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.