Perturbations of discrete elliptic operators

Abstract

Abstract. Given V a finite set, a self-adjoint operator on C(V ), K, is called elliptic if it is positive semi-definite and its lowest eigenvalue is simple. Therefore, there exists a unique, up to sign, unitary function ω ∈ C(V ) satisfying K(ω) = λω and then K is named (λ, ω)-elliptic. Clearly, a (λ, ω)-elliptic operator is singular iff λ = 0. Examples of elliptic operators are the so-called Schrödinger operators on finite connected networks, as well as the signless Laplacian of connected bipartite graphs. If K is a (λ, ω)-elliptic operator, it defines an automorphism on ω ⊥ , whose inverse is called orthogonal Green operator of K. We aim here at studying the effect of a perturbation of K on its orthogonal Green operator. The perturbation here considered is performed by adding a self-adjoint and positive semi-definite operator to K. As particular cases we consider the effect of changing the conductances on semi-definite Schödinger operators on finite connected networks and on the signless Laplacian of connected bipartite graphs. The expression obtained for the perturbed networks is explicitly given in terms of the orthogonal Green function of the original network.

Optical tomography on graphs

“…In [23] we presented an algorithm for solving the forward problem of determining u, given η. Our approach was a perturbative one, making use of known Green's functions for the time-independent diffusion equation (or Schrödinger equation) [3,8,9,7,12,13,14,15,40,42], with η identically zero. The corresponding inverse problem, which we refer to as graph optical tomography, is to recover the potential η from measurements of u on the boundary of the graph.…”