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arXiv · 1610.04954

Trace Formulas for a Class of non-Fredholm Operators: A Review

Alan Carey·Fritz Gesztesy·Harald Grosse·Galina Levitina·Denis Potapov·Fedor Sukochev·Dmitriy Zanin
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Abstract

We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$ acting in $L^2(\mathbb{R}; \mathcal{H})$, where $A$ denotes the operator of multiplication $(A f)(t) = A(t)f(t)$. In this article we review what is known when these operators have some essential spectrum and describe some new results in terms of associated spectral shift functions. We are especially interested in extensions to non-Fredholm situations, replacing the Fredholm index by the Witten index, and use a particular $(1+1)$-dimensional model setup to illustrate our approach based on spectral shift functions.

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Alan Carey, Fritz Gesztesy, Harald Grosse, Galina Levitina, Denis Potapov, Fedor Sukochev, Dmitriy Zanin. 2016-10-17. Trace Formulas for a Class of non-Fredholm Operators: A Review. https://doi.org/10.1142/s0129055x16300028

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