Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization

Generalizations of a metho d for c onstructing first in tegrals of a class of nat ura l Hamilto nians and some remark s ab out quan tization Claudia Chan u Dipartimen to di Matematica e Applicazioni, Univ ers it` a di Milano Bicocca. Milano, via Cozzi 53 , Italia. Luca Degio v anni, Gio v anni Rastelli F ormerly at Dipartimen t o di Matematica, Univ ersit` a di T orino. T orino, via Carlo Alb erto 10, Italia. e-mail: claudia.chan u@unimib.it luca.degio v anni@gmail.com giorast.giorast@alice.it Septem b er 18, 2018 Abstract In previous pap ers w e determined necessary and sufficient conditions for the existence of a class of natural Hamiltonians with non-trivial first integ rals of arbitrarily high degree in the momenta. Such Hamiltoni- ans w ere c haracterized as (n+1)-dimensional extensions of n-dimensional Hamiltonians on constant-curv atu re ( pseudo-)Riemannian manifolds Q. In this pap er, we generalize that approac h in v arious directions, we ob - tain an explicit exp ression for t h e first in tegrals, holding on t he more general case of H amiltonians on P oisson manifolds, and sh ow how the construction of above is made p ossible by th e existence on Q of particu- lar conformal Killing tensors or, equiv alently , particular conformal m aster symmetries of the geod esic equations. Finally , we consider t h e p roblem of Laplace-Beltrami quantization of these first in tegrals when they are of second-degree. 1 In tro duction In re c e n t years, several progress e s hav e b een done in the field o f integrable and sup e rint egrable Hamilto nia n sy stems, b oth classical and q ua n tum, by the in- tro duction of new techniques for the s tudy of higher-degre e po lynomial first int egrals and higher-order symmetry op erators. After resear c hes ex pos ed in [4], 1 [7] and [10] is now p ossible to explicitly build a nd analyze Hamiltonian systems po ssessing symmetries of arbitr arily-high degree. F or a mo r e detailed intro duc- tion see the contribution to the QTS 7 pro ceedings written by W. Miller J r. In several paper s ([4], [5 ], [6]) we developed the analysis o f a class of systems which, in dimension tw o, are a subset of the celebrated T rem blay-T urbiner- Win ternitz (TTW) systems and are strictly rela ted with the Jacobi- Calogero and W olfes three-b o dy sy s tems [4], [6]. In [5] we gene r alized these systems to higher-dimensions by in troducing a ( n + 1)-dimensional extension H of a g iven n -dimensional natur a l Hamiltonian L . W e obtained necessa ry and sufficien t conditions for the exis tence of a fir s t integral of H in a pa rticular form, one necessary condition b eing the co ns tan t curv ature o f the c onfiguration ma nifold on which L is defined (for sup erintegrable systems with higher -degree first in- tegrals on cons ta n t curv ature manifolds see als o [8]). The first integral o f H , which is independent from those of L , is p olyno mial in the momenta a nd can be explicitly constructed thr ough a differential op erator . In the pre s en t pap er, we generalize the analysis do ne in [5] in several directions. In Sec. 2 we extend the construction to non natural Hamiltonians on a genera l Poisson manifolds a nd obtain, also in this cas e, an explicit ex pression for the p olynomial first integral. In Sec. 3 w e restrict ourselves to cotangent bundles of (pseudo-)Riemannian manifolds and consider a wider class of higher-deg ree first integrals, we prov e that a nece s sary condition f or their existence is the presence of a particula r class of conformal Killing t ensors or, equiv alently , of conformal master symmetries of the geo desic equations; we end the sec tion with an example showing how the metho d can provide several independent fir st integrals of degree m . In Sec. 4 we characterize our constr uction in an inv a riant wa y and determine neces sary a nd sufficient co nditions for the constant curv ature or co nfo r mal flatness of the con- figuration manifold o f H , conditions employed in Sec. 5, where the quantization of the second-deg ree fir st integrals obta ined by o ur metho d is co nsidered. 2 Extensions on a Poi sson manifold Let us co nsider a Poisson manifold M a nd a one- dimensional manifold N . F or any Hamiltonia n function L ∈ F ( M ) with Hamiltonian vector field X L , we consider its extension on ˜ M = T ∗ N × M g iven by the Hamiltonia n H = 1 2 p 2 u + α ( u ) L + β ( u ) (1) where ( p u , u ) are canonica l co or dinates o n T ∗ N and α ( u ) 6 = 0. T he Hamiltonian flow o f (1) is X H = p u ∂ ∂ u − ( ˙ αL + ˙ β ) ∂ ∂ p u + αX L , where dots deno tes total deriv ative w.r.t. the (single) v aria ble u . It is immediate to see that any first integral of L is als o a co nstant o f mo tion of H , when co nsidered a s a function on ˜ M . W e r ecall that a function F is a first integral of H if a nd only if X H F = { H, F } = 0. In [5] we deter mined on L , α and β necessary and sufficient conditions for the existence of tw o functions γ ∈ F ( N ) and G ∈ F ( M ) such that, given the differential op erator U = p u + γ ( u ) X L , (2) 2 the function U m ( G ) obtained applying m 6 = 0 times U to G is a no n trivial additional first int egral for H . In pa rticular, if L is a natural Hamiltonia n o n the cotangent bundle of a (pseudo-)Riemannian manifold ( Q, g ) L = 1 2 g ij p i p j + V and α is as s umed to b e not co nstant, an integral o f the for m U m ( G ) e x ists, with G not de p endent on the momenta, if and only if G sa tisfy for so me constant c 6 = 0 the equa tions: ∇ i ∇ j G + mcg ij G = 0 , (3) ∇ i V ∇ i G = 2 mc V G, (4) which are equiv alent to {∇ i Gp i , L } = 2 mcGL, meaning that ∇ i Gp i is a conformal first integral of L . If a solution o f the previous eq uations exists, then the extended Hamiltonian (1) and the differential op erato r (2 ) ta k e the for m H = 1 2 p 2 u + mc S 2 κ ( cu + u 0 ) L (5) U = p u + 1 T κ ( cu + u 0 ) X L , (6) where the trigonometr ic tagged functions (see [3, 9]) are employ ed S κ ( x ) =        sin √ κx √ κ κ > 0 x κ = 0 sinh √ | κ | x √ | κ | κ < 0 T κ ( x ) =        tan √ κx √ κ κ > 0 x κ = 0 tanh √ | κ | x √ | κ | κ < 0 Here we show tha t a n ana lo gous r e sult holds in a genera l situation. Prop osition 1. L et H b e the ext ension (1) of the Hamiltonian L on the Poisson manifold ˜ M , let U the differ ent ial op er ator (2) and G ∈ F ( M ) a function such that X L ( G ) 6 = 0 . Then, U m ( G ) is a first inte gr al for H if and only if G satisfies X 2 L ( G ) + 2 m ( cL + L 0 ) G = 0 c, L 0 ∈ R . (7) and α , β and γ satisfy α = − m ˙ γ , (8) β = mL 0 γ 2 + β 0 , β 0 ∈ R , (9) ¨ γ + 2 cγ ˙ γ = 0 . (10) Pr o of. In [5] it is proved that w e ha v e tha t X H U m ( G ) = 0 for a function G ∈ F ( M ) if a nd only if L , α , β satisfy ( m ˙ γ + α ) X L ( G ) = 0 , (11) αγ X 2 L ( G ) − m ( ˙ αL + ˙ β ) G = 0 . (12) 3 Because X L ( G ) 6 = 0 , from (11) it follows that α = − m ˙ γ (13) and co ndition (12) b eco mes ˙ γ γ X 2 L ( G ) G = m ¨ γ L − ˙ β . Since ˙ γ = − α/ m 6 = 0, we get X 2 L ( G ) G = m ¨ γ γ ˙ γ L − ˙ β γ ˙ γ , (14) which derived with r esp ect to u gives d du  ¨ γ γ ˙ γ  L = d du ˙ β mγ ˙ γ ! . But L is a non-c o nstant function on M , hence the functions ¨ γ and ˙ β m ust b e bo th pr op ortional to γ ˙ γ : ¨ γ = − 2 cγ ˙ γ = − c d du  γ 2  , ˙ β = 2 mL 0 γ ˙ γ = mL 0 d du  γ 2  . By integrating and substituting in (14), we obtain conditions (7) a nd (9). Remark 1. If X L ( G ) = 0 w e tr ivially hav e U m ( G ) = p m u , whic h is a first int egral o f H only if α and β are consta n t. Hence, it is a consta n t of motion functionally dep endent on L and H . Remark 2. The equation (7) is o b viously equiv alent to { L, { L, G } } = − 2 m ( cL + L 0 ) G ; this condition can be int erpreted in terms of master symmetries: the Hamilto- nian vector field X G is a master s ymmetry for the Ha miltonian vector field X L on the hypersurfac e s L = 0 or G = 0. F urther remark s ab out the sp ecial case when L is a natural Hamiltonia n a re at the end o f Sec. 3. By int egrating the equatio ns for α , β and γ in Pro pos ition 1 the ex plic it expression for the extended Hamiltonian H and the differential op erator U can be found. F r om equa tion (7) we hav e [5] Theorem 2. L et H b e the extension (1) of the Hamiltonian L on the Poisson manifold ˜ M , let U t he differ ential op er ator (2) and G ∈ F ( M ) a function satis- fying X L ( G ) 6 = 0 and (7). Then, U m G is a first inte gr al of H if and only if H and U ar e in either one of the two fol lowing forms char acterize d by the value of c in (7) 4 i) for c 6 = 0 H = 1 2 p 2 u + mc S 2 κ ( cu + u 0 ) ( L + V 0 ) + W 0 , (15) U = p u + 1 T κ ( cu + u 0 ) X L , ii) for c = 0 H = 1 2 p 2 u + mA ( L + V 0 ) + B ( u + u 0 ) 2 , (16) U = p u − A ( u + u 0 ) X L , with κ, V 0 , W 0 ∈ R , B = mL 0 A 2 and A 6 = 0 . Pr o of. B y P rop osition 1, α , β , γ must sa tis fy (8), (9 ), (10). In the c a se c 6 = 0 equation (10) be c o mes ˙ γ + c ( γ 2 + κ ) = 0 , whose solutio n is γ = 1 T κ ( cu + u 0 ) . Hence, α = mc S 2 κ ( cu + u 0 ) , β = mcV 0 S 2 κ ( cu + u 0 ) + W 0 , with V 0 = L 0 /c and W 0 = β 0 − mκL 0 . In the c a se c = 0 , equation (1 0) gives ˙ γ + A = 0 with A 6 = 0 in or der to av oid α = 0. Hence, α = mA, β = mAV 0 + B ( u + u 0 ) 2 , γ = − A ( u + u 0 ) , where V 0 is now an arbitrar y constant. Remark 3. The constants u 0 , V 0 and W 0 are not essential. Indeed, H and L are defined up to a dditive co nstant W 0 and V 0 while u 0 can b e eliminated by a translation of u . In the ca s e c 6 = 0, the choice V 0 = W 0 = 0 gives the expressio ns (5) and (6) for H and U obtained in [5]. Moreov er, by including the constant L 0 in the Hamiltonian L , the condition (7) ass umes the simpler form X 2 L ( G ) + 2 mcLG = 0 . Once L and G satisfy condition (7) the first integrals U m ( G ) ar e explicitly determined for any G : M − → R . Theorem 3. Under the hyp othesis of Pr op osition 1 the functions U m G c an b e explicitely writt en as U m G = P m G + D m X L G, (17) 5 wher e P m = [ m/ 2] X k =0  m 2 k  γ 2 k p m − 2 k u ( − 2 m ( cL + L 0 )) k , D m = [ m/ 2] − 1 X k =0  m 2 k + 1  γ 2 k +1 p m − 2 k − 1 u ( − 2 m ( cL + L 0 )) k , m > 1 , wher e [ · ] denotes the inte ger p art and D 1 = γ . Pr o of. F rom equation (7) it follows that for all k ∈ N we hav e X 2 k +1 L G = ( − 2 m ( cL + L 0 )) k X L G, X 2 k L G = ( − 2 m ( cL + L 0 )) k G. (18) By ex panding U m using the binomial fo rmu la U m G = ( p u + γ X L ) m = m X k =0  m k  p k u ( γ X L ) m − k , and separ ating even a nd o dd terms in k , by taking in acco un t rela tions (18) we get eq uation (17 ). The setting describ ed in the prev io us section can be further gener alized a s follows. L et X L be a Hamiltonia n vector field o n a Poisson manifold ˜ M , let on ˜ M X H = Y + f 3 X L , for a vector field Y a nd U = f 1 + f 2 X L , where f i : ˜ M → R . F ollowing the same pro of pro cedure as in [5] we get Prop osition 4. If X L ( f i ) = 0 and [ Y , X L ] = 0 t hen X H U m ( G ) = 0 , i.e. U m ( G ) is a first inte gr al of H , if and only if  f 1 Y + ( mY ( f 2 ) + f 1 f 3 ) X L + f 2 X L Y + f 2 f 3 X 2 L  ( G ) = − mY ( f 1 ) G. (19) Pr o of. If X L ( f i ) = 0 and [ Y , X L ] = 0 ,then { H , L } = 0 , [ X H , U ] = Y ( f 1 ) + Y ( f 2 ) X L , [[ X H , U ] , U ] = 0 . Thu s, X H U m = U m − 1 ( m [ X H , U ] + U X H ) = = U m − 1  mY ( f 1 ) + f 1 Y + ( mY ( f 2 ) + f 1 f 3 ) X L + f 2 X L Y + f 2 f 3 X 2 L  . and the thesis follows. The ana lysis of such a gener alization will not b e co nsidered here. 6 3 Extensions of a natural Hamiltonian In the following sections w e will assume that L is a natura l n -dimensional Hamil- tonian o n M = T ∗ Q for a (pseudo-)riemannian manifold ( Q, g ): L = 1 2 g ij ( q h ) p i p j + V ( q h ) , (20) where g ij are the c ont rav a riant comp onents of the metric tensor and V a scalar po ten tial. This a ssumption, together with the hypo thesis that G is po lynomial of degr e e d in the momen ta ( p i ), allows us to expa nd co ndition (7) into an equality of tw o p olyno mia ls in ( p i ) of degree d +2 that can b e splitted into several differential conditions inv o lving the metric, the p otential and the co efficients of G . Indeed, b eing L a natur al Hamiltonian, we hav e (in [5] the equa tion for X 2 L was mistyped, how ev er, this do es no t a ffects any of the r esults o f the pap er, X L = p i ∇ i − ∇ i V ∂ ∂ p i , X 2 L = p i p j ∇ i ∇ j − ∇ i V ∇ i − 2 p j ∇ i V ∇ j ∂ ∂ p i − p i ∇ i ∇ j V ∂ ∂ p j + ∇ i V ∇ j V ∂ 2 ∂ p i ∂ p j . In [5] we dea lt with the case c 6 = 0, d = 0, i.e. G independent o f momenta, obtaining the conditio ns (3) and (4). The max imal dimension of the space o f solutions o f equation (3) is n + 1 and it is achiev ed o nly if the metric g o n Q has co nstant cur v ature. W e call c omplete the solutions G of (3) satisfying this int egrability c ondition (see [5]). In the follo wing, we analy ze in deta ils the d = 1 case ( G linear in the momenta), in o rder to s how how the pr o ce dur e works. Prop osition 5. L et b e G = λ l ( q i ) p l + W ( q i ) . Then, U m G is a first inte gr al of H if and only if ∇ ( i ∇ j λ l ) + mcg ( ij λ l ) = 0 , (21) ∇ i ∇ j W + mcg ij W = 0 , (22) ∇ i V ( ∇ i λ l + 2 ∇ l λ i ) + λ i ∇ l ∇ i V − 2 mλ l ( cV + L 0 ) = 0 , (23) ∇ i V ∇ i W − 2 m ( cV + L 0 ) W = 0 , (24) Pr o of. F or G linear in the momenta we hav e X L G = p i p l ∇ i λ l − ∇ i V λ i + p i ∇ i W , X 2 L G = p i p j p l ∇ i ∇ j λ l − p l ( ∇ i V ( ∇ i λ l + 2 ∇ l λ i ) + λ i ∇ l ∇ i V ) + + p i p j ∇ i ∇ j W − ∇ i V ∇ i W , and co ndition (7) holds if and o nly if p i p j p l ( ∇ i ∇ j λ l + mcg ij λ l ) + p i p j ( ∇ i ∇ j W + m cg ij W ) − p l ( ∇ i V ( ∇ i λ l + 2 ∇ l λ i ) + λ i ∇ l ∇ i V − 2 mλ l ( cV + L 0 ))+ 2 m ( cV + L 0 ) W − ∇ i V ∇ i W + = 0 , which is equiv alent to eqs. (21, 2 2, 2 3, 2 4). 7 Remark 4. The co efficients of terms w ith even and o dd degree in the momenta are inv olv ed in different equations: eq .s (2 2) a nd (24) contain the ones o f a G independent of p i . Hence, for λ i = 0 we r ecov e r the d = 0 cas e: (22) and (24) are the expansio n in co ordina tes of (3) a nd (4) for G = W . F or W 6 = 0 the compa tible p otentials V hav e to s atisfy b oth conditions (2 4) and (23), th us it is imp ossible to get new p otentials other than thos e compa tible with a G independent of the momenta i.e., satisfying conditions (22 – 2 4). F rom (22) one can derive (see [5]) integrabilit y co nditio ns for W ( R hij k − mc ( g hj g ik − g hk g ij )) ∇ h ln W = 0 . (25) If these equa tions are identically satisfied we hav e complete integrabilit y which is equiv a len t to constant curv a ture o f Q , otherw is e W must satisfy all equa tions (22) a nd (25). F o r exa mple, when Q has dimensio n tw o, we hav e from (25) ( R 1212 − mc det( g ij )) ∇ 1 ln W = 0 , and ( R 2121 − mc det( g ij )) ∇ 2 ln W = 0 . Therefore, beca use of the symmetries of the Riemann tensor, we hav e Theorem 6. If Q has dimension 2 , t hen e quations (22) admit non-c onst ant solutions W only if Q has c onstant curvature . F or each l , the integrability co nditions of (21 ) are w eaker than tho s e for the Hessian equation for G ( q i ) (3) and therefor e the curv ature of Q could b e non-constant. W e give t wo examples in order to illustrate the Pr opo sition 5. Example 1. As shown in [5] and recalled ab ov e, when Q has constant curv ature, equation (3 ), or equiv alen tly eq uation (22), a dmits a solution dep ending on n + 1 rea l parameters ( a i ). L e t G i be a solution determinated b y the choice of a particular set of the ( a i ), let us assume that G i 6 = G j . It is then natural to consider the relations b etw een U m G i and U m G j and see if some choice of the parameters ca n pr ovide new indep endent first integrals of the system. F or example, let L b e the natural Hamiltonian o n the constant curv a ture manifold Q = S 2 with ( q 1 = θ , q 2 = φ ) L = 1 2 ( p 2 θ + 1 sin 2 θ p 2 φ ) + V . (26) A complete solution of a 0th de g ree G ( θ, φ, a 1 , a 2 , a 3 ) has b een computed in [5] G = ( a 1 sin φ + a 2 cos φ ) sin θ + a 3 cos θ. (27) and for a 3 = 0, the integration o f equation (4) – or equiv alen tly (24 ) – gives V = 1 cos 2 θ F (( a 1 sin φ − a 2 cos φ ) tan θ ) . F or different sets o f the para meters ( a k ), U m G i and U m G j are no long er simul- taneously first integrals of H unles s if F = F 0 = constant , and ther e fore V = F 0 cos 2 θ . (28) 8 In this c a se, let b e G 1 = G ( a 1 = 1 , a 2 = 0 , a 3 = 0 ) and G 2 = G ( a 1 = 0 , a 2 = 1 , a 3 = 0 ). Hence, for any extension o f L o f the form (1) with α given b y (8), the five functions L 0 = L , L 1 = p 2 = p φ , H , U m G 1 and U m G 2 are functionally independent fir s t integrals of H . F or m = 2, reca lling that c = K/ m , the curv ature of Q = S 2 is K = 1 and choo sing for the other free parameter s of α the v alues κ = 0 and u 0 = 0, we have α = 4 u 2 , and U 2 G 1 and U 2 G 2 are U 2 G 1 = (sin φ sin θ )  p 2 u − p 2 θ 4 u 2 − F 0 8 u 2 cos 2 θ  + p θ p u 4 u cos θ s in φ + p φ p u 4 u cos φ sin θ − p 2 φ 4 u 2 sin φ sin θ , U 2 G 2 = (cos 4 θ + sin 2 θ − co s 2 θ ) co s φ sin 3 θ  p 2 u − p 2 θ 4 u 2 − F 0 8 u 2 cos 2 θ  + p θ p u 4 u cos θ c o s φ − p φ p u 4 u sin φ sin θ − p 2 φ 4 u 2 cos φ sin θ . Example 2. W e can use a complete solution G ( q i , a k ) of (3) in orde r to con- struct so lutions λ i of (21). Namely , we ca n choos e λ i = G i , i = 1 , . . . , n whe r e G i denotes any par ticular so lutio n of (3). W e r e ma rk that it is not necessary that G i 6 = G j for i 6 = j , or G i 6 = 0 for all i . By substituting the λ i int o (2 3), the equations beco me n second-o rder PDE in V whose s olutions pr ovide examples of compatible p otentials. F or instance, let us co nsider aga in L given by (26) on Q = S 2 . W e ca n cho ose for λ i the particular v alues λ 1 = c o s( θ ), λ 2 = 0 o f (27) as co efficients for a linear homog eneous G . Then, equa tions (23) can b e int egrated yielding, V = c 1 + c 2 sin θ cos 2 θ , which, for c 2 6 = 0 do es no t satisfies (4) with G given by (27), hence, for this po ten tial the construction of U m G is possible only when G depends on the momenta. F or the different choice of λ i , λ 1 = 0, λ 2 = c o s θ , the integration of (23) g ives V = c 1 sin 2 θ , which is compatible w ith G g iv en by (27) for a 1 = a 2 = 0. The expressions of U m G can b e co mputed by using (17). Remark 5. By consider ing the functions λ i as the co mponents o f a vector field Λ , equation (21) can b e written as [ g , [ g , Λ ]] = − mc Λ ⊙ g , (29) where [ · , · ] are the Sc ho uten-Nijenh uis brac kets and ⊙ denotes s ymmetrized tensor pr oduct. This means that [ g , Λ ] is a particular kind of confor mal Killing tensor, or, e q uiv alently , that Λ is a pa rticular conformal master s y mmetry of the geo desic equa tions, wher e the conformal factor is a c o nstant multiple o f Λ , instead of an ar bitrary v ector field. In a s imila r wa y , for G p olynomial in the momenta of degree k with highest degree term given by λ i 1 ...i k p i 1 . . . p i k , it is straightforward to show that a necessa ry c o ndition for U m G to b e fir st int egral of H is still of the form (29), where now Λ is a k -tensor field. In the 0 -th degr ee case G = W ( q i ) eq. (29) b ecomes [ g , ∇ W ] = − mcW g . 9 Definition 1. W e c al l self-confo r mal ( s-conforma l in short) the ( k + 1) -or der c onformal Kil ling ten s or field [ g , Λ] such that [ g , [ g , Λ ]] = C g ⊙ Λ , C ∈ R , is satisfie d. In this c ase, the k -or der t ensor Λ is said to b e a s-confor mal master symmetry of the ge o desic e quations of g . In the cas e o f C = 0 , i.e. c = 0, s-conformal Killing tensors a nd master symmetries b ecome the usual Killing tensors and master symmetrie s . Theorem 7. L et G b e a k -de gr e e p olynomial of de gr e e k in the momenta. A ne c essary c ondition for U m G to b e first int e gr al of H is that the tensor Λ of c omp onent s λ i 1 ...i k given by the c o efficients of the highest-de gr e e term of G is a self-c onformal master symmetry of the ge o desic e quations of g or, e quivalently, that [ g , Λ ] is a self-c onformal tensor field of g with C = − mc . The existence of a complete so lution intro duced in [5] and recalle d a bove can b e r estated as follows Corollary 8. Equation (3 ) admits a c omplete solution G = W ( q i ) if and only if t he dimension of the sp ac e of the s-c onformal Kil ling ve ct ors ∇ G , with C = − mc , is max imal and e qual to n + 1 . 4 In trinsic c haracterisatio n of the extended Hamil- tonians W e show under which geometrical conditions a ( n + 1 )-dimensional natural Hamiltonian can b e written as the extension (15) of a natural Hamiltonian L . Let us c o nsider a na tural Hamilto nia n H = 1 2 ˜ g ab p a p b + ˜ V (30) on a ( n + 1)-dimensional Riemannian manifold ( ˜ Q, ˜ g ) and let X b e a confor mal Killing vector o f ˜ g , that is a vector field s atisfying [ X , ˜ g ] = L X ˜ g = φ ˜ g , where φ is a function on ˜ Q a nd [ · , · ] are the Sc houten-Nijenhuis brack ets. W e denote by X ♭ the corr esp onding 1-for m obta ine d by low er ing the indices b y means of the metric tensor ˜ g . Theorem 9. If on ˜ Q ther e exists a c onformal Kil ling ve ctor field X with c on- formal factor φ such that dX ♭ ∧ X ♭ = 0 , (31) dφ ∧ X ♭ = 0 , (32) d k X k ∧ X ♭ = 0 , (33) X ( ˜ V ) = − φ ˜ V , (34) ˜ R ( X ) = k X , k ∈ R , (35) 10 wher e ˜ R is the Ric ci tensor of the Riemannian manifold, then, ther e ex ist on ˜ Q c o or dinates ( u, q i ) such that ∂ u c oincides up to a r esc aling with X and the natur al H amiltonian (3 0) has the form (15 ) . Pr o of. Co ndition (31) means that X is nor mal i.e., orthogo nally integrable: lo- cally there e x ists a fo liation o f n dimensiona l diffeomor phic manifolds Q such that T P Q = X ⊥ = { v ∈ T ˜ Q | ˜ g ( v , X ) = 0 } ; it follows that ther e exists a co ordi- nate system ( q 0 = u , q i ) for i = 1 , . . . , n such that ∂ u is para llel to X and the comp onents ˜ g 0 i v anish for all i = 1 , . . . , n . F urthermor e, by (3 2) the conformal factor φ is constant on the leav es Q ( v ( φ ) = 0 for all v ∈ X ⊥ ); th us, φ dep ends only on u . By expa nding the condition that X = F ( q a ) ∂ u is a c o nformal Killing vector  1 2 ˜ g 00 ( q a ) p 2 u + 1 2 ˜ g ij ( q a ) p i p j , F ( q a ) p u  = φ ( u )  1 2 ˜ g 00 ( q a ) p 2 u + 1 2 ˜ g ij ( q a ) p i p j  we g et the e quations ˜ g 00 (2 ∂ u F − φ ) − F ∂ u ˜ g 00 = 0 (36) ˜ g hj ∂ h F = 0 j = 1 , . . . , n (37) ˜ g ij φ + F ∂ u ˜ g ij = 0 i, j = 1 , . . . , n (38) By (37), we hav e F = F ( u ), hence due to (38) we get that ∂ u ln ˜ g ij is a function of u , the same function for all i, j . Thus, without los s of genera lit y w e can assume ˜ g ij = g ij ( q h ) α ( u ). More over, Eq. (36) implies that, up to a rescaling o f u , ˜ g 00 is indep enden t of u . B y imp osing X ( ˜ V ) = − φV , we obtain ∂ u ln ˜ V = − φ/F , that means ˜ V = α ( u ) V ( q h ), th us we get H = 1 2 g 00 ( q h ) p 2 u + α ( u )  1 2 g ij ( q h ) p i p j + V ( q h )  . Finally , condition (33) means that the norm of X is constant on Q , that is F ( u ) 2 g 00 ( q i ) is independent o f ( q i ). This shows that up to a r escaling and a change o f s ig n of H we can assume g 00 = 1 and in the co ordinate system ( u, q i ) (30) has the required form (15). By computing a gain the Poisson brack et, we get relations b etw een φ , α and F : α = k ( F ) − 2 and φ = 2 ˙ F with k a real not v anishing co nstant. When X is a prop er confor ma l Killing vector, we ca n assume that α is pr op o rtional to F ( u ) − 2 . T he cov aria n t comp onents of the Ricci tensor of ˜ Q are given in Lemma 10, in particular we hav e for i = 1 , . . . , n ˜ R 00 = n ¨ F F , ˜ R 0 i = 0 . Hence, X = F ( u ) ∂ u is an e igenv ecto r of the Ricci tensor with eigenv a lue ρ = n ¨ F F , which is constant if and o nly if F is pro por tional to S κ ( cu + u 0 ). Remark 6. If φ = 0 (i.e., X is a Killing vector), then α and F ar e necess a rily constant and this gives the geo desic term of the case c = 0, but equa tion (3 4) do es not c haracterize the p otential o f the Hamiltonian (1 6). Remark 7. It is stra ightf orward to chec k that for a Hamilto nia n o f the form H = 1 2 p 2 u + F − 2 ( u ) L with L a natur al n - dimens io nal Hamiltonian, X = F ∂ u is a CKV with conforma l factor φ = 2 ˙ F suc h that X ( F − 2 ( u ) V ) = − φ ( F − 2 ( u ) V ). Hence, conditions of the ab ov e theorem are necess a ry for having an e x tended Hamiltonian of our form. 11 W e wan t now to study the geometric pro per ties of the metric ˜ g o btained by an ex tension o f a metric g , in particular when g is of co nstant curv ature. In the following, we assume α ( u ) = f − 2 in or der to simplify computations. In particular, f is a llow ed to b e pure imaginary . Lemma 10. L et ( g ij ) b e the c omp onent s of a n -dimensional m etric on Q in the c o or dinates ( q i ) . We c onsider the ( n + 1) -dimensional metric on ˜ Q having c omp onent s ˜ g ab ( a, b = 0 , . . . n, i , j = 1 , . . . n ) with r esp e ct to c o or dinates ( q 0 = u, q i ) defin e d as fol lows ˜ g ab =    1 a = b = 0 , 0 a = 0 , b 6 = 0 , f 2 ( u ) g ij ( q h ) a = i, b = j. (39) Then, t he r elations b etwe en the c ovari ant c omp onent s of the Riemann tensors asso ciate d with ˜ g and g ar e for al l h, i, j, k = 1 , . . . , n ˜ R hj kl = f 2 R hj kl − ˙ f 2 f 2 ( ˜ g hk ˜ g j l − ˜ g hl ˜ g j k ) , (40) ˜ R 0 j kl = 0 , (41) ˜ R 0 j 0 l = − ¨ f f ˜ g j l . (42) Mor e over, t he c ovaria nt c omp onent s of t he Ric ci t ensors R ij and ˜ R ab of the two metrics ar e r elate d, for al l h, i , j, k = 1 , . . . , n , by ˜ R 00 = n ¨ f f , (43) ˜ R 0 i = 0 , (44) ˜ R ij = R ij +  f ¨ f + ( n − 1) ˙ f 2  f − 2 ˜ g ij , (45) and t he r elation b etwe en the Ric ci sc alars R and ˜ R is ˜ R = R f 2 + n 2 f ¨ f + ( n − 1) ˙ f 2 f 2 , (46) wher e ˙ f and ¨ f denote the first and se c ond derivative w.r.t. u of f ( u ) . Expressio ns (40), (45), and (46) b ecome simpler when Q is of constant cur- v ature, while the other formulas remain unchanged. Lemma 11 . Under the hyp otheses of L emma 10 with n > 1 , if g is a metric of c onstant curvatur e K , then the non z er o c ovariant c omp onents of the Riemann tensor ˜ R asso ciate d with ˜ g ar e, for al l h, i, j, k = 1 , . . . , n ˜ R hj kl = K − ˙ f 2 f 2 ( ˜ g hk ˜ g j l − ˜ g hl ˜ g j k ) , (47 ) Mor e over, the c ovari ant c omp onents of the Ric ci t en s or ˜ R ij and t he Ric ci sc alar ˜ R ar e, for i, j = 1 , . . . , n , ˜ R ij =  f ¨ f + ( n − 1 )( ˙ f 2 − K )  f − 2 ˜ g ij , (48) ˜ R = n 2 f ¨ f + ( n − 1)( ˙ f 2 − K ) f 2 . (49) 12 Theorem 12 . L et ( Q, g ) b e a n -dimensional Riemannian manifold of c onstant curvatur e K = mc and ( ˜ Q, ˜ g ) the extende d manifold with metric (39), ther efor e i) the metric ˜ g is of c onstant curvatu r e if and only if either n = 1 or m = 1 or K = c = ˙ f = 0 , ii) the metric ˜ g is c onformal ly flat if and only if either n > 2 or ˜ g is of c onstant curvatur e. Pr o of. F or n = 1 the extended metric is, up to a rescaling of q 1 , ˜ g ab =  1 0 0 f 2  , which is of constant cur v ature if and only if ¨ f is prop ortiona l to f which is true if f is any trigonometr ic ta gged function. F or n ≥ 2, due to the B ia nch i ident ities, the metric is of consta n t c urv ature if the ratios ˜ R abcd / ( ˜ g ac ˜ g bd − ˜ g ad ˜ g bc ) are indep enden t of ( a, b, c, d ), tha t is b y (47), (41 ), a nd (42) ¨ f f + K − ˙ f 2 = 0 , (50) which for c 6 = 0, i.e. f 2 = S 2 κ ( K m u + u 0 ) K , b e comes K 2 ( m 2 − 1) m 2 = 0 , which holds only for m = 1 or for K = c = 0, when f is constant (se e Theor em 2) a nd (5 0) holds. F or n = 2 the three- dimens ional extended metric ˜ g is conformally fla t if and only if the W eyl-Schouten tensor ˜ R abc = ˜ ∇ c ˜ R ab − ˜ ∇ ˜ R ac + 1 2 n  ˜ g ac ˜ ∇ b ˜ R − ˜ g ab ˜ ∇ c ˜ R  , where ˜ ∇ denotes the cov ariant der iv ative w.r.t. ˜ g , v anishes. By a pplying the formulas derived in Le mma 11 we hav e that the only non v a nishing comp onents of ˜ R abc are, for i, k = 1 , 2, ˜ R i 0 k = ˙ f f 3 ˜ g ik ( ¨ f f + K − ˙ f 2 ) , which, as shown ab ove, v anish only for m = 1 or in the case when 0 = K = c and f is constant. F or n > 2 the ( n + 1)-dimensiona l extended metr ic ˜ g is conformally flat if and only if the W eyl tenso r ¯ C abcd = ˜ R abcd + 1 n − 1  ˜ g ac ˜ R bd − ˜ g ad ˜ R bc + ˜ g bd ˜ R ac − ˜ g bc ˜ R ad  + + ˜ R n ( n − 1) ( ˜ g ad ˜ g bc − ˜ g ac ˜ g bd ) . v anishes and, by applying Lemma 11, this is true for all manifold Q of c onstant curv ature . 13 5 Quan tization W e consider he r e q uan tization for the case m ≤ 2 only . F or m = 1 , it is well k nown how to asso ciate a first order symmetry op erato r with any co ns tan t of motion linear in the mo menta. In [2] the quantization o f quadr atic in the momenta first integrals o f natural Hamiltonian functions has b een a nalyzed and we r ecall her e the re sults rele v an t for our ca s e. Let ˆ H b e the Hamiltonian op erator a sso ciated with the Hamiltonian H = 1 2 g ij p i p j + V , we hav e ˆ H = − ~ 2 2 ∇ i ( g ij ∇ j ) + V = − ~ 2 2 ∆ + V , where ∆ is the Laplac e-Beltrami op era tor. Let T = 1 2 T ij p i p j + V T be a first int egral o f H , let ˆ T = − ~ 2 2 ∇ i ( T ij ∇ j ) + V T . (51) W e hav e (Prop osition 2.5 of [2]) Prop osition 13. L et b e { H , T } = 0 , then [ ˆ H , ˆ T ] = 0 if and only if δ C = δ ( T R − RT ) = 0 , (52) wher e R is the Ric ci tensor, T a nd R ar e c onsider e d as endomorphisms on ve ctors and one-forms and ( δ A ) ij . ..k = ∇ r A r ij.. .k , is t he diver genc e op er ator for skew-symmetric tensor fields A . F or our pur po s es we need to a pply (52) to the Ricci tensor of the extended metric and to the consta n t of the motion T = U 2 G . By assuming cons tan t the curv ature K of Q , the comp onents of ˜ R ab are g iven by inserting f 2 = 1 K S 2 κ ( K m u + u 0 ) or f 2 = 1 mA in Lemmas 1 0 and 11; the cov aria n t comp onents of the Ricci tensor are given r esp ectively by ˜ R 00 = − n κK 2 m 2 , ˜ R 0 i = 0 , ˜ R ij = K 2 m 2 nκ + ( n − 1)( m 2 − 1) ( T k ( K m u + u 0 )) 2 ! g ij , for K 6 = 0 and ˜ R ab = 0 fo r K = 0 . In or der to make computations easier, we r e mark that for A , B tw o-tensors on a Riemannian ma nifold ( ˜ Q, ˜ g ) we have ( AB − B A ) a c = A a b B b c − B a b A b c = A ad B cd − g ad g ec B db A be . (53) Lemma 14 . F or any symmetric tens or T ij the (1,1) c omp onents of C = T ˜ R − ˜ RT , wher e ˜ R is the Ric ci tensor of ˜ g , ar e C 0 0 = 0 , C i 0 = T 0 i W , C 0 i = − ˜ g ij T 0 j W , C i j = 0 , 14 wher e W = ( n − 1) ¨ f f − ˙ f 2 + K f 2 . (54) Remark 8. W e immediately have that if W = 0 then C = 0 and, by Pro p ositio n 13 , { H , T } = 0 implies [ ˆ H , ˆ T ] = 0. How ever, by Theorem 12, W = 0 if a nd only if either n = 1, m = 1 o r f is cons ta n t, i.e., if and o nly if ˜ g is of co nstant curv ature . Theorem 15. F or m = 2 , { H , T } = 0 implies [ ˆ H , ˆ T ] = 0 if and only if ˜ g is of c onstant curvatur e i.e., if and only if n = 1 or f is c onstant. Pr o of. If K = 0, a nd therefor e c = 0 a nd f is constant, then W = 0. Otherw is e, when K 6 = 0 and c 6 = 0, by computing T = U 2 G and by applying P rop osition 3 we g et T 00 = G, T 0 i = γ ∇ i G, T ij = − K 2 γ 2 Gg ij , where γ is given by γ = ( T κ ( K m u + u 0 )) − 1 , as prov ed in Theorem 2. A straightforward computation gives δ C 0 = γ W  g il ∂ 2 il G + ∂ l G ( ∂ i g il + g il ∂ i ln √ g )  = = γ W ∆ G = − γ n K W G, δ C i = f ∂ i G d du ( γ f W ) , where g = det( g ij ). By ins e rting the expre s sions of γ a nd of f 2 = S 2 κ ( K m u + u 0 ) K we hav e that there are no no n- trivial ( G 6 = const. ) solutions to δ C = 0 other than those such that W = 0, that is, a fter Remark 8, when n = 1 or ˜ Q is of constant curv atur e. In a recent pa p er [1], where particula r conformally flat, non-co nstant cur- v ature manifolds are considered, it is shown that even if the La pla ce-Beltrami quantization o f some first integrals of the Hamiltonian fails, their quantization is somehow made po ssible by considering the co nformal Sc hr¨ odinger op erator instead of the standa r d (Lapla ce-Beltrami) one. The co nformal Schr¨ odinger o p- erator is r elated to the s tandard one by an additiona l term pro po rtional to the scalar c ur v ature. In Theorem 12, w e proved that our extended Hamiltonians for n > 2 hav e alwa ys conforma lly flat configuration manifolds, therefore, the metho d ex pos ed in [1] could b e, at least in principle, a pplicable. If we de no te by ˜ ∆ the Laplace-Be ltr ami op erator of ( ˜ Q, ˜ g ) a nd by ∆ the Laplace-Beltr ami op erator of the c o nstant curv ature manifold ( Q, g ), a direct calculation shows that ˜ ∆ = ∂ 2 u + n ˙ f f ∂ u + K f 2 ∆ , (55) and [ ˜ ∆ , ∆] = 0 . Therefore, being ˆ H = − ~ 2 2 ( ∂ 2 u + n ˙ f f ∂ u ) + K f 2 ˆ L, 15 with ˆ L = − ~ 2 2 ∆ + V , we have Prop osition 16. ˆ L is a symmetry op er ator of ˆ H : [ ˆ H , ˆ L ] = 0 . Since ˆ H and ˆ L hav e common eigenfunctions, from ˆ H ψ = µψ and ˆ Lψ = λψ we o btain for the eigenfunction o f ˆ H the following characteriza tion Prop osition 17. The function ψ ( u, q i ) is an eigenfunction of ˆ H if and only if ψ is an eigenfunction of ˆ L and − ~ 2 2 ( ∂ 2 u ψ + n ˙ f f ∂ u ψ ) +  K λ f 2 − µ  ψ = 0 . ( 56) Ac kno wledgemen ts This research was par tially supp orted (C.C.) by the Europ ean pro gram “Dote ricercato ri” (F.S.E. and Regione Lombardia). 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