On an inverse spectral problem for a quadratic Jacobi matrix pencil ✩

Given two monic polynomials P2n and P2n−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 which P2n(λ) = det λ2In + λB +C , P2n−2(λ) = det λ2In−1 +λB1 + C1 , where B1 and C1 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 P2n and P2n−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.  2004 Elsevier Inc. All rights reserved.