On an inverse spectral problem for a quadratic Jacobi matrix pencil

Given two monic polynomials P-2n 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 x n Jacobi matrices B and C for which P2n (lambda) = det(lambda I-2(n) + lambda B + C), P2n-2(lambda) = det(lambda I-2(n) + lambda B-1 + C-1), where B-1 and C-1 are the (n - 1) x (n - 1) Jacobi matrices obtained from B and C by deleting the last row and the last column. The zeros of P-2n 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. (c) 2004 Elsevier Inc. All rights reserved.