Schrödinger Operators and Subordinate Structures on Infinite Networks

Abstract

Potential theory associated with the Schrödinger operators in a domain ω in the Euclidean space is closely related to the Laplace potential theory in ω. Based on this fact, in an infinite network X two harmonic structures, called the Laplacian and the q-Laplacian, are introduced and the interrelations between the two associated potential theories are investigated. In some sense, the q-Laplace structure is subordinate to the Laplace structure; this leads to the introduction of subordinate harmonic structures in an infinite network which already has a harmonic structure on it. This chapter deals with these subordinate structures in relation to the original structure.