Spectral reciprocity and matrix representations of unbounded operators

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.