Darboux transformation with parameter of generalized Jacobi matrices

Abstract

A monic generalized Jacobi matrix \( \mathfrak{J} \) is factorized into upper and lower triangular two-diagonal block matrices of special forms so that J = UL. It is shown that such factorization depends on a free real parameter d(∈ ℝ). As the main result, it is shown that the matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)}= LU \) is also a monic generalized Jacobi matrix. The matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) is called the Darboux transform of \( \mathfrak{J} \) with parameter d. An analog of the Geronimus formula for polynomials of the first kind of the matrix \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) is proved, and the relations between m-functions of J and \( {\mathfrak{J}}^{\left(\mathbf{d}\right)} \) are found.