  Persson's formula expresses the infimum of the essential spectrum of a
suitable self-adjoint Schr\"odinger operator in $R^n$ in terms of the lower
spectral points of a family of restrictions of the operator to complements of
relatively compact subsets. It has been extended to other situations. We
present an approach based on $C^*$-algebras associated to \'etale groupoids. In
such a setting there are intrinsic versions, referring to self-adjoint elements
of the groupoid $C^*$-algebras. When representations in Hilbert spaces are
considered, the results not always involve only the essential spectrum. The
range of applications is quite distinct from the traditional one. We indicate
examples related to symbolic dynamics and band dominated operators on discrete
metric spaces. The treatment needs only a small amount of symmetry. Even when
group actions are involved, restrictions to non-invariant subsets are needed
and have to be treated carefully.
