Orthogonal polynomials and the finite Toda lattice Alex Kasman Department of Mathematics, University of Georgia ~Received 11 March 1996; accepted for publication 12 August 1996! The choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed. It is then shown that the same induced time dependence for finitely supported distributions which gives the ith KP flow under the dual isomorphism induces the ith flow of the Toda hierarchy on the matrix. The corresponding solution is an N particle, finite, nonperiodic Toda solution where N is the cardinality of the support of c plus the sum of the orders of the highest derivative taken at each point. © 1997 American Institute of Physics. @S0022-2488~97!00501-X# I. INTRODUCTION Recent interest in the Toda lattice1 has stemmed from its role in relating theories of quantum gravity to soliton theory.2 This correspondence is given by a measure dr determined by the partition function ~i.e., the ‘‘specific heat’’! of matrix models which is interpreted as an inner product on time-dependent polynomials in the spectral parameter.3 In that construction, the polynomials are written in terms of an orthogonal basis with respect to this nondegenerate inner product and the Toda lattice is determined as the matrix representing multiplication in the spectral parameter. The present paper replaces integration with respect to the measure dr by an arbitrary finitely supported distribution. It is then shown that the same correspondence between orthogonal polynomials and integrable systems continues to hold in the case of the induced degenerate bilinear form. This relates the Toda hierarchy to techniques for the construction of t -functions of the KP hierarchy4,5 using finitely supported distributions. It should be noted that there is an intersection of the construction developed below and that discussed in the opening paragraph. In particular, finitely supported distributions which are linear combinations of Dirac delta functions can be represented as Stieltjes integrals with respect to Heaviside functions.6 The solutions constructed from such distributions by the method below are known7 and are the same as those which would be given by the corresponding measure. However, finitely supported distributions involving differentiation ~i.e., mi.0) and the Toda lattices which they generate are discussed here for the first time.

II. ASSOCIATING A JACOBI MATRIX TO A DISTRIBUTION Let c be the finitely supported distribution c5(i51 m d li +(j50 mi ai j]z j , ~2.1! where d l is the delta function evaluating its argument at z5l, the constants li P C are distinct, ] z is the differential operator ] /] z , andai j P C withaimi Þ 0. ~In fact, the discussion to follow only depends upon c as determined up to a nonzero constant multiple, and so the coefficients ai j can be viewed as elements of PN 21C.! Then let N be the integer 0022-2488/97/38(1)/247/8/$10.00 J. Math. Phys. 38 (1), January 1997 © 1997 American Institute of Physics 247 N 5m1(i51 m mi , where m and mi are as in ~II.1!. Associated to c we have the symmetric bilinear form on C@z# defined by ^p,q&5c~pq!. Note that given two polynomials p5(i51 n21 aizi, q5(i51 n21 bizi of degree less than n, ^p,q&5~a0 , . . . ,an21!•Tn•S b0 A bn21D where Tn5S c~1! . . . c~zn21! A  A c~zn21! . . . c~z2n21!D. ~2.2! A. The annihilator of c Any function sufficiently differentiable on the support of c acts on the right by composition: c+p~ f !5c~p f !. In particular, we may associate to c its annihilator in C@z#. Definition II.1: For any distribution c, let Ic,C@z# denote the ideal Ic5$pPC@z#uc+p[0%. Lemma II.1: Let c be written in the form (II.1) and let sc~ z !5)i51 m ~ z2li!mi11. Then Ic is the ideal generated by sc(z): Ic5scC@z#. Proof: Since c + sc[0, it is clear that sc(z)C@z#,Ic . Then, let p(z) P Ic be written in the form p~ z !5q~ z !r~ z !, r~ z !5)i51 m ~ z2li!g i, ~2.3! 248 Alex Kasman: Orthogonal polynomials and the finite Toda lattice J. Math. Phys., Vol. 38, No. 1, January 1997 where q(z) P C@z# is such that q(li) Þ 0. Suppose that g j,m j11 for some particular 1<j<m. Then for the polynomial s~ z !5~ z2lj !m j2g j)iÞj ~ z2li!mi11 we have that the distribution c+r+s5kd lj , k5~m j! !a jm j Þ0, is a nonzero distribution evaluating its argument at lj without differentiation. But then, since c + p5c + qr[0 we have that 05c+qr„s~z!…5c+rs„q~z!…5kq~lj!, which implies that q(lj)50, contradicting the assumption. Consequently, each element of Ic written in form ~II.3! has g i>mi11 and Ic,sc(z)C@z#. Since deg sc5Si51 m (mi11)5N , we then have the following. Corollary II.1: There exists a polynomial p P C@z# with deg p5n such that c + p[0 if and only if n>N . B. A basis for C@z # The choice of a generic distribution c uniquely specifies a basis for C@z# as follows. Definition II.2: For any positive integer i, let t i denote the determinant t i5uTiu, where Ti is the symmetric matrix described in ~II.2!. We say that c is regular if t i Þ 0 for i51, . . . ,N . Let P denote the vector space of polynomials of degree less than N . If c is regular, then the Gram–Schmidt orthogonalization specifies a unique basis $p0 , . . . ,pN 21% of P such that pi~ z !5zi1O~ zi21! and which is orthogonal with respect to the form ^•,•&. Furthermore, since t N 21 Þ 0 the form is nondegenerate on P and so ^pi ,pi&Þ0, i50, . . . ,N 21. It will now be supposed that c is in fact regular and that the polynomials pi for i50, . . . ,N 21 have been fixed by the Gram–Schmidt orthogonalization. We may then define pN 1i~ z !5zisc~ z !, i50,1, . . . . By Lemma II.1, pN 1i P Ic , and so it is in the kernel of the form.

Therefore, the basis of monic polynomials $piui>0% for C@z# is orthogonal with respect to the form, but the form is degenerate. C. The tri-diagonal matrix The significance of the basis specified in the preceding section is that multiplication by z is represented as a tri-diagonal Jacobi matrix in terms of this basis. Proposition II.1: There exist numbers ai and bi in C such that zpi5pi111bipi1aipi21 Alex Kasman: Orthogonal polynomials and the finite Toda lattice 249 J. Math. Phys., Vol. 38, No. 1, January 1997 for all i.0. Proof: Since each polynomial pi is monic of degree i, we certainly have zpn5pn111(j50 n a jpj for some constants a j . But then applying the functional ^pi ,•& to zpn yields ^pi ,zpn&5^zpi ,pn&5^pi111bipi1aipi21 ,pn&, which is zero if i,n22. On the other hand, one could also compute this as ^pi ,zpn&5^pi ,aipi&5ai^pi ,pi&. If i,N , then ^pi ,pi& Þ 0 and so ai50. Finally, for i>N , the claim is true by construction since zpi5pi11. Proposition II.2: Denote by An the constant ^pn ,pn&. Then ~i! An5anAn21 , ~i! an Þ 0 for n50, . . . ,N 21, ~iii! An /Ak5an•••ak11 for k,n,N . Proof: The first relationship can be found by using the fact that ^zp,q&5^p,zq& and so ^zpn ,pn21&5^pn11 ,pn21&1bn^pn ,pn21&1an^pn21 ,pn21&5an^pn21 ,pn21& is also equal to ^pn ,zpn21&5^pn ,pn&, producing the desired result. Then, by the nondegeneracy of the bilinear form on P, we have that anAn215An5^pn ,pn& Þ 0 for 0<n<N 21. The final claim clearly follows from the first by an inductive argument. Associate to c the N3N tri-diagonal matrix L5Sb0 1 0 0 0 . . . a1 b1 1 0 0 0 a2 b2 1 0 A    D. Outside of the principal N 3N minor, this matrix is simply the shift matrix with 1s along the super-diagonal and zeroes elsewhere. Note that L corresponds to multiplication by z in C@z# with basis $pi%. This is particularly important in the next result. Notation: Denote by Lj,k i the element in the jth column and kth row of the matrix Li. Note that since L is indexed by N3N, the top left corner is L0,0 i and not L1,1 i as one might expect. Proposition II.3: ^zipk ,pn&5Ln,k i An . Proof: By orthogonality, the only significant term in zipk is the pn term in its expansion in the orthogonal basis. However, this is simply Ln,k i pn . So ^zpk ,pn&5^Ln,k i pn ,pn&5Ln,k i An . By the symmetry of the form used in Proposition II.3, we then also have the following. Corollary II.2: Ln,k i An5Lk,n i Ak . 250 Alex Kasman: Orthogonal polynomials and the finite Toda lattice J. Math. Phys., Vol. 38, No. 1, January 1997 III. TIME DEPENDENCE Now suppose that c is an arbitrary, i.e., not necessarily regular, finitely supported distribution. To it we associate the time-dependent distribution cˆ5c+ exp(i51 ` tizi. Note that N cˆ 5N c and, moreover, scˆ 5sc since neither the support nor the highest derivative taken at each point is affected by this composition.

Whenever t5(t1 ,t2 , . . . ) is chosen such that cˆ is regular, we may associate to it a basis of polynomials and a tri-diagonal matrix by the method of the preceding section. Thus, one is led to consider a basis $pi(z,t)% of polynomials and a time-dependent matrix L(t) which are defined whenever cˆ is regular. Note: This time dependence for distributions was introduced in Ref. 8 because it induces the KP flow on the Sato Grassmannian under the dual isomorphism. In fact, this is a convenient way to prove the next claim: Proposition III.1: The distribution cˆ is regular for almost every value of t5(t1 ,t2 ,•••). Proof: By Corollary II.1 the distributions c+ zi are linearly independent for n50, . . . ,N 21. Then the determinants t n are nonzero, time-dependent functions t n~ t!5Ucˆ ~1! ••• cˆ ~zn!  cˆ ~zn! cˆ ~z2n21!U 5U c~exp (tizi! ••• ]n21/]t1n21c~exp (tizi!  ] n21/] t1 n21c~exp (tizi! . . . ]2n21/]t12n21c~exp (tizi!U. In fact, if we let Vcˆ ,n denote the set of polynomials in the kernel of the distributions cˆ + zi for i50, . . . ,n, then the Hilbert closure of z2nVcˆ ,n is the a point Wn(t) in the Sato Grassmannian Gr and the Wronskian determinant above gives the corresponding tau function for the KP hierarchy.4 So, we can cite Ref. 9 to show that these functions have isolated zeros. The distribution cˆ is then regular on the complement of the zeros of the t -functions t i for i50, . . . ,N 21. Note: Tau functions determined from symmetric Wronskian matrices or Hankel determinants of the form above are known to be associated with finite Toda lattices.10–12 IV. DIFFERENTIAL EQUATIONS This section will determine differential equations satisfied by the matrix L(t) in the temporal variable ti . Throughout the remainder, prime (8) will be used to denote the derivative with respect to this variable. Since the form ^•,•& is now taken to be the time-dependent form specified by cˆ , its derivative is given by the following lemma. Lemma IV.1: ^p,q&85^zip,q&1^p8,q&1^p,q8&. Proof: ^p,q&85„c~eSt jz j pq!…85c„eSt jz j ~zipq1p8q1pq8!)5^zip,q&1^p8,q&1^p,q8&. The leading coefficients of the polynomials pn are constant, and so they satisfy differential equations of the form pn85(k50 n21 Ck npk . ~4.1! Alex Kasman: Orthogonal polynomials and the finite Toda lattice 251 J. Math. Phys., Vol. 38, No. 1, January 1997 Define the time-dependent functions Ck n by this formula. In fact, since scˆ is constant in time, pn850 for n>N . Thus, it is clear that Ck n50 for n>N . Proposition IV.1: The coefficients Ck n for k,n,N in (IV.1) can be written either as Ck n52 An Ak Ln,k i ~4.2! or Ck n52Lk,n i . ~4.3! In particular, Ck n50 if i,n2k. Proof: This can be seen by differentiating the equation ^pn ,pk&50 because then you get ^zipn ,pk&1^pn8 ,pk&1^pk8 ,pn&50 which ~using Proposition II.3! implies that Ck nAk52Ln,k i An . Since k,N , Ak Þ 0 and we may solve for Ck n yielding ~IV.2!.

Then, substituting for An by the formula in Corollary II.2, Ck nAk52Lk,n i Ak which leads to the equivalent form ~IV.3!. Furthermore, it is elementary to determine that Ln,k i 50 if i,n2k merely from the tri-diagonal form of the matrix. The main result of the present paper is the equations of motion satisfied by ai and bi . Theorem IV.1: The dependence of the distribution cˆ on the time variable ti induces the equations of motion bn85an11Ln11,n i 2anLn,n21 i ~4.4! and an85~bn2bn21!Ln21,n i 1Ln21,n11 i 2Ln22,n i . ~4.5! Proof: Since the actions of ] /] ti and multiplication by z commute, we can equate the coefficients of pn in z(pn8) and (] /] ti)(zpn). zpn85z (j50 n21 Cj n pj5(j50 n21 Cj n ~pj111bjpj1ajpj21! and so the coefficient of pn is just Cn21 n . Alternatively, ] ] ti ~pn111bnpn1anpn21!5pn8111bn8pn1bnS(j50 n21 Cj n pj D1an8pn211pn821an 252 Alex Kasman: Orthogonal polynomials and the finite Toda lattice J. Math. Phys., Vol. 38, No. 1, January 1997 and the coefficient of pn is just Cn n 111bn8 . Equating these and making use of ~IV.2! yields the equation of motion ~IV.4!. ~Here we take L0,21 i 50 to handle the boundary case n50.! Similarly, equating the coefficients of pn21 in these same expressions we get that bn21Cn21 n 1Cn22 n 5bnCn21 n 1an81Cn21 n11 . Using the substitution ~IV.3! and solving for an8 gives the desired form ~IV.5!. ~Again, L1,21 i 50 to handle the case n51.! The equations ~IV.4! and ~IV.5! are one form of the Toda hierarchy and can be written in the Lax form ] ] ti L5@L,~Li!2#, where the minus subscript indicates the projection to the lower triangular part. Since the superdiagonal elements are the only nonzero elements outside the principal N 3N minor, this is in fact an N -particle finite nonperiodic Toda lattice. Theorem IV.2: Let c be any finitely supported distribution and cˆ5c + exp Stizi. Then the corresponding matrix L is an N particle finite nonperiodic Toda lattice.

V. REMARKS As usual,3 one may write the functions ai(t) and bi(t) in terms of the t -functions t i(t): ai5 t it i12 t i11 2 , bi5 ] ]t1 log t i11 t i , for i50, . . . ,N 21 where t 0[1. This is an easier way to construct the solution corresponding to a distribution c than determining the orthogonal basis of polynomials as above.

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The points Wi P Gr described in Proposition III.1 are clearly seen to be related by the formula zWi11,Wi and are therefore related by Darboux transformations. As shown in Ref. 10, these are precisely the Darboux transformations which preserve the N-boson form of the corresponding KP solutions. The geometric spectral data is a line bundle over a rational curve with one singularity introduced by bringing together the points on a desingularization with coordinates li and multiplicity mi1i11. The inclusion of the coordinate rings induces covering maps from the more singular to the less singular curves. One may wish to consider the moduli space of all distributions c with some given value of N so as to have a moduli of N -particle nonperiodic Toda solutions. The different forms of c leading to an N -particle system are indexed by the Young diagrams of with N blocks. Given such a Young diagram, a distribution may be specified by attaching a distinct value li P C to each column and a constant ai j P C to the j11st block in the column. The different diagrams lead to qualitatively different behaviors in the corresponding solutions. In particular, the t functions give KP solitons when the Young diagrams consists entirely of columns of length one and, alternatively, they give rational KP solutions when the Young diagram has only one column. ACKNOWLEDGMENTS The author appreciates the advice and assistance of M. Rothstein, M. Bergvelt, and M. Adams.

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