AUTHOR=Kovalyov Ivan , Levina Oleksandra 

TITLE=Darboux transformation of symmetric Jacobi matrices and Toda lattices

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=10

YEAR=2024

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2024.1397374

DOI=10.3389/fams.2024.1397374

ISSN=2297-4687

ABSTRACT=<p>Let <italic>J</italic> be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix <italic>J</italic> to admit factorization <italic>J</italic> = <italic>LU</italic> (or <italic>J</italic> = 𝔘𝔏) with <italic>L</italic> (or 𝔏) and <italic>U</italic> (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of <italic>J</italic> is the symmetric Jacobi matrix <italic>J</italic><sup>(<italic>p</italic>)</sup> = <italic>UL</italic> (or <italic>J</italic><sup>(<italic>d</italic>)</sup> = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, <bold>m</bold>-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.</p>