Spectrum and Generation of Solutions of the Toda Lattice

Abstract

Sufficient conditions for constructing a set of solutions of the Toda lattice are analyzed. First, under certain conditions the invariance of the spectrum of 𝐽(𝑡) is established in the complex case. Second, given the tri-diagonal matrix 𝐽(𝑡) defining a Toda lattice solution, the dynamic behavior of zeros of polynomials associated to 𝐽(𝑡) 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:

𝛼̇_𝑛(𝑡)=𝜆²_𝑛+1(𝑡)−𝜆²_𝑛𝜆̇(𝑡),_𝑛+11(𝑡)=2𝜆_𝑛+1(𝑡)(𝛼_𝑛+1(𝑡)−𝛼_𝑛(𝑡)),𝑛∈ℕ(𝜆₁≡0), (1.1) where 𝜆_𝑛(𝑡),𝛼_𝑛(𝑡) are complex and differentiable functions of one real variable, 𝛼̇_𝑛(𝑡),𝜆̇_𝑛(𝑡) denote its derivatives, and we assume 𝜆_𝑛(𝑡)≠0 for each 𝑡∈ℝ and 𝑛≥2. It is well known (see [1, page 705]) that (1.1) can be expressed in the Lax pair form as

𝐽(̇𝑡)=[𝐽(𝑡),𝐾(𝑡)], (1.2) where [𝐴,𝐵]=𝐴𝐵−𝐵𝐴 is the commutator of the operators 𝐴 and 𝐵, and 𝐽(𝑡),𝐾(𝑡) are the operators for which matrix representation is given, respectively, by

𝛼𝐽(𝑡)=¹(𝑡)𝜆₂𝜆(𝑡)₂(𝑡)𝛼₂(𝑡)𝜆₃𝜆(𝑡)₃(𝑡)𝛼₃1(𝑡)⋱⋱⋱,𝐾(𝑡)=20−𝜆₂𝜆(𝑡)₂(𝑡)0−𝜆₃𝜆(𝑡)₃(𝑡)0⋱⋱⋱ (1.3) with respect to the canonical basis {𝑒_𝑖},𝑖≥0. (In what follows, we identify an operator and its matrix representation respect to the canonical basis.)

In the particular case that 𝜆_𝑛(𝑡),and𝛼_𝑛(𝑡) are real functions, under certain conditions 𝐽(𝑡) 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 𝐽(𝑡),𝑡∈ℝ, was established in [2]. In that paper, the existence of unitary operators 𝑈(𝑡) such that

𝐽(𝑡)=𝑈(𝑡)⁻¹𝐽(0)𝑈(𝑡) (1.4) for each 𝑡∈ℝ was proved. As it is well known that under these conditions the spectrum 𝜎(𝐽(𝑡)) of each operator 𝐽(𝑡) verifies

𝜎(𝐽(𝑡))=𝜎(𝐽(0)),𝑡∈ℝ. (1.5) In other words, the spectrum does not depend on 𝑡∈ℝ. 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 [5–7]). Then, due to some properties of the real Toda lattice (see, e.g., [1]), the associated Cauchy problem can be solved, recovering the solution 𝐽(𝑡) from the initial values defined by 𝐽(0).

If 𝜆_𝑛(𝑡)and𝛼_𝑛(𝑡) are complex functions, then the operator 𝐽(𝑡) 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 𝐽(𝑡) in the complex case, so we would like to establish that result in the more general possible situation, that is, when 𝐽(𝑡) 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 {𝛼_𝑛(𝑡),𝜆_𝑛+1(𝑡)},𝑛∈ℕ be a solution of (1.1) such that the sequence {𝜆_𝑛+1(𝑡)},𝑛∈ℕ, is bounded for each 𝑡∈ℝ. Then one has 𝜎(𝐽(𝑡))=𝜎(𝐽(𝑡₀)),∀𝑡,𝑡₀∈ℝ; (1.6) that is, the spectrum 𝜎(𝐽(𝑡)) of 𝐽(𝑡) is invariant on 𝑡∈ℝ.

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

𝐽̇_𝑛𝑛(𝑡)=𝐽_𝑛,𝑛+1(𝑡)𝐽^𝑝_𝑛,𝑛+1(𝑡)−𝐽_{𝑛−1,𝑛}(𝑡)𝐽^𝑝_{𝑛−1,𝑛}𝐽̇(𝑡),_𝑛,𝑛+11(𝑡)=2𝐽_𝑛,𝑛+1(𝑡)[𝐽^𝑝_{𝑛+1,𝑛+1}(𝑡)−𝐽^𝑝_𝑛,𝑛(𝑡)],𝑛=0,1,…, (1.7) where we denote by 𝐽_𝑖𝑗(𝑡) (resp., 𝐽^𝑝_𝑖𝑗(𝑡)) the entry of 𝐽(𝑡) (resp., 𝐽^𝑝(𝑡)) corresponding to the row 𝑖 and the column 𝑗 (see [8, 9]). The sequence {𝑃_𝑛(𝑡,𝑧)} of polynomials defined by the three-term recurrence relation

𝑃_𝑛+1(𝑡,𝑧)=(𝑧−𝛼_𝑛+1(𝑡))𝑃_𝑛(𝑡,𝑧)−𝜆²_𝑛+1(𝑡)𝑃_𝑛−1𝑃(𝑡,𝑧),𝑛≥0₋₁(𝑡,𝑧)≡0,𝑃₀(𝑡,𝑧)≡1 (1.8) is an important tool in the study of complex solutions of (1.7) (see [10, 11]). From {𝑃_𝑛(𝑡,𝑧)} we can define the sequence {𝑝_𝑛(𝑡,𝑧)} by

𝑝_𝑛𝑃(𝑡,𝑧)=^𝑛(𝑡,𝑧)𝜆₂(𝑡)⋯𝜆_𝑛+1(𝑡),𝑛∈ℕ. (1.9) (Obviously, the zeros of 𝑝_𝑛(𝑡,𝑧) and 𝑃_𝑛(𝑡,𝑧) 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 {𝑝_𝑛(𝑡,𝑧)} plays an important role, and the relevance of finding a point 𝐶∈ℂ which is not a root of any polynomial 𝑝_𝑛(𝑡,𝑧),𝑛∈ℕ,𝑡∈ℝ, was showed. Hence, our interest is in knowing the dynamic behaviour of 𝑝_𝑛(𝑡,𝑧) and, also, some bound for its zeros.

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

𝑃_𝑛(𝑡,𝑧)=det(𝑧𝐼_𝑛−𝐽_𝑛(𝑡)) (1.10) (see, i.e., [13]). Thus, for any 𝑡∈ℝ and 𝑛∈ℕ, the set of zeros of 𝑃_𝑛(𝑡,𝑧) coincides with the spectrum 𝜎(𝐽_𝑛(𝑡)) of 𝐽_𝑛(𝑡). When 𝐽(𝑡) is a self-adjoint operator, then the spectrum 𝜎(𝐽_𝑛(𝑡)) of each main section 𝐽_𝑛(𝑡) is contained in 𝜎(𝐽(𝑡)). So, for this kind of operators, using (1.5) and the relationship between 𝜎(𝐽_𝑛(𝑡)) and 𝜎(𝐽(𝑡)) it is possible to deduce some bound for the set of zeros of 𝑃_𝑛(𝑡,𝑧) in terms of 𝜎(𝐽(0)).

If 𝐽(𝑡) 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 𝜎(𝐽_𝑛(𝑡)) and 𝜎(𝐽(𝑡)) (see [11, 14]). For general banded matrices, the relation between the spectrum 𝜎(𝐴) of a band infinite matrix 𝐴 and the spectrum 𝜎(𝐴_𝑛) of its main sections was analyzed, under certain conditions, in [15]. More precisely, the representation 𝐴=ℜ𝐴+𝑖ℑ𝐴 was used, assuming ℜ𝐴 self-adjoint and ℑ𝐴 bounded. In our case, if we suppose that 𝐽(𝑡) verifies this restriction, then we have

𝐽(𝑡)=ℜ𝐽(𝑡)+𝑖ℑ𝐽(𝑡),𝑡∈ℝ, (1.11) where ℜ𝐽(𝑡) is a self-adjoint operator and ℑ𝐽(𝑡) is bounded. For 𝐶∈ℝ verifying

𝑑(𝐶,𝜎(ℜ𝐽_𝑛(𝑡)))>‖ℑ𝐽_𝑛(𝑡)‖, (1.12) from [15, Lemmas  1, 2] we know 𝑃_𝑛(𝑡,𝐶)≠0 or, what is the same, 𝐶∉𝜎(𝐽_𝑛(𝑡)). Moreover, taking into account that ‖ℑ𝐽(𝑡)‖≥‖ℑ𝐽_𝑛(𝑡)‖, from these results we can deduce 𝑃_𝑛(𝑡,𝐶)≠0 for any 𝑛∈ℕ when 𝐶∈ℂ is such that 𝑑(𝐶,𝜎(ℜ𝐽(𝑡)))>‖ℑ𝐽(𝑡)‖. In this way, the zeros of each sequence of polynomials {𝑃_𝑛(𝑡,𝑧)},𝑛∈ℕ, are located in the neighborhood of 𝜎(ℜ𝐽(𝑡)) given by

{𝑧∶𝑑(𝑧,𝜎(ℜ𝐽(𝑡)))≤‖ℑ𝐽_𝑛(𝑡)‖}. (1.13) Besides the above comments on Theorem 1.1 importance, this fact justifies our interest in obtaining relationship between 𝜎(𝐽(𝑡)) for different values of 𝑡∈ℝ, 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 𝑃_𝑛(𝑡,𝑧).

Theorem 1.2. Let 𝑧_𝑛1(𝑡),𝑧_𝑛2(𝑡),…,𝑧_𝑛𝑛(𝑡) be the roots of 𝑃_𝑛(𝑡,𝑧), nonnecessary distinct. Then one has 𝑧̇_𝑛𝑘(𝑝(𝑡)=^𝑛−1(𝑡,𝑧_𝑛𝑘(𝑡)))²∑^𝑛−1_𝑗=0(𝑝_𝑗(𝑡,𝑧_𝑛𝑘(𝑡)))², (1.14) understanding that 𝑧̇_𝑛𝑘(𝑡)=∞ when the multiplicity of 𝑧_𝑛𝑘(𝑡) as a zero of 𝑃_𝑛(𝑡,𝑧) would be 𝑚(𝑧_𝑛𝑘(𝑡))>1.

We stress that, in Theorem 1.2, we do not need additional conditions about the operator 𝐽(𝑡). 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 𝐶∈ℂ such that 𝑃_𝑛(𝑡,𝐶)≠0 for any 𝑛∈ℕ,𝑡∈ℝ, 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 𝒞 such that 𝒞𝑒_𝑖=𝑒_𝑖 for each vector 𝑒_𝑖,𝑖=0,1,…, in the canonical base. Thus, for any 𝑥∈ℓ² we have

𝑥=_𝑖≥0𝑥_𝑖𝑒_𝑖,𝒞𝑥=_𝑖≥0𝑥_𝑖𝑒_𝑖. (2.1) In [16], antilinear operators were introduced in order to study symmetric complex operators. In our case, we have the following auxiliaryy result for 𝒞, 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 𝐴 be a linear operator and let 𝐴^∗ be the adjoint operator of 𝐴. Then, the matrix representation of 𝒞𝐴^∗𝒞 is 𝐴^𝑇, that is, 𝐴^𝑇=𝒞𝐴^∗𝒞.
(b) 𝐽(𝑡) is a symmetric complex operator, that is, 𝐽(𝑡)=𝒞𝐽(𝑡)^∗𝒞 for each 𝑡∈ℝ.
(c) 𝐾(𝑡) is an antisymmetric operator, that is, 𝐾(𝑡)=−𝒞𝐾(𝑡)^∗𝒞 for each 𝑡∈ℝ.

Proof. Given a linear operator 𝐵, it is obvious that 𝒞𝐵𝒞 is also a linear operator. So, it is sufficient to prove the enunciated equalities for each basic vector 𝑒_𝑖. In (a), for 𝐴=(𝑎_𝑘𝑠)^∞_𝑘,𝑠=0, the column 𝑖 of 𝐴^∗ is given by 𝐴^∗𝑒_𝑖=_𝑘≥0𝑎_𝑖𝑘𝑒_𝑘, (2.2) and therefore, 𝒞𝐴^∗𝒞𝑒_𝑖=_𝑘≥0𝑎_𝑖𝑘𝑒_𝑘 (2.3) is the 𝑖 column of the transpose matrix 𝐴^𝑇. For proving (b) and (c), it is sufficient to take in account the following expressions, 𝐽(𝑡)𝑒_𝑖=𝜆_𝑖+1(𝑡)𝑒_𝑖−1+𝛼_𝑖+1(𝑡)𝑒_𝑖+𝜆_𝑖+2(𝑡)𝑒_𝑖+1,2𝐾(𝑡)𝑒_𝑖=−𝜆_𝑖+1(𝑡)𝑒_𝑖−1+𝜆_𝑖+2(𝑡)𝑒_𝑖+1,𝑖=0,1,…, (2.4) where we understand 𝑒₋₁=0. In other words, (b) and (c) can be obtained directly as a consequence of the structure of the matrices 𝐽(𝑡) and 𝐾(𝑡).

Now, we consider the following matrix initial value problem:

𝑄̇(𝑡)=𝑄(𝑡)𝐾(𝑡)𝑄(0)=𝐼. (2.5) Under the restrictions of Theorem 1.1, the operator 𝐾(𝑡) given by (1.3) is bounded. Hence, we assume that 𝐾(𝑡) is a bounded operator in the rest of the section. Moreover, assuming continuous solutions for the Toda lattice, the operator 𝐾(𝑡) is a continuous function on 𝑡∈ℝ. It is known that we can consider different kinds of continuity for a operator-value function 𝑡↦𝐴(𝑡). In our case, the function 𝑡↦𝐾(𝑡) of a real variable is continuous in norm (see [17, page 152]). Therefore, the existence of a solution 𝑄(𝑡) of (2.5) can be guaranteed (see [18, page 123]).

We have the following auxiliary result.

Lemma 2.2. Let 𝑄(𝑡) be a solution of (2.5). Then 𝑄(𝑡)𝑄(𝑡)^𝑇=𝑄(𝑡)^𝑇𝑄(𝑡)=𝐼; (2.6) that is, 𝑄(𝑡) is an invertible matrix and 𝑄(𝑡)⁻¹=𝑄(𝑡)^𝑇

Proof. Transposing the equations (2.5), since Lemma 2.1 we arrive to 𝑄̇(𝑡)^𝑇=−𝐾(𝑡)𝑄(𝑡)^𝑇𝑄(0)^𝑇=𝐼. (2.7) In other words, 𝑄(𝑡)^𝑇 is a solution of differential equation 𝑅(𝑡)=−𝐾(𝑡)𝑅(𝑡), verifying the same initial condition given by (2.5). To see this one has the following. (1)First of all, we show 𝑄(𝑡)𝑄(𝑡)^𝑇=𝐼. Using (2.5) and (2.7) we obtain 𝑑𝑑𝑡(𝑄(𝑡)𝑄(𝑡)^𝑇)=𝑄̇(𝑡)𝑄(𝑡)^𝑇+𝑄(𝑡)𝑄̇(𝑡)^𝑇=0, (2.8) then 𝑄(𝑡)𝑄(𝑡)^𝑇 is independent on 𝑡∈ℝ. From this fact and 𝑄(0)=𝐼, we deduce 𝑄(0)𝑄(0)^𝑇=𝐼 and the first part of (2.6) is proved. (2)Following [18, pages 123-124], we can write𝑄^𝑇(𝑡)=𝐼−^𝑡₀𝐾(𝜏)𝑑𝜏+_𝑛≥2(−1)^𝑛𝑡₀^𝜏1₀⋯^𝜏𝑛−1₀𝐾(𝜏₁)𝐾(𝜏₂)⋯𝐾(𝜏_𝑛)𝑑𝜏_𝑛⋯𝑑𝜏₂𝑑𝜏₁. (2.9) Due to the continuity in norm of 𝐾(𝑡), the series given in the right-hand side of (2.9) converges in norm. Even more, ‖𝑄^𝑇(𝑡)‖≤𝑒^{𝑡max[0,𝑡]‖𝐾(𝜏)‖}. (2.10) In a similar way, the series given in the right-hand side of 𝑅(𝑡)=𝐼+^𝑡₀𝐾(𝜏)𝑑𝜏+_𝑛≥2(−1)^𝑛0_𝑡^𝜏1_𝑡⋯^𝜏𝑛−1_𝑡𝐾(𝜏₁)𝐾(𝜏₂)⋯𝐾(𝜏_𝑛)𝑑𝜏_𝑛⋯𝑑𝜏₂𝑑𝜏₁ (2.11) converges in norm. Then, the above series defines the bounded operator 𝑅(𝑡) for each 𝑡∈ℝ. From straightforward computations, we obtain 𝑄^𝑇(𝑡)𝑅(𝑡)=𝑅(𝑡)𝑄^𝑇(𝑡)=𝐼,𝑡∈ℝ. (2.12) Then, from 𝑄(𝑡)𝑄^𝑇(𝑡)=𝐼 we get that 𝑅(𝑡)=𝑄(𝑡) and, finally, 𝑄^𝑇(𝑡)𝑄(𝑡)=𝐼.

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

𝑑𝑑𝑡(𝑄(𝑡)𝐽(𝑡)𝑄^𝑇(𝑡))=𝑄̇(𝑡)𝐽(𝑡)𝑄(𝑡)^𝑇+𝑄(𝑡)𝐽(̇𝑡)𝑄(𝑡)^𝑇+𝑄(𝑡)𝐽(𝑡)𝑄̇(𝑡)^𝑇=0. (2.13) Then, taking into account the initial condition in (2.5) and (2.7),

𝑄(𝑡)𝐽(𝑡)𝑄(𝑡)^𝑇=𝑄(0)𝐽(0)𝑄(0)^𝑇=𝐽(0). (2.14) From this and Lemma 2.2,

𝐽(𝑡)=𝑄(𝑡)^𝑇𝐽(0)𝑄(𝑡). (2.15) Therefore, 𝐽(0) and 𝐽(𝑡) are equivalent operators, and we have, as a consequence,

𝜎(𝐽(𝑡))=𝜎(𝐽(0)) (2.16) for each 𝑡∈ℝ. So, 𝜎(𝐽(𝑡)) is independent on 𝑡∈ℝ, as we wanted to prove.

2.2. Proof of Theorem 1.2

Taking 𝑝=1 in (2.6) of [11, Theorem  2], we obtain

𝑃̇_𝑛(𝑡,𝑧)=−𝜆²_𝑛+1(𝑡)𝑃_𝑛−1(𝑡,𝑧) (2.17) for each 𝑛∈ℕ and all 𝑧∈ℂ. Then, writing

𝑃_𝑛(𝑡,𝑧)=^𝑛_𝑖=1(𝑧−𝑧_𝑛𝑖(𝑡)) (2.18) and taking derivatives with respect to 𝑡, we have

𝑃̇_𝑛(𝑡,𝑧)=−^𝑛_𝑖=1𝑧̇_𝑛𝑖(𝑡)_𝑗≠𝑖(𝑧−𝑧_𝑛𝑗(𝑡)). (2.19) With the notation established in Section 1, for each fixed zero 𝑧=𝑧_𝑛𝑘(𝑡) of 𝑃_𝑛(𝑡,𝑧) the right-hand side of (2.17) is not zero. As a matter of fact, we have 𝜆_𝑛+1(𝑡)≠0, and, if we suppose 𝑃_𝑛(𝑡,𝑧_𝑛𝑘(𝑡))=𝑃_𝑛−1(𝑡,𝑧_𝑛𝑘(𝑡))=0, then using the recurrence relation (1.8) we will arrive to 𝑃_𝑛−2(𝑡,𝑧_𝑛𝑘(𝑡))=0 and, iterating, to 𝑃₀(𝑡,𝑧_𝑛𝑘(𝑡))=0, which is not possible being 𝑃₀≡1.

Comparing (2.17) and (2.19) for 𝑧=𝑧_𝑛𝑘(𝑡), we see

𝑧̇^𝑛_{𝑖=1𝑛𝑖}(𝑡)_𝑗≠𝑖(𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑗(𝑡))=𝜆²_𝑛+1(𝑡)𝑃_𝑛−1(𝑡,𝑧_𝑛𝑘(𝑡)),𝑘=1,…,𝑛. (2.20) Moreover, ∏_𝑗≠𝑖(𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑗(𝑡))=0 when 𝑖≠𝑘. Therefore, from (2.20) we have

𝑧̇_𝑛𝑘(𝑡)_𝑗≠𝑘(𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑗(𝑡))=𝜆²_𝑛+1(𝑡)𝑃_𝑛−1(𝑡,𝑧_𝑛𝑘(𝑡)),𝑘=1,…,𝑛, (2.21) and, consequently,

𝑧̇_𝑛𝑘(𝑡)_𝑗≠𝑘(𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑗(𝑡))≠0,𝑘=1,…,𝑛. (2.22) We will take in consideration the-two possible cases following.

(i)If the multiplicity of 𝑧_𝑛𝑘(𝑡) as a zero of 𝑃_𝑛(𝑡,𝑧) is 𝑚_𝑛𝑘(𝑡)>1, then the factor 𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑘(𝑡) is in the left-hand side of (2.22), so 𝑧̇_𝑛𝑘(𝑡)=∞. (ii)If 𝑧_𝑛𝑘(𝑡) is a simple zero of 𝑃_𝑛(𝑡,𝑧), then, from (2.23), we obtain

𝑧̇_𝑛𝑘𝜆(𝑡)=^2𝑛+1(𝑡)𝑃_𝑛−1(𝑡,𝑧_𝑛𝑘(𝑡))∏_𝑗≠𝑘(𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑗.(𝑡)) (2.23)

On the other hand, writing

𝑃_𝑛(𝑡,𝑧(𝑡))=^𝑛_𝑖=1(𝑧−𝑧_𝑛𝑖(𝑡)) (2.24) and taking derivatives with respect to 𝑧,

𝑃^′_𝑛(𝑡,𝑧)=^𝑛_{𝑖=1𝑗≠𝑖}(𝑧−𝑧_𝑛𝑗(𝑡)). (2.25) So,

𝑃^′_𝑛(𝑡,𝑧_𝑛𝑘(𝑡))=_𝑗≠𝑘(𝑧_𝑛𝑘(𝑡)−𝑧_𝑛𝑗(𝑡)). (2.26) Moreover, the following formula is well known:

^𝑛−1_𝑗=0(𝑝_𝑗(𝑡,𝑧_𝑛𝑘(𝑡)))²=𝑃^′_𝑛(𝑡,𝑧_𝑛𝑘(𝑡))𝑃_𝑛−1(𝑡,𝑧_𝑛𝑘(𝑡))(𝜆₂(𝑡)⋯𝜆_𝑛+1(𝑡))² (2.27) (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 𝑧_𝑛𝑘(𝑡) 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 𝑃_𝑛(𝑡,𝑧) depend on 𝑡∈ℝ because its derivatives are not zero. Moreover, in the case of real Toda lattices, that is, when the coefficients 𝛼_𝑛(𝑡),𝜆_𝑛(𝑡) in (1.8) are real functions, we have 𝑧̇_𝑛𝑘(𝑡)>0,𝑘=1,2,…,𝑛. Then, in this case 𝑧_𝑛𝑘(𝑡),𝑘=1,2,…,𝑛, are monotonically increasing functions of 𝑡∈ℝ. For each fixed 𝑛,𝑧_𝑛𝑘(𝑡),𝑘=1,2,…,𝑛, are simple zeros of 𝑃_𝑛(𝑡,𝑧). Then, 𝑧_𝑛𝑘(𝑡)≠𝑧_𝑛𝑘′(𝑡) for 𝑘≠𝑘^′,𝑘,𝑘^′=1,…,𝑛, and, therefore, the curves {𝑧_𝑛𝑘(𝑡)∶𝑘=1,2,…,𝑛} have no points in common.
(ii) Let 𝐽(𝑡) be a bounded operator. It is a consequence of Theorem 1.1 that ‖𝐽(𝑡)‖ is independent on 𝑡∈ℝ. Then, for each 𝑛∈ℕ,
|𝑧_𝑛𝑘(𝑡)|≤‖𝐽(𝑡)‖≤𝑀,𝑘=1,…,𝑛,𝑛∈ℕ. (2.28) From this fact and (i), we deduce lim_𝑡→∞𝑧_𝑛𝑘(𝑡)=𝑚_𝑘∈ℝ,𝑘=1,…,𝑛; (2.29) that is, each curve 𝑧_𝑛𝑘(𝑡),𝑡∈ℝ, has an asymptotic line 𝑧=𝑚_𝑘 in the (𝑡,𝑧)—plane.
(iii) When the entries of 𝐽(𝑡) are not real functions, then we do not know the multiplicity of 𝑧_𝑛𝑘(𝑡) as a zero of 𝑃_𝑛(𝑡,𝑧). 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):

𝛼_𝑛(𝑡)=𝑒^𝑡𝜆+𝑛−1,_𝑛(𝑡)=(𝑛−1)𝑒^𝑡,𝑛∈ℕ. (3.1) With the notation employed in the above sections, we have

𝑒𝐽(𝑡)=^𝑡√𝑒^𝑡√𝑒^𝑡𝑒^𝑡+1√2𝑒^𝑡√2𝑒^𝑡𝑒^𝑡+2⋱⋱⋱. (3.2) Since ∑_𝑛≥1(1/√𝑛𝑒^𝑡) is a divergent series, the Carleman condition ([19, page 59]) indicates that 𝐽(𝑡) is a self-adjoint operator. Moreover, it is easy to see that det(𝐽_𝑛(𝑡))=𝑒^𝑛𝑡,𝑛∈ℕ, and therefore, 𝐽(𝑡) is a positive-definite operator. From both issues, we can get that

𝜎(𝐽_𝑛(𝑡))⊂[0,+∞). (3.3) Then, (3.1) is an example of solution of (1.1) for which the associated polynomials {𝑃_𝑛(𝑡,𝑧)} have all their zeros in [0,+∞). 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 𝐶∈ℂ⧵[0,+∞) and we apply the method given in [11]. Here, we explain and illustrate that method. Let

𝐽⁽¹⁾𝑒(𝑡)∶=^𝑡𝑒^𝑡1𝑒^𝑡+12𝑒^𝑡1𝑒^𝑡+2⋱⋱⋱. (3.4) Due to the fact that 𝑃_𝑛(𝑡,𝐶)≠0, we have

det(𝐽_𝑛(𝑡)−𝐶𝐼_𝑛)=det(𝐽_𝑛⁽¹⁾(𝑡)−𝐶𝐼_𝑛)≠0. (3.5) Thus, 𝐽⁽¹⁾(𝑡)−𝐶𝐼 admits the formal representation given by

𝐽⁽¹⁾(𝑡)−𝐶𝐼=𝐿(𝑡)𝑈(𝑡), (3.6) ([20, Theorem  1, page 35]), where

𝑙𝐿(𝑡)∶=¹¹𝑙(𝑡)₂₁(𝑡)𝑙₂₂𝑙(𝑡)₃₂(𝑡)𝑙₃₃(𝑡)⋱⋱,𝑈(𝑡)∶=1𝑢₁₂(𝑡)1𝑢₂₃(𝑡)1𝑢₃₄(𝑡)⋱⋱. (3.7) (Despite the fact that the entries in both matrices depend on 𝐶, in order to simplify our notation we do not write down explicitly this dependence.) More precisely, for each 𝑚∈ℕ we obtain

𝑙_𝑚𝑚(𝑡)=𝑒^𝑡+𝑚−1−𝐶−(𝑚−1)𝑒^𝑡𝑒^𝑡+𝑚−2−𝐶−(𝑚−2)𝑒^𝑡/⋱−𝑒^𝑡/(𝑒^𝑡,𝑙−𝐶)_𝑚+1,𝑚(𝑡)=1,𝑢_𝑚,𝑚+1(𝑡)=𝑚𝑒^𝑡𝑙_𝑚𝑚.(𝑡) (3.8) Then, the new obtained solution, generated from 𝐽(𝑡) and 𝐶, is given as {𝛼_𝑛(𝑡),𝜆_𝑛+1(𝑡)},𝑛∈ℕ, being

𝛼𝑈(𝑡)𝐿(𝑡)∶=¹(𝑡)−𝐶(𝜆₂(𝑡))²1𝛼₂(𝑡)−𝐶(𝜆₃(𝑡))²1𝛼₃(𝑡)−𝐶⋱⋱⋱. (3.9) In other words, for each 𝐶∈ℂ⧵[0,+∞) a new solution of the Toda lattice can be generated from the product 𝑈(𝑡)𝐿(𝑡). In this way, a sequence of solutions can be obtained iterating this process. In our example, the new complex solution is given by

𝛼₁(𝑡)=𝑒^𝑡+𝑒^𝑡𝑒^𝑡,𝛼−𝐶₂(𝑡)=𝑒^𝑡𝑒+1−^𝑡𝑒^𝑡+−𝐶2𝑒^𝑡𝑒^𝑡+1−𝐶−𝑒^𝑡/(𝑒^𝑡𝜆−𝐶),⋯₂𝑒(𝑡)=^𝑡𝑒^𝑡−𝐶(𝑒^𝑡𝑒+1−𝐶−^𝑡𝑒^𝑡𝜆−𝐶),₃(𝑡)=2𝑒^𝑡𝑒^𝑡+1−𝐶−𝑒^𝑡/(𝑒^𝑡−𝐶)(𝑒^𝑡+2−𝐶2𝑒^𝑡𝑒^𝑡+1−𝐶−𝑒^𝑡/(𝑒^𝑡−𝐶)),⋯. (3.10) Because our initial solution {𝛼_𝑛(𝑡),𝜆_𝑛+1(𝑡)},𝑛∈ℕ, is a real solution, we know 𝜎(𝐽(𝑡))=𝜎(𝐽(𝑡₀)) for any 𝑡,𝑡₀∈ℝ. Moreover,

𝜎(𝐽(̃𝑡))⧵{𝐶}=𝜎(𝐽(𝑡))⧵{𝐶} (3.11) (see [21, Proposition  3.6, page 225]). Thus

𝜎(𝐽(̃𝑡))⧵{𝐶}=𝜎(𝐽(𝑡₀))⧵{𝐶}. (3.12) Because {𝜆_𝑛(𝑡)},𝑛∈ℕ, 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.

Acknowledgments

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.