On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy

ON THE EQUIV ALENCE OF DIFFERENT LAX P AIRS F OR THE KA C–V AN MOER BEKE HIERAR CHY JOHANNA MICHOR AND GERALD TESCHL Abstract. W e give a simple algebraic pro of that the tw o differen t Lax pairs for the Kac–v an Mo erb eke hierarc hy , construct ed f rom Jacobi respectively super- symmetric Dirac-type difference operators, gi ve rise to the same hier- arc hy of evolution equations. As a byproduct w e obtain so me new recu rsions for computing these equations. 1. Introduction There are tw o differen t Lax equatio ns for the Kac–v an Mo erb eke equatio n: The original one of Kac and v an Moer b eke [4 ] based on a Jacobi matrix with zero diagonal elements and its skew-symmetrized s quare a nd the second one based on sup e r-symmetric Dirac-type matrice s . Both appro aches can b e genera lized to give corres p o nding hiera rchies of evolution equatio ns in the usual way and b oth reveal a close co nnection to the T oda hie r arch y . In fact, the first approach sho ws that the Kac–v an Mo erb eke hierarch y (KM hier arch y ) is contained in the T o da hierar ch y by setting b = 0 in the o dd equa tio ns. The second one relates b oth hiera rchies via a B¨ acklund trans fo rmation since the Dirac-type difference op er a tor g ives rise to tw o Ja cobi op er ators by taking s quares (res p e c tively factorizing p ositive Jaco bi op erator s to obtain the o ther direction). Both wa y s of in tro ducing the KM hierarch y hav e its merits, how ever, tough it is obvious that bo th pr o duce the same hierarch y by lo oking at the first few equa tions, we c o uld not find a formal pro of in the literature. The purp ose of this short note is to g ive a simple algebr a ic pro of for this fact. As a b ypro duct we will als o obtain some new recur sions for c omputing the equa tions in the KM hiera rch y . In Section 2 we revie w the recur sive construction of the T o da hierar chy via Lax pairs in volving Ja c o bi o p erator s and obtain the K M hierarch y by setting b = 0 in the o dd equations. In Sec tio n 3 we in tro duce the KM hiera rch y via Lax pair s in volving Dirac-type difference op era tors. In Section 4 we show that b o th co nstructions pro duce the s a me equatio ns . Finally , we recall how to identify Jacobi op era tors with b = 0 in Section 5. 2. The To da hierarchy In this sectio n we intro duce the T oda hierarc hy using the standar d Lax formalism following [2] (see also [7]). W e will o nly consider b ounded solutions and hence req uire 2000 M athematics Subject Classific ation. Pri mary 47 B36, 37K15; Secondary 81U40, 39A10. Key wor ds and phr ases. Kac–v an Mo erb eke hierarch y , Lax pair , T o da hierarch y . W ork supp orted by the Austrian Science F und (FWF) under Grant s No. Y330 and J2655. 1 2 J. MICHOR AND G. TESCHL Hyp othesi s H. 2.1. Supp ose a ( t ) , b ( t ) satisfy a ( t ) ∈ ℓ ∞ ( Z , R ) , b ( t ) ∈ ℓ ∞ ( Z , R ) , a ( n, t ) 6 = 0 , ( n, t ) ∈ Z × R , and let t 7→ ( a ( t ) , b ( t )) b e differ ent iable in ℓ ∞ ( Z ) ⊕ ℓ ∞ ( Z ) . Asso ciated with a ( t ) , b ( t ) is a J acobi op er a tor (2.1) H ( t ) = a ( t ) S + + a − ( t ) S − + b ( t ) in ℓ 2 ( Z ), where S ± f ( n ) = f ± ( n ) = f ( n ± 1) are the usual shift op erators and ℓ 2 ( Z ) denotes the Hilb ert space o f sq uare s ummable (co mplex-v alued) sequences over Z . Moreov er, cho ose consta nt s c 0 = 1, c j , 1 ≤ j ≤ r , c r +1 = 0 , and set g j ( n, t ) = j X ℓ =0 c j − ℓ h δ n , H ( t ) ℓ δ n i , h j ( n, t ) = 2 a ( n, t ) j X ℓ =0 c j − ℓ h δ n +1 , H ( t ) ℓ δ n i + c j +1 . (2.2) The sequences g j , h j satisfy the recur s ion relations g 0 = 1 , h 0 = c 1 , 2 g j +1 − h j − h − j − 2 bg j = 0 , 0 ≤ j ≤ r, h j +1 − h − j +1 − 2 ( a 2 g + j − ( a − ) 2 g − j ) − b ( h j − h − j ) = 0 , 0 ≤ j < r . (2.3) Int ro ducing (2.4) P 2 r +2 ( t ) = − H ( t ) r +1 + r X j =0 (2 a ( t ) g j ( t ) S + − h j ( t )) H ( t ) r − j + g r +1 ( t ) , a stra ightforw a rd computation shows that the Lax equation (2.5) d dt H ( t ) − [ P 2 r +2 ( t ) , H ( t )] = 0 , t ∈ R , is equiv alent to (2.6) TL r ( a ( t ) , b ( t )) =   ˙ a ( t ) − a ( t )  g + r +1 ( t ) − g r +1 ( t )  ˙ b ( t ) −  h r +1 ( t ) − h − r +1 ( t )    = 0 , where the dot deno tes a deriv ative with r esp ect to t . V arying r ∈ N 0 yields the T o da hierar ch y TL r ( a, b ) = 0. The corr esp onding homogeneous quantities obta ined by taking all summation c onstants equal to zero , c ℓ ≡ 0 , ℓ ∈ N , a re denoted by ˆ g j , ˆ h j , etc., res p. (2.7) c TL r ( a, b ) = TL r ( a, b )   c ℓ ≡ 0 , 1 ≤ ℓ ≤ r . Next we show that we ca n set b ≡ 0 in the o dd equatio ns of the T o da hierarch y . Lemma 2.2. L et b ≡ 0 . Then the homo gene ous c o efficients satisfy ˆ g 2 j +1 = ˆ h 2 j = 0 , j ∈ N 0 . Pr o of. W e us e induction on the re c ur sion re la tions (2.3). The claim is true for j = 0. If ˆ h 2 j = 0 then ˆ g 2 j +1 = 0 , and ˆ h 2 j = 0 follows from the last equation in (2.3).  EQUIV ALENCE OF LAX P AIRS FOR THE KAC–V AN MOERBEKE HIERARC HY 3 In particular , if we choose c 2 ℓ = 0 in TL 2 r +1 , then we ca n s et b ≡ 0 to o btain a hierarch y of evolution equations for a alone. In fact, set (2.8) G j = ˆ g 2 j , K j = ˆ h 2 j +1 , in this case. Then they satisfy the r ecursion G 0 = 1 , K 0 = 2 a 2 , 2 G j +1 − K j − K − j = 0 , 0 ≤ j ≤ r, K j +1 − K − j +1 − 2 ( a 2 G + j − ( a − ) 2 G − j ) = 0 , 0 ≤ j < r , (2.9) and TL 2 r +1 ( a, 0) = 0 is equiv a lent to the KM hier a rch y defined as (2.10) KM r ( a ) = ˙ a − a ( G + r +1 − G r +1 ) , r ∈ N 0 . 3. The K ac–v an Moerbeke hierarchy as a modified Toda hierarchy In this section we review the constructio n of the K M hier arch y a s a mo dified T o da hierarch y . W e refer to [2], [7 ] for further details. Suppo se ρ ( t ) sa tisfies Hyp othesi s H. 3.1. L et (3.1) ρ ( t ) ∈ ℓ ∞ ( Z , R ) , ρ ( n, t ) 6 = 0 , ( n, t ) ∈ Z × R and let t 7→ ρ ( t ) b e differ ent iable in ℓ ∞ ( Z ) . Define the “even” and “ o dd” pa rts of ρ ( t ) by (3.2) ρ e ( n, t ) = ρ (2 n, t ) , ρ o ( n, t ) = ρ (2 n + 1 , t ) , ( n, t ) ∈ Z × R , and consider the b ounded op erator s (in ℓ 2 ( Z )) (3.3) A ( t ) = ρ o ( t ) S + + ρ e ( t ) , A ( t ) ∗ = ρ − o ( t ) S − + ρ e ( t ) . In addition, we s et (3.4) H 1 ( t ) = A ( t ) ∗ A ( t ) , H 2 ( t ) = A ( t ) A ( t ) ∗ , with (3.5) H k ( t ) = a k ( t ) S + + a − k ( t ) S − + b k ( t ) , k = 1 , 2 , and a 1 ( t ) = ρ e ( t ) ρ o ( t ) , b 1 ( t ) = ρ e ( t ) 2 + ρ − o ( t ) 2 , (3.6) a 2 ( t ) = ρ + e ( t ) ρ o ( t ) , b 2 ( t ) = ρ e ( t ) 2 + ρ o ( t ) 2 . (3.7) Now we define o p erator s D ( t ), Q 2 r +2 ( t ) in ℓ 2 ( Z , C 2 ) as fo llows, D ( t ) =  0 A ( t ) ∗ A ( t ) 0  , (3.8) Q 2 r +2 ( t ) =  P 1 , 2 r +2 ( t ) 0 0 P 2 , 2 r +2 ( t )  , r ∈ N 0 . (3.9) Here P k, 2 r +2 ( t ), k = 1 , 2 are defined as in (2 .4), that is, (3.10) P k, 2 r +2 ( t ) = − H k ( t ) r +1 + r X j =0 (2 a k ( t ) g k,j ( t ) S + − h k,j ( t )) H k ( t ) j + g k,r +1 , 4 J. MICHOR AND G. TESCHL { g k,j ( n, t ) } 0 ≤ j ≤ r , { h k,j ( n, t ) } 0 ≤ j ≤ r +1 are defined as in (2.2). Moreov er, we choose the same integration cons tants in P 1 , 2 r +2 ( t ) and P 2 , 2 r +2 ( t ) (i.e., c 1 ,ℓ = c 2 ,ℓ ≡ c ℓ , 1 ≤ ℓ ≤ r ). Analogous to equa tio n (2.5) one obtains that (3.11) d dt D ( t ) − [ Q 2 r +2 ( t ) , D ( t )] = 0 is equiv alent to KM r ( ρ ) = (KM r ( ρ ) e , KM r ( ρ ) o ) =  ˙ ρ e − ρ e ( g 2 ,r +1 − g 1 ,r +1 ) ˙ ρ o + ρ o ( g 2 ,r +1 − g + 1 ,r +1 )  = 0 . (3.12) As in the T o da c ontext (2.6), v arying r ∈ N 0 yields the KM hierar ch y which we denote by (3.13) KM r ( ρ ) = 0 , r ∈ N 0 . The homoge neous KM hier arch y is denoted by (3.14) d KM r ( ρ ) = KM r ( ρ )   c ℓ ≡ 0 , 1 ≤ ℓ ≤ r . One lo ok at the transformations (3.6) , (3.7) verifies that the equations for ρ o , ρ e are in fact one equation for ρ . More explicitly , co mbining g k,j , resp. h k,j , into one sequence (3.15) G j (2 n ) = g 1 ,j ( n ) G j (2 n + 1) = g 2 ,j ( n ) , resp. H j (2 n ) = h 1 ,j ( n ) H j (2 n + 1) = h 2 ,j ( n ) , we can rewrite (3.12) a s (3.16) KM r ( ρ ) = ˙ ρ − ρ ( G + r +1 − G r +1 ) . F rom (2 .3) we see that G j , H j satisfy the recur s ions G 0 = 1 , H 0 = c 1 , 2 G j +1 − H j − H −− j − 2 ( ρ 2 + ( ρ − ) 2 ) G j = 0 , 0 ≤ j ≤ r, H j +1 − H −− j +1 − 2 (( ρρ + ) 2 G + j − ( ρ − ρ ) 2 G −− j ) − ( ρ 2 + ( ρ − ) 2 )( H j − H −− j ) = 0 , 0 ≤ j < r . (3.17) The homoge neous quantities are deno ted by ˆ G j , ˆ H j , etc., as b efor e. As a simple consequence o f (3.1 1) we hav e (3.18) d dt D ( t ) 2 − [ Q 2 r +2 ( t ) , D ( t ) 2 ] = 0 and obser ving (3.19) D ( t ) 2 =  H 1 ( t ) 0 0 H 2 ( t )  yields the implication (3.20) KM r ( ρ ) = 0 ⇒ TL r ( a k , b k ) = 0 , k = 1 , 2 , that is, given a so lution ρ of the KM r equation (3.13), o ne o btains tw o solutions, ( a 1 , b 1 ) and ( a 2 , b 2 ), o f the TL r equations (2.6 ) related to ea ch other by the Miura- t yp e transforma tions (3.6), (3.7). F or mor e information we refer to [3], [6], [7], and [8]. EQUIV ALENCE OF LAX P AIRS FOR THE KAC–V AN MOERBEKE HIERARC HY 5 4. Equiv alence of both constructions In this section we want to show that the constructions of the KM hiera rch y out- lined in the previo us t wo sections yield in fact the same set o f evolution equations. This will follow once we show that G j defined in (2.8) is the same as G j defined in (3.15). It will b e sufficient to consider the homogeneous quantities, how ever, we will omit the additional hats for notational simplicity . Moreover, we will denote the sequence G j defined in (2.8) by ˜ G j to distinguish it from the one defined in (3.15). Since b o th ar e defined recurs ively via the re cursions (2 .9) for ˜ G j , K j resp ec- tively (3.17) for G j , H j our first a im is to elimina te the additiona l sequences K j resp ectively H j and to get a recur sion for ˜ G j resp ectively G j alone. Lemma 4.1. The c o efficients g j ( n ) satisfy the fol lowing line ar r e cu rs ion g + j +3 − g j +3 = ( b + 2 b + ) g + j +2 − (2 b + b + ) g j +2 − (2 b + b + ) b + g + j +1 + b (2 b + + b ) g j +1 + k + j +1 + k j +1 + b ( b + ) 2 g + j − b + b 2 g j − b k + j − b + k j , (4.1) wher e (4.2) k j = a 2 g + j − ( a − ) 2 g − j , j ∈ N . Pr o of. It s uffices to co nsider the homog eneous case g j ( n ) = h δ n , H j δ n i . Then (compare [7, Sect 6 .1]) g ( z , n ) = h δ n , ( H − z ) − 1 δ n i = − ∞ X j =0 g j ( n ) z j +1 satisfies [7, (1.10 9)] ( a + ) 2 g ++ − a 2 g z − b + + a 2 g + − ( a − ) 2 g − z − b = ( z − b + ) g + − ( z − b ) g , and the claim follows after comparing c o efficients.  Corollary 4.2. F or j ∈ N 0 , the se qu enc es ˜ G j , define d by (2.8) and c orr esp onding to the TL hier ar chy with b ≡ 0 , satisfy ˜ G + j +1 − ˜ G j +1 = ( a + ) 2 ˜ G ++ j + a 2 ( ˜ G + j − ˜ G j ) − ( a − ) 2 ˜ G − j . (4.3) 6 J. MICHOR AND G. TESCHL The c orr esp onding se quenc es G j for the KM hier ar chy define d in (3.15) satisfy G j +3 − G ++ j +3 =  ( a − ) 2 + a 2  2  ( a + ) 2 + ( a ++ ) 2  G j + ( a −− ) 2 ( a − ) 2 G −− j +1 + a 2 ( a + ) 2 G j +1 +  ( a + ) 2 + ( a ++ ) 2  2( a − ) 2 + 2 a 2 + ( a + ) 2 + ( a ++ ) 2  G ++ j +1 +  2( a − ) 2 + 2 a 2 + ( a + ) 2 + ( a ++ ) 2  G j +2 −  ( a − ) 2 + a 2  ( a + ) 2 + ( a ++ ) 2  2 G ++ j −  ( a + ) 2 + ( a ++ ) 2  ( a −− ) 2 ( a − ) 2 G −− j − a 2 ( a + ) 2 G ++ j  −  ( a − ) 2 + a 2  a 2 ( a + ) 2 G j − ( a ++ ) 2 ( a +++ ) 2 G ++++ j  −  ( a − ) 2 + a 2  ( a − ) 2 + a 2 + 2 ( a + ) 2 + 2 ( a ++ ) 2  G j +1 − a 2 ( a + ) 2 G ++ j +1 − ( a ++ ) 2 ( a +++ ) 2 G ++++ j +1 −  ( a − ) 2 + a 2 + 2 ( a + ) 2 + 2 ( a ++ ) 2  G ++ j +2 . (4.4) Pr o of. Use (4.1) with b ≡ 0 for (4.3) resp. (3.6), (3.7) with a = ρ for (4 .4).  Lemma 4.3. F or al l n ∈ Z , (4.5) ˜ G j ( n ) = G j ( n ) , j ∈ N 0 . Pr o of. O ur a im is to s how that ˜ G j satisfy the linea r recur s ion r elation (4.4) for ˆ G j . W e start with (4.3), ˜ G j +3 − ˜ G + j +3 + ˜ G + j +3 − ˜ G ++ j +3 = − ( a + ) 2 ˜ G ++ j +2 + a 2 ( ˜ G j +2 − ˜ G + j +2 ) + ( a − ) 2 ˜ G − j +2 − ( a ++ ) 2 ˜ G +++ j +2 + ( a + ) 2 ( ˜ G + j +2 − ˜ G ++ j +2 ) + a 2 ˜ G j +2 , (4.6) and observe that the rig ht hand side of (4.4) only involv es e ven shifts of G j . Hence we sys tema tically replace in (4 .6) o dd shifts of ˜ G j by (4.3), ˜ G j = ( G 1 ,j := ˜ G + j − ( a + ) 2 ˜ G ++ j − 1 + a 2 ( ˜ G j − 1 − ˜ G + j − 1 ) + ( a − ) 2 ˜ G − j − 1 G 2 ,j := ˜ G − j + a 2 ˜ G + j − 1 + ( a − ) 2 ( ˜ G j − 1 − ˜ G − j − 1 ) − ( a −− ) 2 ˜ G −− j − 1 , as follows: ˜ G +++ j +2 → G +++ 2 ,j +2 , ˜ G + j +2 → xG + 1 ,j +2 + (1 − x ) G + 2 ,j +2 , ˜ G − j +2 → G − 1 ,j +2 , with x = ( a − ) 2 + a 2 + ( a ++ ) 2 a 2 − ( a + ) 2 . In the resulting eq uation we repla ce ˜ G +++ j +1 → G +++ 2 ,j +1 , ˜ G + j +1 → y G + 1 ,j +1 + (1 − y ) G + 2 ,j +1 , ˜ G − j +1 → G − 1 ,j +1 , where y = ( a − ) 2 ( a ++ ) 2 + a 2 ( a ++ ) 2 a 2 ( a ++ ) 2 − ( a − ) 2 ( a + ) 2 . This gives (4.4) fo r ˜ G j .  Hence b oth cons tructions for the KM hiera rch y are equiv alent a nd we hav e Theorem 4.4. L et r ∈ N 0 . Then (4.7) TL 2 r +1 ( a, 0) = KM r ( a ) . pr ovide d c TL 2 j +1 = c KM j and c TL 2 j = 0 for j = 0 , . . . , r . EQUIV ALENCE OF LAX P AIRS FOR THE KAC–V AN MOERBEKE HIERARC HY 7 Remark 4.5. As p ointe d out by M. Gekhtman to us, an alternate way of pr oving e quivalenc e is by showing that (in the semi-infinite c ase, n ∈ N ) b oth c onstru ctions give rise to the same set of evolutions for the moments of the underlying sp e ctr al me asur e (c omp ar e [1] ). Our pur ely algebr aic appr o ach has the advantage t hat it do es neither r e quir e the semi-infinite c ase nor self-adjoi ntness. 5. A ppendix: Ja cobi opera to rs with b ≡ 0 In o rder to ge t solutions for the Kac–v an Mo e rb eke hiera rch y o ut of solutions of the T o da hierar ch y o ne clearly needs to iden tify those cases which le ad to Jacobi op erator s with b ≡ 0. F or the s a ke of completeness we recall some folklo re results here. Let H b e a Jacobi op era to r ass o ciated with the sequences a , b as in (2.1). Reca ll that under the unitary op era tor U f ( n ) = ( − 1) n f ( n ) our Ja cobi ope rator transfor ms according to U − 1 H ( a, b ) U = H ( − a, b ), where we write H ( a, b ) in o rder to display the dep endence of H on the sequences a and b . Hence, in the sp e cial case b ≡ 0 we infer that H and − H ar e unitarily equiv alent, U − 1 H U = − H . In particular , the spec tr um is symmetr ic with resp ect to the reflection z → − z a nd it is not surprising, that this symmetry plays an imp orta nt role. Denote the dia gonal and first off-diagonal o f the Green’s function of a Ja cobi op erator H by g ( z , n ) = h δ n , ( H − z ) − 1 δ n i , h ( z , n ) = 2 a ( n ) h δ n +1 , ( H − z ) − 1 δ n i − 1 . (5.1) Then we hav e Theorem 5.1. F or a given Jac obi op er ator, b ≡ 0 is e qu ivalent to g ( z , n ) = − g ( − z , n ) and h ( z , n ) = h ( − z , n ) . Pr o of. Set ˜ H = − U − 1 H U , then the co rresp onding dia gonal and first off-diag o nal elements a re re la ted via ˜ g ( z , n ) = − g ( − z , n ) and ˜ h ( z , n ) = h ( − z , n ). Hence the claim follows since g ( z , n ) and h ( z , n ) uniquely determine H (see [7, Sect. 2 .7] resp ectively [5] for the unbounded c ase).  Note that one could a lternatively use recursio ns: Since g j ( n ) and h j ( n ) ar e just the co e fficients in the a symptotic expansions of g ( z , n ) resp ectively h ( z , n ) around z = ∞ (see [7, Chap. 6]), our c la im is equiv alent to g 2 j +1 ( n ) = 0 and h 2 j ( n ) = 0. Similarly , b ≡ 0 is equiv a le nt to m ± ( z , n ) = − m ± ( − z , n ), where (5.2) m ± ( z , n ) = h δ n ± 1 , ( H ± ,n − z ) − 1 δ n ± 1 i are the W eyl m - functions. Here H ± ,n are the tw o half-line o p er ators obtained from H by impo sing an additiona l Dirichlet b oundary condition at n . The co rresp o nding sp ectral mea sures ar e of co urse symmetric in this cas e. F or a quas i-p erio dic algebro -geometric so lution (see e.g. [7, Chap. 9]), this implies b ≡ 0 if a nd o nly if b oth the sp e ctrum and the Dirichlet divisor ar e s ymmetric with resp ect to the reflection z → − z . F or an N so lito n solution this implies b ≡ 0 if and only if the eigenv alues co me in pairs, E and − E , and the nor ming constants asso cia ted with each eigenv a lue pair ar e equa l. 8 J. MICHOR AND G. TESCHL Ackno wledgments W e thank Michael Gekhtman and F ritz Gesztesy for v alua ble discussions on this topic and hints with res pe ct to the liter ature. References [1] Y. Berezansky and M. Shmois h, Nonisosp e ct r al flows on semi- infinite Jac obi matric e s , J. Nonli near Math. Phys. 1 , no. 2, 116–146 (1994). [2] W. Bulla, F. Gesztesy , H. Holden, and G. T eschl, Algebr o-Ge ometric Q uasi-Perio dic Finite-Gap Solutions of the T o da and Kac-van Mo erb eke H ier ar chies , Mem. Amer. Math. Soc. 135-64 1 , (1998). [3] F. Gesztesy , H. Holden, B. Simon, and Z. Zhao, On the T o da and Kac-van Mo erb e ke systems , T rans. Amer. M ath. Soc. 33 9 , 849–868 (1993). [4] M. Kac and P . v an M oerb eke, On an explicitly soluble system of nonline ar differ ential e quations, r elate d to c ertain T o da lattic es , Adv. Math. 16 , 160–169 (1975). [5] G. T eschl, T r ac e formulas and inverse sp e ctr al the ory for Jac obi op er ators , Comm. Math. Ph ys. 1 96 , 175–202 (1998). [6] G. T eschl, O n the T o da and Kac–van Mo erb e k e hier ar chies , Math. Z. 231 , 325–344 (1999). [7] G. T eschl, Jac obi Op er ators and Completely Inte gr able Nonline ar L att ic es , Math. Surv. and M on. 72 , Amer. Math. So c., Rho de Isl and, 2000. [8] M. T oda and M. W adati, A c anonic al t ra nsformation for t he ex p onential lattic e , J. Phys. Soc. Jpn. 39 , 1204–1211 (1975). Imperial College, 180 Queen’s Gat e, London SW7 2BZ, and I nterna tional Er win Schr ¨ odinger Institute for Ma them a tical Physics, Bol tzmanngasse 9, 1090 W ien, Aus- tria E-mail addr ess : Johanna.Michor @esi.ac.at URL : http://w ww.mat.un ivie.ac.at/~jmichor/ F acul ty of Ma thema tics, Nordbergstrasse 15, 1090 Wien, Austria, and Interna tional Er win S chr ¨ odinger Institute for Ma thematic al Physics, Bol tzmanngasse 9, 10 90 Wien, Aust ria E-mail addr ess : Gerald.Teschl@ univie.ac.at URL : http://w ww.mat.un ivie.ac.at/~gerald/