Discrete applied mathematics

Vol. 263, p. 130-139

DOI: 10.1016/j.dam.2019.03.005

Date of publication: 2019-06-30

Abstract:

We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm–Liouville boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout a discrete Schrödinger operator and their solutions can be computed using recent advances in the study of linear difference equations. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provide the entries of the inverse matrix]]>

Discrete applied mathematics

Vol. 263, p. 22-34

DOI: 10.1016/j.dam.2018.10.004

Date of publication: 2018-01-01

Abstract:

We aim here at determining the Green function for general Schrödinger operators on product networks. The first step consists in expressing Schrödinger operators on a product network as sum of appropriate Schrödinger operators on each factor network. Hence, we apply the philosophy of the separation of variables method in PDE, to express the Green function for the Schrödinger operator on a product network using Green functions on one of the factors and the eigenvalues and eigenfunctions of some Schrödinger operator on the other factor network. We emphasize that our method only needs the knowledge of eigenvalues and eigenfunctions of one of the factors, whereas other previous works need the spectral information of both factors. We apply our results to compute the Green function of Pm × Sh, where Pm is a Path with m vertices and Sh is a Star network with h + 1 vertices.

We aim here at determining the Green function for general Schrödinger operators on product networks. The first step consists in expressing Schrödinger operators on a product network as sum of appropriate Schrödinger operators on each factor network. Hence, we apply the philosophy of the separation of variables method in PDE, to express the Green function for the Schrödinger operator on a product network using Green functions on one of the factors and the eigenvalues and eigenfunctions of some Schrödinger operator on the other factor network. We emphasize that our method only needs the knowledge of eigenvalues and eigenfunctions of one of the factors, whereas other previous works need the spectral information of both factors. We apply our results to compute the Green function of Pm×Sh , where Pm is a Path with m vertices and Sh is a Star network with h+1 vertices.]]>

Discrete applied mathematics

Vol. 160, num. 1-2, p. 24-37

DOI: 10.1016/j.dam.2011.09.017

Date of publication: 2012-01

Discrete applied mathematics

Vol. 156, num. 18, p. 3443-3463

Date of publication: 2008-11

Discrete applied mathematics

Vol. 83, num. 1-3, p. 31-39

DOI: 10.1016/S0166-218X(97)00102-9

Date of publication: 1998-03