The Loring--Schulz-Baldes Spectral Localizer Revisited

The spectral localizer, introduced by Loring in 2015 and Loring and Schulz-Baldes in 2017, is a method to compute the (infinite volume) topological invariant of a quantum Hamiltonian on ${\text{ZZ}}^d$, as the signature of the (finite) localizer matrix. We present a direct and elementary spectral-theoretic proof treating the $d=1$ and $d=2$ cases on an almost equal footing. Moreover, we re-interpret the localizer as a higher-dimensional topological insulator via the bulk-edge correspondence.