Sufficient conditions for constructing a set of solutions of the Toda lattice are analyzed. First, under certain conditions the invariance of the spectrum of is established in the complex case. Second, given the tri-diagonal matrix defining a Toda lattice solution, the dynamic behavior of zeros of polynomials associated to is analyzed. Finally, it is shown by means of an example how to apply our results to generate complex solutions of the Toda lattice starting with a given solution.

1. Introduction

We consider the Toda lattice given by the following equations:

where are complex and differentiable functions of one real variable, denote its derivatives, and we assume for each and It is well known (see [1, page 705]) that (1.1) can be expressed in the Lax pair form as

where is the commutator of the operators and , and are the operators for which matrix representation is given, respectively, by

with respect to the canonical basis (In what follows, we identify an operator and its matrix representation respect to the canonical basis.)

In the particular case that are real functions, under certain conditions is a self-adjoint operator. This property has several consequences in the study of system (1.1). For instance, in this situation the unitary equivalence between operators was established in [2]. In that paper, the existence of unitary operators such that

for each was proved. As it is well known that under these conditions the spectrum of each operator verifies

In other words, the spectrum does not depend on . This fact permits to use the self-adjoint operator theory to analyze the integrability of system (1.1) (see [3, 4]). These tools can be used, also, in more general systems (see [57]). Then, due to some properties of the real Toda lattice (see, e.g., [1]), the associated Cauchy problem can be solved, recovering the solution from the initial values defined by .

If are complex functions, then the operator given in (1.3) is not any longer a self-adjoint operator. Therefore, some of the assumptions of [2] are not verified. To the best of our knowledge there is no proof of the invariance of the spectrum of in the complex case, so we would like to establish that result in the more general possible situation, that is, when is not necessarily a bounded operator. However, we think that some advance, in this sense, is a relevant contribution in the study of solutions of the Toda lattice. This is related with our first result.

Theorem 1.1. Let be a solution of (1.1) such that the sequence , is bounded for each . Then one has that is, the spectrum of is invariant on

System (1.1) is a particular case of the generalized Toda lattice of order ;

where we denote by (resp., ) the entry of (resp., ) corresponding to the row and the column (see [8, 9]). The sequence of polynomials defined by the three-term recurrence relation

is an important tool in the study of complex solutions of (1.7) (see [10, 11]). From we can define the sequence by

(Obviously, the zeros of and are the same.) Beside some other results, the bases of a method for obtaining new solutions of (1.7) from a given solution were established in [11]. In that paper, the location of zeros of the sequence plays an important role, and the relevance of finding a point which is not a root of any polynomial was showed. Hence, our interest is in knowing the dynamic behaviour of and, also, some bound for its zeros.

Denote by the finite-dimensional matrix of order defined by the first rows and columns of (see [12]). From (1.8), it can be easily established that

(see, i.e., [13]). Thus, for any and , the set of zeros of coincides with the spectrum of . When is a self-adjoint operator, then the spectrum of each main section is contained in . So, for this kind of operators, using (1.5) and the relationship between and it is possible to deduce some bound for the set of zeros of in terms of .

If is not a self-adjoint, to get some knowledge about the behaviour of solutions of (1.7) as well as (1.1) is very difficult. This is due to the lack of a general result about the relationship between and (see [11, 14]). For general banded matrices, the relation between the spectrum of a band infinite matrix and the spectrum of its main sections was analyzed, under certain conditions, in [15]. More precisely, the representation was used, assuming self-adjoint and bounded. In our case, if we suppose that verifies this restriction, then we have

where is a self-adjoint operator and is bounded. For verifying

from [15, Lemmas  1, 2] we know or, what is the same, . Moreover, taking into account that , from these results we can deduce for any when is such that . In this way, the zeros of each sequence of polynomials are located in the neighborhood of given by

Besides the above comments on Theorem 1.1 importance, this fact justifies our interest in obtaining relationship between for different values of , because bounding the zeros in a certain region of the complex plane permits to work with the method given in [11] in the complement of the region zeros free.

On the other hand, in conditions under which there are not any information about the dynamic behaviour of the spectrum, our following result gives complementary information about the knowledge and the dynamic behavior of zeros of .

Theorem 1.2. Let be the roots of , nonnecessary distinct. Then one has understanding that when the multiplicity of as a zero of would be .

We stress that, in Theorem 1.2, we do not need additional conditions about the operator . Theorems 1.1 and 1.2 are complementary results, in the sense that both can be used for determining conditions to obtain some new solutions of (1.1) and (1.7).

Section 2 is devoted to prove Theorems 1.1 and 1.2. After the existence of such that for any , can be guaranteed, we will show, in Section 3, how to construct a new solution of (1.1) from a given solution.

2. Invariance of Spectrum versus Variation of Zeros of Polynomials

2.1. Proof of Theorem 1.1

We define the antilinear operator such that for each vector in the canonical base. Thus, for any we have

In [16], antilinear operators were introduced in order to study symmetric complex operators. In our case, we have the following auxiliaryy result for , which justifies the definition of transpose operator (see [16, page 2]). We recall that we identify an operator with its matrix representation.

Lemma 2.1. (a) Let be a linear operator and let be the adjoint operator of . Then, the matrix representation of is , that is, .
(b) is a symmetric complex operator, that is, for each .
(c) is an antisymmetric operator, that is, for each .

Proof. Given a linear operator , it is obvious that is also a linear operator. So, it is sufficient to prove the enunciated equalities for each basic vector . In (a), for , the column of is given by and therefore, is the column of the transpose matrix . For proving (b) and (c), it is sufficient to take in account the following expressions, where we understand . In other words, (b) and (c) can be obtained directly as a consequence of the structure of the matrices and .

Now, we consider the following matrix initial value problem:

Under the restrictions of Theorem 1.1, the operator given by (1.3) is bounded. Hence, we assume that is a bounded operator in the rest of the section. Moreover, assuming continuous solutions for the Toda lattice, the operator is a continuous function on . It is known that we can consider different kinds of continuity for a operator-value function . In our case, the function of a real variable is continuous in norm (see [17, page 152]). Therefore, the existence of a solution of (2.5) can be guaranteed (see [18, page 123]).

We have the following auxiliary result.

Lemma 2.2. Let be a solution of (2.5). Then that is, is an invertible matrix and

Proof. Transposing the equations (2.5), since Lemma 2.1 we arrive to In other words, is a solution of differential equation , verifying the same initial condition given by (2.5). To see this one has the following. (1)First of all, we show . Using (2.5) and (2.7) we obtain then is independent on . From this fact and , we deduce and the first part of (2.6) is proved. (2)Following [18, pages 123-124], we can write Due to the continuity in norm of , the series given in the right-hand side of (2.9) converges in norm. Even more, In a similar way, the series given in the right-hand side of converges in norm. Then, the above series defines the bounded operator for each . From straightforward computations, we obtain Then, from we get that and, finally, .

Now, we will finish the proof of Theorem 1.1. For this purpose, take the solution of (2.5). Using (1.2), (2.5), and (2.7), we immediately arrive to

Then, taking into account the initial condition in (2.5) and (2.7),

From this and Lemma 2.2,

Therefore, and are equivalent operators, and we have, as a consequence,

for each . So, is independent on , as we wanted to prove.

2.2. Proof of Theorem 1.2

Taking in (2.6) of [11, Theorem  2], we obtain

for each and all . Then, writing

and taking derivatives with respect to , we have

With the notation established in Section 1, for each fixed zero of the right-hand side of (2.17) is not zero. As a matter of fact, we have , and, if we suppose , then using the recurrence relation (1.8) we will arrive to and, iterating, to , which is not possible being .

Comparing (2.17) and (2.19) for , we see

Moreover, when . Therefore, from (2.20) we have

and, consequently,

We will take in consideration the-two possible cases following.

(i)If the multiplicity of as a zero of is , then the factor is in the left-hand side of (2.22), so . (ii)If is a simple zero of , then, from (2.23), we obtain

On the other hand, writing

and taking derivatives with respect to ,


Moreover, the following formula is well known:

(see [13, page 24]).

Finally, from (2.23), (2.26), and (2.27) we arrive to (1.14).

We point out that (2.27) also holds in the case (i) when is not a simple zero and the denominator in (1.14) is zero.

Remark 2.3. (i) It follows, from Theorem 1.2, that the zeros of each polynomial depend on because its derivatives are not zero. Moreover, in the case of real Toda lattices, that is, when the coefficients in (1.8) are real functions, we have . Then, in this case , are monotonically increasing functions of . For each fixed are simple zeros of . Then, for , and, therefore, the curves have no points in common.
(ii) Let be a bounded operator. It is a consequence of Theorem 1.1 that is independent on . Then, for each ,
From this fact and (i), we deduce that is, each curve , has an asymptotic line in the —plane.
(iii) When the entries of are not real functions, then we do not know the multiplicity of as a zero of . Therefore, in the complex case it is possible that (i) and (ii) are not longer true.

3. Obtaining Some New Solutions of Toda Lattice

Consider the following solution of (1.1):

With the notation employed in the above sections, we have

Since is a divergent series, the Carleman condition ([19, page 59]) indicates that is a self-adjoint operator. Moreover, it is easy to see that , and therefore, is a positive-definite operator. From both issues, we can get that

Then, (3.1) is an example of solution of (1.1) for which the associated polynomials have all their zeros in . The dynamic behavior of these zeros was determined in Theorem 1.2 and Remark 2.3.

From (3.1), it is possible to obtain some complex solutions of (1.1). For this purpose we take and we apply the method given in [11]. Here, we explain and illustrate that method. Let

Due to the fact that , we have

Thus, admits the formal representation given by

([20, Theorem  1, page 35]), where

(Despite the fact that the entries in both matrices depend on , in order to simplify our notation we do not write down explicitly this dependence.) More precisely, for each we obtain

Then, the new obtained solution, generated from and , is given as being

In other words, for each a new solution of the Toda lattice can be generated from the product . In this way, a sequence of solutions can be obtained iterating this process. In our example, the new complex solution is given by

Because our initial solution is a real solution, we know for any . Moreover,

(see [21, Proposition  3.6, page 225]). Thus

Because is not a bounded sequence, we cannot apply Theorem 1.1. However, from (3.12) we conjecture that Theorem 1.1 could be extended to a more general situation.


The work of the first author was supported in part by Dirección General de Proyectos de Investigación, Ministerio de Ciencia e Innovación, under Grant MTM2006-13000-C03-02. This author also thank Universidad Politécnica de Madrid for the financial support received through the research group Teoría de Aproximación Constructiva y Aplicaciones.