Julian Grossmann | Searcharxiv

arXiv2017

Oscillation theory for the density of states of high dimensional random operators

Sturm-Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Pruefer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

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arXiv2015

Index pairings in presence of symmetries with applications to topological insulators

In a basic framework of a complex Hilbert space equipped with a complex conjugation and an involution, linear operators can be real, quaternionic, symmetric or anti-symmetric, and orthogonal projections can furthermore be symplectic. This paper investigates index pairings of projections and unitaries submitted to such symmetries. Various scenarios emerge: Noether indices can take either arbitrary integer values or only even integer values or they can vanish and then possibly have secondary $\mathbb{Z}_2$-invariants. These general results are applied to prove index theorems for the strong invariants of topological insulators. The symmetries come from the Fermi projection ($K$-theoretic part of the pairing) and the Dirac operator ($K$-homological part of the pairing depending on the dimension of physical space).

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