Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graphtheoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on 2(X), and the energy space HE . In particular, we prove that these operators are always essentially self-adjoint on 2(X), but may fail to be essentially self-adjoint on HE . In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme. © 2011 Elsevier Inc. All rights reserved