Quantum B"acklund Transformations: some ideas and examples

Quan tum B¨ ac klund T ransformations: some ideas and examples 1 Orlando Ragnisco † , F ederico Zullo ‡ †‡ Dipartimen to di Fisica, Univ ersit` a di Roma T re † Istituto Nazionale di Fisica Nucleare, sezione di Roma T re Via V asca Nav ale 84, 00146 Ro ma , Italy E-mail: ragnisco@fis.uniroma3.it, zullo@fis.uniroma3.it Abstract In this work w e give a mec hanical (Hamiltonian) in terpretation of the s o called sp e c tr ality pr op erty int r o duced b y Sk lyanin and Kuznetso v in the con text of B¨ ac k- lund transf orm ations (BTs) for finite dimensional in tegrable systems. The pr op- ert y turns out to b e deeply c onn ected with the Hamilto n-Jacobi separation of v ariables and can lead to the explicit int egration of the underlyin g mo del through the expression of the BTs. O nce suc h construction is given, it is sh o wn, in a sim- ple example, that it is p ossible to interpret the Baxter Q op erator defining the quan tum BTs us the Green’s f unction, or propagator, of the time d ep endent Sc hr¨ odinger equation for the interp olating Hamiltonian. KEYW ORDS: Quan tum B¨ ac klund transformations, sp ectrality prop erty In tegrable maps, Quantum pro pagator 1 In tro ductio n Starting from the last decade of the previous millennium, a num b er of results on discretization of finite dimensional in tegrable systems app eared: w e quote for instance the Euler top [1], the Lagrange top [2], the rational Gaudin magnet [5 ] t he Ruijsenaars- Sc hneider mo del [10], the Henon-Heiles, Garnier and Neu mann systems [12], [13], [4] and others (see also [17] and the references therein). It turned out [8] that all the exact discretizations for these systems a sso ciate new solutions to a give n one: they are BTs for suc h systems . These dev elopmen ts suggested to Skly anin and Kuznetso v [6] that the concept of B¨ acklund transformations s hould b e revised in order to enligh ten some new (and old) asp ects of the sub ject. Actually , these t w o authors had previously obtained fundamen tal r esults ab out ”separation of v ariables” and their connection with the new tec hniques of classical and quan tum inv erse scattering metho d. These findings pa ve d the w ay t o a b etter understanding of the ro le of BTs in the context 1 “This paper is a con tribution to Soli tons in 1+1 and 2+1 dimensions. DS, KP and a l l that , conference in honor of the 7 0 th birthday of Marco Bo iti and Flora Pempinelli, Lecce, 2011 September 13-14 .” 1 of finite dimensional in tegrable system s a nd to un veiling the ir deep and fruitf ul links with Hamiltonian dynamics and s eparatio n of v ariables (for a c haracterization of the BTs f o r fin ite dimensional in tegrable systems b y a geometrical po int of view see [8]). Let us briefly remem b er some of these findings. Assume that the dynamical system under consideration is defined b y a Lax pair with a sp ectral parameter, sa y L ( λ ) and M ( λ ). The trace of the p ow ers of L ( λ ) will be conserv ed quantities, so also the determinan t of the Lax matrix will b e conserv ed. The BTs, in the case of finite dimensional systems , are c anonic al tr ansform ations pr eserv i n g the alge br aic form of the inte gr als , so if ( p i , q i ) n i =1 is the set of the dynamical v ariables, one has:  p i , q i  n i =1 B T = ⇒  ˜ p i , ˜ q i  n i =1 L ( λ, p k , q k ) B T = ⇒ ˜ L ( λ ) = L ( λ, ˜ p k , ˜ q k ) (1) { ˜ p i , ˜ p j } = { p i , p j } ; { ˜ q i , ˜ q j } = { q i , q j } ; { ˜ p i , ˜ q j } = { p i , q j } = δ ij . (2) Since the algebraic form of the in tegra ls is preserv ed, also the determinan t of the Lax matrices L ( λ ) a nd ˜ L ( λ ) is the same: H i ( p k , q k ) = ˜ H i ( ˜ p k , ˜ q k ) = ⇒ det( L ( λ )) = det( ˜ L ( λ )) Hence, the existe nce of a BT s for the given system en tails that the t w o Lax matrices L ( λ ) a nd ˜ L ( λ ) a re connected by a similarit y tra nsformation, pro vided by the so called dressing matrix D ( λ ): ˜ L ( λ ) D ( λ ) = D ( λ ) L ( λ ) (3) Ob viously the dressing matrix is not unique, b ecause one can hav e different BTs f or t he same syste m. F urthermore, if the transformatio ns are not explicit, D ( λ ) might dep end on b oth the sets ( p i , q i ) n i =1 and ( ˜ p i , ˜ q i ) n i =1 More in teresting are para metric and explicit BTs, that is BTs dep ending on one or more parameters, arbitrary (or restricted to some range of v alues) suc h that the express ion of the v ariables ( ˜ p i , ˜ q i ) n i =1 is kno wn explicitly in terms of the v ariables ( p i , q i ) n i =1 . A sufficient condition to obtain parametric and explicit BTs is to find a dressing matrix D ( λ ) that i) whose manifo ld coincides with a symplectic leaf of the same Poiss on brac k et as those satisfied by L ( λ ) [16 ], and ii) suc h that its determinant has a non dynamical zero , i.e. det ( D ( λ )) = 0    λ = µ , where µ is a para meter not dep ending on the dynamical v a riables. The condition i) ensures that the transformatio ns a r e canonical ( see [16]), the condition ii) ensures that are explicit and parametric. Supp ose indeed t hat the par ticular symplectic leaf defined b y D ( λ ) is one dimensional, so that D ( λ ) will dep end on another (up to now f r ee) v ariable, sa y a . Then, since det( D ( µ )) = 0, D ( µ ) will hav e a kernel, sa y | Ω i , that in gen eral also will dep end on µ and a . But from 3 w e see also that | Ω i is an eigen v ector of L ( µ ); indeed: ˜ L ( µ )  D ( µ ) | Ω( µ ) i | {z } =0  = D ( µ )  L ( µ ) | Ω( µ ) i  ⇒ L ( µ ) | Ω( µ ) i = γ ( µ ) | Ω( µ ) i (4) The la st equiv alence can be read as an equation for a t ha t, if solv ed, will giv e to a a dep endence o n µ a nd on the dynamical v ariables of the system (only one set of them, in this case the “un tilded o ne”). So, returning to 3 , one obtains para metric ( µ ) and explicit transforma t io ns. 2 Dep ending on the dimension of the leaf defined b y D ( λ ) it is p ossible to obtain multi- parametric BTs . Als o, one can obtain m ulti-parametric BTs through rep eat ed itera- tions of one parameter transformatio ns, since BTs for finite dimensional systems alw ays comm ute [16]. F or simplicit y in the rest of the pap er w e ta k e L ( λ ) to b e a sl (2) matrix, also if our results can b e easily generalized to L ( λ ) ∈ gl (2). F or BTs fo und through a N x N Dar- b oux matr ix see [1 0], [16]. Please note that in the case L ( λ ) ∈ sl ( 2) the function γ ( µ ) in 4 satisfies γ 2 ( µ ) + det( L ( µ )) = 0 and is the generating function of the conserv ed quan tities of the system. An a dditio nal prop erty of BTs w as in tro duced in [6], namely the “sp ectralit y pro p ert y”. In order to understand t his prop ert y remem b er that BTs are canonical transformations, implying that it exists a generating function F 1 ( ˜ q , q , µ ) suc h t ha t: p i = ∂ F 1 ∂ q i ˜ p i = − ∂ F 1 ∂ ˜ q i The spectrality prop ert y [6] sa ys that there exists a function f 1 satisfying the follo wing equation: ∂ F 1 ( ˜ q , q , µ ) ∂ µ = − f 1 ( γ ( µ ) , µ ) (5) where γ ( µ ) is just the function in 4 : it satisfies the relation det( L ( µ ) − v ( µ )1 ) = 0, that can b e seen as the s e p ar ation e quation , in t he sense of Hamilton- Ja cobi separabilit y , for the dynamical sys tem; b y a classical p oin t of view one obtains the quadrature of the equations, by a quantum p oint of view the factorization of t he eigenfunctions [15] [6], [8]. Starting b y t his p oin t w e sho w ho w the sp ectralit y prop ert y can lead to the “lineariza- tion” of the maps defined by the BTs, that, in turns, giv e the general solution of the equations of motion. This construction g ive also us the p o ssibilt y to prop erly interpre t the Baxter Q o p erator (represen ting the quan tum BTs) as the G r een’s function of the Sc hr¨ odinger equation defined b y the Hamiltonian interpola t ing the discrete flo w giv en b y the BTs. 2 Sp ectralit y prop erty and separation of v ariables. First of all w e w an t to mak e some commen ts ab o ut the equation 5 defining the sp ectral- it y prop ert y . Supp ose that it is possible to choose the parameter µ in such a manner that the dressing matrix D ( λ, µ ) is prop ortional to the iden tit y when µ = 0. Expanding equation 3 a r o und µ = 0, if D ( λ, µ ) . = k (1 + µD 0 ( λ ) + O ( µ 2 )), one o btains: ˜ L ( λ ) = L ( λ ) + µ [ D 0 ( λ ) , L ( λ )] + O ( µ 2 ) (6) so that in the limit µ → 0, b y defining ˙ L . = lim µ → 0 ˜ L − L µ , t he BTs define an Hamilto nia n flo w: in this sense µ can b e considered an ev olutio n parameter. Equation 5 tha n resem bles the Hamilton-Jacobi equation with r espect to the time µ , since the function γ ( µ ) contains all the conserv ed quan tities of the system. This p oint will b e made 3 clearer in the next lines. Let us assume to hav e a set of BTs, with the parameter µ playing the role of a time, in the sense giv en b efore: ( ˜ p i = ˜ p i ( p k , q k , µ ) with ˜ p i   µ =0 = p i ˜ q i = ˜ q i ( p k , q k , µ ) with ˜ q i   µ =0 = q i (7) These transformations can b e also rewritten as (assuming obv iously the implicit func- tion theorem can b e applied): ( ˜ p i = ˜ p i ( ˜ q k , q k , µ ) p i = p i ( ˜ q k , q k , µ ) (8) Since 7 and 8 are canonical transformatio ns, there exists the resp ectiv e generating func- tions, say F 0 ( p, q , µ ) a nd F 1 ( ˜ q , q , µ ), solving the cor r esp o nding system of differen tial equations:            p i − n X k =1 ˜ p k ∂ ˜ q k ∂ q i = ∂ F 0 ∂ q i n X k =1 ˜ p k ∂ ˜ q k ∂ p i = − ∂ F 0 ∂ p i        p i = ∂ F 1 ∂ q i ˜ p i = − ∂ F 1 ∂ ˜ q i (9) No w w e assume that the transformations p ossess the sp ectralit y prop erty b oth with resp ect to F 0 ( p, q , µ ) and with resp ect to F 1 ( ˜ q, q , µ ). So there exist t w o functions, sa y f and g , suc h that: ∂ F 0 ∂ µ = − f ( γ ( µ ) , µ ) ∂ F 1 ∂ µ = − g ( γ ( µ ) , µ ) (10) This assumption has non trivial consequences as will b e sho wn in the following tw o the- orems. The result will b e that, under the assumption, t he BTs can b e re-parametrized so to represe nt the solution of t he Hamilton- Jacobi equation for the Hamiltonian in- terp olating the flo w defined by 6. Theorem 1 Supp ose that the sp e ctr ality pr op erty holds true b oth for F 1 and F 0 . Then P n k =1 ˜ p k ∂ ˜ q k ∂ µ is giv e n by ∂ F 1 ∂ µ − ∂ F 0 ∂ µ , that is P n k =1 ˜ p k ∂ ˜ q k ∂ µ is a function of only γ ( µ ) and µ . Pro of. The tw o generating functions F 0 and F 1 are related b y [3]: F 0 ( p, q , µ ) = F 1 ( ˜ q ( p, q , µ ) , q , µ ) (11) So, utilizing equation 11 and ˜ p i = − ∂ F 1 ∂ ˜ q i it is p ossible to write: ∂ F 0 ∂ µ = n X k =1 ∂ F 1 ∂ ˜ q k ∂ ˜ q k ∂ µ + ∂ F 1 ∂ µ    ˜ q = const. ⇒ n X k =1 ˜ p k ∂ ˜ q k ∂ µ = f ( γ , µ ) − g ( γ , µ ) ✷ 4 Theorem 2 Supp ose that the assumption of The or em 1 holds, then givin g to µ a dep en d enc e o n the c onstants of mo tion thr ough any o f the r o ot(s) of the e quation µ = h ( g ( µ, γ ( µ )) ) , wher e h i s an arbitr ary function, one obtains again a set o f BTs. The gener ating function of the new tr a nsformations is given by: F ( p, q , µ ( k ) ) = F 0 ( p, q , µ ( k ) ) + Z µ ( k ) g ( γ ( µ ) , µ ) d µ (12) wher e µ ( k ) = h ( g ( µ ( k ) , γ ( µ ( k ) ))) is any of the r o o t of µ = h ( g ( µ, γ ( µ )) ) Pro of. Supp ose to giv e to µ a dep endence to the dynamical v a riables ( p i , q i ) n i =1 . W e ask if it is p ossible to c ho ose some function µ ( p, q ) in such a w a y that the transformations ˜ p i = ˜ p i ( p, q , µ ( p, q )) and ˜ q i = ˜ q i ( p, q , µ ( p, q )) obtained inse rting the function µ ( p, q ) in the expressions 8 are a g ain canonical. If the new transformations are canonical they p ossess a generating f unction, sa y F , suc h that:            p i − n X k =1 ˜ p k ∂ ˜ q k ∂ q i = ∂ F ∂ q i n X k =1 ˜ p k ∂ ˜ q k ∂ p i = − ∂ F ∂ p i (13) Inserting the ansatz F = F 0 + A ( p, q ) into the previous system and taking in to account that b y Theorem 1 g = − ∂ F 0 ∂ µ − P n k =1 ˜ p k ∂ ˜ q k ∂ µ , one readily finds the eq uatio ns that has to b e satisfied by A ( p, q ):        ∂ A ∂ q i = g ∂ µ ∂ q i i = 1 ..n ∂ A ∂ p i = g ∂ µ ∂ p i i = 1 ..n (14) Indeed from F = F 0 + A and from 13 w e ha v e: ∂ F ∂ p i = ∂ F 0 ∂ p i    µ = const. + ∂ F 0 ∂ µ ∂ µ ∂ p i + ∂ A ∂ p i = − n X k =1 ˜ p k ∂ ˜ q k ∂ p i    µ = const. − n X k =1 ˜ p k ∂ ˜ q k ∂ µ ∂ µ ∂ p i (15) When µ is constan t the BTs are canonical transformations with generating function F 0 , so it holds the equation: ∂ F 0 ∂ p i    µ = const. = − n X k =1 ˜ p k ∂ ˜ q k ∂ p i    µ = const. Inserting this equiv alence in 15, we are left with: ∂ F 0 ∂ µ + n X k =1 ˜ p k ∂ ˜ q k ∂ µ ! ∂ µ ∂ p i + ∂ A ∂ p i = 0 5 But from Theorem 1 we ha v e that ∂ F 0 ∂ µ + P n k =1 ˜ p k ∂ ˜ q k ∂ µ = − g , so finally: ∂ A ∂ p i = g ∂ µ ∂ p i i = 1 ..n In the same manner, by the equation for ∂ F ∂ q i it is p ossible to show that: ∂ A ∂ q i = g ∂ µ ∂ q i i = 1 ..n The compatibility equations fo r the f unction A ( p, q ), that is ∂ 2 A ∂ p k ∂ q i = ∂ 2 A ∂ q i ∂ p k giv e (b y 14): ∂ g ∂ p k ∂ µ ∂ q i = ∂ g ∂ q i ∂ µ ∂ p k i, k = 1 ..n (16) These equations means that µ has to b e an arbitrary function of g , that is: µ = h ( g ( µ, γ ( µ ))) Note that this (set of ) implicit equation for µ fixes the dep endence of µ b y t he dynamical v ariables ( p i , q i ) n i =1 through its r o ots. At an y of the ro ot of this (set of ) equation will corresp ond a function g and a function µ , b o th dep ending only on the dynamical v ariables ( p i , q i ) n i =1 , satisfying the set of differen tial equations 16. Indicating the k th ro ot of µ = h ( g ( µ, γ ( µ ))) as µ ( k ) and returning to the system 14, one finds: A ( p, q ) = Z µ ( k ) g ( γ ( µ ) , µ ) d µ (17) ✷ Let us no w make some remarks that will b e useful in what fo llo ws. Remark 1 The gener ating function F ( p, q , µ ) as a function of µ sa tisfi e s the e quation: ∂ F ( p, q , µ ) ∂ µ = − n X k =1 ˜ p ∂ ˜ q ∂ µ This is jus t a trivial calculation on ∂ F ( p,q ,µ ) ∂ µ for the function F ( p, q , µ ) as giv en in 12 and taking into accoun t the Theorem 1. Remark 2 By adding a dep end enc e on an extr a c onstant p ar ame ter, say T , to the function µ , that is by c onsidering µ = µ ( k ) ( T ) , the c anonicity of the tr ansformations is pr eserve d, so again o ne has a set of BTs. This remark gives the p ossibility to obtain a new parametric fa mily of BTs by letting to µ ( k ) to dep end also b y the parameter T . Indeed it is p ossible to repeat the same line of reasoning of the T heorem 2 b y conside ring, fro m the beginning, µ as a function of the dynamical v ariables ( p i , q i ) n i =1 and the new constan t parameter T . Ob viously now also the generating function F will got a dep endence o n the parameter T . 6 The ke y p oint of all this construction is that, assuming the sp ectrality prop erty holds b oth for F 0 and F 1 , it is p ossible, at least in principle, t o obtain a new, larger, family of parametric BTs b y giving t o µ a dep endence on the constan ts of motion and on the parameter T : as we will sho w now, to this freedom to obtain new BTs it corresp onds the p ossibilit y to get, with a suitable c hoice of the function µ ( p, q , T ), the canonical transformation from the v ariables ( p i , q i ) n i =1 to the new v ariables ( ˜ p i , ˜ q i ) n i =1 suc h that the Hamilton-Jacobi equation, where the parameter T plays the ro le of “time” and the in terp olating Hamiltonian the role o f its conjuga t ed v ar ia ble, is identically solv ed. Summarizing, giv en a pa rametric BTs ( ˜ p i ( p k , q k , µ ) , ˜ p i ( p k , q k , µ )) ha ving the sp ectral- it y prop erty b oth for F 0 and F 1 , the goal w ould b e to find a function µ fitting the Hamiltonian-Jacobi equation for t he flux describ ed by the BTs: H + ∂ F ∂ µ ∂ µ ∂ T = 0 where with the italic H w e mean the in terp o la ting Ha milto nian of the BTs. By Remark 1 the previous equation can b e b etter rewritten as: H − ∂ µ ∂ T n X k =1 ˜ p ∂ ˜ q ∂ µ = 0 (18) Indeed now b y Theorem 1 the s um P n k =1 ˜ p ∂ ˜ q ∂ µ is a function of µ a nd of the dynamical v ariables only through the constan ts of motion; ob viously also the in terp olating Hamil- tonian H w ill dep end only on s ome com binations of the c onstant of motions. So the equation 18 is indeed s olv able in terms of µ . Inserting the solution µ ( p, q , T ) in to t he expressions of BTs w e obta in, b y construction, the gener al solution of the equations of motion with resp ect the in terp olating Hamiltonian H . Let us make another remark. It seems that there is an am biguity in the c hoice of the function µ solving equation 18 b ecause, of course, the solution dep ends also on an a r - bitrary function of the constan ts of motion. But now we will sho w that the freedom to c ho ose the v alue of this ar bitr ary function corresp onds to the freedom in the choice of the initial conditions in the g eneral solution of our equations of motion: so it represen ts a shift in the time T . Ind eed, t aking in to account Theorem 1 and the equations 10, the solution of the Ha milto n-Jacobi equation 18 can b e implicitly written a s: T + 1 H Z µ 0 ( g ( γ ( η ) , η ) − f ( γ ( η ) , η )) dη + F ( H i ) = 0 (19) where, by 10 w e recall tha t the functions f and g ar e respectiv ely give n b y − ∂ F 0 ∂ µ and − ∂ F 1 ∂ µ and F is an arbitra ry function of the Hamiltonians of the system H i . It is no w eviden t fr om equation 1 9 tha t i) t he a r bitrary function F appear s as an additive quan tit y with r esp ect to the time par a meter T and ii) if o ne w an ts to retain the initia l conditions as giv en in 7, that is ˜ p i   T =0 = p i and ˜ q i   T =0 = q i , then one ha s to choose F = 0. In the next section we will giv e a simple application o f the theorems just seen with the BTs for the one dimensional harmonic o scillator. 7 3 A tutorial example: t he harmonic os cillator A Lax represen tation with sp ectral parameter for the one dimensional harmonic oscil- lator is giv en b y: L ( λ ) =  1 p − i q λ p +i q λ − 1  , M = i 2  1 0 0 − 1  , ˙ L ( λ ) = [ L, M ] In this simple case the sp ectral curv e is simply given b y: γ 2 ( λ ) = − det( L ( λ )) = 1 + p 2 + q 2 λ 2 In the follo wing for brevit y w e p ose a . = p − i q and a ∗ . = p + i q . A class of BTs can b e obtained b y a dr essing matrix D ( λ ) parametrized as follows: D ( λ ) =  1 α λ β λ 1  As p ointed out in t he in tro duction, to find e xplicit transformations w e imp ose that det( D ( λ = µ ) ) = 0. This constrain means that αβ = µ 2 , so w e can rewrite, p osing β = µζ : D ( λ ) =  1 µ λζ µζ λ 1  Ob viously when λ = µ the dressing matrix p o ssess es a k ernel | Ω( µ ) i . Explic itly it is giv en by: | Ω( µ ) i =  1 − ζ  Then the eigenv ector relatio n: L ( µ ) | Ω( µ ) i = γ ( µ ) | Ω( µ ) i giv es ζ as a function of a and a ∗ : ζ = µ (1 − γ ( µ )) a = − a ∗ µ (1 + γ ( µ ) ) No w we a re able to write out the explicit form of BTs:        ˜ a = a γ ( µ ) + 1 γ ( µ ) − 1 ˜ a ∗ = a ∗ γ ( µ ) − 1 γ ( µ ) + 1 with γ ( µ ) 2 = 1 + | a | 2 µ 2 Note that when µ is equal to zero the tra nsfor ma t io ns reduce to the identit y m ap. What we wan t to c hec k no w is whether these canonical transformatio ns p ossess the 8 sp ectralit y property b oth for F 0 and for F 1 . The generating function in the v ar iables a, a ∗ is giv en b y F 0 ( a, a ∗ , µ ) = − 2 µ 2 ( γ ( µ ) + 1), while the inv erse transformatio ns:        ˜ a = 4 µ 2 a ∗ (˜ a ∗ − a ∗ ) 2 a = 4 µ 2 ˜ a ∗ (˜ a ∗ − a ∗ ) 2 are generated b y the func tion F 1 ( a ∗ , ˜ a ∗ , µ ) = 4 µ 2 a ∗ ˜ a ∗ − a ∗ . By differentiating F 0 and F 1 one finds:            ∂ F 0 ∂ µ = − 2 µ p γ ( µ ) + 1 p γ ( µ ) ! 2 ∂ F 1 ∂ µ    ˜ a ∗ = a ∗ γ ( µ ) − 1 γ ( µ )+1 = − 4 µ ( γ ( µ ) + 1) so tha t b oth ∂ F 0 ∂ µ and ∂ F 0 ∂ µ are only functions of the integral ( p 2 + q 2 ) and of the parameter µ . The results of the previous section can b e applied. In the o ne dimensional case the Theorem 2 simply implies that µ can get an arbitrar y dep endence on the Hamiltonian and the BTs are again canonical transformations. So b y now w e consider µ as a function of | a | a nd of a new pa rameter T , t ha t is µ = µ ( | a | , T ). Again fro m Theorem 2 the new transformations a re again canonical with the generating function: F = F 0    µ = µ ( | a | ) + 4 Z | a | µ ( γ ( µ ) + 1 ) ∂ µ ( x, T ) ∂ x dx In order to fix the dep endence of µ by | a | and T w e need to find the interpola ting Hamil- tonian. As exp lained at the b eginning of the s ection 2 the in terp olating Hamiltonian of the flow with respect to µ can b e extrapolat ed considering the re latio n ˜ LD = D L in the limit µ → 0: D = 1 − µD 0 + O ( µ 2 ) ⇒ ˙ L ( λ ) . = lim µ → 0 ˜ L − L µ = [ L ( λ ) , D 0 ( λ )] (20) In our case the matrix D 0 ( λ ) is given b y: D 0 ( λ ) = 0 1 λ p a a ∗ 1 λ q a ∗ a 0 ! The con tinuous flow giv en by 2 0 is gov erned b y the Hamiltonian H = 2i | a | . Note that the factor “i” app ear b ecause a and a ∗ are not canonically conjuga t e. No w we hav e all the ingredien ts of the recip e. So w e can lo ok at the Hamilton-Jacobi equation: H − ˜ a ∂ ˜ a ∗ ∂ µ ∂ µ ∂ T = 0 → 2i | a | + 2 | a | 2 p µ 2 + | a | 2 ∂ µ ∂ T = 0 → → µ ( | a | , T ) = − i | a | sin( T | a | + F ( | a | )) (21) 9 As noted at the end of s ection 2 , one can c ho ose the arbitrary function F ( | a | ) to b e zero by requiring that µ    T =0 = 0. Inserting the formula µ ( | a | , T ) = − i | a | sin( T | a | ) into the expressions of BTs 3 and r e- turning to the ph ysical v aria bles p = a ∗ + a 2 and q = a ∗ − a 2i one finds:          ˜ p = p cos( 2 T p p 2 + q 2 ) − q sin ( 2 T p p 2 + q 2 ) ˜ q = q cos( 2 T p p 2 + q 2 ) + p sin( 2 T p p 2 + q 2 ) that is the general solutio n o f the equations of mo t ion gov erned by the Hamiltonian H = 2 p p 2 + q 2 . Note that w e now omit the factor “i” f rom the in terp olating Hamiltonia n b ecause w e are considering directly the flo w with resp ect to T and not with resp ect to µ (when T go es to zero µ ∼ − i T ). Note that the parameter T is a “linearizing para meter” for the flow . F or completeness w e ha v e to cite [14], where ano t her w a y to find the functional relatio n b et w een µ and T (if any!) has b een give n. Ho w ev er in [14] there w ere no definite statemen t ab out the necessary and sufficien t condition for suc h relation to exist, neither its connections with analytic mec hanics w as p ointed out. Nev ertheless that p o in t of view can b e v ery useful b ecause to the explicit kno wledge of the BTs do es not alw a ys corresp ond the explicit kno wledge of the generating functions F 0 and F 1 of the transformations. It can b e also sho wn that applying the machine ry describ ed in [14], one obtains exactly the result µ ( | a | , T ) = − i | a | sin( T | a | ), as in 21. The reader is referred to this work for ot her examples o f linearization of BTs. 4 Quan tum B¨ ack lu nd transformations : some h in ts and insights. By a classical p oin t of view the BTs a sso ciate new solutions of t he equations o f motion to a given one. So, in general, if g ( p ( t ) , q ( t )) is a ph ysical observ able connected to the curv e in the phase space ( p ( t ) , q ( t )), g ( ˜ p ( t ) , ˜ q ( t )) will be the same ph ysical observ able connected to t he curv e( ˜ p ( t ) , ˜ q ( t )). By a quan tum p o in t of view the BTs are represen ted b y a unita r y o p erator Q (1) µ ( µ b eing the (set of ) parameter(s) of the BTs) realized as an inte gra l op erato r on the space of eigenfunctions [11], [6]: Q (1) µ : ψ ( q ) → Z f (1) ( ˜ q , q ) ψ ( q ) dq (22) The similarit y tra nsfor ma t io ns induced b y Q (1) µ are the equiv alen t of the classical canon- ical transformations and the k ernel f (1) ( ˜ q , q ) is given , in the semiclassical appro xima- tion, b y: f (1) ( ˜ q, q ) ∼ exp( − i ~ F 1 ( ˜ q, q )) , ~ → 0 (23) 10 As far as w e kno w, the explicit construction of the Q op erato r , known in lit era t ur e as B axter op er ator , and its relations with BTs, ha ve b een p oin ted out only for the T o da lattice [11], [1 6] and for the discrete self-trapping (DST) mo del [7]. By o ur p oint of v iew, when the BTs can lead to the in tegration of the equations through the con- struction given in the previous sections, the k ernel f (1) ( ˜ q , q ) has to be iden tified with the propa g ator or the Green’s function for the time de p enden t Sc hr¨ oding er equation, the Hamiltonian b eing t ha t interpolating t he flow represen ted by the BTs. Indeed, in this case, to the classical time shift o n the tra j ectory sp ecified b y t he initial conditions represen ted by the BTs it corresp onds, as it is clear also from eq. 22 , a quantum pro b- abilit y a mplitude fo r the particle with p osition eigen v alue q at the time 0 to b e found at a later time µ in ˜ q . In the follo wing we mak e a c hec k on the Baxter operato r for the harmonic oscillator b y sho wing that indeed its ke rnel giv es exactly the w ell-known propagator of the cor- resp onding Schr¨ odinger equation. In order to compare the results we make a sligh t mo dification in the c hoice of the function µ in 21, by p osing T = φ | a | 2 . No w φ pla y the role of the time. With this c hoice indeed the in terp ola t ing Hamiltonian is just the ph ysical o ne, and ob viously the BTs giv e the form ulae: ( ˜ p = p cos( φ ) + q sin( φ ) ˜ q = q cos( φ ) − p sin( φ ) In order to find the generating function F 1 ( ˜ q , q ) we express ˜ p and p in terms of q and ˜ q :        ˜ p = q − ˜ q cos( φ ) sin( φ ) p = q cos( φ ) − ˜ q sin( φ ) F rom dF 1 = pdq − ˜ pd ˜ q one readily finds: F 1 ( ˜ q, q ) = ( ˜ q 2 + q 2 ) cos( φ ) − 2 q ˜ q 2 sin( φ ) + w ( φ ) where w ( φ ) is an arbitrary function of φ . No w w e hav e to find the Q op erator, o r , tha t is the same, it s ke rnel f 1 ( ˜ q, q ). I n the h bra | k et i notation the k ernel f 1 ( ˜ q , q ) is g iv en, in the q repre sen tatio n, b y h ˜ q | q i . So, b y the following form of the BTs: ( q = ˜ q cos( φ ) + ˜ p sin( φ ) ˜ q = q cos ( φ ) − p sin( φ ) it is easily found t ha t f 1 ( ˜ q , q ) solv es the system:        q f (1) = ˜ q f (1) cos( φ ) − i ~ ∂ f (1) ∂ ˜ q sin( φ ) ˜ q f ∗ (1) = q f ∗ (1) cos( φ ) + i ~ ∂ f ∗ (1) ∂ ˜ q sin( φ ) 11 where  ∗ means complex conjug a tion. The solution of the previous system is: f (1) ( ˜ q , q ) = C ( φ ) exp  − i ~  ( ˜ q 2 + q 2 ) cos( φ ) − 2 q ˜ q 2 sin( φ )  in exact a greemen t with 23. F or completeness w e recall the classical results on the propagator of the Schr¨ odinger equation for the ha rmonic oscillator. In general the pro pagator can b e fo r ma lly written as: K ( ˜ q , q ) = h ˜ q | exp( − i H t ~ ) | q i = X n h ˜ q | n ih n | q i exp( − i E n t ~ ) In the case of the harmonic oscillator h q | n i are the Hermite p olynomials, so that: K ( ˜ q , q ) = X n e −  ˜ q 2 + q 2 2 ~  2 n n ! √ π ~ H n ( ˜ q √ ~ ) H n ( q √ ~ ) e − i ( n + 1 2 ) φ It is p ossible to sum up the series by using the so called ” Meheler’s fo r mula” (see e.g. [9]): X n w n 2 n n ! H n ( x ) H n ( y ) = 1 √ 1 − w 2 exp  2 xy w − ( x 2 + y 2 ) w 2 1 − w 2  (24) obtaining: K ( ˜ q , q ) = 1 p 2 π i ~ sin( φ ) exp  − i ~  ( ˜ q 2 + q 2 ) cos( φ ) − 2 q ˜ q 2 sin( φ )  This form ula agrees with that o f f (1) ( ˜ q , q ) iden tifying C ( φ ) with 1 √ 2 π i ~ sin( φ ) . On the same line of reasoning we can also ask for the quan tum pro ba bilit y amplitude for the particle with mo mentum eigen v alue p at the t ime 0 to p ossess at a later t ime µ the momen tum ˜ p . Ag a in there is the corresp ondence b etw een the classical generating function, no w give n b y d F 2 = ˜ q d ˜ p − q d p , a nd the k ernel o f the corresp onding Baxter op erator Q (2) µ : f (2) ( ˜ p, p ) ∼ exp( − i ~ F 2 ( ˜ p, p )) , ~ → 0 (25) F rom the equations:        ˜ q = ˜ p cos( φ ) − p sin( φ ) q = ˜ p − p cos ( φ ) sin( φ ) one readily finds: F 2 ( ˜ p, p ) = ( ˜ p 2 + p 2 ) cos( φ ) − 2 p ˜ p 2 sin( φ ) + η ( φ ) where again η is an arbitrary function. Whereas from the fo llo wing form of the BTs: ( ˜ p = p cos( φ ) + q sin( φ ) p = ˜ p cos( φ ) − ˜ q sin( φ ) 12 one finds, recalling that in the p represen ta t ion q = i ~ ∂ ∂ p , the equations to b e satisfied b y f (2) as:        pf (2) = ˜ p f (2) cos( φ ) − i ~ ∂ f (2) ∂ ˜ p sin( φ ) ˜ pf ∗ (2) = pf ∗ (2) cos( φ ) + i ~ ∂ f ∗ (2) ∂ ˜ p sin( φ ) so that: f (2) ( ˜ p, p ) = V ( φ ) exp  − i ~  ( ˜ p 2 + p 2 ) cos( φ ) − 2 p ˜ p 2 sin( φ )  again in exact agreemen t with 2 5 . 5 Conclus ions In this w o rk w e tried to shed some light on the sp ectralit y prop erty of BTs, by sho wing ho w it can lead, thro ugh the constructions of t he section 3, to the explicit in tegration of the underlying equations o f motion. It could b e in teresting to apply these results to many -b o dy systems p ossessing know n BTs with spectrality prop ert y , suc h a s, for example, the T o da latt ice. Also the interpretation of the Baxter op erat o r for suc h systems as the propagator for the corresp onding Sc hr¨ odinger o p erator could lead to in teresting summation formulae for orthogonal p olynomials, similar to the Meheler’s form ula 24. As a final remark w e p oin t out that the non- ob vious relation among B¨ ac k- lund transformations and t he Green’s function of the Sc hr¨ odinger o p erator pro vides a bridge b et w een the theory of suc h transformations and the F eynman path in tegral approac h, op ening unexplored p ersp ective s in statistical mec hanics and quantum field theory . References [1] Bob enk o A.I., Lorb eer B., Suris Y u.B. In tegrable discretizations o f the Euler top, J. Math. Ph ys., 3 9 6668- 6683, 1998 [2] Bob enk o A.I., Suris Y u.B. Discrete Time Lagrangian Mec hanics on Lie Groups, with an Application to the Lagrange T o p, Commun. Math. Ph ys., 2 04, 147-188, 1999 [3] A. F asano, S. 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