A class of complex solutions to the finite Toda lattice

Mathematical and Computer Modelling 57 (2013) 1190–1202

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A class of complex solutions to the finite Toda lattice Gusein Sh. Guseinov Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey

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Article history: Received 22 November 2011 Received in revised form 13 October 2012 Accepted 15 October 2012 Keywords: Toda lattice Jacobi matrix Difference equation Spectral data Inverse spectral problem

abstract In this paper, a class of complex-valued solutions to the finite Toda lattice is constructed by using the inverse spectral method. The corresponding Lax operator is a finite complex Jacobi matrix. As the initial values there are taken such complex numbers that the corresponding Jacobi matrix has a simple spectrum. Some examples are given. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction As is known, for the (open) finite Toda lattice the corresponding Lax operator is a finite order Jacobi matrix (tri-diagonal symmetric matrix). Therefore, the initial value problem for the finite Toda lattice can be solved by means of the inverse spectral method using solutions of the inverse spectral problems for finite Jacobi matrices. If the initial values are real, then the corresponding Jacobi matrix is real (has real entries) and the inverse spectral method yields all real solutions to the Toda lattice. However, if the initial values are complex, then the corresponding Jacobi matrix is complex and therefore the inverse spectral method allows to construct all complex solutions to the Toda lattice. In the case of real entries the finite Jacobi matrix is selfadjoint and its eigenvalues are real and distinct. In the complex case the Jacobi matrix is, in general, no longer selfadjoint and its eigenvalues may be complex and multiple. This circumstance gives rise to significant complication of solution of both the inverse spectral problem and the complex Toda lattice. In the present paper, we consider the finite complex Toda lattice and we put an additional simplifying condition: we suppose that the spectrum of the Jacobi matrix associated with the complex-valued initial data is simple (i.e., we suppose that the complex Jacobi matrix has only eigenvalues of multiplicity 1). This extra condition simplifies the solutions of the inverse spectral problem and the Toda lattice and allows to get more specific results. We introduce the concept of spectral data for finite order complex Jacobi matrices with a simple spectrum and present a solution of the inverse problem of recovering the matrix with a simple spectrum from its spectral data. We give an application to the solving of the finite Toda lattice by the method of inverse spectral problem and construct a class of complex solutions. This class of solutions contains all real solutions together with some non-real solutions to the finite Toda lattice. A distinguishing feature of the Jacobi matrices from other matrices is that they are related to certain three-term recursion equations (second order linear difference equations). Therefore these matrices can be viewed as the discrete analogue of Sturm–Liouville operators and their investigation has many similarities with Sturm–Liouville theory. There are three kinds of the Toda lattices: finite, semi-infinite, and doubly-infinite Toda lattices. A huge number of papers have been devoted to the investigation of the Toda lattices and their various generalizations, from which we indicate here only [1–12]. In the present paper we deal with the finite Toda lattice.

E-mail address: [email protected]. 0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.10.022

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This paper is organized as follows. In Section 2 we introduce the finite Toda lattice and its Lax representation. In Section 3, following the author’s paper [13], we introduce the concept of spectral data for finite complex Jacobi matrices with a simple spectrum and solve the inverse problem which consists of recovering the matrix from its spectral data. The time evolution of the spectral data of the time-dependent Jacobi matrix associated with the solution of the Toda lattice is computed in Section 4. In Section 5, the solving procedure of the Toda lattice by the method of inverse spectral problem is presented and illustrated in the two particles case N = 2. Finally, in Section 6, we discuss complex solutions to the Toda lattice which have real energy. 2. The finite Toda lattice The (open) finite Toda lattice is a nonlinear Hamiltonian system which describes the motion of N particles moving in a straight line, with ‘‘exponential interactions’’. Adjacent particles are connected by strings. Let the positions of the particles at time t be q0 (t ), q1 (t ), . . . , qN −1 (t ), where qn = qn (t ) is the displacement at the moment t of the n-th particle from its equilibrium position. We assume that each particle has mass 1. The momentum of the n-th particle at time t is therefore pn = q˙ n . The Hamiltonian function is taken to be N −1 1

H =

2 n =0

p2n +

N −2 

eqn −qn+1 .

n =0

The Hamiltonian system q˙ n =

∂H , ∂ pn

p˙ n = −

∂H ∂ qn

becomes q˙ n = pn ,

n = 0, 1, . . . , N − 1,

p˙ 0 = −eq0 −q1 , p˙ n = eqn−1 −qn − eqn −qn+1 , p˙ N −1 = e

qN −2 −qN −1

n = 1, 2, . . . , N − 2,

,

where the dot denotes differentiation with respect to t. Let us set 1 (qn −qn+1 )/2 e , n = 0, 1, . . . , N − 2, 2 1 bn = − pn , n = 0, 1, . . . , N − 1. 2

an =

Then the above system can be written in the form a˙ n = an (bn+1 − bn ),

b˙ n = 2(a2n − a2n−1 ),

n = 0, 1, . . . , N − 1,

(1)

with the boundary conditions a−1 = aN −1 = 0.

(2)

If we define the N × N matrices J and A by

b

0

 a0 0  . J =  ..  0  0 0

0 a0 0  . A =  ..  0  0 0

a0 b1 a1

.. .

··· ··· ··· .. .

0 0 0

0 0 0

··· ··· ···

b N −3 aN − 3 0

0 a1 b2

.. .

0 0 0

−a 0

0 0 0

.. .

0 0 0

.. .

.. .

aN − 3 bN −2 aN − 2

0 aN − 2 b N −1

     ,   

(3)

0 a1

0 −a 1 0

.. .

··· ··· ··· .. .

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

··· ··· ···

0

−a N − 3

aN − 3 0

0 aN − 2

0 −a N − 2 0

.. .

.. .

.. .

.. .

     ,   

(4)

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then the system (1) with the boundary conditions (2) is equivalent to the Lax equation [3,9] d dt

J = [J , A] = JA − AJ .

(5)

The system (1), (2) is considered subject to the initial conditions an (0) = a0n ,

bn (0) = b0n ,

n = 0, 1, . . . , N − 1,

(6)

where a0n , b0n are given complex numbers such that a0n ̸= 0 (n = 0, 1, . . . , N − 2), = 0. Existence of the Lax representation (5) allows us to find solutions to the nonlinear initial-boundary value problem (1), (2), (6) by means of the inverse spectral problem method. Note that Eq. (1) subject to the boundary condition (2) is related to the so-called open finite Toda lattice (i.e., the finite Toda lattice with free ends). Whereas Eq. (1) subject to the boundary condition a−1 = aN −1 equals a nonzero constant is related to the so-called periodic infinite Toda lattice. In the case of real solutions the integration of the open Toda lattice was fulfilled in [6,10] by the inverse spectral method, using orthogonal polynomials in [6] and continued fractions in [10]. The integration of the periodic Toda lattice is much more difficult and uses algebraic geometric methods (see [7,8,12]). In the present paper we deal only with the open Toda lattice. a0N −1

3. Inverse problem for finite complex Jacobi matrices with a simple spectrum In this section we follow the author’s paper [13]. An N × N complex Jacobi matrix is a matrix of the form

b

0

 a0 0  .

J =  ..  0  0 0

a0 b1 a1

.. .

··· ··· ··· .. .

0 0 0

0 0 0

··· ··· ···

bN −3 aN − 3 0

0 a1 b2

.. .

0 0 0

0 0 0

.. .

.. .

aN − 3 b N −2 aN − 2

0 0 0

.. .

0 aN − 2 b N −1

     ,   

(7)

where for each n, an and bn are arbitrary complex numbers such that an is different from zero: an , bn ∈ C,

an ̸= 0.

(8)

Given a Jacobi matrix J of the form (7) with the entries (8), consider the eigenvalue problem Jy = λy for a column vector y = {yn }Nn=−01 , that is equivalent to the second order linear difference equation (three-term recursion equation)

 an−1 yn−1 + bn yn + an yn+1 = λyn , yn Nn=−1 ,

for { }

n ∈ {0, 1, . . . , N − 1},

(9)

with the boundary conditions

y−1 = yN = 0, where

 a− 1 =  aN −1 = 1,

 an = an for all n ∈ {0, 1, . . . , N − 2}.

Denote by {Pn (λ)}Nn=−1 and {Qn (λ)}Nn=−1 the solutions of Eq. (9) satisfying the initial conditions P−1 (λ) = 0, Q−1 (λ) = −1,

P0 (λ) = 1; Q0 (λ) = 0.

(10) (11)

For each n ≥ 0, Pn (λ) is a polynomial of degree n and is called a polynomial of the first kind and Qn (λ) is a polynomial of degree n − 1 and is known as a polynomial of the second kind. The equality det (J − λI ) = (−1)N a0 a1 · · · aN −2 PN (λ)

(12)

holds so that the eigenvalues of the matrix J coincide with the zeros of the polynomial PN (λ). Let R(λ) = (J − λI )−1 be the resolvent of the matrix J (by I we denote the identity matrix of needed dimension) and e0 be the N-dimensional column vector with the components 1, 0, . . . , 0. The rational function

  w(λ) = − ⟨R(λ)e0 , e0 ⟩ = (λI − J )−1 e0 , e0 ,

(13)

we call the resolvent function of the matrix J, where ⟨·, ·⟩ stands for the standard inner product in CN . This function is known also as the Weyl–Titchmarsh function of J.

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In [13] it is shown that the entries Rnm (λ) of the matrix R(λ) = (J − λI )−1 (resolvent of J) are of the form Rnm (λ) =



Pn (λ)[Qm (λ) + M (λ)Pm (λ)], Pm (λ)[Qn (λ) + M (λ)Pn (λ)],

0 ≤ n ≤ m ≤ N − 1, 0 ≤ m ≤ n ≤ N − 1,

where M (λ) = −

QN (λ) PN (λ)

.

Therefore according to (13) and using initial conditions (10), (11), we get

w(λ) = −R00 (λ) = −M (λ) =

QN (λ) PN (λ)

.

(14)

As is shown in [13], the equations PN −1 (λ)QN (λ) − PN (λ)QN −1 (λ) = 1,

(15)

N −1

PN −1 (λ)PN′ (λ) − PN (λ)PN′ −1 (λ) =



Pn2 (λ)

(16)

n =0

hold, where the prime denotes the derivative with respect to λ. Lemma 1. If PN (λ0 ) = 0, then {Pn (λ0 )}Nn=−01 is an eigenvector of J corresponding to the eigenvalue λ0 and any eigenvector of J corresponding to the eigenvalue λ0 is a constant multiple of {Pn (λ0 )}Nn=−01 . Proof. It is easy to see that an arbitrary solution {yn }Nn=−1 of Eq. (9) satisfying the initial condition y−1 = 0 has the form yn = α Pn (λ) (n = −1, 0, 1, . . . , N ) with some constant α . This solution is nontrivial for α ̸= 0 and satisfies the condition yN = 0 if and only if PN (λ) = 0. Hence the statements of the lemma follow.  Lemma 2. If PN (λ0 ) = 0, then PN −1 (λ0 ) ̸= 0. Proof. Suppose that PN (λ0 ) = 0. Then putting λ = λ0 in (15) we get that PN −1 (λ0 ) ̸= 0.



Definition 3. We will say that a Jacobi matrix J of the form (7), (8) has a simple spectrum if it has precisely N distinct (complex) eigenvalues. Note that, as is well known (see, e.g. [13,14]), any real N × N Jacobi matrix has precisely N real and distinct eigenvalues and, therefore, has a simple spectrum. However, some non-real Jacobi matrices also may have a simple (in general complex) spectrum. Since the PN (λ) is a polynomial of degree N, it has N zeros λ1 , . . . , λN (which coincide by (12) with the eigenvalues of the matrix J), PN (λ) = c (λ − λ1 ) · · · (λ − λN ), where c is a nonzero constant. If the matrix J has a simple spectrum, then the zeros λ1 , . . . , λN of the polynomial PN (λ) must be distinct. Thus the Jacobi matrix J has a simple spectrum if and only if the polynomial PN (λ) has only simple zeros. Hence we have the following statement. Lemma 4. If the Jacobi matrix J has a simple spectrum and PN (λ0 ) = 0, then PN′ (λ0 ) ̸= 0. The following lemma presents another characterization of the matrix J with a simple spectrum. Lemma 5. If the Jacobi matrix J has a simple spectrum and λ0 is an eigenvalue of J, then N −1 

Pn2 (λ0 ) ̸= 0.

n =0

Proof. The eigenvalue λ0 of the matrix J is, by (12), a zero of the polynomial PN (λ) : PN (λ0 ) = 0. Putting λ = λ0 in (16) and using PN (λ0 ) = 0, we get PN −1 (λ0 )PN′ (λ0 ) =

N −1 

Pn2 (λ0 ).

(17)

n =0

The left-hand side of (17) is different from zero by Lemmas 2 and 4. This completes the proof.



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Next we will use the following known simple useful lemma (for its proof see [14, Lemma 2.1]). Lemma 6. Let A(λ) and B(λ) be polynomials with complex coefficients and deg A < deg B. Next, suppose that B(λ) = b(λ − z1 ) · · · (λ − zN ), where z1 , . . . , zN are distinct complex numbers and b is a nonzero complex number. Then there exist uniquely determined complex numbers a1 , . . . , aN such that A(λ) B(λ)

=

N 

ak

k=1

λ − zk

for all values of λ different from z1 , . . . , zN . The numbers ak are given by the equation ak = lim (λ − zk ) λ→zk

A(λ)

=

B(λ)

A(zk ) B′ (zk )

,

k ∈ {1, . . . , N }.

Denote by λ1 , . . . , λN all the zeros of the polynomial PN (λ) (which coincide by (12) with the eigenvalues of the matrix J and which are distinct by the condition of simplicity of the spectrum of J): PN (λ) = c (λ − λ1 ) · · · (λ − λN ), where c is a nonzero constant. Therefore applying Lemma 6 to (14) we can get for the resolvent function w(λ) the following decomposition:

w(λ) =

N  k=1

βk , λ − λk

(18)

where

βk =

QN (λk ) PN′ (λk )

.

(19)

Further, putting λ = λk in (15) and (16) and taking into account that PN (λk ) = 0, we get PN −1 (λk )QN (λk ) = 1,

(20)

N −1

PN −1 (λk )PN′ (λk ) =



Pn2 (λk ),

(21)

n=0

respectively. It follows from (20) that QN (λk ) ̸= 0 and therefore βk ̸= 0. Comparing (19)–(21), we find that

βk =

 N −1 

−1 Pn2

(λk )

.

(22)

n =0

Since {Pn (λk )}Nn=−01 is an eigenvector of the matrix J corresponding to the eigenvalue λk , it is natural, according to the formula (22), to call βk the normalizing number of the matrix J corresponding to the eigenvalue λk . Definition 7. We call the collection of the eigenvalues and normalizing numbers

{λk , βk (k = 1, . . . , N )}

(23)

of the matrix J of the form (7), (8) with a simple spectrum, the spectral data of this matrix. Determination of the spectral data of a given Jacobi matrix is called the direct spectral problem for this matrix. Thus, the spectral data consist of the eigenvalues and associated normalizing numbers derived by decomposing the resolvent function (Weyl–Titchmarsh function) into partial fractions using the eigenvalues. The resolvent function w(λ) of the matrix J can be constructed by using Eq. (14). Another convenient formula for computing the resolvent function is (see [13])

w(λ) = −

det(J1 − λI )

, (24) det(J − λI ) where J1 is the first truncated matrix (with respect to the matrix J) and is obtained by deleting the first row and first column of the matrix J. It follows from (24) that λw(λ) tends to 1 as λ → ∞. Therefore multiplying (18) by λ and passing then to the limit as λ → ∞, we find N 

βk = 1.

k=1

The inverse spectral problem is stated as follows: (i) To see if it is possible to reconstruct the matrix J with a simple spectrum, given its spectral data (23). If it is possible, to describe the reconstruction procedure.

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(ii) To find the necessary and sufficient conditions for a given collection (23) to be spectral data for some matrix J of the form (7), (8) with a simple spectrum. The solution of this problem follows from [13] as follows. Given a collection (23) define the numbers sl =

N 

βk λlk ,

l = 0, 1, 2, . . . ,

(25)

k=1

and using these numbers introduce the determinants

 s0 s 1 Dn =  .  .. s

.. .

··· ··· .. .

s n +1

···

s1 s2

n

sn 

  sn+1  ..  , .   s

n = 0, 1, 2, . . . .

(26)

2n

Lemma 8. Let Dn be the determinant defined by (26) and (25), where the complex numbers λ1 , . . . , λN are distinct and β1 , . . . , βN are different from zero. Then DN −1 ̸= 0. Proof. Assume the contrary; let DN −1 = 0. Then the columns of the determinant DN −1 are linearly dependent, i.e., there are not all zero complex numbers c0 , c1 , . . . , cN −1 such that N −1 

cm sj+m = 0,

j = 0, 1, . . . , N − 1.

m=0

Substituting for sj+m its expression (25), we get N 

 j k k

βλ

k=1

N −1 

 λ

cm m k

= 0,

j = 0, 1, . . . , N − 1.

m=0

Setting G(λ) =

N −1 

cm λm ,

m=0

we can write the last equations in the form N 

βk λjk G(λk ) = 0,

j = 0, 1, . . . , N − 1.

k=1

These relations can be considered as a homogeneous system of linear algebraic equations with respect to βk G(λk ) (k = 1, . . . , N ) with the determinant

  1  λ  1 V =  ..  .  N −1 λ1

1

··· ··· .. .

λ2N −1

···

λ2 .. .

1 

 λN  ..  .   λNN −1

that is the Vandermonde determinant and hence different from zero. Therefore G(λk ) = 0 for k = 1, . . . , N. This means that the polynomial G(λ) of degree ≤N − 1 has N distinct zeros. Then G(λ) ≡ 0 and hence its coefficients c0 , c1 , . . . , cN −1 must be zero, which is a contradiction. Thus we have proved that DN −1 ̸= 0.  Lemma 9. Let Dn be the determinant defined by (26) and (25), where λ1 , . . . , λN and β1 , . . . , βN are arbitrary complex numbers. Then Dn = 0 for n ≥ N. For a proof of Lemma 9 see [14, Lemma 2.2]. Now from [13, Theorem 6] and the above Lemmas 8 and 9 we get the following result. Theorem 10. Let an arbitrary collection (23) of complex numbers be given, where λ1 , . . . , λN are distinct and β1 , . . . , βN are different from zero. In order for this collection to be the spectral data for some Jacobi matrix J of the form (7), (8) with a simple spectrum, it is necessary and sufficient that the following two conditions be satisfied: (i) k=1 βk = 1; (ii) Dn ̸= 0, for n ∈ {1, 2, . . . , N − 2}, where Dn is the determinant defined by (26), (25).

N

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Under the conditions of the theorem we have D0 = 1, DN −1 ̸= 0 and the entries an and bn of the matrix J with a simple spectrum for which the collection (23) is spectral data, are recovered by the formulae

√ ± Dn−1 Dn+1

an =

Dn

∆n

bn =

Dn

−

∆n−1 Dn−1

,

,

n ∈ {0, 1, . . . , N − 2}, D−1 = 1, n ∈ {0, 1, . . . , N − 1}, ∆−1 = 0, ∆0 = s1 ,

(27) (28)

where Dn is defined by (26) and (25), and ∆n is the determinant obtained from the determinant Dn by replacing in Dn the last column by the column with the components sn+1 , sn+2 , . . . , s2n+1 . It follows from the above solution of the inverse problem that the matrix (7) is not uniquely restored from the spectral data. This is linked with the fact that the an are determined from (27) uniquely up to a sign. To ensure that the inverse problem is uniquely solvable, we have to specify additionally a sequence of signs + and −. Namely, let {σ0 , σ1 , . . . , σN −2 } be a given finite sequence, where for each n ∈ {0, 1, . . . , N − 2} the σn is + or −. We have 2N −1 such different sequences. Now to determine an uniquely from (27) for n ∈ {0, 1, . . . , N − 2} we can choose the sign σn when extracting the square root. In this way we get precisely 2N −1 distinct Jacobi matrices possessing the same spectral data. The inverse problem is solved uniquely from the data consisting of the spectral data and a sequence {σ0 , σ1 , . . . , σN −2 } of signs + and −. Thus, we can say that the inverse problem with respect to the spectral data is solved uniquely up to signs of the off-diagonal elements of the recovered Jacobi matrix. Note that in the case of arbitrary real distinct numbers λ1 , . . . , λN and positive numbers β1 , . . . , βN condition (ii) of Theorem 10 is satisfied automatically and in this case we have Dn > 0, for n ∈ {1, 2, . . . , N − 1}; see [14, Lemma 2.2]. In the case N = 2, the collection (23) becomes

{λ1 , λ2 , β1 , β2 }. Let us check the conditions of Theorem 10 in this case. We must assume that λ1 , λ2 , β1 , β2 are arbitrary complex numbers such that

λ1 ̸= λ2 ,

β1 ̸= 0,

β2 ̸= 0,

β1 + β2 = 1.

Further, sl = β1 λl1 + β2 λl2 ,

l = 0, 1, 2, . . . ,

and we have D−1 = 1,

D0 = s0 = 1,



s D1 =  0 s1



s1  = β1 β2 (λ1 − λ2 )2 ̸= 0. s2 

We see that all the conditions of Theorem 10 are satisfied. Next, we have

∆−1 = 0, ∆ = s1 = β1 λ1 + β2 λ2 ,   0 s0 s2   = β1 β2 (λ1 + λ2 )(λ1 − λ2 )2 , ∆1 =  s1 s3  and the entries of the desired matrix

 J =

b0 a0

a0 b1



are found by a0 = b0 = b1 =

√ ± D −1 D 1

 = ±(λ1 − λ2 ) β1 β2 ,

D0

∆0 D0

∆1 D1

−

∆ −1

−

∆0

D −1 D0

= ∆0 = β1 λ1 + β2 λ2 ,

= λ1 + λ2 − β1 λ1 − β2 λ2 = β2 λ1 + β1 λ2 .

In the case N = 3 the collection (23) becomes

{λ1 , λ2 , λ3 , β1 , β2 , β3 } and it is assumed that λ1 , λ2 , λ3 , β1 , β2 , β3 are arbitrary complex numbers such that

λ1 = ̸ λ2 , β1 = ̸ 0,

λ1 ̸= λ3 , β2 ̸= 0,

λ2 ̸= λ3 , β3 ̸= 0, β1 + β2 + β3 = 1.

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Further, sl = β1 λl1 + β2 λl2 + β3 λl3 ,

l = 0, 1, 2, . . . ,

and it is not difficult to show that

 s D1 =  0 s1



s1  s2 

= β1 β2 (λ1 − λ2 )2 + β1 β3 (λ1 − λ3 )2 + β2 β3 (λ2 − λ3 )2 ,    s0 s1 s2    D2 = s1 s2 s3  = β1 β2 β3 (λ1 − λ2 )2 (λ1 − λ3 )2 (λ2 − λ3 )2 , s s s  2

3

4

∆0 = s1 = β1 λ1 + β2 λ2 + β3 λ3 ,   s s2   = β1 β2 (λ1 + λ2 )(λ1 − λ2 )2 + β1 β3 (λ1 + λ3 )(λ1 − λ3 )2 + β2 β3 (λ2 + λ3 )(λ2 − λ3 )2 , ∆1 =  0 s1 s3        1 1 1  1 1 1  s0 s1 s3       ∆2 = s1 s2 s4  = β1 β2 β3 λ1 λ2 λ3  λ1 λ2 λ3  . s s s  2 2 2 3 3 3 λ λ λ  λ λ λ  2

3

5

1

2

3

1

2

3

We see that the condition D1 ̸= 0 is not satisfied automatically and therefore one must require D1 ̸= 0 as a condition. For example, if take

√ β1 = β2 = β3 =

1 3

,

λ1 =

1±i 3 2

,

λ2 = 1,

λ3 = 0,

then we get D1 = 0. 4. Time evolution of the spectral data The following theorem is a standard result of the Lax representation (5) of problem (1), (2). Theorem 11. Let {an (t ), bn (t )} be a solution of (1), (2) and J = J (t ) be the Jacobi matrix defined by this solution according to (3). Then there exists an invertible N × N matrix-function X (t ) such that X −1 (t )J (t )X (t ) = J (0)

for all t .

(29)

Proof. Let A(t ) be defined according to (4). Then the Lax equation (5) holds. Denote by X = X (t ) the matrix solution of the initial value problem X˙ (t ) = −A(t )X (t ),

(30)

X (0) = I .

(31)

Such a solution X (t ) exists and is unique. From the Liouville formula t

 

det X (t ) = exp −

trA(τ )dτ



0

it follows that det X (t ) ̸= 0 so that the matrix X (t ) is invertible. Using the formula dX −1 dt

= −X −1

dX −1 X , dt

Eq. (30), and the Lax equation (5), we have d  −1  dX −1 dJ dX X JX = JX + X −1 X + X −1 J dt dt dt dt dJ = X −1 AXX −1 JX + X −1 X − X −1 JAX dt   dJ = X −1 − [JA − AJ ] X = 0. dt Therefore X −1 (t )J (t )X (t ) does not depend on t and by initial condition (31) we get (29). The proof is complete.



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It follows from (29) that for any t the characteristic polynomials of the matrices J (t ) and J (0) coincide. Therefore we arrive at the following statement. Corollary 12. The eigenvalues of the matrix J (t ), as well as their multiplicities, do not depend on t. Remark 13. From the skew-symmetry AT = −A of the matrix A(t ) defined by (4) it follows that for the solution X (t ) of (30), (31) we have X −1 (t ) = X T (t ), where T stands for the matrix transpose. Indeed, by Eq. (30), d  dt

dX T

XT X =



dt

dX

X + XT

dt

= −X T AT X − X T AX

= X T AX − X T AX = 0. Therefore X T (t )X (t ) does not depend on t and by initial condition (31) we get X T (t )X (t ) = X T (0)X (0) = I. By Corollary 12 the eigenvalues λk of the matrix J (t ) do not depend on t. However, the normalizing numbers βk of the matrix J (t ) will depend on t. The following theorem describes the time dependence of the normalizing numbers of the matrix J (t ). Theorem 14. For the normalizing numbers βk (t ) (k = 1, . . . , N ) of the matrix J (t ) the following time evolution holds:

βk (t ) =

e2λk t

βk (0),

(32)

e2λj t βj (0).

(33)

S (t )

where S (t ) =

N  j =1

Proof. Denote by Pn (λ, t ) the first kind polynomial of the matrix J (t ). Then the column vector P = P (λk , t ) = {Pn (λk , t )}Nn=−01 will be an eigenvector of the matrix J (t ) corresponding to the eigenvalue λk : JP = λk P. Differentiating this equation with respect to t, we get

˙ + J P˙ = λk P˙ . JP Substituting here J˙ = [J , A] = JA − AJ, according to the Lax equation (5), and taking into account that AJP = λk AP, we obtain J (P˙ + AP ) = λk (P˙ + AP ). This means that together with P the vector P˙ + AP also is an eigenvector of J (t ) corresponding to the eigenvalue λk . Since the eigenspaces of J are one-dimensional by our assumption on simplicity of the spectrum of J, we have P˙ (λk , t ) + AP (λk , t ) = ck (t )P (λk , t ),

(34)

where ck (t ) is an indeterminate scalar function. Taking the ‘‘scalar product ’’ of (34) with P (λk , t ), in the sense

⟨x, y⟩0 =

N −1 

xn yn ,

x, y ∈ C N

n =0

(without the complex conjugation of yn ), we can write



P˙ (λk , t ), P (λk , t )

 0

+ ⟨AP (λk , t ), P (λk , t )⟩0 = ck (t ) ⟨P (λk , t ), P (λk , t )⟩0 .

(35)

Further, taking into account (22) and the skew-symmetry of A, we find

⟨P (λk , t ), P (λk , t )⟩0 =

1

⟨AP (λk , t ), P (λk , t )⟩0 = 0,     1 d 1 d 1 ⟨P (λk , t ), P (λk , t )⟩0 = P˙ (λk , t ), P (λk , t ) 0 = . 2 dt 2 dt βk (t ) βk (t )

,

Therefore (35) yields ck (t ) =

1 d 2 dt

 ln

1

βk (t )



.

(36)

G.Sh. Guseinov / Mathematical and Computer Modelling 57 (2013) 1190–1202

1199

Let us write (34) for the first coordinate: P˙ 0 (λk , t ) − a0 (t )P1 (λk , t ) = ck (t )P0 (λk , t ). Since P0 (λk , t ) = 1,

P1 (λk , t ) =

λk − b0 (t ) , a0 (t )

the last equation gives

−λk + b0 (t ) = ck (t ). Therefore (36) yields



1 d 2 dt



1

ln

= −λk + b0 (t ).

βk (t )

Hence ln

1

βk (t )

= −2λk t + 2

t



b0 (τ )dτ + ln 0

1

βk (0)

and we find that

βk (t ) =

1 S (t )

βk (0)e2λk t ,

(37)

where S ( t ) = e2

t

0 b0 (τ )dτ

.

Thus (32) is established. Finally, summing (37) over k ∈ {1, . . . , N } and taking into account that N 

βk (t ) = 1

k=1

(see above the first condition in Theorem 10) we arrive at (33). The theorem is proved.



5. The procedure of constructing complex solutions to the Toda lattice By virtue of the results established in the previous sections we get the following procedure for solving the problem (1), (2), (6). We take the initial values {an (0), bn (0)} such that the Jacobi matrix

b (0) 0 a0 (0)  0   . J (0) =  ..   0 

a0 (0) b1 (0) a1 (0)

0 a1 (0) b2 (0)

.. .

··· ··· ··· .. .

0 0 0

0 0 0

··· ··· ···

0 0

.. .

0 0 0

0 0 0

0 0 0

bN −3 (0) aN −3 (0) 0

aN −3 (0) bN −2 (0) aN −2 (0)

0 aN −2 (0) bN −1 (0)

.. .

.. .

.. .

        

has a simple spectrum and determine its spectral data

{λk , βk (0) (k = 1, . . . , N )}. This is the same that we take arbitrary distinct complex numbers λ1 , . . . , λN and arbitrary nonzero complex numbers β1 (0), . . . , βN (0) satisfying the conditions (i) and (ii) of Theorem 10. Then we calculate for each t ∈ R the numbers

βk (t ) =

e2λk t S (t )

βk (0)

(38)

dependent on t, where S (t ) =

N 

e2λj t βj (0).

(39)

j =1

Finally, solving the inverse spectral problem with respect to

{λk , βk (t ) (k = 1, . . . , N )},

(40)

1200

G.Sh. Guseinov / Mathematical and Computer Modelling 57 (2013) 1190–1202

we construct a Jacobi matrix J (t ). The entries {an (t ), bn (t )} of the matrix J (t ) give a solution of problem (1), (2), (6). We can write the explicit expressions for an (t ), bn (t ) through the ‘‘moments’’ sl (t ) =

N 

βk (t )λlk ,

l = 0, 1, 2, . . . ,

(41)

k=1

using (26), (27), (28) (sl in them should be replaced by sl (t )). Note that from the continuity of the function S (t ) defined by (39) and S (0) = 1 it follows that S (t ) ̸= 0 for values of t in some neighborhood of t = 0. For these values of t the function βk (t ) is well-defined by (38) and is continuous. Obviously, N 

βk (t ) = 1.

k=1

Next, from the continuity of the functions Dn (t ) defined by (26) (in which sl are replaced by sl (t )) and Dn (0) ̸= 0 for n ∈ {1, 2, . . . , N − 1} it follows that Dn (t ) ̸= 0 for n ∈ {1, 2, . . . , N − 1} and values of t in some neighborhood of t = 0. Therefore the time-dependent collection (40) will satisfy the conditions of Theorem 10 (on solvability of the inverse problem) for values of t in some neighborhood of t = 0. The fact that the constructed functions an (t ), bn (t ) satisfy the Toda equations in (1) can be proved as in [1]. In the case N = 2, Eqs. (1) and (2) become a˙ 0 = a0 (b1 − b0 ),

b˙ 0 = 2(a20 − a2−1 ),

a˙ 1 = a1 (b2 − b1 ),

b˙ 1 = 2(a21 − a20 ),

a−1 = a1 = 0. This system in turn is equivalent to a˙ 0 = a0 (b1 − b0 ),

b˙ 0 = 2a20 ,

b˙ 1 = −2a20 ,

(42)

with respect to a0 , b0 , and b1 . For an arbitrarily given 2 × 2 initial Jacobi matrix J (0) =



b0 (0) a0 (0)

a0 (0) b1 (0)



with a simple spectrum, its spectral data may consist of two arbitrary distinct complex eigenvalues λ1 , λ2 , and two corresponding normalizing numbers β1 , β2 which may be arbitrary nonzero complex numbers with β1 + β2 = 1. Starting with these data {λ1 , λ2 , β1 , β2 } we construct the time dependent normalizing numbers β1 (t ), β2 (t ) according to (38), (39):

β1 (t ) =

e2λ1 t S (t )

β1 ,

β2 (t ) =

e2λ2 t S (t )

β2 ,

where S (t ) = e2λ1 t β1 + e2λ2 t β2 . Next we solve the inverse problem with respect to {λ1 , λ2 , β1 (t ), β2 (t )}. To this end we construct the quantities sl (t ) = β1 (t )λl1 + β2 (t )λl2 ,

l = 0, 1, 2, . . .

and compute the determinants D−1 (t ) = 1,

D0 (t ) = s0 (t ) = 1,

 s (t ) D1 (t ) =  0 s1 (t )

s1 (t ) = β1 (t )β2 (t )(λ1 − λ2 )2 , s2 (t )

∆ −1 ( t ) = 0 ,  s (t ) ∆1 (t ) =  0 s1 ( t )

∆0 (t ) = s1 (t ) = β1 (t )λ1 + β2 (t )λ2 ,  s2 (t ) = β1 (t )β2 (t )(λ1 + λ2 )(λ1 − λ2 )2 . s3 (t )



Therefore, the desired solution of (42) is found by

√  ± D−1 (t )D1 (t ) = ±(λ1 − λ2 ) β1 (t )β2 (t ) D0 (t )  e(λ1 +λ2 )t = ±(λ1 − λ2 ) β1 β2 2λ t , 1 e β1 + e2λ2 t β2

a0 (t ) =

G.Sh. Guseinov / Mathematical and Computer Modelling 57 (2013) 1190–1202

b0 (t ) =

1201

∆−1 (t ) ∆0 (t ) − = ∆0 ( t ) D0 (t ) D −1 ( t )

= β1 (t )λ1 + β2 (t )λ2 =

e2λ1 t β1 λ1 + e2λ2 t β2 λ2 e2λ1 t β1 + e2λ2 t β2

,

∆1 (t ) ∆0 (t ) − = λ1 + λ2 − β1 (t )λ1 − β2 (t )λ2 D1 (t ) D0 (t ) e2λ1 t β1 λ1 + e2λ2 t β2 λ2 = λ1 + λ2 − b0 (t ) = λ1 + λ2 − . e2λ1 t β1 + e2λ2 t β2

b1 (t ) =

These give a class of solutions of (42), where λ1 , λ2 are arbitrary two distinct complex numbers and β1 , β2 are arbitrary two nonzero complex numbers such that β1 + β2 = 1. Now using the equations a0 =

1 (q0 −q1 )/2 e ,

q˙ 0 = p0 = −2b0 ,

2

q˙ 1 = p1 = −2b1

and the last expressions of a0 (t ), b0 (t ), and b1 (t ) we find easily that q0 (t ) = c − ln(e2λ1 t β1 + e2λ2 t β2 ),

q1 (t ) = c − ln[4(λ1 − λ2 )2 β1 β2 ] − 2(λ1 + λ2 )t − ln(e2λ1 t β1 + e2λ2 t β2 ),

where as c = q0 (0) one can take an arbitrary complex number. 6. Complex solutions with real energy In [15] it is noted that the complex solutions of a dynamical system having real constants of motion, such as the energy, can be viewed as being physical. In the case N = 2, the Toda lattice system has the form q¨ 0 = −eq0 −q1 ,

q¨ 1 = eq0 −q1

(43)

with the Hamiltonian H =

1 2

(˙q20 + q˙ 21 ) + eq0 −q1 .

On the complex solutions {q0 (t ), q1 (t )} of (43), constructed in the previous section, we have H = 2(b20 + b21 ) + 4a20

= 2{[β1 (t )λ1 + β2 (t )λ2 ]2 + [β2 (t )λ1 + β1 (t )λ2 ]2 } + 4(λ1 − λ2 )2 β1 (t )β2 (t ) = 2(β12 + β22 )(λ21 + λ22 ) + 8β1 β2 λ1 λ2 + 4(λ1 − λ2 )2 β1 β2 = 2(λ21 + λ22 )(β12 + β22 + 2β1 β2 ) = 2(λ21 + λ22 )(β1 + β2 )2 = 2(λ21 + λ22 ) because β1 (t ) + β2 (t ) = 1. Hence H is real and positive if λ1 and λ2 are taken to be real. In general, on any complex solution {qn (t ), pn (t )} of the Toda lattice system we have H =

N −1 1

2 n =0

=2

N −1 

p2n +

N −2 

eqn −qn+1

n =0

b2n

+4

N −2 

n =0

 a2n

=2

n=0

N −1  n =0

b2n

+2

N −2 

 a2n

n=0

= 2trJ = 2(λ + · · · + λ ), 2

2 1

2 N

and therefore the energy H will be real and positive if the eigenvalues λ1 , . . . , λN of the initial complex Jacobi matrix J (0) are taken to be real (while the normalizing numbers of J (0) are taken to be complex). References [1] Yu.M. Berezanskii, The integration of semi-infinite Toda chain by means of inverse spectral problem, Rep. Math. Phys. 24 (1986) 21–47. [2] Yu.M. Berezanskii, A.A. Mokhonko, Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices, Funct. Anal. Appl. 42 (2008) 1–18. [3] H. Flaschka, The Toda lattice I, Phys. Rev. B 9 (1974) 1924–1925. [4] H. Flaschka, The Toda lattice II, Progr. Theoret. Phys. 51 (1974) 703–716. [5] M. Henon, Integrals of the Toda lattice, Phys. Rev. B 9 (1974) 1921–1923.

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[6] M. Kac, P. van Moerbeke, On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math. 16 (1975) 160–169. [7] M. Kac, P. van Moerbeke, On some periodic Toda lattices, Proc. Natl. Acad. Sci. USA 72 (1975) 1627–1629. [8] M. Kac, P. van Moerbeke, A complete solution of the periodic Toda problem, Proc. Natl. Acad. Sci. USA 72 (1975) 2879–2880. [9] S.V. Manakov, Complete integrability and stochastization in discrete dynamic media, Sov. Phys. JETP 40 (1975) 269–274. [10] J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975) 197–220. [11] M. Toda, Waves in nonlinear lattice, Progr. Theoret. Phys. Suppl. 45 (1970) 174–200. [12] M. Toda, Theory of Nonlinear Lattices, Springer, New York, 1981. [13] G.Sh. Guseinov, Inverse spectral problems for tridiagonal N by N complex Hamiltonians, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009) 1–28, Paper 018. [14] G.Sh. Guseinov, On a discrete inverse problem for two spectra, Discrete Dyn. Nat. Soc. 2012 (2012) 1–14, Article ID 956407. [15] C.M. Bender, D.D. Holm, D.W. Hook, Complexified dynamical systems, J. Phys. A Math. Theor. 40 (2007) F793–F804.