations. The first one is the corresponding one to a 1-dimensional system. The second one however is a bit different because includes the contribution of the angular momentum. If a new parameter ν = e− µ is introduced the second equation becomes ( 1 + Λy2 ) Y ′′ + 2ΛyY ′ − (1 + Λ− 2Λµ) y2 1 + Λy2 Y + 2νY = 0, i.e. defining G2 µ = 1 + (1− 2µ)Λ, ( 1 + Λy2 ) Y ′′ + 2ΛyY ′ −G2 µ y2 1 + Λy2 Y + 2νY = 0, and then writing Y (Λ, µ) = q(y,Λ) ( 1 + Λy2 )−Gµ/(2Λ) , it becomes( 1 + Λy2 ) q′′ + 2(λ−Gµ)yq′ + (2ν −Gµ)q = 0 which is a deformation of the Hermite equation. Assuming the power expansion q(y,Λ) = ∞∑ n=0 cn(Λ)yn we obtain the recursion relation cn+2 = − cn (n+ 2)(n+ 1) (Λn(n− 1)−Gµ(2n+ 1) + 2ν) . Therefore the general solution is determined by the values of the coefficients c0 and c1. In particular we can write the solutions q0 and q1 determined by c0 = 1, c1 = 0 and c0 = 1, c1 = 0, respectively. The solution q0 is an even function while the q1 function is odd. The general solution is q = c0q0 + c1q1. The radius of convergence of both power series defining q0 and q1 is R = 1√ \|Λ\| . The solution reduces to a polynomial of degree n if one of the coefficients is zero and there exists an integer number n related with ν as follows: 2ν = Gµ(2n+ 1)− n(n+ 1)Λ. One can study the properties of these polynomial solutions which have a lot of similarities with the Hermite polynomials. The associated Sturm–Liouville problem plays a relevant rôle. 22 J.F. Cariñena, M.F. Rañada and M. Santander In summary, the bound states for this system have energies em,n = µm + νn = (m+ n+ 1) ( 1− 1 2 (m+ n)λ ) and wave functions given by Ψm,n(z, y) = Zm(z)Yn(y), z = x√ 1 + Λy2 . We finally mention that the properties of these λ-dependent Hermite polynomials are dis- cussed in [40, 44]. It is proved the orthogonality as well as the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polyno- mials of different orders. 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Phys. 322 (2007), to appear. http://arxiv.org/abs/math-ph/0108015 http://arxiv.org/abs/math-ph/0211012 http://arxiv.org/abs/math-ph/0505081 http://arxiv.org/abs/math-ph/0512084 http://arxiv.org/abs/math-ph/0406002 http://arxiv.org/abs/math-ph/0409062 http://arxiv.org/abs/hep-th/0210260 http://arxiv.org/abs/hep-th/9309088 http://arxiv.org/abs/hep-th/0501106 http://arxiv.org/abs/math-ph/0604008 http://arxiv.org/abs/math-ph/9910020 1 Super-integrable systems 2 Some one-dimensional classical systems 2.1 The harmonic oscillator 2.2 The isotonic harmonic oscillator 2.3 A 1-dimensional nonlinear oscillator 2.4 A deformed isotonic oscillator 3 Some two-dimensional classical systems 3.1 The harmonic oscillator 3.2 The Smorodinsky-Winternitz system 3.3 A nonlinear oscillator in two dimensions 3.4 Nonlinear Smorodinsky-Winternitz system 4 Hamilton-Jacobi super-separability of the previous systems 4.1 Super-separability of the nonlinear oscillator 4.2 Super-separability of the deformed S-W system 5 A geometric interpretation 6 The one-dimensional quantum nonlinear oscillator 7 Factorization method and shape-invariance 8 Quantization of the 2-dimensional nonlinear harmonic oscillator References

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

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