This article is about a local formula for the index pairing between an invertible operator on a Hilbert space and an unbounded operator with compact resolvent. The point is to compute this index locally, using the signature of a finite-dimensional matrix. This matrix depends on a tuning parameter \(\kappa>0\), which must be sufficiently small, and a size parameter \(\varrho>0\), which must be sufficiently large depending on \(\kappa\). In the allowed region, the signature of the resulting finite-dimensional matrix does not depend on the parameters.
The authors have already proven the index formula in [T. A. Loring and H. Schulz-Baldes, New York J. Math. 23, 1111–1140 (2017; Zbl 1384.46048)]. Here they give a shorter proof for it that is based on properties of the spectral flow. And the statement here is more general because the unbounded operator for the pairing is no longer required to have a special form.
The index theorem facilitates numerical computations and is particularly remarkable because Fredholm operators with non-zero index can only exist on infinite-dimensional Hilbert spaces. The finite-dimensional matrices whose signature gives the index are called the spectral localiser. They allow to talk about the spectrum of a Hamiltonian and its topological space in suitable regions in position space, which is very desirable for the study of topological phases in mathematical physics.