Related papers: The spectral shift function and spectral flow
Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.
In this article we continue our analysis of Schroedinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in…
We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by…
Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed. Examples comprise heat-kernels, determinants and spectral sums needed for the analysis of…
Methods from scattering theory are introduced to analyze random Schroedinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the…
We show that a recent spectral flow approach proposed by Berkolaiko-Cox-Marzuola for analyzing the nodal deficiency of the nodal partition associated to an eigenfunction can be extended to more general partitions. To be more precise, we…
We review the spectral flow techniques for computing the index of the overlap Dirac operator including results relevant for SUSY Yang-Mills theories. We describe properties of the overlap Dirac operator, and methods to implement it…
It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in…
We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index…
An analytic definition of a $\mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings…
We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We…
Spectral flow was first studied by Atiyah and Lusztig, and first appeared in print in the work of Atiyah-Patodi-Singer (APS). For a norm-continuous path of self-adjoint Fredholm operators in the multiplier algebra $\mathcal{M}(\mathcal{B})$…
We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von…
A new construction of a semifinite spectral triple on an algebra of holonomy loops is presented. The construction is canonically associated to quantum gravity and is an alternative version of the spectral triple presented in…
Placing a Dirac-Schr\"odinger operator along the orbit of a flow on a compact manifold \(M\) defines an \(\R\)-equivariant spectral triple over the algebra of smooth functions on \(M\). We study some of the properties of these triples,…
For a continuous curve of families of Dirac type operators we define a higher spectral flow as a $K$-group element. We show that this higher spectral flow can be computed analytically by $\heta$-forms, and is related to the family index in…
This paper is devoted to the study of generalized subdifferentials of spectral functions over Euclidean Jordan algebras. Spectral functions appear often in optimization problems playing the role of "regularizer", "barrier", "penalty…
We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd…
The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of…
We consider general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$. A distribution matrix-valued function $\Si(s), \; s\in\bR,$ is called a…