Darboux transformation of symmetric Jacobi matrices and Toda lattices

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, theDarboux 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.