  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.
