Spectral reciprocity and matrix representations of unbounded operators

Abstract

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic 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 ℓ 2 ( X ) , and the energy space H E . In particular, we prove that these operators are always essentially self-adjoint on ℓ 2 ( X ) , but may fail to be essentially self-adjoint on H E . 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 H E operators with the use of a new approximation scheme.