Orthogonal polynomials and the finite Toda lattice - Alex Kasman

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 d r 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 d r 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., m i .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 mi

m

c5

( d l + j50 ( a i j ] zj ,

i51

i

~2.1!

where d l is the delta function evaluating its argument at z5l, the constants l i P C are distinct, ] z is the differential operator ] / ] z , and a i j P C with a i m i Þ 0. ~In fact, the discussion to follow only depends upon c as determined up to a nonzero constant multiple, and so the coefficients a i j can be viewed as elements of PN 21 C.! 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

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Alex Kasman: Orthogonal polynomials and the finite Toda lattice

m

N 5m1

( mi ,

i51

where m and m i 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 n21

p5

(

i51

n21

a iz , i

q5

(

i51

of degree less than n,

b iz i

S D b0

^ p,q & 5 ~ a 0 , . . . , a n21 ! •T n •

A

b n21

where

T n5

S

c~ 1 !

...

A

c~ z

n21

!

...

c ~ z n21 ! A c~ z

2n21

!

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 I c ,C@ z # denote the ideal I c 5 $ pPC@ z # u c+p[0 % . Lemma II.1: Let c be written in the form (II.1) and let m

s c~ z ! 5

) ~ z2l i ! m 11 . i

i51

Then I c is the ideal generated by s c (z): I c 5 s c C@ z # . Proof: Since c + s c [0, it is clear that s c (z)C@ z # ,I c . Then, let p(z) P I c be written in the form m

p ~ z ! 5q ~ z ! r ~ z ! ,

r~ z !5

) ~ z2l i ! g , i

i51

J. Math. Phys., Vol. 38, No. 1, January 1997

~2.3!


249

where q(z) P C@ z # is such that q(l i ) Þ 0. Suppose that g j , m j 11 for some particular 1< j m i 11 and I c , s c (z)C@ z # . Since deg s c 5S m i51 ( m i 11)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 i5 u T iu , where T i 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 $ p 0 , . . . , p N 21 % of P such that p i ~ z ! 5z i 1O ~ z i21 ! and which is orthogonal with respect to the form ^ •,• & . Furthermore, since t N 21 Þ 0 the form is nondegenerate on P and so

^ p i , p i & Þ0,

i50, . . . ,N 21.

It will now be supposed that c is in fact regular and that the polynomials p i for i50, . . . ,N 21 have been fixed by the Gram–Schmidt orthogonalization. We may then define p N 1i ~ z ! 5z i s c ~ z ! ,

i50,1, . . . .

By Lemma II.1, p N 1i P I c , and so it is in the kernel of the form. Therefore, the basis of monic polynomials $ p i u i>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 a i and b i in C such that zp i 5 p i11 1b i p i 1a i p i21


250


for all i.0. Proof: Since each polynomial p i is monic of degree i, we certainly have n

zp n 5 p n11 1

(

j50

a jp j

for some constants a j . But then applying the functional ^ p i ,• & to zp n yields

^ p i ,z p n & 5 ^ zp i ,p n & 5 ^ p i11 1b i p i 1a i p i21 ,p n & , which is zero if i,n22. On the other hand, one could also compute this as

^ p i ,zp n & 5 ^ p i , a i p i & 5 a i ^ p i ,p i & . If i,N , then ^ p i , p i & Þ 0 and so a i 50. Finally, for i>N , the claim is true by construction since z p i 5p i11 . Proposition II.2: Denote by A n the constant ^ p n ,p n & . Then ~i! A n 5a n A n21 , ~i! a n Þ 0 for n50, . . . ,N 21, ~iii! A n /A k 5a n •••a k11 for k,n,N . Proof : The first relationship can be found by using the fact that ^ zp,q & 5 ^ p,zq & and so

^ z p n , p n21 & 5 ^ p n11 , p n21 & 1b n ^ p n ,p n21 & 1a n ^ p n21 ,p n21 & 5a n ^ p n21 ,p n21 & is also equal to

^ p n ,zp n21 & 5 ^ p n ,p n & , producing the desired result. Then, by the nondegeneracy of the bilinear form on P , we have that a n A n21 5A n 5 ^ p n , p n & Þ 0 for 0N . Proposition IV.1: The coefficients C nk for k,n,N in (IV.1) can be written either as C nk 52

An i L A k n,k

~4.2!

or C nk 52L ik,n .

~4.3!

In particular, C nk 50 if i,n2k. Proof: This can be seen by differentiating the equation

^ p n ,p k & 50 because then you get

^ z i p n ,p k & 1 ^ p 8n ,p k & 1 ^ p 8k ,p n & 50 which ~using Proposition II.3! implies that C nk A k 52L in,k A n . Since k,N , A k Þ 0 and we may solve for C nk yielding ~IV.2!. Then, substituting for A n by the formula in Corollary II.2, C nk A k 52L ik,n A k which leads to the equivalent form ~IV.3!. Furthermore, it is elementary to determine that L in,k 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 a i and b i . Theorem IV.1: The dependence of the distribution cˆ on the time variable t i induces the equations of motion b 8n 5a n11 L in11,n 2a n L in,n21

~4.4!

i 2L in22,n . a 8n 5 ~ b n 2b n21 ! L in21,n 1L n21,n11

~4.5!

and

Proof: Since the actions of ] / ] t i and multiplication by z commute, we can equate the coefficients of p n in z(p 8n ) and ( ] / ] t i )(z p n ). n21

z p 8n 5z

(

j50

n21

C nj p j 5

(

j50

C nj ~ p j11 1b j p j 1a j p j21 !

and so the coefficient of p n is just C nn21 . Alternatively,

] 8 1b 8n p n 1b n 1b n p n 1a n p n21 ! 5 p n11 ~p ] t i n11

S( D n21 j50

8 an C nj p j 1a 8n p n21 1 p n21



253

and the coefficient of p n is just C n11 1b 8n . Equating these and making use of ~IV.2! yields the n equation of motion ~IV.4!. ~Here we take L i0,21 50 to handle the boundary case n50.! Similarly, equating the coefficients of p n21 in these same expressions we get that b n21 C nn21 1C nn22 5b n C nn21 1a 8n 1C n11 n21 . Using the substitution ~IV.3! and solving for a 8n gives the desired form ~IV.5!. ~Again, L i1,21 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

] L5 @ L, ~ L i ! 2 # , ]ti 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 cˆ 5c + 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 a i (t) and b i (t) in terms of the t -functions t i (t): a i5

t i t i12 , t 2i11

b i5

] t i11 log , ]t1 ti

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. The points W i P Gr described in Proposition III.1 are clearly seen to be related by the formula zW i11 ,W i 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 l i and multiplicity m i 1i11. 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 l i P C to each column and a constant a i 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.


254


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