Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of "sparse potentials," suggested by the authors for the lattice ℤd, d gt; 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D gt; 2. For the Schrödinger operator - Δ - αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(-Δ - αV) of negative eigenvalues of - Δ - αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(-Δ - αV) under very mild regularity assumptions. A similar construction works also for the lattice ℤ2, where D = 2. Bibliography: 13 titles. © 2011 Springer Science+Business Media, Inc.