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Quantum uncertainty and the spectra of symmetric operators

In certain circumstances, the uncertainty, $ΔS [ϕ]$, of a quantum observable, $S$, can be bounded from below by a finite overall constant $ΔS>0$, \emph{i.e.}, $ΔS [ϕ] \geq ΔS$, for all physical states $ϕ$. 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, $t=\langle ϕ, S ϕ\rangle$, through a function $ΔS_t$ of $t$, \emph{i.e.}, $ΔS [ϕ]\ge ΔS_t$, for all physical states $ϕ$ with $\langle ϕ, S ϕ\rangle=t$. 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 $ΔS_t$. We also discuss potential applications in quantum and classical information theory.