Abstract
Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J(p) = UL (or J(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.
1 Introduction
Let a sequence of real numbers be associated with a measure μ on (−∞, +∞), i.e.
However, in the general case, is associated with a linear functional 𝔖 by
We consider the sequence such that
where . Note, if Dn > 0 for all n ∈ ℕ, then there exists measure μ associated with , otherwise, the sequence is associated with only linear functional 𝔖.
On the other hand (see [1, 2]), the real sequence is associated with the symmetric Jacobi matrix J and the sequence of orthogonal polynomials of the first kind , which can defined by
[3, 4] Moreover, the sequence satisfies a three-term recurrence relation
with the initial conditions
In the short form we can rewrite Equation (1.2) as
where and the symmetric Jacobi matrix J is defined by
On the other hand, the symmetric Jacobi matrix J is associated with the moment sequence , the following relation holds (see [2, 5])
where and m– function of Jacobi matrix is found by
There exist two type transformations of orthogonal polynomials, which are the Christoffel and Geronimus transformations. One are studied in the paper Zhedanov [6]. The Christoffel transformation is defined by
where and α is arbitrary parameter. Moreover, Equation (1.7) can be rewritten as follows:
Theorem 1.1. ([7, Theorem 1.5]) Let be the sequence of the orthogonal polynomials associated with Equation (1.2). Then the Christoffel–Darboux formula takes the following form
The second transformation is a Geronimus transformation of the orthogonal polynomials [6], one is defined by
Toda lattice. The Toda lattice is a system of differential equations
which was introduced in Toda [8].
We study the semi-infinite system with x−1 = −∞. [9, 10] Flaschka variables are defined by
Therefore, we obtain the following system in terms of Flaschka variables
Hence, the semi-infinite Toda lattice is associated with the symmetric Jacobi matrix J and Lax pair (J, A), such that
where the matrix A = J+ − J−, where J+ and J− are upper and lower triangular part of J, respectively and
As is known (see [8, 11]), the system (1.11) is equivalent to the following
Darboux transformation of the monic classical and generalized Jacobi matrices were studied in Bueno and Marcellán [12], Derevyagin and Derkach [13], and Kovalyov [14, 15]. Darboux transformation involves finding a factorization of a matrix from a certain class such that the new matrix is from the same class. There are two types of Darboux transformation: transformation with and without parameter. Jacobi matrix is associated with many objects. There are moment sequence, measure, linear functional orthogonal polynomials and Toda lattice. Hence, in the current paper, we study not only Darboux transformation of the symmetric Jacobi matrices, but we also study the transformation of the associated objects. Hence, we investigate the Darboux transformation of the symmetric Jacobi matrices J and find relations between associated Toda lattice, orthogonal polynomials, moment sequences and m–functions. We obtain that the Darboux transformation without parameter of the symmetric Jacobi matrices has more additional existence conditions in contrast to case of the monic Jacobi matrices. On the other hand, the Darboux transformation with parameter of the symmetric Jacobi matrices is generated more easily. The results obtained can be applied for further research related to symmetric Jacobi matrices, Toda lattices and inverse problems. Of course, it can also be applied to the Toda lattice hierarchy.
Now, briefly describe the content of the paper. Section 2 contains Darboux transformation without parameter of the symmetric Jacobi matrix J. We find LU–factorization of J and the transformed matrix J(p). Relation between Toda lattices, moment sequences and m–functions associated with the Jacobi matrices was obtained. In this case, the orthogonal polynomials are transformed by the Christoffel formula (1.7). In Section 3, we study the Darboux transformation with parameter of the symmetric Jacobi matrix J. We find 𝔘𝔏–factorization of J and transformed matrix J(d). Moreover, the relations between orthogonal polynomials, m–functions, moment sequence and Toda lattices are found according to explicit formulas.
2 Darboux transformation without parameter of symmetric Jacobi matrix
Now we study a Darboux transformation without parameter of symmetric Jacobi matrix J. The goal is to find the transformations of polynomials of the first kind, m-functions, measure, moment sequence and Toda lattice, which are associated with the transformed Jacobi matrix.
2.1 LU–factorization
Lemma 2.1. Let J be a symmetric Jacobi matrix. Then J admits LU–factorization
where L and U are lower and upper triangular matrices, respectively, which are defined by
if and only if the following system is solvable
Proof. Let us calculate the product LU
Comparing the product LU with the Jacobi matrix J
we obtain the system (2.3).
If the system (2.3) is solvable, then J admits the factorization J = LU of the form (2.1–2.3), where L and U are found uniquely. Conversely, if J admit LU—factorization then the system (2.3) is solvable. This completes the proof.
Lemma 2.2. Let J be the symmetric Jacobi matrix and let J = LU be its LU– factorization of the form (2.1–2.3). Let Pj be the polynomials of the first kind associated with the matrix J. Then
Proof. Let J admit the LU–factorization of the form (2.1–2.3). Setting λ = 0 in Equation (1.2), we obtain
By induction, we prove Equation (2.4).
1. Let n = 0, then
and due to the initial condition (1.3) and (2.3), we get
2. Let n = 1, then
and by Equation (2.3), we have
3. Let Equation (2.4) hold for n = k − 1.
4. Let us prove Equation (2.4) for n = k, we obtain
Consequently
= {by Section (2.3)}
So, Equation (2.4) is proven. This completes the proof.
Corollary 2.3. Let J be the symmetric Jacobi matrix and let J = LU be its LU–factorization of the form (2.1–2.3). Let Pj be the polynomials of the first kind associated with the matrix J. Then
Proof. Let J admit the LU–factorization of the form (2.1–2.3) and let Pj be the polynomials of the first kind associated with J. By Lemma 2.2, Equation (2.4) holds and we obtain
So, Equation (2.5) is proven. This completes the proof.
Corollary 2.4. Let J be the symmetric Jacobi matrix and let J = LU be its LU–factorization of the form (2.1–2.3). Let Pj be the polynomials of the first kind associated with the matrix J. Then
Proof. Let J admit the LU–factorization of the form (2.1–2.3). By Lemma 2.2, we obtain
Hence, Equation (2.6) is proven. This completes the proof.
Theorem 2.5. Let J be the symmetric Jacobi matrix and let Pj be the polynomials of the first kind associated with J. Then J admits LU—factorization of the form (2.1–2.3) if and only if
Furthermore,
Proof. Let Pj(0) ≠ 0 for all j ∈ ℤ+. By Lemma 2.2 the system (2.8) is equivalent to the system (2.3). Consequently, by Lemma 2.1 the Jacobi matrix J admits LU—factorization of the form (2.1–2.3). Conversely, if the Jacobi matrix J admits LU—factorization of the form (2.1–2.3), then by Lemma 2.1 and Lemma 2.2 the polynomials of the first kind Pj satisfy (2.7). This completes the proof.
2.2 Transformed Jacobi matrix J(p) = UL
Definition 2.6. Let the symmetric Jacobi matrix J admit LU—factorization of the form (2.1–2.3). Then a transformation
is called a Darboux transformation without parameter of the symmetric Jacobi matrix J.
Theorem 2.7. Let J be the symmetric Jacobi matrix (1.4) and let J = LU be its LU–factorization of the form (2.1–2.3). Then the Darboux transformation without parameter of the matrix J is the symmetric Jacobi matrix
if and only if
Proof. Calculating UL, we obtain
= {by Equation (2.3)}
Consecuently, J(p) is the symmetric Jacobi matrix if and only if
Comparing Equation (2.3) with Equation (2.11), we get
By Equation (2.3), uj + vjlj = bj for all j ∈ ℕ, we obtain Equations (2.9, 2.10) and J(p) is the symmetric Jacobi matrix. This completes the proof.
Theorem 2.8. Let the symmetric Jacobi matrix J satisfy (2.7) and let J = LU be its LU–factorization of the form (2.1–2.3). Let J(p) = UL be the Darboux transformation without parameter of J. Then the polynomials of the first kind associated with J(p) can be found by Christoffel–Darboux formula
where Pj are the polynomials of the first kind associated with the symmetric Jacobi matrix J.
Proof. Let the Jacobi matrix J satisfy (2.7) and admit LU–factorization of the form (2.1–2.3). Calculating the inverse matrix of L, we obtain
On the other hand,
Consequently, we obtain the relation between the polynomials of the first kind
By Corollary 2.4, we obtain
However, we can rewrite Equation (2.13) and by Christoffel–Darboux formula (1.8), we obtain
Hence, Equation (2.12) holds. This completes the proof.
In the following statements we find the connection between orthogonal polynomials, moment sequences, measures, linear functionals, m–functions and Toda lattices according to the transformation Darboux transformation without parameter of the symmetric Jacobi matrix.
Proposition 2.9. Let the symmetric Jacobi matrix J admit LU– factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J(p) = UL be the Darboux transformation without parameter of J. Let and be the moment sequences associated with the matrices J and J(p), respectively. Then the moment sequence can be found by the following formula
Proof. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J(p) = UL be its Darboux transformation without parameter. By Equation (1.5), we obtain
Consequently, the moments can be found by Equation (2.14). This completes the proof.
Corollary 2.10. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1)–(2.3) and let the symmetric Jacobi matrix J(p) = UL be the Darboux transformation without parameter of J. Let 𝔖 and 𝔖(p) be the linear functionals associated with the matrices J and J(p), respectively. Then
Proof. Let 𝔖 and 𝔖(p) be the linear functionals associated with the symmetric Jacobi matrices J = LU and J(p) = UL, respectively, where L and U are defined by Equations (2.1–2.3). By Equation (1.1), we obtain
Consequently, Equation (1.19) holds. This completes the proof.
Corollary 2.11. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J(p) = UL be the Darboux transformation without parameter of J. Let dμ and dμ(p) be the measures associated with the matrices J and J(p), respectively. Then
Proof. Let μ and μ(p) be the measures associated with the symmetric Jacobi matrices J = LU and J(p) = UL, respectively, where L and U are defined by Equation (2.1–2.3). Then
Consequently, we find transformation of the measure and Equation (2.16) holds. This completes the proof.
Proposition 2.12. Let the symmetric Jacobi matrix J admit LU– factorization of the form (2.1–2.3) and let the symmetric Jacobi matrix J(p) = UL be the Darboux transformation without parameter of J. Let m and m(p) be the m–functions associated with the matrices J and J(p), respectively. Then
Proof. By Equation (1.6)
Hence, m–function is transformed by Equation (2.17). This completes the proof.
Toda latice. The last statement is the following theorem of this section. One is described the Toda lattice associated with the symmetric Jacobi matrices J(p).
Theorem 2.13. Let the symmetric Jacobi matrix J admit LU–factorization of the form (2.1–2.3) and J be associated with the Toda lattice (1.9–1.11). Let the symmetric Jacobi matrix J(p) = UL be the Darboux transformation without parameter of J. Then J(p) is associated with the following Toda lattice
Furthermore, the matrix A does not change.
Proof. Let the symmetric Jacobi matrix be associated be associated with the Toda (1.9–1.11) and let J = LU, where L and U are defined by Equations (2.2, 2.3, 2.10). Consequently, the symmetric Jacobi matrix J(p) = UL is the Darboux transformation without parameter of J. By Equation (2.9), we obtain , and the matrix A does not change in the Lax pair, i.e.
Moreover, similar to Equation (1.9–1.11), the symmetric Jacobi matrix J(p) = UL is associated with the Toda lattice (2.18–2.20). This completes the proof.
3 Darboux transformation with parameter of the Jacobi matrix
The next step is the Darboux transformation with parameter of the symmetric Jacobi matrix J. We study the transformations of the polynomials of the first kind, m–functions, measure, moment sequence and Toda lattice, which are associated with the transformed Jacobi matrix.
3.1 𝔘𝔏–factorization
Theorem 3.1. Let J be the symmetric Jacobi matrix and let S0 be a some real parameter. Then J admits the following 𝔘𝔏–factorization
where 𝔏 and 𝔘 are lower and upper triangular matrices, respectively, which are defined by
if and only if the following system is solvable
Proof. Let J be the Jacobi matrix. Let 𝔏 and 𝔘 are defined by Equation (3.2), where the parameter S0 ∈ ℝ \ {0, −b0}. Calculating the product 𝔘𝔏, we obtain
Comparing the product 𝔘𝔏 with the Jacobi matrix J, we obtain the system (3.3). This completes the proof.
3.2 Transformed Jacobi matrix J(d) = 𝔘𝔏
Definition 3.2. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3). Then a transformation
is called a Darboux transformation with parameter of the Jacobi matrix J.
Theorem 3.3. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) with parameter S0 ∈ ℝ\{0, −b0}. Then the Darboux transformation with parameter of the Jacobi matrix J is the symmetric Jacobi matrix
if and only if
Proof. Let J admit 𝔘𝔏—factorization of the form (3.1–3.3). Calculating the product 𝔏𝔘, we obtain
Hence, J(d) is the symmetric Jacobi matrix if and only if
On the other hand, by Equation (3.3), we know
Consequently, we obtain Equation (3.5). This completes the proof.
Theorem 3.4. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let J(d) = 𝔏𝔘 be its Darboux transformation with parameter. Then the polynomials of the first kind transform by the Geronimus formula
where Pi and are polynomials of the first kind associated with the matrix J and J(d), respectively.
Proof. Let J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let J(d) = 𝔏𝔘 be its Darboux transformation with parameter. Then
where
So, the polynomials of the first kind are transformed by the Geronimus formula and Equation (3.6) holds. This completes the proof.
Proposition 3.5. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let and be the moment sequences associated with the matrices J and J(d), respectively. Then the moment sequence can be found by
Proof. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be its Darboux transformation with parameter. By Equation (1.5), we obtain
and
Hence, Equation (3.7) holds. This completes the proof.
Corollary 3.6. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let and be the moment sequences associated with the matrices J and J(d), respectively. Then
Proof. By Equation (3.7) and s0 = 1, we obtain
So, Equation (3.8) holds. This completes the proof.
Corollary 3.7. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let 𝔖 and 𝔖(d) be the linear functionals associated with the matrices J and J(d), respectively. Then
Proof. Let 𝔖 and 𝔖(d) be the linear functionals associated with the symmetric Jacobi matrices J = 𝔘𝔏 and J(d) = 𝔏𝔘, respectively, where 𝔏 and 𝔘 are defined by Equations (3.2, 3.3). By Equation (1.1), we obtain
Consequently, Equation (3.9) holds. This completes the proof.
Corollary 3.8. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3). and let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let dμ and dμ(d) be the measures associated with the matrices J and J(d), respectively. Then
Proof. Let J = 𝔘𝔏 and J(d) = 𝔏𝔘, where the matrices 𝔏 and 𝔘 are defined by Equations (3.2, 3.3, 3.5). The measures dμ and dμ(d) are associated with the matrices J and J(d), respectively. Then
Consequently,
Hence, Equation (3.10) holds. This completes the proof.
Proposition 3.9. Let the symmetric Jacobi matrix J admit 𝔘𝔏—factorization of the form (3.1–3.3) and let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be the Darboux transformation with parameter of J. Let m and m(d) be m–functions associated with the matrices J and J(d), respectively. Then
Proof. Let J = 𝔘𝔏 and J(d) = 𝔏𝔘, where the matrices 𝔏 and 𝔘 are defined by Equations (3.2, 3.3, 3.5). Then m–functions of the matrices J and J(d) are related by
On the other hand
By Equation (3.7), and Equation (3.11) holds. This completes the proof.
Toda latice. There is the last target of our investigation.
Theorem 3.10. Let the symmetric Jacobi matrix J admit 𝔘𝔏–factorization of the form (3.1–3.3) and J be associated with the Toda lattice (1.9–1.11). Let the symmetric Jacobi matrix J(d) = 𝔏𝔘 be the Darboux transformation without parameter of J. Then J(d) is associated with the following Toda lattice
Furthermore, the matrix A does not change.
Proof. Let the symmetric Jacobi matrix J be associated with the Toda lattice (1.9–1.11) and let J = 𝔘𝔏, where 𝔏 and 𝔘 are defined by Equations (3.2, 3.3, 3.5). Consequently, the symmetric Jacobi matrix J(d) = 𝔏𝔘 is the Darboux transformation with parameter of J. By Equation (3.4), we obtain , and the matrix A does not change in the Lax pair, i.e.
Moreover, similar to Equations (1.9–1.11), the symmetric Jacobi matrix J(d) = 𝔏𝔘 is associated with the Toda lattice (3.12–3.14). This completes the proof.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
IK: Investigation, Writing – original draft. OL: Investigation, Writing – original draft.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. IK gratefully acknowledges financial support by the German Research Foundation (DFG, grant 520952250).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.
AkhiezerNI. The Classical Moment Problem. Edinburgh: Oliver and Boyd (1965).
2.
SimonB. The classical moment problem as a self-adjoint finite diference operator. Adv Math. (1998) 137:82–203. 10.1006/aima.1998.1728
3.
BerezanskyYMDudkinME. Jacobi Matrices and the Moment Problem, Springer, Series Operator Theory: Advances and Applications. Cham: Birkhauser (2023), IX+487. 10.1007/978-3-031-46387-7
4.
BerezanskiYM. The integration of semi-infinite Toda chain by means of inverse spectral problem. Rep Math Phys. (1986) 24:21–47. 10.1016/0034-4877(86)90038-8
5.
GesztesyFSimonB. m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J Anal Math. (1997) 73:267–97. 10.1007/BF02788147
6.
ZhedanovA. Rational spectral transformations and orthogonal polynomials. J Comput Appl Math Vol. (1997) 85:67–86. 10.1016/S0377-0427(97)00130-1
7.
SuetinPK. Classical Orthogonal Polynomials, 2nd rev. ed. Moscow: Nauka (1979).
8.
TodaM. Theory of Nonlinear Lattices. Springer Series in Solid-State Sciences, 20, 2nd ed.Berlin: Springer (1989). 10.1007/978-3-642-83219-2
9.
Van AsscheW. Orthogonal polynomials, toda lattices and Painlevé equations. Physica D. (2022) 434:133214. 10.1016/j.physd.2022.133214
10.
BerezanskyYMDudkinMEeditors. Applications of the spectral theory of Jacobi matrices and their generalizations to the integration of nonlinear equations. In: Jacobi Matrices and the Moment Problem Operator Theory: Advances and Applications, Vol. 294. Cham: Birkhauser (2023). p. 413–65. 10.1007/978-3-031-46387-7_9
11.
TeschlG. Jacobi Operator and Completely Integrable Nonlinear Lattices. Providence, RI: American Mathematical Society (2001).
12.
BuenoMIMarcellánF. Darboux transformation and perturbation of linear functionals. Linear Algebra Appl. (2004) 384:215–42. 10.1016/j.laa.2004.02.004
13.
DerevyaginMDerkachV. Darboux transformations of Jacobi matrices and Páde approximation. Linear Algebra Appl. (2011) 435:3056–84. 10.1016/j.laa.2011.05.035
14.
KovalyovI. Darboux transformation of generalized Jacobi matrices. Methods Funct Anal Topol. (2014) 20:301–20.
15.
KovalyovI. Darboux transformation with parameter of generalized Jacobi matrices. J Math Sci. (2017) 222:703–22. 10.1007/s10958-017-3326-3
Summary
Keywords
Jacobi matrix, Darboux transformation, orthogonal polynomials, moment problem, Toda lattice
Citation
Kovalyov I and Levina O (2024) Darboux transformation of symmetric Jacobi matrices and Toda lattices. Front. Appl. Math. Stat. 10:1397374. doi: 10.3389/fams.2024.1397374
Received
07 March 2024
Accepted
25 April 2024
Published
16 May 2024
Volume
10 - 2024
Edited by
Marina Chugunova, Claremont Graduate University, United States
Reviewed by
Youssri Hassan Youssri, Cairo University, Egypt
Dudkin Mykola, Kyiv Polytechnic Institute, Ukraine
Updates
Copyright
© 2024 Kovalyov and Levina.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ivan Kovalyov i.m.kovalyov@gmail.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.