On two superintegrable nonlinear oscillators in N dimensions

On t w o sup erin tegrable nonlinear oscillators in N dimensions ´ Angel Ballesteros a , Alb erto Enciso b , F rancisco J. Herranz a , Orlando Ragnisco c and Danilo Riglioni c a Departamen to de F ´ ısica, Univ ersidad de Burgos, 09001 Burgos, Spain E-mail: angelb@ubu.es fjherranz@ubu.es b Departamen to de F ´ ısica T e´ orica I I, Univ ersidad Complutense, 28040 Madrid, Spain E-mail: aenciso@fis.ucm.es c Dipartimen to di Fisica, Universit` a di Roma T re and Istituto Nazionale di Fisica Nucleare sezione di Roma T re, Via V asca Na v ale 84, 00146 Roma, Italy E-mail: ragnisco@fis.uniroma3.it riglioni@fis.uniroma3.it Abstract W e consider the classical superintegrable Hamiltonian system giv en b y H λ = T + U = p 2 2(1 + λ q 2 ) + ω 2 q 2 2(1 + λ q 2 ) , where U is kno wn to b e the “in trinsic” oscillator p oten tial on the Darb oux spaces of nonconstan t curv ature determined by the kinetic energy term T and parametrized b y λ . W e sho w that H λ is St¨ ac k el equiv alent to the free Euclidean motion, a fact that directly pro vides a curved F radkin tensor of constants of motion for H λ . F urthermore, w e analyze in terms of λ the three different underlying manifolds whose geo desic motion is provided b y T . As a consequence, w e find that H λ comprises three different nonlinear ph ysical models that, b y constructing their radial effective p otentials, are shown to be t wo differen t nonlinear oscillators and an infinite barrier p otential. The quantization of these t w o oscillators and its connection with spherical confinement mo dels is briefly discussed. P ACS: 02.30.Ik 05.45.-a 45.20.Jj KEYW ORDS: sup erintegrabilit y , deformation, h yp erb olic, spherical, curv ature, effective p o- ten tial, St¨ ac kel transform 1 1 In tro duction Let us consider the N -dimensional ( N D) classical Hamiltonian defined b y H λ = T ( q , p ) + U ( q ) = p 2 2(1 + λ q 2 ) + ω 2 q 2 2(1 + λ q 2 ) , (1) where λ and ω are real parameters, and q , p ∈ R N are conjugate co ordinates and momenta with canonical Poisson brack et { q i , p j } = δ ij . The mathematical and physical relev ance of this system rely on t w o main prop erties [1]: (i) H λ is a maximally sup erintegrable (MS) Hamiltonian, since it is endow ed with the max- im um p ossible num b er of 2 N − 1 functionally indep endent integrals of motion; and (ii) the cen tral potential U ( q ) can be in terpreted as the “in trinsic” oscillator on the underlying curv ed manifold defined through the kinetic term T . In particular, T determines the geo desic mo- tion of a particle with unit mass on a conformally flat space which w as constructed in [2, 3] and is the N D spherically symmetric generalization of the Darb oux surface of type I I I [4, 5]. The corresp onding metric and scalar curv ature dep end on λ and are giv en b y d s 2 = (1 + λ q 2 )d q 2 , R ( q ) = − λ ( N − 1)  2 N + 3( N − 2) λ q 2  (1 + λ q 2 ) 3 . (2) F rom this viewp oin t, H λ can b e regarded as a MS “ λ -deformation” of the N D isotropic harmonic oscillator with frequency ω since lim λ → 0 H λ = 1 2 p 2 + 1 2 ω 2 q 2 . W e recall that H λ can be iden tified as a particular case within other framew orks suc h as: (i) the “3D multifold Kepler” Hamiltonians [6, 7] (which generalize the MIC–Kepler and T aub- NUT systems); (ii) the “3D Bertrand systems” [8, 9, 10] (coming from a generalization of the classical Bertrand’s theorem [11] to curved spaces); and (iii) the “ N D p osition-dep endent mass systems” [12, 13, 14, 15, 16, 17, 18, 19, 20] (see also references therein) pro vided that the conformal factor of the metric (2) is identified with the v ariable mass function m ( q ) = 1 + λ q 2 . The aim of this pap er is tw ofold. On one hand, in the next section w e provide a deep er insigh t in the set of integrals of motion of H λ giv en in [1] by applying the so-called St¨ ac kel transform or coupling constant metamorphosis [21, 22, 23, 24, 25]. In this wa y , w e obtain the corresp onding λ -deformation of the F radkin tensor of in tegrals of motion [26] for the isotropic harmonic oscillator. On the other hand, w e explicitly sho w that H λ giv es rise, in fact, to thr e e differ ent physical mo dels. F or this latter (and main) purp ose, w e present in section 3 which are the underlying manifolds that come out according to the v alues of λ . This analysis leads to thr e e t yp es of manifolds whic h, in turn, corresp ond to t wo nonlinear oscillator systems plus a barrier-lik e one, whic h are studied in section 4 by constructing their asso ciated effective p oten tial. The final result is that the Hamiltonian H λ comprises the hyp erb olic oscillator ( λ > 0), the spheric al one (the “in terior” space with λ < 0) and an infinite p otential b arrier (the “exterior” space with λ < 0). Remark ably enough, the effectiv e oscillator p otentials are, in this order, h ydrogen-like and oscillator-lik e, whic h means that the quan tization of H λ w ould provide different types of spherical confinemen t models like, for instance, [27, 28]. First results in this direction [29] are briefly sketc hed. 2 2 Sup erin tegrabilit y and the St¨ ac k el transform The MS prop erty of H λ is c haracterized b y the following statement. Theorem 1. (i) The Hamiltonian H λ (1), for any r e al value of λ , is endowe d with the fol lowing c onstants of motion. • (2 N − 3) angular momentum inte gr als: C ( m ) = X 1 ≤ i 0 was considered. Ho w ever, the same algebraic results do hold for λ < 0, and this possibility enable us to get other ph ysical systems different from the one with λ > 0 that w as solved in [1]. W e also remark that the existence of a (curv ed) F radkin tensor (4) is what makes H λ (1) a distinguished Hamiltonian, that is, a MS one which can be regarded as the “closest neigh b our of nonconstant curv ature” to the harmonic oscillator system, which is obtained in the limit λ → 0. It is also w orth stressing that theorem 1 can also b e pro ven b y relating H λ with the fr e e Euclide an motion through a St¨ ac kel transform [21, 22, 23, 24, 25] as follows. Let H b e an “initial” Hamiltonian, H U an “intermediate” one and ˜ H the “final” system giv en b y H = p 2 µ ( q ) + V ( q ) , H U = p 2 µ ( q ) + U ( q ) , ˜ H = H U = p 2 ˜ µ ( q ) + ˜ V ( q ) , (5) suc h that ˜ µ = µU, ˜ V = V /U. (6) Then, each se c ond-or der in tegral of motion (symmetry) S of H leads to a new one ˜ S corre- sp onding to ˜ H through an “intermediate” symmetry S U of H U . In particular, if S and S U are giv en b y S = N X i,j =1 a ij ( q ) p i p j + W ( q ) = S 0 + W ( q ) , S U = S 0 + W U ( q ) , (7) 3 then w e get a second-order symmetry of ˜ H in the form ˜ S = S 0 − W U U H + 1 U H . (8) In our case, w e consider as the initial Hamiltonian H (5) the free one on the N D Euclidean space min us a real constant α (related with λ and ω ): H = 1 2 p 2 − α, 2 λα = ω 2 . (9) And our aim is to p erform a St¨ ac kel transform to the Hamiltonian H λ (1) but written in “final” form as ˜ H = H λ − α = 1 2  p 2 − 2 α 1 + λ q 2  . (10) Th us it can b e chec k ed that the transformation works provided that µ = 2 , V = − α, ˜ µ = 2(1 + λ q 2 ) , ˜ V = − α 1 + λ q 2 , U = (1 + λ q 2 ) , (11) and the intermediate Hamiltonian is the N D istropic harmonic oscillator H U = 1 2 p 2 + λ q 2 + 1 . (12) Next w e consider the symmetries S of H (9) whic h is clearly MS and endo wed with 2 N − 1 functionally indep endent functions. Some of them are exactly (3): S ( m ) = X 1 ≤ i 0 The Darb oux space is the complete Riemannian manifold M N = ( R N , g ), with metric g ij := (1 + λ q 2 ) δ ij . The scalar curv ature R ( r ) ≡ R ( | q | ) (2) is alwa ys a negativ e increasing function suc h that lim r →∞ R = 0 and it has a minimum at the origin R (0) = − 2 λN ( N − 1) , whic h is exactly the scalar curv ature of the N D hyp erb olic sp ac e with negative constan t sectional curv ature equal to − 2 λ . 3.2 T yp e I I : λ < 0 restricted to the in terior space In this case we consider the in terior Darb oux space defined by M N = ( B r c , g ) such that g ij := (1 − | λ | q 2 ) δ ij , B r c = [0 , r c ) , r c = | q | c = 1 / p | λ | , that is, B r c denotes the ball cen tered at 0 of radius r c whic h is the critical or singular v alue for whic h R ( r ) div erges and lim g r → r c − = 0. It is clear that M N is incomplete as a Riemannian manifold. Notice also that R (0) = 2 | λ | N ( N − 1) , whic h coincides with the the scalar curv ature of the N D spheric al sp ac e with p ositiv e constan t sectional curv ature equal to 2 | λ | . The b eha vior of R ( r ) dep ends on the dimension N as follows. • When 2 ≤ N ≤ 6, the scalar curv ature is a p ositiv e increasing function such that lim r → r − c R ( r ) = + ∞ . • If 7 ≤ N , there is a p ositiv e maxim um for R ( r ) corresp onding to r max = s N + 2 2( N − 2) | λ | , R ( r max ) = 4 | λ | ( N − 1)( N − 2) 3 ( N − 6) 2 , and lim r → r − c R ( r ) = −∞ . 5 3.3 T yp e I I I : λ < 0 restricted to the exterior space T o consider the exterior Darb oux space, M N = ( R N \ B r c , g ), requires to c hange the sign of b oth the metric and scalar curv ature (2): g ij := ( | λ | q 2 − 1) δ ij , R N \ B r c = ( r c , ∞ ) , R ( q ) = | λ | ( N − 1)  2 N − 3( N − 2) | λ | q 2  ( | λ | q 2 − 1) 3 . Note that M N is again incomplete. According to the dimension N , the function R ( r ) behav es as follo ws: • F or N = 2, this is a p ositive decreasing function suc h that lim r → r + c R ( r ) = + ∞ and lim r →∞ R ( r ) = 0. • If 3 ≤ N ≤ 5, the scalar curv ature has a negative minim um r min = s N + 2 2( N − 2) | λ | , R ( r min ) = − 4 | λ | ( N − 1)( N − 2) 3 ( N − 6) 2 , with lim r → r + c R ( r ) = + ∞ and lim r →∞ R ( r ) = 0. • When 6 ≤ N , R ( r ) is a negativ e increasing function with lim r → r + c R ( r ) = −∞ and lim r →∞ R ( r ) = 0. 4 Three radial systems and their effectiv e p otentials Firstly , we remark that H λ can also b e expressed in terms of hyperspherical co ordinates r, θ j , and canonical momenta p r , p θ j , ( j = 1 , . . . , N − 1) defined by q j = r cos θ j j − 1 Y k =1 sin θ k , 1 ≤ j < N , q N = r N − 1 Y k =1 sin θ k , (16) so, r = | q | . Thus the Hamiltonian (1) reduces to a 1D radial system: H λ ( r , p r ) = p 2 r + r − 2 L 2 2(1 + λr 2 ) + ω 2 r 2 2(1 + λr 2 ) = T ( r, p r ) + U ( r ) , (17) where L 2 ≡ C ( N ) ≡ C ( N ) is the total angular momentum giv en by L 2 = N − 1 X j =1 p 2 θ j j − 1 Y k =1 1 sin 2 θ k . (18) No w, the geometric analysis p erformed in the previous section indicates that w e must deal with thr e e different physical systems that, for the types I and I I we name nonline ar hyp erb olic oscil lator and nonline ar spheric al oscil lator , resp ectively . In these t wo cases the generic expression for the Hamiltonian (1) is k ept (with the boundary r c for t yp e II), while for 6 t yp e I I I the sign of the Hamiltonian has to b e rev ersed, th us ensuring a positive kinetic term (and provided that the corresp onding restriction on the domain is considered). In particular, as far as the nonlinear radial potential U ( r ) (17) is concerned w e point out the follo wing facts: • Nonlinear h yp erb olic oscillator. When λ > 0, the p oten tial is a p ositive incr e asing function, suc h that U ( r ) = ω 2 r 2 2(1 + λr 2 ) , U (0) = 0 , and lim r →∞ U ( r ) = ω 2 2 λ . (19) • Nonlinear spherical oscillator. If λ < 0 and r < r c , the p otential is also a p ositive incr e asing function v erifying U ( r ) = ω 2 r 2 2(1 − | λ | r 2 ) , U (0) = 0 , and lim r → r c − U ( r ) = + ∞ . (20) • Exterior p otential. When λ < 0 and r c < r w e imp ose the change of the sign of the Hamiltonian. In this wa y the p otential b ecomes a p ositive de cr e asing function: U ( r ) = ω 2 r 2 2( | λ | r 2 − 1) , lim r → r c + U ( r ) = + ∞ , lim r →∞ U ( r ) = ω 2 2 | λ | . (21) But it is essen tial to stress that each of the ab ov e p otentials has to b e considered on the corresp onding curved space describ ed in section 3. In this resp ect, the complete classi- cal system can b e b etter understoo d by introducing an effectiv e p otential (EP) that takes in to account each curv ed background. This can b e achiev ed by applying a 1D canonical transformation [29] P = P ( r, p r ) , Q = Q ( r ) , { Q, P } = 1 , on the 1D radial Hamiltonian (17) yielding H λ ( Q, P ) = 1 2 P 2 + U eff ( Q ) . Next w e presen t suc h an effective p oten tial for the three ab ov emen tioned systems. 4.1 The nonlinear hyperb olic oscillator The 1D canonical transformation is defined by P ( r, p r ) = p r √ 1 + λr 2 , Q ( r ) = 1 2 r p 1 + λr 2 + arcsinh( √ λr ) 2 √ λ , (22) whic h implies that Q ( r ) has a unique (contin uously differentiable) in verse r ( Q ), on the whole p ositiv e semiline, that is, b oth r, Q ∈ [0 , ∞ ); note that d Q ( r ) = √ 1 + λr 2 d r . This transfor- mation yields the EP U eff ( Q ( r )) = c N 2(1 + λr 2 ) r 2 + ω 2 r 2 2(1 + λr 2 ) , (23) where c N ≥ 0 is the v alue of the integral of motion corresp onding to the square of the total angular momentum C ( N ) ≡ L 2 (18). Hence the radial motion of the system can b e describ ed as the 1D problem given b y the p otential U eff ( Q ( r )). 7 0 2 4 6 8 10 r 5 10 15 20 25 30 35 Hyperbolic  Spherical U eff Figure 1: The effectiv e nonlinear h yp erb olic and spherical oscillator p oten tials (23) and (29) for λ = ± 0 . 02, c N = 100 and ω = 1. The minimum of the hyperb olic p oten tial (red curve) is lo cated at r min = 3 . 49 with U eff ( r min ) = 8 . 2 and U eff ( ∞ ) = 25, while the minimum of the spherical one (blue curv e) is located at r min = 2 . 86 with U eff ( r min ) = 12 . 2 and r c = 7 . 07. The dashed line corresp onds to the effective p oten tial of the harmonic oscillator with λ = 0 with minim um U eff ( r min ) = 10 at r min = 3 . 16. In fact, U eff is alw a ys p ositiv e and it has a minimum located at r min suc h that r 2 min = λc N + q λ 2 c 2 N + ω 2 c N ω 2 , U eff ( Q ( r min )) = − λc N + q λ 2 c 2 N + ω 2 c N . (24) Therefore, r min and U eff ( Q ( r min )) are, resp ectively , greater and smaller than those corre- sp onding to the isotropic harmonic oscillator, which are λ = 0 → r 2 min = √ c N /ω , U eff ( Q ( r min )) = ω √ c N . (25) This EP has tw o represen tativ e limits: lim r → 0 U eff ( Q ( r )) = + ∞ , lim r →∞ U eff ( Q ( r )) = ω 2 / (2 λ ) , (26) the latter b eing coincident with (19). Thus, this EP is h ydrogen-like (see fig. 1). 4.2 The nonlinear spherical oscillator In this case, the canonical transformation is given b y P ( r, p r ) = p r p 1 − | λ | r 2 , Q ( r ) = 1 2 r p 1 − | λ | r 2 + arcsin( p | λ | r ) 2 p | λ | , (27) so that Q ( r ) has a unique inv erse r ( Q ) on the in terv als r ∈ [0 , r c ) , r c = 1 p | λ | ; Q ∈ [0 , Q c ) , Q c = π 4 p | λ | . (28) 8 The EP reads U eff ( Q ( r )) = c N 2(1 − | λ | r 2 ) r 2 + ω 2 r 2 2(1 − | λ | r 2 ) , (29) whic h is alw ays p ositiv e and it has a minimum lo cated at r min suc h that r 2 min = −| λ | c N + q λ 2 c 2 N + ω 2 c N ω 2 , U eff ( Q ( r min )) = | λ | c N + q λ 2 c 2 N + ω 2 c N . (30) But now r min and U eff ( Q ( r min )) are, resp ectiv ely , smaller and greater than those corresp ond- ing to the isotropic harmonic oscillator (25). This EP has again t wo c haracteristic limits: lim r → 0 U eff ( Q ( r )) = + ∞ , lim r → r c − U eff ( Q ( r )) = + ∞ , (31) whic h means that w e hav e a deformed oscilator p otential that go es smo othly to an infinite barrier as r approaches r c − (see fig. 1). 4.3 The exterior p otential The canonical transformation for the third system turns out to b e P ( r, p r ) = p r p | λ | r 2 − 1 , Q ( r ) = 1 2 r p | λ | r 2 − 1 − ln  2  | λ | r + p | λ | p | λ | r 2 − 1  2 p | λ | (32) and Q ( r ) has a unique inv erse r ( Q ) on the interv als r ∈ [ r c , ∞ ) , r c = 1 p | λ | ; Q ∈ [ Q c , ∞ ) , Q c = − ln  2( p | λ | r  2 p | λ | . (33) The EP is U eff ( Q ( r )) = c N 2( | λ | r 2 − 1) r 2 + ω 2 r 2 2( | λ | r 2 − 1) . (34) The function U eff is again positive but, unlike the tw o previous systems, it has no minim um; this fulfils the same limits (21) so EP is an infinite (left) potential barrier whic h is represen ted in fig. 2. Finally , some remarks concerning the quantization of these systems are in order. The nonlinear h yp erb olic oscillator ( λ > 0) has b een fully quantized in [29] and its discrete sp ectrum is given b y E n = − ~ 2 λ  n + N 2  2 + ~  n + N 2  s ~ 2 λ 2  n + N 2  2 + ω 2 . (35) The corresp onding stationary states hav e b een obtained in analytic form. Note that the limit n → ∞ of E n is just the asymptotic v alue ω 2 / 2 λ , as exp ected. In view of the shap e of the effectiv e p otential (see fig. 1), the quantum spherical os- cillator ( λ < 0) should pro vide a new radial confinement mo del that could b e useful as a p osition-dep endent-mass mo del for spherical quan tum dots [30]. The exact solution of the corresp onding Schr¨ odinger problem is still in progress. 9 0 5 10 15 20 25 30 35 r 20 40 60 80 100 Exterior U eff Figure 2: The effectiv e nonlinear exterior oscillator p otential (34) for λ = − 0 . 02, c N = 100 and ω = 1. The critical p oin t is r c = 7 . 07 and U eff ( ∞ ) = 25 Ac kno wledgmen ts This w ork w as partially supp orted by the Spanish MICINN under gran ts MTM2010-18556 and FIS2008-00209, b y the Jun ta de Castilla y Le´ on (pro ject GR224), b y the Banco Santander– UCM (grant GR58/08-910556) and by the Italian–Spanish INFN–MICINN (pro ject A CI2009- 1083). F.J.H. is very grateful to W. Miller for helpful suggestions on the St¨ ack el transform. References [1] Ballesteros A., Enciso A., Herranz F.J., Ragnisco O.: Physica D 237 , (2008) 505 [2] Ballesteros A., Enciso A., Herranz F.J., Ragnisco O.: Phys. Lett. B 652 , (2007) 376 [3] Ballesteros A., Enciso A., Herranz F.J., Ragnisco O.: Ann. Phys. 324 , (2009) 1219 [4] Ko enigs G.: in Le¸ cons sur la th´ eorie g ´ en´ erale des surfaces, v ol. 4, ed. Darb oux G., Chelsea, New Y ork (1972) 368 [5] Kalnins E.G., Kress J.M., Miller W. Jr., Winternitz P .: J. Math. Ph ys. 44 , (2003) 5811 [6] Iw ai T., Katay ama N.: J. Math. 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