 Essential-spectrum invariance under relatively compact perturbations

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.