Spectral localizer for unbounded topological Hamiltonians

Abstract

In a series of joint papers with Loring, it was shown how index parings of $K$-theory classes with Fredholm modules can be computed as the half-signature of a suitably constructed finite-dimensional matrix, called the spectral localizer. After offering a novel perspective on earlier results, it is shown how the framework can be extended to unbounded gapped Hamiltonians given as second-order linear differential operators (in particular, matrix-valued Schrödinger and regularized Dirac operators). Important insights are the coercive phase space localization and discretizations by means of wavelets.