Integration of a modified double-infinite Toda lattice by using the inverse spectral problem
An approach to finding a solution of the Cauchy problem for a modified double-infinite Toda lattice by using the inverse spectral problem is given.
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| author | Berezansky, Yu. M. Березанський, Ю. М. |
| author_facet | Berezansky, Yu. M. Березанський, Ю. М. |
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| description | An approach to finding a solution of the Cauchy problem for a modified double-infinite Toda lattice by using the inverse spectral problem is given. |
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UDС 530.1
Yu. M. Berezansky (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
INTEGRATION OF THE MODIFIED
DOUBLE-INFINITE TODA LATTICE
WITH THE HELP OF INVERSE SPECTRAL PROBLEM∗
IНТЕГРУВАННЯ ЗМIНЕНОГО
ДВОСТОРОННЬО НЕСКIНЧЕННОГО ЛАНЦЮЖКА ТОДИ
ЗА ДОПОМОГОЮ ОБЕРНЕНОЇ СПЕКТРАЛЬНОЇ ЗАДАЧI
The approach to finding the solution of the Cauchy problem for the indicated Toda lattice by means of
inverse spectral problem is given.
Викладено пiдхiд до знаходження розв’язку задачi Кошi для вказаного ланцюжка Тоди за допомогою
оберненої спектральної задачi.
1. Introduction. The classical method of investigation of the Cauchy problem for
the KdV equation via an application of inverse spectral problem for Sturm – Liouville
equation (I. Gelfand, B. Levitan, V. Marchenko, M. Krein; account see in the book [1])
can be adjusted for the Toda semi-infinite lattice
α̇n(t) =
1
2
αn(t)(βn+1(t)− βn(t)),
β̇n(t) = α2
n(t)− α2
n−1(t), n = 0, 1, . . . , t ∈ [0, T ]; α−1 = 0.
Using some results for a finite Toda lattice [2, 3], this approach was proposed in the
articles [4, 5] of author. There the role of the Sturm – Liouville equation was played by
the simpler spectral theory of Jacobi matrices.
But finding solutions of the double-infinite Toda lattice (when n = . . . ,−1, 0, 1, . . .)
is a more difficult problem. In the periodic case, this equation was integrated in terms of
theta-functions in [6] (see also [7]); the inverse scattering problem method (for difference
equations) was applied in [8 – 10]; the method similar to that of [4, 5] was used in [11 –
13] in the case when initial data tend to zero when |n| → ∞; new classes of solutions
were found in [14]; see also [15 – 17].
In [18], Ch. 7, the author proposed to investigate spectral problems for double-infinite
Jacobi matrices by doubling such a matrix and reducing the problem to the case of block
one-sided Jacobi matrices with (2× 2)-matrix blocks (the spectral theory of such block
Jacobi matrices was proposed in [19] and developed in [18], Ch. 7; in this theory, instead
of ordinary scalar-valued spectral measure the matrix-valued spectral measure appeared).
In [20] author and M. Gekhtman tried to apply the inverse spectral theory of (2 × 2)-
block one-sided Jacobi matrices for integration of double-infinite Toda lattice, but the
corresponding differential equation for (2× 2)-matrix-spectral measure was impossible
to solve and therefore this approach to solving the Cauchy problem was ineffective.
In this paper, the author applies an analogue of the approach in [20], but instead
of (2 × 2)-block Jacobi matrices here the dimensions of blocks are changed: the first
∗ This work was partly supported by the DFG436 UKR 113/78/0-1 and by the Program of National
Academy of Sciences of Ukraine (project № 0107U002333).
c© YU. M. BEREZANSKY, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 453
454 YU. M. BEREZANSKY
diagonal block is 1 × 1, i.e., scalar, and therefore, instead of (2 × 2)-matrix-spectral
measure we have an ordinary scalar spectral measure (such idea has appeared in the
works [21 – 23]). As a result, we integrate the double-infinite Toda chain as in [4, 5],
but with an additional condition:
α0(t)α−1(t) = 0, t ∈ [0, T ].
Thus, we can integrate an appropriately modified double-infinite Toda chain when functi-
ons α0(t), α−1(t), t ∈ [0, T ], are given.
2. The spaces and the corresponding block Jacobi matrix. We will investigate
an operator on the complex Hilbert space
l2 = H0 ⊕H1 ⊕H2 ⊕ . . . , H0 = C1 := C, H1 = H2 = . . . = C2. (1)
Vectors f from l2 have a form f = (fn)∞n=0 where fn ∈ Hn; so f0 = f0e0, fn =
= fn;0en;0 + fn;1en;1 =: (fn;0, fn;1) where e0 = 1 and en = (en;0, en;1), n ∈ N =
= {1, 2, . . .}, form the standard basis in C1 and Hn = C2 respectively.
By lfin we denote the linear space of finite vectors from l2 and by l2(p) we denote
the corresponding weighted space of vectors for which
‖f‖2l2(p) =
∞∑
n=0
‖fn‖2Hn
pn <∞, (f, g)l2(p) =
∞∑
n=0
(fn, gn)Hn
pn. (2)
Here p = (pn)∞n=0, pn > 0, is a given sequence of weights. In what follows, pn ≥ 1
and
∑∞
n=0
p−1
n < ∞, therefore the imbedding of the positive space l2(p) ⊂ l(p) is
quasinuclear. The corresponding negative space is l2(p−1), p−1 := (p−1
n )∞n=0. As a
result, we construct the quasinuclear rigging (see, e.g., [24], Ch. 15)
l = (lfin)′ ⊃ (l2(p−1)) ⊃ l2 ⊃ (l2(p)) ⊃ lfin (3)
(l denotes the space of all sequences f = (fn)∞n=0, fn ∈ Hn are arbitrary).
In the space (1), consider a Hermitian matrix J = (Jj,k)∞j,k=0 with operator (real
matrix) — valued elements Jj,k : Hk → Hj , Jj,k = (Jj,k;α,β)1α,β=0, of the following
block Jacobi structure:
b0 : R1 → R1,
J =
b0 a0 0 0 0 . . .
a∗0 b1 a1 0 0 . . .
0 a1 b2 a2 0 . . .
0 0 a2 b3 a3 . . .
...
...
...
...
...
. . .
,
a0 = [a0;0,0a0;0,1] : R2 → R1, a∗0 =
[
a0;0,0
a0;0,1
]
: R1 → R2, (4)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 455
an = a∗n =
[
an;0,0 an;0,1
an;1,0 an;1,1
]
: R2 → R2,
bn = b∗n =
[
bn;0,0 bn;0,1
bn;1,0 bn;1,1
]
: R2 → R2, n ∈ N.
By assumption, all elements of matrix J (4) are real and uniformly bounded.
Therefore the operator J constructed in a usual way from the matrix J is a bounded
self-adjoint operator acting in the space l2. It is connected with the chain (3) in a standard
way.
3. The spectral theory of the operator J. Basic constructions. Now we will use
the result from [18], Ch. 5, and [24], Ch. 15, on the generalized eigenvector expansion
for a bounded self-adjoint operator standardly connected with the chain (3). For our
operator J we have the representation
Jf =
∫
R
λΦ(λ)dσ(λ)f, f ∈ l2, (5)
where Φ(λ) : l2(p) → l2(p−1) is a generalized projection operator and dσ(λ) is a spectral
measure (with a bounded support). For all f, g ∈ lfin we have the Parseval equality
(f, g)l2 =
∫
R
(Φ(λ)f, g)l2dσ(λ); (6)
and, after extending by continuity, the equality (6) takes place for all f, g ∈ l2.
Let us denote by πn the operator of orthogonal projection in l2 on Hn, n ∈ N0 =
= {0, 1, 2, . . .}. Hence for all f = (fn)∞n=0 ∈ l2 we have fn = πnf. This operator acts
analogously in the space l2(p) and l2(p−1) but possibly with norm which is not equal
to one.
Let us consider the operator matrix (Φj,k(λ))∞j,k=0 where
Φj,k(λ) = πjΦ(λ)πk : l2 → Hj (or Hk → Hj). (7)
The Parseval equality (6) can be rewritten as follows: ∀f, g ∈ l2
(f, g)l2 =
∞∑
j,k=0
∫
R
(Φ(λ)πkf, πjg)l2dσ(λ) =
=
∞∑
j,k=0
∫
R
(πjΦ(λ)πkf, g)l2dσ(λ) =
∞∑
j,k=0
∫
R
(Φj,k(λ)fk, gj)l2dσ(λ). (8)
In what follows we will assume that all matrices an, n ∈ N, are invertible and
a0 6= 0. The difference equation Jϕ(λ) = λϕ(λ), ϕ(λ) = (ϕn(λ))∞n=0 ∈ (lfin)′ = l has
the following form: let ϕ0(λ) = ϕ0 be independent of λ :
b0ϕ0 + a0ϕ1(λ) = λϕ0,
i.e.,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
456 YU. M. BEREZANSKY
b0ϕ0 + (a0;0,0ϕ1;0(λ) + a0;0,1ϕ1;1(λ)) = λϕ0, ϕ−1 = 0,
a0ϕ0 + b1ϕ1(λ) + a1ϕ2(λ) = λϕ1(λ),
a1ϕ1 + b2ϕ2(λ) + a2ϕ3(λ) = λϕ2(λ), (9)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
an−1ϕn−1 + bnϕn(λ) + anϕn+1(λ) = λϕn(λ),
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
n ∈ N.
Let us explain that above we have assumed that, as in the classical theory of Jacobi
matrices, the initial value ϕ0(λ) of an eigenvector ϕ(λ) =
(
ϕ0(λ), ϕ1(λ), ϕ2(λ), . . .) ∈
∈ (lfin
)′ = l is ϕ0 ∈ R and does not depend on λ (we will often assume ϕ0 = 1).
The expression Pm(λ) =
∑m
j=0
λjAj , λ ∈ R, where Aj are linear operators
acting: R2 → R2 and Am is invertible, we will call an operator polynomial of degree
m ∈ N0 w.r.t. λ. For fixed λ it is an operator, acting: R2 → R2. If the coefficients
A0, . . . , Am : C2 → C2, such operator polynomial will be called a complex operator
polynomial.
The system (9) is a system of recurrence relations: starting from some ϕ1(λ) ∈ R2
we can find, step by step, ϕ2(λ), ϕ3(λ), . . . ; since the matrices a1, a2, . . . are invertible
we have (below 1 is the identity operator in R2)
ϕ2(λ) = a−1
1
(
(λ1− b1)ϕ1(λ)− a0ϕ0
)
=: Q2(λ)(ϕ0, 0) =: Q2(λ)ϕ0,
ϕ3(λ) = a−1
2
(
(λ1− b2)ϕ2(λ)− a1ϕ1(λ)
)
=: Q3(λ)(ϕ0, 0) =: Q3(λ)ϕ0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ϕn+1(λ) = a−1
n
(
(λ1− bn)ϕn(λ)− an−1ϕn−1(λ)
)
=
= a−1
n
(
(λ1− bn)Qn(λ)− an−1Qn−1(λ)
)
=:
=: Qn+1(λ)(ϕ0, 0) =: Qn+1(λ)ϕ0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
n ∈ N.
(10)
The first vector ϕ1(λ) ∈ R2 cannot be found uniquely from the first equation in (9).
Our way around this is as follows: we fix some θ1, ω1 6= 0, then after easy calculations
we get
ϕ1(λ) = (λ− b0)
[
θ−1
1 0
0 ω−1
1
]
(ϕ0, 0) =: Q1(λ)(ϕ0, 0) =: Q1(λ)ϕ0, (11)
where θ1, ω1 are some solutions of one equation with two unknowns
a0;0,0θ
−1
1 + a0;0,1ω
−1
1 = 1 (12)
(these solutions exist because the matrix a0 : R2 → R is nonzero). These θ1, ω1 will be
fixed in Sections 3 – 5 of the paper.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 457
Thus, we start from Q1(λ)ϕ0 of the form (11) with condition (12) and then fi-
nd Q2(λ)ϕ0, Q3(λ)ϕ0, . . . . It is easy to understand that every Qn(λ) is an operator
polynomial of degree n ∈ N. The solution of the difference equation (9) has a form
ϕ(λ) = (ϕn(λ))∞n=0, ϕn(λ) = Qn(λ)ϕ0, n ∈ N0;
Q0(λ) := 1, Q0(λ)ϕ0 := ϕ0.
(13)
It is possible to find the representation of elements (7) of matrix (Φj,k(λ))∞j,k=0 by
operator polynomials Q0(λ), Q1(λ), . . . (compare with [18], Ch. 7, and [21, 22]).
Lemma 1. For every fixed j, k ∈ N0 the operator (7) Φj,k(λ) has the following
representation:
Φj,k(λ) = Qj(λ)Φ0,0(λ)(Qk(λ))∗ (14)
where Φ0,0(λ) ≥ 0 is understood as an operator of multiplication by the scalar Φ0,0(λ).
Proof. For a fixed k ∈ N0, the vector (with a fixed λ ∈ R and f ∈ lfin) ϕ(λ) =
= (ϕj(λ)∞j=0), where
ϕj(λ) = Φj,k(λ)f = πjΦ(λ)πkf ∈ Hj , λ ∈ R, (15)
is a generalized solution, in l = (lfin)′, of the equation Jϕ(λ) = λϕ(λ), since Φ(λ)
is a projector onto a generalized eigenvector of the self-adjoint operator J with the
corresponding generalized eigenvalue λ. Therefore for all g ∈ lfin we have (ϕ(λ), Jg)l2 =
= λ(ϕ(λ), g)l2 . Transferring the finite difference Hermitian expression J to ϕ(λ)
we get (Jϕ(λ), g)l2 = λ(ϕ(λ), g)l2 . Hence, it follows that ϕ(λ) ∈ l2(p−1) exists
as a usual solution of the difference equation Jϕ = λϕ with the initial condition
ϕ0(λ) = π0Φ(λ)πkf ∈ H0.
The uniqueness of the solution of the Cauchy problem for the difference equation (9)
(with a condition ϕ−1(λ) = 0 and finding θ1, ω1, according to (12)), i.e., for Jϕ = λϕ,
ensures that the solution (15)
(
ϕ0(λ), ϕ1(λ), . . .
)
and the solution (13) with the initial
condition ϕ0(λ) = π0Φ(λ)πkf are the same. Vector f ∈ lfin is arbitrary, therefore we
obtain
Φj,k(λ) = Qj(λ)(π0Φ(λ)πk) = Qj(λ)Φ0,k(λ), j ∈ N0. (16)
The operator Φ(λ) : l2(p) → l2(p−1) is formally self-adjoint on l2 (since it is equal
to the derivative of the resolution of identity of the operator J in l2 with respect to the
spectral measure). Hence, according to (7), we get(
Φj,k(λ)
)∗ = (πjΦ(λ)πk)∗ = πkΦ(λ)πj = Φk,j(λ), j, k ∈ N0. (17)
For a fixed j ∈ N0, it follows from (16) and the previous discussion that the vector
ψ(λ) = (ψk(λ))∞k=0, ψk(λ) = Φk,j(λ)f, k ∈ N0, f ∈ lfin,
is the usual solution of the difference equation Jψ = λψ with the initial condition
ψ0(λ) = Φ0,j(λ)f = (Φj,0(λ))∗f.
Again as above, we obtain the representation of the type (16)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
458 YU. M. BEREZANSKY
Φj,k(λ) = Qk(λ)(π0Φ(λ)πj) = Qk(λ)Φ0,j(λ), k ∈ N0. (18)
Taking into an account (16) with j (k) replaced by 0 (k), we get
Φ0,k(λ) = (Φk,0(λ))∗ = (Qk(λ)(π0Φ(λ)π0))∗ = Φ0,0(λ)(Qk(λ))∗ (19)
(here we used the fact that the scalar Φ0,0(λ) ≥ 0; this inequality follows from (6) and
(8)). Substituting (19) into (16), we get for all j, k ∈ N0 : Φj,k(λ) = Qj(λ)Φ0,k(λ) =
= Qj(λ)Φ0,0(λ)(Qk(λ))∗.
The lemma is proved.
It will be essential for us to rewrite the Parseval equality (6), (8) in the form that
involves the operator polynomials Q0(λ), Q1(λ), . . . introduced above.
Using Parseval equality (8) and representation (14) we get: ∀f, g ∈ lfin
(f, g)l2 =
∞∑
j,k=
∫
R
(Φj,k(λ)fk, gj)l2dσ(λ) =
=
∞∑
j,k=0
∫
R
(Qj(λ)Φ0,0(λ)Q∗k(λ)fk, gj)l2dσ(λ) =
=
∞∑
j,k=0
∫
R
(Q∗k(λ)fk, Q
∗
j (λ)gj)l2dρ(λ) =
=
∫
R
( ∞∑
k=0
Q∗k(λ)fk
)
,
∞∑
j=0
Q∗j (λ)gj
C2
dρ(λ). (20)
Here dρ(λ) = Φ0,0(λ)dσ(λ) is the spectral measure of our operator J, it is a probability
Borel measure on R with a bounded support.
Remark 1. The operator polynomials Q1(λ), Q2(λ), . . . form a solution of equati-
ons (9) – (11) with real coefficients. Therefore they are real and the star * means
transposed matrix.
Introduce the Fourier transform ̂ induced by self-adjoint bounded operators J in the
space lfin :
l2 ⊃ lfin 3 f = (fn)∞n=0 7→ f̂(λ) = f0 +
∞∑
n=1
Q∗n(λ)fn =
=
∞∑
n=0
Q∗n(λ)fn ∈ L2
(
R, dρ(λ)
)
. (21)
Hence, (20) gives the Parseval equality in the final form: ∀f, g ∈ lfin
(f, g)l2 =
∫
R
f̂(λ)ĝ(λ)dρ(λ). (22)
Extending (22) by continuity, it becomes valid for all f, g ∈ l2.
We find now the orthogonal properties of operator polynomials Q∗n(λ), n ∈ N0.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 459
Remind, that the zeroth polynomial Q∗0(λ) is equal to 1. The orthonormal properties
of these polynomials (which are some operators: R2 → R2, polynomially depending
on λ ∈ R) are easy to find. Namely, recall that every C2 3 fk, k ∈ N, has a 2-vector
form: fk = (fk;0ek;0, fk;1ek;1) where ek = (ek;0, ek;1) is a fixed standard basis in the
space C2. In the case k = 0 we can view the value f0 ∈ C as vector (f0;0) ∈ C2.
Therefore, taking in (21) the vectors f, g of the form f = (0, . . . , 0, fk;β , 0, . . .),
g = (0, . . . , 0, gj;α, 0, . . .), we conclude that f̂(λ) =
(
Q∗k(λ)fk;βek;β
)
α
=
= (Q∗k(λ))α,βfk;β = (Qk(λ))β,αfk;β , ĝ(λ) = (Q∗j (λ)gj;αej;α)β = (Q∗j (λ))α,βgj;α =
= (Qj(λ))β,αgj;α and than the Parseval equality (22) gives∫
R
(Qk(λ))β,αfk;β(Qj(λ))β,αgj;αdρ(α) = fk;βgj;α, j, k ∈ N0, α, β = 0, 1,
or in a more symmetric form∫
R
(Qj(λ))α,β(Qk(λ))∗β,αdρ(α) = δj,k(eα, eβ)C2 , j, k ∈ N0, α, β = 0, 1, (23)
where dρ(λ) is a probability Borel measure on R.
It is easy enough to find the elements of matrix J in terms of operator polynomials
Q0(λ), Q1(λ), . . . . For this, we take f, g ∈ lfin. The representation (5) and (8), (20)
give
(Jf, g)l2 =
(
Jf, g
)
l2
=
∫
R
λ(Φ(λ)f, g)l2dσ(λ) =
=
∞∑
j,k=0
∫
R
λ(Φj,k(λ)fk, gj)l2dσ(λ) =
∞∑
j,k=0
∫
R
λ(Q∗k(λ)fk, Q
∗
j (λ)gj)l2dρ(λ).
Setting in this equality f(k) = (fn)∞n=0 ∈ lfin for which fn = 0 except for fk = ek and,
similarly, g(j) = (gn)∞n−0 ∈ lfin composed from zeros except for gj = ej (j, k ∈ N0 are
fixed) we find: ∀j, k ∈ N0
Jj,k = (Jf(k), g(j))l2 =
∫
R
λ(Q∗k(λ)ek, Q
∗
j (λ)ej)C2dρ(λ) =
=
∫
R
λ(Qj(λ)Q∗k(λ)ek, ej)C2dρ(λ). (24)
Taking into an account that Jj,k is a matrix (Jj,k;α,β)1α,β=0, we can rewrite (24) in
the form
(Jj,k;α,β)1α,β=0 =
∫
R
λ(Qj(λ)Q∗k(λ))1α,β=0dρ(λ), j, k ∈ N0 (25)
(
in particular, b0 =
∫
R
λdρ(λ)
)
.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
460 YU. M. BEREZANSKY
4. The inverse spectral problem for operator J. Basic constructions. Our goal
now is to prove that the spectral measure dρ(λ) of operator J (a probability Borel measure
on R, with a bounded support supp dρ(λ)), allows one to reconstruct the matrix J (4).
We first recall some constructions (more general results can be found in [18], Ch. 7).
Consider the space F of all matrix-valued (i.e., operator valued of the type: C2 → C2)
continuous functions R 3 λ 7→ F (λ) : C2 → C2 and introduce for such functions the
operator-valued scalar product {·, ·} putting for all F, G ∈ F
{F,G} =
∫
R
(F (λ))∗G(λ)dρ(λ) : C2 → C2. (26)
Some examples of functions F (λ) were introduced earlier: the operator polynomials
Qm(λ). Of course, it is possible to perform some procedure of completion, but for
our purposes it is not essential. We only note that for an arbitrary operator “scalar”
C : C2 → C2 we have
{FC,G} = C∗{F,G}, {CF,G} = {F,C∗G}, {F,GC} = {F,G}C,
(27)
{F,G}∗ = {G∗, F}, {F, F} ≥ 0, F, G ∈ F .
Let x, y ∈ C2, than for all F, G ∈ F
(
{F,G}x, y
)
C2 =
∫
R
(
(F (λ))∗G(λ)x, y
)
C2dρ(λ) =
∫
R
(
G(λ)x, F (λ)y
)
C2dρ(λ).
(28)
For us will be essential the procedure of normalization of vectors F from F . If for
F ∈ F the operator {F, F}−1 : C2 → C2 exists, than we have from (27)
{FC,FC} = C∗{F, F}C = 1 for C = {F, F}−1/2, (29)
i.e., FC is “normalized” F.
Lemma 2. Assume additionally, that supp dρ(λ) contains infinitely many different
points. Then every complex operator polynomial Pm(λ) ∈ F , m ∈ N0, can be norma-
lized.
Proof. According to (29), it is only necessary to prove that
{
Pm(λ), Pm(λ)
}−1 :
C2 → C2 exists. Taking into an account that
{
Pm(λ), Pm(λ)
}
is an operator on
finite dimensional space C2, the existence of
{
Pm(λ), Pm(λ)
}−1
is equivalent to the
following assertion: if for some x ∈ C2
({
Pm(λ), Pm(λ)
}
x, x
)
C2 = 0, then x = 0.
Using (28), we assume that
0 =
({
Pm(λ), Pm(λ)
}
x, x
)
C2 =
∫
R
‖Pm(λ)x‖2C2 dρ(λ).
The expression ‖Pm(λ)x‖2C2 is an ordinary polynomial of degree 2m, therefore the last
equality gives ‖Pm(λ)x‖2C2 = 0. But this equality shows that x = 0 because the higher
coefficient of Pm(λ) is an invertible operator.
The lemma is proved.
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INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 461
In what follows, the following special generalization of the usual Schmidt orthogonali-
zation procedure will be very essential (see, e.g., [18], Ch. 7).
Lemma 3. Let supp dρ(λ) contains infinitely many different points. Denote by
Pm(λ) some complex operator polynomial of degree m ∈ N0. Assume that we have a
system P1(λ), . . . , Pn(λ), n ∈ N, of such polynomials with properties:{
P1(λ), P1(λ)
}
= . . . = {Pn(λ), Pn(λ)} = 1,{
Pj(λ), Pk(λ)
}
= 0, j 6= k, j, k = 1, . . . , n.
(30)
Then one can be construct a complex operator polynomial Pn+1(λ) of degree n+ 1 for
which{
Pn+1(λ), Pn+1(λ)
}
= 1,
{
Pn+1(λ), Pj(λ)
}
= 0, j = 1, . . . , n. (31)
Proof. Let Dn+1(λ) be some fixed arbitrary complex operator polynomial of degree
n+ 1. We will find polynomial Pn+1(λ) at first in a non-normalized form (denote it by
Qm+1(λ)). Put
Qn+1(λ) = Dn+1(λ)−
n∑
j=1
Pj(λ)Cj (32)
where Cj : C2 → C2 are some unknown operator coefficients.
Multiply (32) on Pk(λ), k = 1, . . . , n, and using (27), (30) obtain
0 =
{
Qn+1(λ), Pk(λ)
}
=
{
Dn+1(λ), Pk(λ)
}
−
n∑
j=1
{
Pj(λ)Cj , Pk(λ)
}
=
=
{
Dn+1(λ), Pk(λ)
}
−
n∑
j=1
C∗j
{
Pj(λ), Pk(λ)
}
=
{
Dn+1(λ), Pk(λ)
}
− C∗k , (33)
i.e., Ck =
{
(Pk(λ))∗, Dn+1(λ)
}
, k = 1, . . . , n.
Last equalities and (32) show that the following operator polynomial of degree n+1:
Qn+1(λ) = Dn+1(λ)−
n∑
j=1
Pj(λ)
{
(Pj(λ))∗, Dn+1(λ)
}
(34)
is orthogonal (w.r.t. {·, ·}) to all P1(λ), . . . , Pn(λ).
According to Lemma 2, this polynomial can be normalized. We get as a result the
normalized polynomial Pn+1(λ) which is equal to Qn+1(λ)C where C =
{
Qn+1(λ),
Qn+1(λ)
}−1/2
. The conditions (33) of orthogonality are not violated because for all
k = 1, . . . , n
{
Pn+1(λ), Pk(λ)
}
=
{
Qn+1(λ)C,Pk(λ)
}
= C∗
{
Qn+1(λ), Pk(λ)
}
= 0.
The lemma is proved.
A few simple remarks are in order.
Remark 2. The results of Lemma 3 are true for (real) operator polynomials.
Remark 3. Introduce a vector-valued polynomial of degree m ∈ N0 as a function
on R such that:
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462 YU. M. BEREZANSKY
R 3 λ 7→
m∑
j=0
λjaj ,
where aj ∈ R2 are real coefficients; am 6= 0. It is obvious that the set
{
Pm(λ)x | x ∈
∈ R2
}
, where Pm(λ) is an operator polynomial of degree m, is a subset of a set of such
polynomials with the highest coefficient am = Amx.
Remark 4. Consider a fixed system
{
P1(λ)}, . . . , {Pm(λ)
}
, m ∈ N, of operator
polynomials with orthogonality properties (30) for which every polynomial Pn(λ), n =
= 2, . . . ,m − 1, is constructed according to the rules of Lemma 3; where polynomial
Dn+1(λ) from (32) has a form Dn+1(λ) = λn+1An+1, λ ∈ R, An+1 : R2 → R2 is
a fixed invertible operator. Operator polynomial P1(λ) is fixed of the form P1(λ) =
= λA1 +A0, where A1, A0 : R2 → R2 are invertible; P0(λ) = 1.
Introduce a set of vector-valued polynomials
R 3 λ 7→
m∑
j=0
Pj(λ)xj , (35)
where xj ∈ R2 are arbitrary. Then this set coincides with the set of all vector-valued
polynomials R 3 λ 7→ P (λ) of degree ≤ m.
This assertion immediately follows from the proof of Lemma 3 and Remark 2.
Remark 5. The union of all the sets (35) (m = 0, 1, . . .) is equal to the set P(R)
of all vector-valued polynomials on R.
Remark 6. In Section 7, in the study of Toda lattices, it will be essential to use in
Remarks 3 – 5 the invertible operators A2, A3, . . . : R2 → R2 with matrices[
θn 0
0 ωn
]
, θn, ωn > 0, n = 2, 3, . . . . (36)
Let dρ(λ) be the spectral measure of our operator J; assume that its supp dρ(λ)
contains infinitely many different points. Consider the linear space P(R) of all vector-
valued (real) polynomials on R. Every vector p(λ) from P(R) can be viewed as a
function of λ ∈ R with values in R2 for which
∫
R
‖p(λ)‖2R2 dρ(λ) <∞. If we introduce
the real Hilbert space L2
(
R,R2; dρ(λ)
)
of functions R 3 λ 7→ f(λ) ∈ R2 with a
scalar product
(
f(λ), g(λ)
)
L2(R,R2;dρ(λ))
=
∫
R
(
f(λ), g(λ)
)
R2dρ(λ), then we can say
that P(R) ⊂ L2
(
R,R2; dρ(λ)
)
is dense in the latter space. Similarly, it is possible to
introduce the complex Hilbert space L2
(
R,C2; dρ(λ)
)
.
We will fix some normalized operator polynomial P1(λ) ∈ L2
(
R,R2; dρ(λ)
)
of
degree one and after this we apply the Lemma 3 (see Remark 2) and construct by
orthogonalization the normalized operator polynomials P2(λ), P3(λ), . . . ;
{
Pj(λ),
Pk(λ)
}
= 1δj,k, j, k = 1, 2, . . . . The form of this polynomials depend on initial
operator polynomials D2(λ), D3(λ), . . . . We will take, in agreement with Remark 6:
Dn(λ) = λn
[
θn 0
0 ωn
]
, θn, ωn > 0, n = 2, 3, . . . . (37)
Lemma 4. Assume that matrices an in (4) have a form
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INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 463
an =
[
θ−1
n 0
0 ω−1
n
]
, θn, ωn > 0, n ∈ N. (38)
For j = 2, 3, . . . the polynomials
(
Qj(λ)
)∗
, constructed as solutions of equations (9),
(10) (see (13)), are equal to operator polynomials Pj(λ) constructed via the procedure
of orthogonalization with fixed P1(λ) equal to Q1(λ) from (13), (11), and Dn(λ) of the
form (37).
Thus, these
(
Qj(λ)
)∗
can be found via a procedure of orthogonalization, described
in Lemma 3.
Proof. According to Remark 5, the set of all vectors (35) coincides with P(R) and
is dense in the space L2
(
R,R2; dρ(λ)
)
(we first fixed an operator polynomial P1(λ) =
= λA1 +A0, where A1 : R2 → R2 is invertible, P0(λ) = 1).
Thus, we have, in the real Hilbert space L2
(
R,R2; dρ(λ)
)
{
set F1 of all vector-
valued polynomials of degree ≤ 1
}
, the orthogonal systems
{
Pn(λ)xn ∈ R2
}
, n ∈
∈ 2, 3, . . . , constructed by means of polynomials (37).
On the other hand from formulas (10) and (38) we see, that the operator polynomials
Q2(λ), Q3(λ), . . . (and, therefore the polynomials Q∗2(λ), Q∗3(λ), . . .) have the leading
coefficients, i.e., the coefficients near λ2, λ3, . . . , that coincide with those for P2(λ),
P3(λ), . . . (their corresponding leading terms are (37)). As we proved in Section 3
(see (23)) the system of vectors Q∗2(λ)ϕ0, Q
∗
3(λ)ϕ0, . . . from L2
(
R,R2; dρ(λ)
)
{F1}
is also orthogonal and gives the basis in the latter space. Since the coefficients of highest
degree λ of these two systems are equal, these systems must coincide: Q∗2(λ) = P2(λ),
Q∗3(λ) = P3(λ), . . . .
The lemma is proved.
5. The direct and inverse spectral problem for the Jacobi matrix considered.
Formulation of results. We can now collect constructions of last two sections and
formulate the main results. Thus, we consider on the Hilbert space l2 (1) the operator
J generated by Jacobi matrix J of the form (4) with the following additional condition:
the matrices an, n ∈ N, are diagonal of the form (38):
an =
[
θ−1
n 0
0 ω−1
n
]
, θn, ωn > 0, n ∈ N. (39)
Recall that all elements of matrix J are real and uniformly bounded. The matrices a0
and a∗0 will be denoted now by
a0 =
[
α0 α−1
]
6= 0, a∗0 =
[
α0
α−1
]
, α0, α−1 ∈ R. (40)
This matrix J generates (starting from its action on finite sequence lfin) a bounded
self-adjoint operator J in the space l2. Its spectral measure dρ(λ) is a bounded Borel
measure with a bounded supp dρ(λ), that, we assume in addition, contains infinitely
many different points.
Theorem 1. The full system of generalized eigenvectors ϕ(λ) of the operator J
has the following form: for all λ from the generalized spectrum of J
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464 YU. M. BEREZANSKY
ϕ(λ) =
(
ϕ0 = 1, ϕ1(λ), ϕ2(λ), . . .) ∈ l2(p−1
)
, (41)
where ϕn(λ), n ∈ N, are the solutions (10), (11) of difference equations (9).
The corresponding Fourier transform f̂(λ) of a vector f = (fn)∞n=0 ∈ lfin is defined
by the formula
lfin 3 f = (fn)∞n=0 7→ f̂(λ) =
∞∑
n=0
Q∗n(λ)fn ∈ L2(R,C2; dρ(λ)) (42)
and maps lfin onto right-hand side in (42) isometrically. The closure of (42) by
continuity gives the unitary operator between l2 and the right-hand side of (42).
The inverse spectral problem consist in the following. Assume that we know an
element a0 =
[
α0 α−1
]
6= 0 of the matrix J and the spectral measure dρ(λ) of
the corresponding operator J. Then we can find all elements of the matrix J in the
following manner. Construct by (11) the normalized operator polynomial Q1(λ) with
fixed θ1, ω1 6= 0 and b0 =
∫
R
λρ(λ). Then apply the procedure of orthogonalization
described in Lemma 3 and Remark 2 with initial operator polynomials Dn(λ), n =
= 2, 3, . . . , given by (37). As a result, we obtain the orthonormal sequence of operator
polynomials Q1(λ), Q2(λ), . . . . The elements of the matrix J (with an, n ∈ N, of the
form (39)) are reconstructed by formulas (25).
Remark 7. The conditions (39) on the blocks an are not essential: using in the
Lemma 3 instead Dn(λ) of the form (37) more complicated expressions (connected with
matrices an from (4)), we can treat the case of general matrix J of type (4).
6. The Lax equation corresponding to Jacobi matrices of our type. Assume
that elements an;α,β , bn;α,β , n ∈ N0, α, β = 0, 1, of matrix (4) are once continuously
differentiable uniformly bounded real functions of t ∈ [0, T ], T <∞. Denote this matrix
by J(t). Let A(t), t ∈ [0, T ], be some other matrix of the same type as J(t). The Lax
equation connected with these two matrices J(t) and A(t), has a form(
dJ
dt
)
(t) =: J̇(t) =
[
J(t), A(t)
]
:= J(t)A(t)−A(t)J(t), t ∈ [0, T ]. (43)
When elements of matrix A(t) do not depend on elements of J(t) the equation
J̇(t) =
[
J(t), A(t)
]
= 0, t ∈ [0, T ], (44)
is a linear differential equation w.r.t. matrix J(t). But if these elements depend on
elements of J(t), the system (44) is a system of nonlinear differential-difference equation
for the elements of matrix J(t). As in [1, 25, 4] and in an extensive list of other works,
we will discuss precisely this situation.
We will use for elements of matrix A(t) the same notations as for elements of J(t),
but with tildes: ãn(t), b̃n(t), n ∈ N0.
For every t ∈ [0, T ], we construct a bounded self-adjoint operator J(t) using the
matrix J(t). The spectrum of J(t), t ∈ [0, T ], is located in a bounded segment [a, b] ⊂ R
and therefore we can apply to our case the scheme of [23], § 3. Thus, in our case the
essential role will be played by the Weyl function
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INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 465
m(z; t) =
b∫
a
dρ(λ; t)
λ− z
, t ∈ [0, T ], z ∈ C\[a, b], (45)
where dρ(λ; t) is the spectral probability measure of the bounded self-adjoint operator
J(t). This operator is weekly continuously differentiable w.r.t. t ∈ [0, T ].
We will use a matrix A(t) similar to the matrix used in the article [23], § 21 . Namely,
we assume that analogically to (29), [23],
(ã0)∗(t) = 0 and ã0(t) = −a0(t), t ∈ [0, T ]. (46)
Then mimicking calculations from [23], § 3, we can prove an analogue of Theorem 3
from [23], which gives the following differential equation for the Weyl function (45):
ṁ(z; t) = (z − b0(t))m(z; t) + 1, z ∈ C\[a, b], (47)
and the representation of spectral measure dρ(λ; t) of the following form:
dρ(λ; t) = C(t)eλtdρ(λ; 0), λ ∈ [a, b], t ∈ [0, T ]. (48)
Here dρ(λ; 0) is the spectral measure of initial operator J(0) and C(t) > 0 is a factor
normalizing the measure (48) to a probability measure, i.e., the equality ρ([a, b]; t) = 1,
t ∈ [0, T ] must hold.
For the particular case of conditions (46), namely, if
ãn(t) = 0, b̃n(t) = −1
2
bn(t), ãn(t) = −an(t), n ∈ N0, t ∈ [0, T ], (49)
it is easy to write analogously to § 2, 3 from [23] (see [23], (33)) the following form of
Lax equations: ∀t ∈ [0, T ]
ȧn(t) =
1
2
(an(t)bn+1(t)− bn(t)an(t)), n ∈ N0,
ḃn(t) = a2
n(t)− a2
n−1(t), n = 2, 3, . . . ,
ḃ1(t) = a2
1(t)− (a0(t))∗a0(t), ḃ0(t) = a0(t)(a0(t))∗.
(50)
Thus, we have the system (50) of nonlinear differential-difference equation with
respect to matrix unknowns a0(t), a1(t), . . . ; b0(t), b1(t), . . . , t ∈ [0, T ]. For this
system we formulate the Cauchy problem: for given a0(0), a1(0), . . .; b0(0), b1(0), . . . ,
find the solution a0(t), a1(t), . . .; b0(t), b1(t), . . . for an arbitrary t ∈ [0, T ].
By using the inverse spectral problem we can formulate the following result:
Theorem 2. Assume that the matrices an(t) from (4) have the following form:
∀t ∈ [0, t]
1 When comparing these formulas with those of [23], it is necessary take into an account that the role
of matrices a0, a1, . . . was played by matrices c0, c1, . . . in [23].
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466 YU. M. BEREZANSKY
an(t) =
[
θ−1
n (t) 0
0 ω−1
n (t)
]
, θn(t), ωn(t) > 0, n ∈ N,
a0(t) =
[
α0(t) α−1(t)
]
6= 0.
(51)
Consider the Cauchy problem for (50) formulated above.
Suppose that for this problem, we know the part of its solutions: α0(t), α−1(t) for
every t ∈ [0, t]. Then the full solution can be reconstructed through following procedure.
Find the spectral measure dρ(λ; 0) of the initial operator J(0) and then construct,
by formula (48), the spectral measure dρ(λ; t) for J(t), t ∈ (0, T ]. Using the second
part of Theorem 1, find the sequence of orthonormal operator polynomials Q1(λ; t),
Q2(λ; t), . . . ;Q1(λ; t) is constructed by rules (11), (12), where a0(t) has the
form (51) and θ1(t), ω1(t) 6= 0 depend on t ∈ [0, T ] in a continuously differentiable
manner. Using these polynomial find, according to formula (25), all the elements of the
Jacobi matrix J(t), t ∈ [0, T ], whose elements constitute the full solution of our Cauchy
problem.
Proof. It is only necessary to check that the formulas (25) give the matrices an(t),
bn(t), t ∈ [0, T ], which are the solution of our Cauchy problem for the system (50). But
this assertion is a partial case of a more general Theorem 4 from [23].
The theorem is proved.
7. The integration of the modified double-infinite Toda lattice. The double-
infinite Toda lattice has the form
α̇n =
1
2
αn(βn+1 − βn), β̇n = α2
n − α2
n−1, n ∈ Z = {. . . ,−1, 0, 1, . . .}, (52)
where αn = αn(t), βn = βn(t) are real once continuously differentiable functions of
t ∈ [0, T ]. (52) is a differential-difference nonlinear equation and for (52) it is possible
to consider the Cauchy problem: we know initial data αn(0), βn(0), n ∈ Z, and it is
necessary to find the solution αn(t), βn(t), n ∈ Z, for t > 0.
The equation (52) is connected with the special case of the equation (50). Namely,
we take in (50) matrices
an(t) =
[
αn(t) 0
0 α−n−1(t)
]
, bn(t) =
[
βn(t) 0
0 −β−n(t)
]
, n ∈ N,
a0(t) =
[
α0(t) α−1(t)
]
, b0(t) =
[
β0(t)
]
, t ∈ [0, T ],
(53)
with positive uniformly bounded once continuously differentiable functions αm(t),
βm(t), m ∈ Z. Then equations (50) transform into a system:[
α̇n(t) 0
0 α̇−n−1(t)
]
=
1
2
([
βn+1(t) 0
0 −β−n−1(t)
][
αn(t) 0
0 α−n−1(t)
]
−
−
[
αn(t) 0
0 α−n−1(t)
][
βn(t) 0
0 −β−n(t)
])
, n ∈ N;
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INTEGRATION OF THE MODIFIED DOUBLE-INFINITE TODA LATTICE ... 467
[
α̇0(t) α̇−1(t)
]
=
=
1
2
([
α0(t) α−1(t)
] [β1(t) 0
0 −β−1(t)
]
−
[
β0(t)
] [
α0(t) α−1(t)
])
, n ∈ N;
[
β̇n(t) 0
0 −β̇−n(t)
]
= (54)
=
[
αn(t) 0
0 α−n−1(t)
]2
−
[
αn−1(t) 0
0 α−n(t)
]2
, n = 2, 3, . . . ;
[
β̇1(t) 0
0 −β̇−1(t)
]
=
[
α1(t) 0
0 α−2(t)
]2
−
[
α0(t)
α−1(t)
][
α0(t) α−1(t)
]
=
=
[
α1(t) 0
0 α−2(t)
]2
−
[
α2
0(t) α0(t)α−1(t)
α−1(t)α0(t) α2
−1(t)
]
;
[
β̇0(t)
]
=
[
α0(t) α−1(t)
][ α0(t)
α−1(t)
]
=
[
α2
0(t) + α2
−1(t)
]
.
From (54) we conclude: ∀t ∈ [0, T ]
α̇n(t) =
1
2
αn(t)
(
βn+1(t)− βn(t)
)
,
α̇−n−1(t) =
1
2
α−n−1(t)(−β−n−1(t)− β−n(t)), n ∈ N;
α̇0(t) =
1
2
α0(t)
(
β1(t)− β0(t)
)
,
α̇−1(t) =
1
2
α−1(t)
(
− β−1(t)− β0(t)
)
;
β̇n(t) = α2
n(t)− α2
n−1(t), (55)
− β̇−n(t) = α2
−n−1(t)− α2
−n(t), n = 2, 3, . . . ;
β̇1(t) = α2
1(t)− α2
0(t), α0(t)α−1(t) = 0,
− β̇−1(t) = α2
−2(t)− α2
−1(t),
β̇0(t) = α2
0(t) + α2
−1(t).
Above mentioned results give the possibility to present, instead of the Cauchy
problem for (52), the solution of the following problem. Consider the “modified”
double-infinite Toda lattice: we take the equation (55) and denote α0 = ϕ, α−1 = ψ.
Then for all t ∈ [0, T ]
α̇n(t) =
1
2
αn(t)(βn+1(t)− βn(t)), n ∈ Z \ {−1};
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468 YU. M. BEREZANSKY
β̇n(t) = α2
n(t)− α2
n−1(t), n ∈ Z \ {−1, 0, 1};
(56)
ψ̇(t) =
1
2
ψ(t)(−β0(t)− β−1(t)), β̇1(t) = α2
1(t)− ϕ2(t),
β̇0(t) = ϕ2(t) + ψ2(t), β̇−1(t) = ψ2(t)− α2
−2(t).
Here ϕ(t), ψ(t) are given real continuously differentiable functions for which
ϕ(t)ψ(t) = 0, t ∈ [0, T ], and ∀t ∈ [0, T ] ϕ(t) 6= 0 or ψ(t) 6= 0. (57)
For system (56) with condition (57) and unknowns αn(t), βn(t), n ∈ Z, we state
the Cauchy problem: for known initial data αn(0), n ∈ Z \ {−1}; βn(0), n ∈ Z, find
the solution of (56).
Theorem 3. The Cauchy problem formulated above has the solution that can be
found as follows.
Find the initial spectral measure dρ(λ; 0) of the operator J(0) which is constructed
from the initial matrix J(0) in the space l2 (1), where an(0), bn(0), n ∈ N, are given
by (53) and a0(0) = [ϕ(0)ψ(0)], b0(0) = β0(0). By formula (48), construct the spectral
measure dρ(λ; t), t ∈ (0, T ]. Using the second part of Theorem 1 and Theorem 2, find
the sequence of orthonormal operator polynomials Q1(λ; t), Q2(λ; t), . . . ; Q1(λ; t) has
the form (11), (12) where a0(t) = [ϕ(t)ψ(t)] and θ1(t), ω1(t) 6= 0 depend on t ∈ [0, T ]
in a continuously differentiable manner. Then elements of matrix J(t), i.e., the solution
of our Cauchy problem, can be found according to formula (25).
The proof of this theorem follows from Theorem 2.
Acknowledgment. This article was prepared in Bonn, in November of 2007. Author
thanks the University of Bonn for the hospitality and M. I. Gekhtman, I. Ya. Ivasyuk for
help.
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Received 23.01.08
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
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| id | umjimathkievua-article-3168 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:31Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-31682020-03-18T19:47:27Z Integration of a modified double-infinite Toda lattice by using the inverse spectral problem Інтегрування змiненого двосторонньо нескiнченного ланцюжка Тоди за допомогою оберненої спектральної задачi Berezansky, Yu. M. Березанський, Ю. М. An approach to finding a solution of the Cauchy problem for a modified double-infinite Toda lattice by using the inverse spectral problem is given. Викладено пiдхiд до знаходження розв’язку задачi Кошi для вказаного ланцюжка Тоди за допомогою оберненої спектральної задачi. Institute of Mathematics, NAS of Ukraine 2008-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3168 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 4 (2008); 453–469 Український математичний журнал; Том 60 № 4 (2008); 453–469 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3168/3083 https://umj.imath.kiev.ua/index.php/umj/article/view/3168/3084 Copyright (c) 2008 Berezansky Yu. M. |
| spellingShingle | Berezansky, Yu. M. Березанський, Ю. М. Integration of a modified double-infinite Toda lattice by using the inverse spectral problem |
| title | Integration of a modified double-infinite Toda lattice by using the inverse spectral problem |
| title_alt | Інтегрування змiненого двосторонньо нескiнченного ланцюжка Тоди за допомогою оберненої спектральної задачi |
| title_full | Integration of a modified double-infinite Toda lattice by using the inverse spectral problem |
| title_fullStr | Integration of a modified double-infinite Toda lattice by using the inverse spectral problem |
| title_full_unstemmed | Integration of a modified double-infinite Toda lattice by using the inverse spectral problem |
| title_short | Integration of a modified double-infinite Toda lattice by using the inverse spectral problem |
| title_sort | integration of a modified double-infinite toda lattice by using the inverse spectral problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3168 |
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