Quantum uncertainty and the spectra of symmetric operators

Abstract

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.