On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

Non-selfadjoint tridiagonal matrices play a role in the discretization and truncation of the Schrödinger equation in some extensions of quantum mechanics, a research field particularly active in the last two decades. In this article, we consider an inverse eigenvalue problem that consists of the reconstruction of such a real nonselfadjoint matrix from its prescribed eigenvalues and those of two complementary principal submatrices. Necessary and sufficient conditions under which the problem has a solution are presented, and uniqueness is discussed. The reconstruction is performed by using a modified unsymmetric Lanczos algorithm, designed to solve the proposed inverse eigenvalue problem. Some illustrative numerical examples are given to test the efficiency and feasibility of our reconstruction algorithm.