Quantum uncertainty and the spectra of symmetric operators

In certain circumstances, the uncertainty, $\mathrm{\Delta }S[\varphi ]$, of a quantum observable, $S$, can be bounded from below by a finite overall constant $\Delta S>0$,\mathit{\text{i.e.}},$\Delta S [\phi] \geq \Delta S$,forallphysicalstates$\phi$. 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, $t=⟨\varphi ,S\varphi ⟩$, through a function $\Delta S_t$of$t$,\mathit{\text{i.e.}},$\Delta S [\phi]\ge \Delta S_t$, for all physical states $\varphi$ with $⟨\varphi ,S\varphi ⟩=t$. 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 $\mathrm{\Delta }{S}_t$. We also discuss potential applications in quantum and classical information theory.