Spectral reciprocity and matrix representations of unbounded operators

Abstract

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical conformal Laplacians and Hamiltonians from statistical mechanics. 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 ${\ell }^2(X)$, and the energy space ${\mathcal{H}}_{\mathcal{E}}$. In particular, we prove that these operators are always essentially self-adjoint on ${\ell }^2(X)$, but may fail to be essentially self-adjoint on ${\mathcal{H}}_{\mathcal{E}}$. 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 ${\mathcal{H}}_{\mathcal{E}}$ operators with the use of a new approximation scheme.

arXiv comment20 pages, 1 figure. To appear: Journal of Functional Analysis