Witten index, spectral shift function and spectral flow

Abstract

The notion of spectral flow has been introduced by Atiyah and Lustzig and is an important tool in geometry. In 1976 Atiyah, Patodi and Singer suggested that a path of elliptic operators on odd dimensional compact manifolds the spectral flow can be computed via the Fredholm index of so-called `suspension', which is a first order elliptic operator on a manifold of one higher dimension and the well-known ``Fredholm index=spectral flow" theorem has appeared for the first time. Later, Robbin and Salamon provided an abstract framework for ``Fredholm index=spectral flow" theorem with a crucial assumption that the operators in the path have discrete spectra and the endpoints are boundedly invertible, the assumption which is usually violated in the setting of differential operators coming from mathematical physics. In 2008 Pushnitski added a new ingredient to this equality, the Krein spectral shift function. With this new ingredient the operators in the path are allowed to have some essential spectral away from zero. If one removes the assumption that the endpoints are boundedly invertible, then the suspension is not necessarily a Fredholm operator. The latter assumption was omitted in the works by Carey and Gesztesy and their collaborators, where the Fredholm index was replaced by Witten index. However, the framework of this new equality ``Witten index=spectral shift function" does not cover yet differential operators on locally compact manifolds even in dimension 1. The present thesis provides a complete framework for the ``index=spectral shift function" theorem, which is suitable for differential operators on locally compact manifolds in all dimensions at once. When specialised to the classical situation (with discrete spectra) our result recovers classical results of Atiyah, Patodi and Singer. In addition, whenever the spectral flow for the path is well-defined we establish an extension of Robbin-Salamon type theorem which is suitable for differential operators with some essential spectrum away from zero in any dimension.