Operators with Singular Continuous Spectrum, Vi. Graph Laplacians and Laplace-beltrami Operators

Examples are constructed of Laplace-Beltrami operators and graph Laplacians with singular continuous spectrum. In previous papers in this series [6,3,5,2,8], we have explored the occurrence of singular continuous spectrum, particularly for Schrodinger operators and Jacobi matrices. In this note, we'll construct examples of graphs whose Laplacians and Riemannian manifolds whose Laplace-Beltrami operators have singular continuous spectrum. The idea will be to take models with considerable symmetry which reduce to Jacobi matrices or Schrodinger operators, and so reduce these two types of models to known cases. The analysis is simple, almost trivial. However, since these are the first examples I know of graph Laplacians or Laplace-Beltrami operators with singular continuous spectrum, it seems worthwhile to make it explicit. By a graph G = (V, I ) , we will mean a countable set of vertices V and an incidence matrix Iij of zeros and ones obeying (i) I..I . . . 21 3 2 1 Iii = 0. (ii) Ni = {j / Iij = 1) is finite for each i. Let r(i) = #(iVi) = Iij, the j coordination number of r. (iii) sup r(,i) < w. i The graph is called regular if r(i) is a constant. Given a graph G, we can define two bounded operators on &(V) = ( { U ( ~ ) ) ~ ~ T . ? / Received by the editors October 7, 1994. 1991 hfathematzcs Subject Classzficatzon. Primary 47B39. 05C50, 35P05, 58C40. This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material. 0 1 9 9 6 Barry Simon