hbar-Dependent KP hierarchy

~ -dep enden t KP hierarc h y Kanehisa T ak asaki Graduat e Sc ho ol of Human a nd Environmental Studies, Ky oto Uni v ersity , Y oshida, Sa k y o, Kyoto, 606-8 501, Ja pan e-mail : tak asaki @math. h.kyoto-u.ac.jp T ak ashi T akeb e F acult y o f Mathem atics, Natio nal Researc h Uni v ersity – Higher Scho ol of Econom ics, V a vil ov a Street , 7, Moscow, 117 312, Russia e-mail : ttakebe@ hse. ru Abstract This is a summary of a recursiv e construction of solutions of the ~ -dep endent KP hierarc h y . W e giv e r ecursion relations for the coeffi- cien ts X n of an ~ -expansion of the op erator X = X 0 + ~ X 1 + ~ 2 X 2 + · · · for whic h the dressing op erator W is expressed in the exp onen tial form W = exp( X/ ~ ). The wa ve function Ψ asso ciated w ith W turns out to ha v e the WKB form Ψ = exp( S/ ~ ), and the co efficien ts S n of the ~ -expansion S = S 0 + ~ S 1 + ~ 2 S 2 + · · · , to o, are d etermin ed b y a set o f rec urs ion relations. This WKB form is u sed to s h o w that the asso ciated tau fu nction has an ~ -expansion of the form log τ = ~ − 2 F 0 + ~ − 1 F 1 + F 2 + · · · . 1 0 In tro d uction The ~ -dep enden t formulation o f the KP hierarc hy w as in tro duced to study the disp ersionless KP hierarc hy [K G], [K r], [TT1] as a classical limit (i.e., the low est order of the ~ -expansion) of t he KP hierarc h y . This p oin t of view turned out to b e v ery useful for understanding v arious features of the disp ersionless KP hierarc hy . In t his pap er, w e return to the ~ -dep enden t KP hierarch y itself, and consider all orders of the ~ -expansion. W e first address the issue of solving a Riemann-Hilb ert prob- lem for the pair ( L, M ) o f Lax and Orlov-Sc hulman o p erators [O S]. This is a kind of “quan tisation” of a Riemann-Hilb ert problem that solv es the disp ersionless KP hierarc h y [TT1]. In this paper, we set- tle this issue b y an ~ -expansion of the dressing o p erator W , whic h is assumed to hav e t he exp onen tial form W = exp( X/ ~ ) with an op erator X of negative order. Roughly sp eaking, the co efficien ts X n , n = 0 , 1 , 2 , . . . , of the ~ -expansion of X are sho wn to b e deter- mined recursiv ely from the lo w est order term X 0 (in other w ords, from a solution of the dispersionless KP hierarch y). W e next con v ert this result to the languag e of the w av e func- tion Ψ . Namely , giv en the dressing op erator in the expo nen tial form W = exp( X/ ~ ), we show that the asso ciated w av e function has the WKB form Ψ = exp( S/ ~ ) with a phase function S expanded into nonnegativ e p ow ers of ~ . Borro wing an idea f rom Aoki’s “exp o- nen tial calculus” o f micro differential o p erators [A], w e show that dressing op erators of the form W = exp ( X/ ~ ) a nd wa ve functions of the form Ψ = exp( S/ ~ ) are determined from eac h other b y a set of recursion relat io ns f o r the co efficien t s of their ~ - expansion. Con- 2 sequen tly , the w av e function of the solution of the aforemen tioned Riemann-Hilb ert problem, t o o, are r ecursiv ely determined b y the ~ -expansion. Ha ving the ~ -expansion of the w a v e function, w e can readily deriv e an ~ -expansion of the tau function as stated in our previous w ork [TT2]. Details ar e found in [TT3 ] and shall b e published elsew here. 1 ~ -dep end en t KP hierarc h y: review In this section w e recall sev eral facts on the KP hierarc h y dep ending on a formal parameter ~ in [TT2], § 1.7. The ~ - dep enden t KP hierarc hy is defined b y the Lax represen- tation (1.1) ~ ∂ L ∂ t n = [ B n , L ] , B n = ( L n ) ≥ 0 , n = 1 , 2 , . . . , where the L ax op er ator L is a micro differen tial o p erator of the form (1.2) L = ~ ∂ + ∞ X n =1 u n +1 ( ~ , x, t )( ~ ∂ ) − n , ∂ = ∂ ∂ x , and “( ) ≥ 0 ” stands for the pro jection on to a differen t ia l opera t or dropping negativ e p ow ers of ∂ . The co efficien ts u n ( ~ , x, t ) of L a re formally regular with resp ect t o ~ . W e in tro duce the notion of the ~ -or der defined b y ord ~  X a n,m ( x, t ) ~ n ∂ m  def = max { m − n | a n,m ( x, t ) 6 = 0 } . In particular, ord ~ ~ = − 1, o rd ~ ∂ = 1, ord ~ ~ ∂ = 0. The regularity condition whic h w e imp osed on the co efficien ts u n ( ~ , x, t ) can b e 3 restated as ord ~ ( L ) = 0. The princ ip al symb ol o f a micro differen- tial op erator A = P a n,m ( x, t ) ~ n ∂ m with resp ect t o the ~ -order is σ ~ ( A ) def = P m − n =ord( A ) a n,m ( x, t ) ξ m . As in the usual KP theory , the Lax op erator L is expressed b y a dr e ssing op er ator W : (1.3) L = Ad W ( ~ ∂ ) = W ( ~ ∂ ) W − 1 The dressing o p erator W should hav e a sp ecific form: (1.4) W = exp( ~ − 1 X ( ~ , x, t, ~ ∂ ))( ~ ∂ ) α ( ~ ) / ~ , where X ( ~ , x, t, ~ ∂ ) = P ∞ k =1 χ k ( ~ , x, t )( ~ ∂ ) − k is a 0-th order op er- ator, ord ~ ( X ) = 0, and and α ( ~ ) is a constant with resp ect to x and t with ~ -order 0, ord ~ α ( ~ ) = 0. The wav e function Ψ( ~ , x, t ; z ) is defined b y (1.5) Ψ( ~ , x, t ; z ) = W e ( xz + ζ ( t,z )) / ~ , where ζ ( t, z ) = P ∞ n =1 t n z n . It is a solution of linear equations L Ψ = z Ψ, ~ ∂ Ψ ∂ t n = B n Ψ ( n = 1 , 2 , . . . ) and has the WKB form as w e shall show in Section 3. Moreo v er it is express ed b y means of the tau function τ ( ~ , t ) as follows: (1.6) Ψ( ~ , x, t ; z ) = τ ( t + x − ~ [ z − 1 ]) τ ( t ) e ~ − 1 ζ ( t,z ) , where t + x = ( t 1 + x, t 2 , t 3 , . . . ) and [ z − 1 ] = (1 / z , 1 / 2 z 2 , 1 / 3 z 3 , . . . ). W e shall study the ~ -expansion of the tau function in Section 4. The Orlo v - Schulman op er ator M is defined b y (1.7) M = Ad  W exp  ~ − 1 ζ ( t, ~ ∂ )  x = W ∞ X n =1 nt n ( ~ ∂ ) n − 1 + x ! W − 1 4 where ζ ( t, ~ ∂ ) = P ∞ n =1 t n ( ~ ∂ ) n . It is easy to see t ha t M has a form (1.8) M = ∞ X n =1 nt n L n − 1 + x + α ( ~ ) L − 1 + ∞ X n =1 v n ( ~ , x, t ) L − n − 1 , and satisfies ord ~ ( M ) = 0, t he canonical commutation relation [ L, M ] = ~ a nd the same Lax equations as L : ~ ∂ M ∂ t n = [ B n , M ], n = 1 , 2 , . . . . The f o llo wing prop osition (Prop osition 1.7.11 of [TT2]) is a Riemann-Hilb ert type construction of solutions of the ~ -KP hi- erarc h y . Prop osition 1.1. (i) L et f ( ~ , x, ~ ∂ ) and g ( ~ , x, ~ ∂ ) b e 0 -th or- der op er ators ( ord ~ f = ord ~ g = 0 ) and c anonic al ly c omm uting, [ f , g ] = ~ , op er ators L and M have the form (1.2) and (1.8) r esp e c- tively and c ommute c ano nic al ly, [ L, M ] = ~ . Supp os e f ( ~ , M , L ) and g ( ~ , M , L ) ar e di ff er ential op er ators: ( f ( ~ , M , L )) < 0 = ( g ( ~ , M , L )) < 0 = 0 , wher e ( ) < 0 is the pr oje ction to the ne gative or der p art: P < 0 := P − P ≥ 0 . T hen L is a solution of the KP hier ar chy (1.1) and M is the c orr esp ondin g Orlov- Schulman op er ator. (ii) Conve rs e ly, for any so l ution ( L, M ) ther e exists a p air ( f , g ) satisfying the c onditions in (i). The leading term of this system with respect to the ~ -order giv es the disp ersion l e ss KP hie r ar chy . Namely , (1.9) L := σ ~ ( L ) = ξ + ∞ X n =1 u 0 ,n +1 ξ − n , ( u 0 ,n +1 := σ ~ ( u n +1 )) satisfies the dispersionless Lax ty p e equations (1.10) ∂ L ∂ t n = {B n , L} , B n = ( L n ) ≥ 0 , n = 1 , 2 , . . . , 5 where ( ) ≥ 0 is the truncation of La uren t series to its p olynomial part and { , } is the P oisson brack et defined by { a ( x, ξ ) , b ( x, ξ ) } = ∂ a ∂ ξ ∂ b ∂ x − ∂ a ∂ x ∂ b ∂ ξ . The dressing op eration (1.3) f or L b ecomes the follow ing dress- ing op eration for L : L = exp  ad { , } X 0  ξ , where X 0 = σ ~ ( X ) and ad { , } ( f )( g ) := { f , g } . The principal sym b o l of the Orlov-Sc hulman op erator is (1.11) M = ∞ X n =1 nt n L n − 1 + x + α 0 L − 1 + ∞ X n =1 v 0 ,n L − n − 1 ( v 0 ,n = σ ~ ( v n ), α 0 = σ ~ ( α )), whic h is equal to M = exp  ad { , } X 0  exp  ad { , } α 0 log ξ  exp  ad { , } ζ ( t, ξ )  x , where ζ ( t, ξ ) = P ∞ n =1 t n ξ n . The series M satisfies the canonical commutation relatio n {L , M} = 1, and the Lax type equations: ∂ M ∂ t n = {B n , M} , n = 1 , 2 , . . . . The Riemann- Hilb ert type construction o f the solution is essen- tially the same as Prop osition 1.1. Prop osition 1.2. (i) L et f 0 ( x, ξ ) and g 0 ( x, ξ ) b e functions c a non- ic al l y c omm uting, { f 0 , g 0 } = 1 , functions L and M have the form (1.9) and (1.11) r esp e ctively. Supp ose f 0 ( M , L ) and g 0 ( M , L ) do not c ontain ne gative p owe rs of ξ , ( f 0 ( M , L )) < 0 = ( g 0 ( M , L )) < 0 = 0 , wher e ( ) < 0 is the pr oje ction to the ne gative de g r e e p art: P < 0 := P − P ≥ 0 . Th e n L is a so l ution of the disp ersion l e s s KP hier ar chy (1.10) and M is the c orr esp onding Orlov-Schulman function. (ii) Conversely, for any solution ( L , M ) ther e e xists a p air ( f 0 , g 0 ) satisfying the c onditions in (i). If f , g , L and M are as in Prop osition 1.1, then f 0 = σ ~ ( f ), g 0 = σ ~ ( g ) , L = σ ~ ( L ) a nd M = σ ~ ( M ) satisfy the conditions in 6 Prop osition 1.2. In other w ords, ( f , g ) and ( L, M ) are quan tisation of the canonical transformations ( f 0 , g 0 ) and ( L , M ) resp ectiv ely . (See, for example, [S] for quan tised canonical transformations.) 2 Recursiv e constru ction of the dre ss- ing o p er ator In this section w e prov e t ha t the solution of the KP hierarch y cor- resp onding to the quan tised canonical transformation ( f , g ) is re- cursiv ely constructed from its leading term, i.e., the solution of the disp ersionless KP hierarc h y corresp onding to the R iemann- Hilb ert data ( σ ~ ( f ) , σ ~ ( g ) ) . Giv en the pair ( f , g ), we hav e to construct the dressing op erator W , or X and α in (1.4), suc h that opera t ors (2.1) f ( ~ , M , L ) = Ad  W exp  ~ − 1 ζ ( t, ~ ∂ )  f ( ~ , x, ~ ∂ ) g ( ~ , M , L ) = Ad  W exp  ~ − 1 ζ ( t, ~ ∂ )  g ( ~ , x, ~ ∂ ) are b oth differen tial op erators (cf. Prop o sition 1.1). Let us expand X and α with resp ect to the ~ - o rder as follows: (2.2) X ( ~ , x, t, ~ ∂ ) = ∞ X n =0 ~ n X n ( x, t, ~ ∂ ) , X n ( x, t, ~ ∂ ) = ∞ X k =1 χ n,k ( x, t )( ~ ∂ ) − k , α ( ~ ) = P ∞ n =0 ~ n α n , where χ n,k and α n do not depend on ~ . Assume that the solution ( L , M ) of t he disp ersionless KP hier- arc h y corresp onding t o ( σ ~ ( f ) , σ ~ ( g ) ) is given. Namely , ( σ ~ ( f )( M , L ) , σ ~ ( g ) ( M , L )) do not con tain negative p ow ers of ξ : (2.3)  σ ~ ( f )( M , L )  < 0 =  σ ~ ( g ) ( M , L )  < 0 = 0 . Let ( X 0 , α 0 ) b e corresponding dressing functions. 7 W e are to construct X n and α n recursiv ely , starting from X 0 and α 0 . The explicit pro cedure is as follows . • ( St ep 0) Assume X 0 , . . . , X i − 1 and α 0 , . . . , α i − 1 are giv en. Set X ( i − 1) := P i − 1 n =0 ~ n X n , α ( i − 1) := P i − 1 n =0 ~ n α n . • ( St ep 1) Set P ( i − 1) := Ad  exp( ~ − 1 X ( i − 1) )( ~ ∂ ) α ( i − 1) / ~  f t , (2.4) Q ( i − 1) := Ad  exp( ~ − 1 X ( i − 1) )( ~ ∂ ) α ( i − 1) / ~  g t . ( 2 .5) Expand P ( i − 1) and Q ( i − 1) as P ( i − 1) = P ∞ k =0 ~ k P ( i − 1) k , Q ( i − 1) = P ∞ k =0 ~ k Q ( i − 1) k . (ord ~ P ( i − 1) k = ord ~ Q ( i − 1) k = 0 .) • ( St ep 2) Put P 0 := σ ~ ( P ( i − 1) 0 ), Q 0 := σ ~ ( Q ( i − 1) 0 ), P ( i − 1) i := σ ~ ( P ( i − 1) i ), Q ( i − 1) i := σ ~ ( Q ( i − 1) i ) and define a constan t α i and a series ˜ X i ( x, t, ξ ) = P ∞ k =1 ˜ χ i,k ( x, t ) ξ − k b y (2.6) α i log ξ + ˜ X i := Z ξ  ∂ Q 0 ∂ ξ P ( i − 1) i − ∂ P 0 ∂ ξ Q ( i − 1) i  ≤− 1 dξ . The in tegral constan t of the indefinite in tegral is fixed so that the righ t hand side a grees with the left hand side. • ( St ep 3) Define a series X i ( x, t, ξ ) = P ∞ k =1 χ i,k ( x, t ) ξ − k b y X i = ˜ X ′ i − 1 2 { σ ~ ( X 0 ) , ˜ X ′ i } + ∞ X p =1 B 2 p (2 p )! (ad { , } ( σ ~ ( X 0 ))) 2 p ˜ X ′ i , ˜ X ′ i := α i log ξ + ˜ X i ( x, ξ ) − exp(ad { , } σ ~ ( X 0 ))( α i log ξ ) . (2.7) Here B 2 p ’s are the Bernoulli num b ers. • ( St ep 4) The op erato r X i ( x, t, ~ ∂ ) is defined as the op erator with the principal sym b ol X i : X i = P ∞ k =1 χ i,k ( x, t )( ~ ∂ ) − k . 8 The main theorem is the follo wing: Theorem 2.1. Assume that X 0 and α 0 satisfy (2.3) and c on- struct X i ’s and α i ’s by the ab ove pr o c e dur e r e cursively. Then W = exp( X/ ~ )( ~ ∂ ) α/ ~ is a dr essing op er ator of the ~ -de p endent KP hi- er ar chy c orr esp onding to ( f , g ) by Pr op osition 1.1. 3 Asymptotics of the w a v e funct ion In this section we prov e that the dressing op erato r of the form W ( ~ , x, t, ~ ∂ ) = exp( X ( ~ , x, ~ ∂ ) / ~ ), ord ~ X ≦ 0, or d X ≦ − 1, giv es a wa ve function of the form Ψ( ~ , x, t ; z ) = W e ( xz + ζ ( t,z )) / ~ = exp( S ( ~ , x, t, z ) / ~ ), S ( ~ , x, t ; z ) = P ∞ n =0 ~ n S n ( x, t ; z )+ ζ ( t, z ), ζ ( t, z ) := P ∞ n =1 t n z n , and vice v ersa. Since the time v ariables t n do not pla y a n y role in this section, w e set them to zero. As the factor ( ~ ∂ ) α/ ~ in (1.4) b ecomes a constan t factor z α/ ~ when it is a pplied to e xz / ~ , w e a lso omit it here. Let A ( ~ , x, ~ ∂ ) = P n a n ( ~ , x )( ~ ∂ ) n b e a micro differen tial o p- erator. The total symb ol o f A is a p o w er series of ξ defined b y σ tot ( A )( ~ , x, ξ ) := P n a n ( ~ , x ) ξ n , or, equiv alen tly defined by the form ula Ae xz / ~ = σ tot ( A )( ~ , x, z ) e xz / ~ . Prop osition 3.1. L et X = X ( ~ , x, ~ ∂ ) b e a micr o d iffer ential op er- ator such that ord X = − 1 and ord ~ X = 0 . Then the total symb ol of e X/ ~ has s uch a form as σ tot (exp( ~ − 1 X ( ~ , x, ~ ∂ ))) = e S ( ~ ,x, ξ ) / ~ , wher e S ( ~ , x, ξ ) is a p ower series of ξ − 1 without non-ne gative p ow - ers of ξ and has an ~ -exp ansion S ( ~ , x, ξ ) = P ∞ n =0 ~ n S n ( x, ξ ) . 9 Mor e over, the c o efficient S n is determine d by X 0 , . . . , X n in the ~ -exp a nsion o f X = P ∞ n =0 ~ n X n . W e omit the explicit form ula for S n . (See [TT3 ].) Prop osition 3.2. L et S = P ∞ n =0 ~ n S n b e a p ower series of ξ − 1 without non-ne gative p owers of ξ . T h en ther e exists a micr o differ- ential op e r ator X ( ~ , x, ~ ∂ ) such that ord X ≦ − 1 , ord ~ X ≦ 0 and σ tot (exp( ~ − 1 X ( ~ , x, ~ ∂ ))) = e S ( ~ ,x, ξ ) / ~ . Mor e over, the c o efficient X n ( x, ξ ) in the ~ -exp a n sion X = P ∞ n =0 ~ n X n of the total symb ol X = X ( ~ , x, ξ ) is determine d by S 0 , . . . , S n in the ~ - exp ansio n of S . W e omit the explicit form ula for X n . ( See [TT3].) Com bining these prop o sitions with the results in Section 2, w e can, in principle, mak e a recursion f orm ula fo r S n ( n = 0 , 1 , 2 , . . . ) of the wa ve function of the solution of the KP hierarc h y corresp ond- ing to the quan tised cano nical transformation ( f , g ) as follows: let S 0 , . . . , S i − 1 b e giv en. 1. By Prop osition 3.2 w e hav e X 0 , . . . , X i − 1 . 2. W e ha v e a recurs ion form ula fo r X i b y Theorem 2.1. 3. Prop osition 3 .1 giv es a form ula f o r S i . If w e tak e the factor ( ~ ∂ ) α/ ~ in to accoun t, this pro cess b ecomes a little bit complicated, but essen tially the same. 4 Asymptotics of the tau funct ion In this section we deriv e an ~ -expansion log τ ( ~ , t ) = P ∞ n =0 ~ n − 2 F n ( t ) of the tau function (cf. (1 .6)). Note that w e hav e suppresse d the 10 v ariable x , whic h is understoo d t o b e absorb ed in t 1 . The log a rithmic deriv ation of (1.6) giv es (4.1) − ~ D ′ ( z ) log τ ( t ) = ~ − 1  ∂ ∂ z + ~ D ′ ( z )  ˆ S ( t ; z ) , where ˆ S ( t ; z ) = S ( t ; z ) − ζ ( t, z ) and D ′ ( z ) := − P ∞ j = 1 z − j − 1 ∂ ∂ t j . By substituting the ~ -expansions log τ ( t ) = P ∞ n =0 ~ n − 2 F n ( t ), ˆ S ( t ; z ) = P ∞ n =0 ~ n S n ( t ; z ), and expanding S n ( t ; z ) as S n ( t ; z ) = − P ∞ k =1 z − k k v n,k , we ha v e the equations (4.2) ∂ F n ∂ t j = v n,j + X k + l = j k ≥ 1 ,l ≥ 1 1 l ∂ v n − 1 ,l ∂ t k ( v − 1 ,j = 0) . This system determine s F n up to in tegration constants . Com bining this with the results in Section 2, Section 3, if F 0 , whic h determines a solution of t he disp ersionless KP hierarc h y , and the corresp onding quan tised canonical transformation ( f , g ) are giv en, w e can construct F n recursiv ely and consequen tly the tau function of a solution of the ~ - KP hierarc h y . A cknow le dgmen ts T he autho r s ar e g rateful to Professor Aki- hiro Tsuc hiy a for dra wing our at t en tion to this sub ject. This w ork is partly support ed b y Grants-in-Aid for Scien tific R esearc h No. 19540179 a nd No. 22540 186 from the Japan So ciety for the Promo- tion of Science. TT is pa r t ly supp orted b y the gran t o f the National Researc h Univ ersit y – Higher School of Economics, Russia, for the Individual Research Pro ject 1 0-01-00 4 3 (20 10). This is a con tribution to t he Pro ceedings of the “In ternational W orkshop on Classical and Q uan tum In tegrable Systems 2011” 11 (Jan uary 24-2 7, 201 1 Protvino, Russia). TT thanks t he organisers of the w orkshop for hospitality . References [A] Aoki, T., Ann. Inst. F ourier (G renoble) 36 (1 9 86), 143–1 65. [K G] Ko dama, Y., Phy s. Lett. 129A (1988) , 2 2 3–226; K o dama, Y., and Gibb ons, J., Ph ys. Lett. 135A (19 8 9), 167–170. [Kr] Kric hev er, I.M., Commu n. Math. Phy s. 143 (1991 ), 415–426. [OS] Orlo v, A. Y u. and Sc h ulman, E. I., Lett. Math. Ph ys. 12 (19 86), 171– 179; Orlov , A. Y u., in: Plasma The ory and Nonline ar a nd T urbulent Pr o c esses in Physics (W orld Scien tific, Singap o re, 1988) ; G rinevic h, P . G ., and Orlov , A. Y u., in: Pr oblems of Mo dern Quantum Field The ory (Springer-V erlag, 1989 ) . [S] Sc hapira, P ., Micro differen tial systems in the complex domain, Grundlehren der mathematisc hen Wis sensc haften 269 , Springer- V erlag, Berlin-New Y ork, (1985 ) [TT1] T ak asaki, K., and T akebe, T., In t. J. Mo d. Phy s. A7S1B (1992), 889-922 . [TT2] T ak asaki, K., and T a k eb e, T., R ev. Math. Phys. 7 (1995), 743 -803. [TT3] T ak asaki, K., and T a kebe, T., ~ -expansion of KP hierarc h y: Recursiv e construction o f solutions, arXiv: 0912.4867 12