  In certain circumstances, the uncertainty, $\Delta S [\phi]$, of a quantum
observable, $S$, can be bounded from below by a finite overall constant $\Delta
S>0$, \emph{i.e.}, $\Delta S [\phi] \geq \Delta S$, for all physical states
$\phi$. For example, a finite lower bound to the resolution of distances has
been used to model a natural ultraviolet cutoff at the Planck or string scale.
In general, the minimum uncertainty of an observable can depend on the
expectation value, $t=\langle \phi, S \phi\rangle$, through a function $\Delta
S_t$ of $t$, \emph{i.e.}, $\Delta S [\phi]\ge \Delta S_t$, for all physical
states $\phi$ with $\langle \phi, S \phi\rangle=t$. An observable whose
uncertainty is finitely bounded from below is necessarily described by an
operator that is merely symmetric rather than self-adjoint on the physical
domain. Nevertheless, on larger domains, the operator possesses a family of
self-adjoint extensions. Here, we prove results on the relationship between the
spacing of the eigenvalues of these self-adjoint extensions and the function
$\Delta S_t$. We also discuss potential applications in quantum and classical
information theory.
