Superintegrable 3-body systems on the line

Sup erin tegrable 3-b o dy systems on the line Claudi a Chanu Luca Degiov anni G iov anni Rastelli Diparti mento di Matem atica , Uni v ersit ` a di T orino . T orino , via Car lo Alb erto 10 , Ital ia. e-mail : claudia mari a.chan u@unito.it luca.deg iov anni @gmai l.com gior ast.gi orast @alice.it No vem b er 1, 2018 Abstract W e consider classical three-b o dy in teractions on a Euclidean line dep endin g on the recipro cal distance of th e particle s and admitting four functionally indep end en t quadratic in the momen ta first in tegrals. These systems are multiseparable, sup erintegrable and equiv alen t (up to rescalings) to a one-particle system in the th ree-dimensional Eu- clidean space. C ommon features of the dynamics are discuss ed . W e sho w how to determine quan tum symmetry op erators asso ciated w ith the first in tegral s consider ed here but do not a nalyze the corresp ond ing quan tum dy n amics. T he conformal multiseparabilit y is discussed and examples of conformal first in tegrals are giv en. The systems consid- ered h ere in generalit y in clud e the C aloge ro, W olfes, and other three- b o dy in teractions widely studied in mathematical physics. I. INTRODUCTION The analysis of dynamical systems of points o n one-dimensional manifolds is a classical sub j ect whic h has assumed a particular relev ance in t he last 1 y ears, in mathematics as w ell a s in ph ysics, with a sp ecial attention to the in tegrable cases. An ov erview can b e found in [23, 24]. Prominent amo ng all is the Calogero- Marc hioro-Moser system [7, 19, 22], that is also maximally sup erin tegrable, i.e. admitting 2 n − 1 functionally indep enden t global first in tegrals [32], where n denotes the degrees of freedom, whic h prov ides mat- ter of study since its disco v ery . L ess kno wn is the W olfes system [33], where the p ot ential is the sup erp osition of the Calogero system and of a g en uine three-b o dy in teraction. The W olfes system is kno wn to b e in tegrable a nd sep- arable. It has b een generalized in [26]. Due to the fact that it is equiv alent to t he Calo g ero system (see Section V), it is also maximally sup erintegrable. Sup erin tegrability and maximal sup erinte gra bility are ob jects of g reat in ter- est in mo dern mathematics and ph ysics , not o nly b ecause additional first in tegrals can allow the determination of the tra jectories of the system s, but also b ecause differen tial and algebraic relations b etw ee n t he first integrals themselv es enlighten deep features of integrabilit y of differential systems in general. In man y cases, sup erintegrable systems are obta ined from m ultisep- arable ones, i.e. systems whose Hamilton-Jacobi equation admits separation of v ariables in sev eral distinct co ordinate systems. I ndeed, for systems with n degrees o f freedom t he separabilit y in more than one co ordinate system of- ten implies t he existence of more than n functionally indep enden t quadratic first integrals. F o r example, in [6] the Benen ti system s, a sub class of the separable systems, are systematically emplo y ed for generating maximally su- p erin tegrable systems in an y finite dimension. The relations b etw een the systems of ab ov e and those considered here hav e not y et b een inv es tiga t ed. Another p ossible application of m ultiseparability consists in setting b ound- ary conditions for m ultiseparable quan tum systems by using the co ordinate surfaces of differen t co ordinate systems f o r the same problem. F ro m the classical multis eparable systems it is alw a ys possible, at least in Euc lidean spaces and for natural Hamiltonians, to obta in m ultiseparable quan tum sys- tems. Both the Calogero and W olfes p oten tials are particular case s of the general o ne considered in the presen t pa p er: a three-b o dy p oten tial on the line with in teractions dep ending on the r ecipro cal distance of the p o ints. In general, natural Ha milto nian n -b o dy systems on the line can b e represen ted as o ne- p oin t Hamiltonian systems in the n - dimensional Euclidean space. In the presen t pap er we adopt this p oin t of view (differen tly fr om [18, 25]), and w e restrict our analysis to a three-b o dy p oten tial dep ending on the recipro cal distances only . Moreo ve r, we require that the p otential has the form obtained in [2 , 1 4, 28, 29]. A p oten tial in this form w as sho wn in the cited pap ers to 2 b e separable in fiv e distinct co ordinate sys tems and to a dmit fo ur f unction- ally indep enden t first inte grals, quadratic in t he momen ta. By writing the p oten tial as a function of the distance s b et w een the p oints on the line, w e find the p otential obta ined in [18, 25] and we prov e t ha t t he W olfes system is not only completely in tegrable but also superintegrable and multis eparable. F urthermore, it app ears t o b e related to a larg er class of similar systems . In Sec tion I I the definitions of sup erin tegrability and multise parability are giv en and basic prop erties o f separable systems are recalled. In Section I I I w e briefly describ e a fundamen tal sup erintegrable and multisep ara ble p oten- tial in E 3 . In Section IV w e deriv e t he three-b o dy interactions o n a line described b y a p o ten tial equiv alent to the previous one and, consequen tly , m ultiseparable and sup erin tegrable. In Section V sev eral w ell known p oten- tials are obtained as part icular cases of this fundamen tal p otential and some new are explicited. In Section VI conformal multise parability is discussed for the systems under consideration. In Section VI I w e consider very shortly the quan tum sys tem corresp onding to the classical general one studied in the previous Sections. Some o f the results presen ted here are kno wn in the literature; w e pro vide here a unified appro ac h to the matter with particular emphasis on multise parability and sup erintegrabilit y . I I. SUPERI N TEGRABI LI T Y AN D MUL TISEP ARABI LITY W e a dopt the following definition of sup erin tegrability (see also [16, 32]), Definition 1. A n -de gr e es of fr e e dom Hamiltonian system is sup erinte- grable if it is Liouvil le in te g r able and admits mor e than n functional ly inde- p endent first inte gr als. Mor e over, the system is c al le d maximally sup erin- tegrable if i t admi ts 2 n − 1 in d ep endent first inte gr als a nd quasi maximally sup erin tegrable if ther e ar e 2 n − 2 functional ly indep endent first inte gr als of it. W e require that the Hamiltonian and the first in tegrals of the ab ov e def- inition are glob al ly defined, i.e. they are defined ev erywhe re on the configu- ration manifold with the exception of a subse t of singular p o in ts whic h can b e assumed closed and of zero-measure. Since t he Hamiltonians and the first in tegrals considered in this pap er a r e p olynomial in the momen ta, the sim- ple definition of globalit y of ab o v e satisfies all our needs. F or example, the Kepler Hamiltonian a nd its first in tegrals are fo r us globally defined in E 3 . F o r n = 3 quasi maximally sup erin tegrable systems are also called mini- mally sup er-in tegrable [12]. 3 W e call separable those Hamiltonians and Hamiltonian systems whose Hamilton-Jacobi equation is in tegrable b y additiv e separation of v ariables. As in the classical St¨ ac k el theory of separato n of v ariables, we consider here p oin t-tr ansformations of co ordinates only . If the Hamiltonian H is nat ural (kinetic energy plus a scalar p oten tial), then a n -dimensional system is sep- arable in o rthogonal co ordinates if and o nly if it is a St¨ ac k el system [30, 3 1]. St¨ ac k el systems of natural Hamiltonians are geometrically c haracterized b y symmetric Killing tw o- tensors asso ciated with first in tegrals quadratic in the momen ta and in in v olution according to t he follo wing theorem [15, 1, 11]: Theorem 1. The sep ar abili ty of a n -d i m ensional natur al Hamiltonian sys- tem with p otential V in ortho gonal c o o r dinates is e quivalent to the exis- tenc e on the c onfigur ation Riemannian manifold, of a n -dimens i o nal sp ac e K of symmetric Kil lin g two-tensors K , p airwise c ommuting w ith r esp e ct to Schouten br ackets, with c ommon eigenve ctors and such that d ( K · dV ) = 0 . The ortho g o nal c o or dinates ar e determine d by the eigenve ctors of the K . W e recall that the Sc houten bra c k ets [ , ] o f t w o con tra v a r ia n t tensors A and B of order a, b resp ectiv ely is the symmetric tensor of o rder a + b − 1 whose comp onents are defined b y [ A, B ] ij ...k m = 1 a A ( li...j ∂ l B k ...m ) − 1 b B ( li...j ∂ l A k ...m ) where ( , ) denote the symmetrization of the indices. The separable co ordi- nate h yp ersurfaces are orthogonal to the eigen v ectors of K (the existence of these surfaces is equiv alen t to the normalit y of the eigen v ectors). The set of the co ordinate hy p ersurfaces is called an orthogonal separable w eb . An y parametrization of it lo cally defines orthogo nal separable co ordinates. Separable w ebs can b e group ed in families, according to their geometrical features. In the Euclidean space E 3 , for example, orthogo nal separable w ebs are all made of confo cal quadrics. W e consider equiv alent t w o w ebs made of the same kind of confo cal quadrics, in this w a y E 3 admits 11 distinct separable orthogonal webs [1 1]. Definition 2. T h e Hamiltonia n H is m ultiseparable if it is sep ar able in at le ast two distinct webs. The separabilit y of a n -dimensional system in sev eral distinct orthogona l separable w ebs implies t he existenc e of n +1 ≤ r ≤ 2 n − 1 linearly indep enden t 4 Killing tensors. Eac h of t hese Killing t ensors K = K ij ∂ i ⊙ ∂ j (where ⊙ denotes the symmetrized tensorial pro duct) generates a lo cal quadratic fir st in tegral H K = 1 2 K ij p i p j + V K , where dV K = K · dV ( K is considered as a a linear op erator on one-forms). If more than n of them are functionally indep enden t a nd globally defined, the sy stem is superintegrable. Most of the kno wn sup erin tegrable systems with fir st in tegrals that ar e p olynomials of second degree in the momen ta, are obtained as multisep ara ble systems (they are often called quadratically sup erin tegrable ). I I I . A N OTE W OR TH Y “SUPER” POTENTIA L Let ( x, y , z ) b e Cartesian co o r dinates in E 3 . Let us consider t he Hamiltonian (1) H = 1 2 ( p 2 x + p 2 y + p 2 z ) + V where (2) V = F ( y /x ) x 2 + y 2 . It is kno wn [2, 28 , 12] that H is separable in a ll rotational se parable w ebs around the z axis, namely circular cylindical, spherical, parab olic, spheroidal prolate and spheroidal oblate [21]. This prop erty follo ws from Prop osition 2. In a n y r otational ortho gonal c o or dinate system ( q 1 , q 2 , q 3 ) with r otational axis z and angle of r otation ψ = q 3 , the p otential (2) takes the form of a S t¨ ackel multiplier (3) V = g 33 F ( q 3 ) , wher e g ii ar e the c omp onents of the metric tensor in ( q 1 , q 2 , q 3 ) . Pr o of. Ev ery rotational co ordinate system ( q 1 , q 2 , q 3 ) around the z axis can b e transformed to co ordinates ( x, y , z ) b y the c hange of v ariables    x = f 1 ( q 1 , q 2 ) cos q 3 y = f 1 ( q 1 , q 2 ) sin q 3 z = f 2 ( q 1 , q 2 ) . W e immediately ha v e g 33 = 1 / f 1 2 , and V = F (tan q 3 ) /f 1 2 . Since F is a generic function, V can b e written as (3). 5 The co ordinate systems in whic h the Hamiltonian (1) is separable are generated by a 5-dimensional linear space of Killing tensors. Correspond- ingly , fiv e quadratic first inte gra ls can b e constructed and result globally defined [2, 28]. How ev er, only four of them are functionally indep enden t (in [2, 28 ] it is erroneously rep orted that the fiv e quadrat ic first integrals are all functionally independent [29]). The sys tem with Hamiltonian (1) is hence quasi maximally superinte grable; it is unkno wn if the system is maximally sup erin tegrable for all F . F o r some particular forms of F a fifth functionally indep enden t constan t o f motio n can b e constructed, th us making the system maximally sup erinte gra ble (see, for example, Remark 6 and [14, 20]). By using cylindrical co ordinates ( r, ψ , z ), with r o tational a xis z , and by indicating with ( p r , p ψ , p z ) t heir conjugate momen ta, the p oten tial (2) b e- comes (4) V = F ( ψ ) r 2 and the fiv e quadratic first in tegrals tak e the form [2, 28] H = H 0 = 1 2  p 2 r + 1 r 2 p 2 ψ + p 2 z  + F ( ψ ) r 2 , H 1 = 1 2 p 2 ψ + F ( ψ ) , H 2 = 1 2 p 2 z , H 3 = 1 2  ( r p z − z p r ) 2 +  1 + z 2 r 2  p 2 ψ  +  1 + z 2 r 2  F ( ψ ) , H 4 = 1 2  z p 2 r + z r 2 p 2 ψ − r p r p z  + z r 2 F ( ψ ) . It is easy to chec k their p olynomial dep endence [29] (5) H 0 ( H 3 − H 1 ) − H 2 H 3 − H 2 4 = 0 . The first t hr ee integrals allo w the separation of the system in cylindrical co ordinates. In particular, the conserv ation o f H 2 implies tha t p z is constan t. By setting H i = h i w e obt a in the three differential equations: ˙ z 2 = p 2 z = 2 h 2 ˙ r 2 = p 2 r = 2( h 0 − h 2 ) − 2 h 1 r 2 , (6) ˙ ψ 2 = 1 r 4 p 2 ψ = 2 h 1 − F ( ψ ) r 4 . 6 Necessarily h 2 ≥ 0, and the motion of the system o ccurs only for ( r, ψ ) suc h that ( h 0 − h 2 ) r 2 ≥ h 1 ≥ F ( ψ ) The additiona l first in tegrals H 3 and H 4 pro vide a relatio n b etw een r and z : h 2 r 2 = ( h 0 − h 2 ) z 2 − 2 h 4 z + h 3 − h 1 that allows to find the ra dia l law of motion without integrating (6), for initia l conditions suc h that h 2 6 = 0. When h 2 = 0, the motio n b ecomes planar. Then, from ( 6 ) w e o bta in (7) ˙ r 2 = 2 h 0 − 2 h 1 r 2 that determines the radial comp onen t of the motion. It is remark able that the radial motion is the same fo r all p o t entials (2) and that ˙ r = 0 for r 2 = h 1 /h 0 only , where a simple zero o ccurs. Therefore, no closed tra jectory is p ossible except for the particular case h 0 = h 1 = h 2 = 0 when the tra jectory is on a circle. IV. THREE-BODY SYSTEMS ON THE LIN E W e consider a natural Hamiltonian sys tem of three p oints on a line, with p ositions x i and momen ta p i . Without loss of generality we can assume that the p oin ts hav e the same mass. This system is equiv alent to t he natural Hamiltonian system in E 3 with co ordinates ( x i ) of the p oint x = ( x 1 , x 2 , x 3 ). W e require that the in teraction depends only on the distance b etw een the p oin ts, including in particular three-b o dy in teractions. The p oten tial has, consequen tly , the form (8) V = U ( X i ) , i = 1 , 2 , 3 where X i = x i − x i +1 , i = 1 , 2 , 3 (mo d 3) . Therefore, the system is in v arian t under the translation ω = (1 , 1 , 1 ), as a consequenc e of the in v ariance of the momen tum of the cen ter of mass. The form (4) V = F ( ψ ) r 2 7 of V can b e referred to an y rota tional co ordinate system where ψ represen ts the angle of the rotation and r is the distance from the axis of rotation. In the following, the axis of rotation is parallel to the v ector ω and the trans- formation b et we en ( x i ) and the cylindrical coor dina t es ( r, ψ , z ) is therefore giv en b y (9)      r cos ψ = 1 √ 2 ( x 1 − x 2 ) r sin ψ = 1 √ 6 ( x 1 + x 2 − 2 x 3 ) z = 1 √ 3 ( x 1 + x 2 + x 3 ) . W e w an t to determine the function U in (8 ) suc h that the p oten tial is o f the form (4). Theorem 3. T h e mos t gener al function V of the form (4 ) c an b e written as (10) V = X i 1 X 2 i F i  X i +1 X i , X i +2 X i  i = 1 , . . . , 3 (mo d 3) , wher e F i ar e ge neric functions of two variables. Pr o of. Let ω b e a unit v ector para llel to the axis of rotation, and let r = x − ( x · ω ) ω ω 2 b e the radial v ector from the axis of rotation to the p oint x . The p oten tial V is manifestly inv arian t with resp ect to ω , i.e. it do es not dep end on z . Therefore, the requiremen t that V in ( 8 ) has the form (4) means that r 2 U dep ends only on t he a ngle, i.e. it is in v arian t with resp ect to r . Since r ( r 2 ) = 2 r 2 , w e hav e r ( r 2 U ) = 2 r 2 U + r 2 r ( U ) and the in v ariance condition is r ( U ) = − 2 U. In the v ariables X k the previous equation is equiv a len t to X i ∂ ∂ X i U ( X j ) = − 2 U ( X j ) whose solution is of the form U = 1 X 2 1 F 1 ( X 2 X 1 , X 3 X 1 ) that without loss of generalit y can b e written mor e symmetrically as (10). 8 Remark 1. Due to the r elation X 1 + X 2 + X 3 = 0 the function (10) c an b e r ewritten in the two e quivalent f o rms V = 1 X 2 1 F  X 2 X 1  , V = F ( X 2 /X 1 ) X 2 1 + X 2 2 . Remark 2. Potentials of form (10) ar e known to b e inte gr able by quadr atur es only for zer o values of total ener gy and momentum sinc e the p ap er by Kozlov and Kolesnikov [18, 25]. Therefore, Theorem 4. The thr e e-b o dy system on the line with Hamiltonian H = 1 2 ( p 2 1 + p 2 2 + p 2 3 ) + X i 1 X 2 i F i  X i +1 X i , X i +2 X i  , i = 1 , . . . , 3 (mo d 3) , is quasi maximal ly sup erinte gr able and sep ar able in fi ve typ es of r otational c o or dinate systems. Pr o of. It follo ws from Theorem 3 and the prop erties of p oten tial (2). Remark 3. The p otential is also sep ar able in any other sep ar able r otational c o or dinate system with the sam e axis an d arbitr ary origin [2]. Remark 4. If the p oints have p ositive mas s e s m i , a r esc aling y i = √ m i x i make the m e tric Euclide an. Then, by setting X i = y i − y i +1 , i = 1 , 2 , 3 (mo d 3 ), and ω = (1 / √ m 1 , 1 / √ m 2 , 1 / √ m 3 ) the ab ove pr o c e dur e c an b e r ep e ate d [2]. V. EXAMPLES There are sev eral w ell-kno wn examples o f p oten tials represen ting a t hree- b o dy in teraction on the line that can b e written in the form (10): for instance the Calogero inv erse square po ten tial [7, 2 ] V I = k 1 ( x 1 − x 2 ) 2 + k 2 ( x 2 − x 3 ) 2 + k 3 ( x 3 − x 1 ) 2 = 3 X i =1 k i X 2 i , k i ∈ R . 9 Another in teresting example is the W olfes p oten tial describing a gen uine three-b o dy interaction [33], V I I = k 1 ( x 1 + x 3 − 2 x 2 ) 2 + k 2 ( x 2 + x 1 − 2 x 3 ) 2 + k 3 ( x 3 + x 2 − 2 x 1 ) 2 = X i k i ( X i − X i +1 ) 2 = X i 1 X 2 i k i +1  X i +1 X i − X i +2 X i  − 2 , i = 1 , . . . , 3 (mo d 3) . Remark 5. In the Calo ger o and Wolfe s p otentials usual ly c onsider e d in liter- atur e we have k 1 = k 2 = k 3 and thes e functions a r e o f ten c onsider e d to gether in a singl e p otential. In this c ase, written in cylindric al c o or dinates ( r , ψ , z ) the functions F ( ψ ) c orr esp onding to V I and V I I ar e r esp e ctive l y k I sin − 2 3 ψ and k I I cos − 2 3 ψ , for suitable c onstants k I , k I I . T he two ki n ds of systems ther efor e c oincide after a r otation in the thr e e-dimen sional sp ac e. Remark 6. I t is e as y to sh o w that, at le ast for n = 0 , . . . , 4 , systems with F ( ψ ) = k sin − 2 2 nψ ( n 6 = 0 ) and F ( ψ ) = k sin − 2 (2 n + 1) ψ adm it ad di- tional irr e ducib le p olynomial first inte gr als of de gr e e 2 n + 1 in ( p r , p ψ ) , an d the same for F ( ψ ) = k sin − 2 ψ / 2 n ( n 6 = 0 ), F ( ψ ) = k sin − 2 ψ / (2 n + 1) (as se en in the pr evious r emark, the functions sin and cos ar e inter change- able). The c o efficients of the p olynomials ar e fairly r e gular and this al lows to c onje ctur e that the pr op erties of ab ove hold for al l inte gers n . F or exam- ple, for n ≥ 0 a fifth functional ly indep endent first inte g r a l fo r the p otential V = k ( r sin(2 n + 1) ψ ) − 2 se ems to b e H 5 = n X σ =0 2 σ +1 X i =0 A i σ r 2 n +1 − i  2 k sin 2 (2 n + 1) ψ  n − σ d 2 σ +1 − i (cos(2 n + 1) ψ ) dψ 2 σ +1 − i p i r p 2 σ +1 − i ψ with A i σ = ( − 1) 2 n − σ (2 n + 1) 2 σ +1 − i  2 n + 1 i   [(2 n + 1 − i ) / 2] [(2 σ + 1 − i ) / 2]  , wher e ( a b ) = a ! b !( a − b )! denotes the Newton binom i a l symb ol and [ a ] the gr e atest inte ger ≤ a . This would pr ovide a se quenc e of maximal ly sup erinte gr able systems with non trivial p olynomial first inte gr als o f any o dd d e gr e e. The analysis of thes e c ases is in pr o gr ess. 10 The p otential V I I I = c 1 x 2 + c 2 y 2 where ( x, y ) are Cartesian co ordinates in the plane, is separable in Cartesian, P olar, Elliptic-Hyp erb o lic co ordinates and in the corresponding cylindrical systems of the space ( [2 0] and references therein). The function V I I I repre- sen ts a sup erin tegrable in teraction on a line. Indeed, b y a c hange of coor- dinates from ( x, y , z ) to ( x 1 , x 2 , x 3 ), where t he z -axis coincides with the line for the origin a nd parallel to (1 , 1 , 1), the p oten tial can b e written as V I I I = c 1 2( x 1 − x 2 ) 2 + c 2 6 1 ( x 1 + x 2 − 2 x 3 ) 2 = c 1 2 X 2 1 + c 2 6 1 ( X 2 − X 3 ) 2 = 1 X 2 1 " c 1 2 + c 2 6  X 2 X 1 − X 3 X 1  − 2 # . It can be considered as a mixture of Calog ero and W olfes in teractions. F or the p oten tials V I and V I I I , cubic fir st in tegrals functionally indep enden t of the quadratic ones are known [27, 20, 32], th us making these systems maximally sup erin tegrable. After Theorem 4 it is not difficult to build quasi ma ximally separable p oten tials on the line, for example V I V = 3 X i =1 k i X 2 i + X 2 i +1 = 3 X i =1 k i X 2 i +2  X 2 i X 2 i +2 + X 2 i +1 X 2 i +2  − 1 (mo d 3) . Moreo v er, b y starting from the expression (4) of the p otential V it is p ossible to write it as the p oten tial of an interaction among three b o dies on the line. Indeed, from (9) we obtain tan ψ = 1 √ 3  2 X 2 X 1 + 1  and r 2 = 4 3 ( X 2 1 + X 1 X 2 + X 2 2 ) . F o r example, with reference to Remark 6, b y expanding sin 2 9 ψ in p ow ers o f tan ψ w e ha v e, after some computations, 1 r 2 sin 2 9 ψ = 3( X 2 1 + X 1 X 2 + X 2 2 ) 8 ( X 2 − X 1 ) 2 (2 X 1 + X 2 ) 2 (2 X 2 + X 1 ) 2 W 1 W 2 , 11 where W 1 = ( X 3 2 − 3 X 1 X 2 − 6 X 2 1 X 2 − X 3 1 ) 2 and W 2 = ( X 3 2 + 6 X 1 X 2 + 3 X 2 1 X 2 − X 3 1 ) 2 . VI. CONF ORMAL MUL TISEP A RABILITY A nat ur a l Hamilto nian H = G + V , where G is the geo desic term, is con- formally separable for the v alue 0 of the energy if the Hamiltonian H /V is separable in the usual sense . The separation is then asso ciated with confo r - mal Killing t w o- tensors of G and conformal first in tegrals of H in a w a y pretty similar to the standard separation. Recall that a conformal first in tegral of H is a function K suc h that { H , K } = f H for some function f . If f = 0 the separation is the standard one. All in tegral curv es of H = 0 coincide with the in tegral curv es o f H /V = − 1 up to a t ime rescaling (Jacobi transformation) [5]. A neces sary and sufficien t condition for the conformal separability o f H = G + V , where G is the Euclidean geo desic term, in orthogonal confor- mally separable co ordinates of E 3 , is that in these co ordinates the p otential tak es the form of a pseudo-St¨ ac k el m ultiplier: V = g ii φ i ( q i ). Indeed, the Hamiltonian (1) is confo rmally separable in all orthogonal conformally sepa- rable r o tational co ordinate systems of the t hree-dimensional Euclidean space with axis of rotation z . In fact, (2) coincides in these systems with g 33 F ( ψ ). The rotatio nal orthogo na l conformally separable co ordinate systems of E 3 are: ta ngen t spheres, cardioids, in v erse o blate, in v erse prolate, bi-cyclide, flat-ring-cyclide, disk-cyclides [21]. More details ab out conformal separation of Hamilton- Jacobi and Schr¨ odinger equations can b e found in [5, 9, 8]. In the case of gen uine confo r ma l separation, the set of zero energy (o r of the v a lue of the energy for whic h confor mal separation holds) is w ell defined b e- cause, if V is a pseudo-St¨ ack el multiplier, then V + h is not a pseudo-St¨ ac k el m ultiplier fo r a n y h 6 = 0, ot herwise the separation would b e the standard one [5]. F or confo rmal separation the p oten tial function cannot b e c hosen up to additiv e constan ts. F o r instance, the conformal first in tegral corresp onding to separation in cardioids co ordinates is H c = z ( z 2 − r 2 ) p 2 z + 2 z r 2 p 2 r + r (3 z 2 − r 2 ) p r p z and { H , H c } = 2 H [4 z r p r + (3 z 2 − r 2 ) p z ] , whic h is zero for in tegral curv es contained in the h yp ersurface H = 0. The four functions ( H 1 , H 2 , H 4 , H c ) are functionally dep enden t on H = 0, with 2 H 4 ( H 1 H 2 + H 2 4 ) + H 2 2 H c = 0, but any three of them together with 12 the Hamiltonia n are not. It is p ossible that there are sys tems with more functionally indep enden t conformal first in tegrals for one fixed v alue of t he energy than for t he others, but this seems not to b e the case. F or a ll t hese conformally separable sys tems, H , H 1 and H c are the quadratic conformal first in tegrals asso ciated with confo r ma l separation in cardioids co ordinates. W e remark that H c is the same for all scalar potentials of the form (2) b ecause it do es no t dep end on F . Some of the natural systems with p oten tial (2) admit a wider set of separable coo r dina t es. In six-spheres co ordinat es ( u, v , w ) (obtained as inv ersion of the Cartesian co o rdinates), for example, the p otential V I I I b ecomes V I I I = ∆ 2  c 1 u 2 + c 2 v 2  , ∆ = u 2 + v 2 + w 2 and, b ecause g uu = g vv = g w w = ∆ 2 , it is a pseudo-St¨ ac k el multiplie r. Then, the p oten tial is conformally separable in these co ordinates [9]. The same prop ert y holds in the co ordinate systems obtained by inv ersion f r o m the circular and elliptic cylindrical co ordinates (the ot her co or dina t es that allo w separation of v ariables). Ev en if conformal m ultiseparabilit y do not enlighten new features of the po t en tials w e ar e considering, the p ossibilit y of separation of v ariables in these new systems can b e useful if one wishes to consider p erturbations of the p otentials. F or example, b y a dding to them for each of the conformally separable co ordinate system a p erturbativ e term in the form of f H , where f is a pseudo-St¨ ack el m ultiplier, one o btains different in tegrable systems whose dynamics coincide with the original one for H = 0. Vice- v ersa, if f is a ny function, the dynamics o n H = 0 could b e a multis eparable appro ximation o f the p erturb ed one for ”small v alues” of H . VI I. QUANTI ZA TION F o r H , H 1 , . . . , H 4 it is easy to build corresp onding quan tum symmetry op er- ators b y follow ing quan tization rules give n for example in [3, 4, 10]. At a clas- sical leve l, symme tric Killing tw o -tensors K k with comp onen ts K ij k are asso- ciated to quadratic first in tegrals of the Hamiltonian H = 1 2 G ij p i p j + V of the form H k = 1 2 K ij k p i p j + V k , provided some compatibilit y conditions in v olving V are satisfied, fo r suitable functions V k . Correspo ndingly , self-adjo int differ- en tial op erators of order tw o ˆ H k are defined b y ˆ H k φ = 1 2 ∇ i ( K ij k ∇ j φ ) + V k φ for an y wa v e function φ . The op erator corresp onding to t he Hamiltonian H is the Sc hr¨ odinger (Laplace-Beltrami) op erator ˆ H . Since in Euclidean spaces 13 the Ricci tensor is null, the Ro b ertson condition holds. Therefore, the dif- feren tial op erators asso ciated to Hamiltonians in inv olution do comm ute. It follo ws tha t a sy mmetry op erator of ˆ H corresp onds to eac h quadratic first in tegral of H . If the first in tegrals are in inv olution, then the correspond- ing differen tial op erators comm ute and ˆ H is multiplic atively separable in the same co ordinat es as H . Th us, in analo g y with the classical m ultiseparability previously sho wn for H , we ha v e Theorem 5. The Schr¨ odinger o p er ator ˆ H asso ciate d to H admits four dis- tinct se c ond or der symm etry op e r ators and is sep ar a b le in five d i s tinc t webs. W e do not consider here the problem of quantum sup erinte grability , more on this topic can b e found in [1 3, 17] and r eferences therein. F or higher order first integrals the pro cedure of quantiz atio n is less understoo d. F or cubic first inte gra ls a quan tization rule is giv en in [10] as follows P = P j k l 3 p j p k p l + P j 1 p j 7− → ˆ P = i 2  − ( ∇ j P j k l 3 ∇ k ∇ l + ∇ j ∇ k P j k l 3 ∇ l ) + P j 1 ∇ j + ∇ j P j 1  . W e do not deve lop here the analysis o f the quantum systems corresp onding to (10). P articular cases are discussed for example in [7, 33, 24, 26]. VI I I. CONCLUSION Sev eral w ell kno wn integrable systems are considered as examples of a more general system represen ting three p oints on a line whose dynamics is quasi maximally sup erintegrable a nd mu ltiseparable, and some new are giv en. Conformal m ultiseparabilit y is discussed for the general system considered. A quan tum system correspo nding t o the classical one is sho wn to be m ultisep- arable. The analysis of thr ee-b o dy systems on the line done here is extensible to n -b o dy interactions on the line or on higher dimens ional Euclidean man- ifolds. Again, quasi maximally sup erintegrable and m ultiseparable systems are obtained. These systems will b e considered in a f o rthcoming pap er. A CKNOWLEDGMENTS W e wish t o tha nk Pa v el Win ternitz for useful discussions on the to pic of this article. This researc h has b een partially supp orted by the pro ject of the Ministero dell’Univ ersit´ a e Ricerca: PRIN 200 6 -2008. 14 References [1] S. Benen ti, J. 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