= Essential-spectrum invariance under relatively compact perturbations
{title2=$\sigma_{\mathrm{ess}}(A+V)=\sigma_{\mathrm{ess}}(A)$}
Let $A$ be a <closed operator> and let $V:D(A)\to H$ be <compact> when the domain carries its <graph norm>. If $A+V$ is closed on the same domain, <compact perturbation invariance of Fredholm operators> applied to the maps $D(A)\to H$ proves invariance of the Fredholm <essential spectrum>. For self-adjoint operators this is also the <essential spectrum> obtained by deleting isolated finite-multiplicity <eigenvalues>. A bounded potential tending to zero at spatial infinity is relatively compact with respect to a second-order constant-coefficient operator: local <Rellich-Kondrachov compactness theorem> gives compactness on bounded intervals and the small tails give compactness globally.
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