On an inverse spectral problem for a quadratic Jacobi matrix pencil

Abstract

Given two monic polynomials P 2n and P 2n−2 of degree 2n and 2n − 2 (n 2) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n × n Jacobi matrices B and C for whichwhere B 1 and C 1 are the (n − 1) × (n − 1) Jacobi matrices obtained from B and C by deleting the last row and the last column. The zeros of P 2n and P 2n−2 are the eigenvalues of the quadratic Jacobi matrix pencils on the right-hand side of the equalities, whence the title of the paper. The problem is formulated and solved in a slightly more general form.