arXiv 1503.028972015-03math.APmath.DG

Boundary value problems with Atiyah-Patodi-Singer type conditions and spectral triples

U. Battisti, J. Seiler

备注：: 27 pages, to appear in Journal of Noncommutative Geometry
发表信息：: J. Noncommut. Geom. 11 (2017), no. 3, 887-017
DOI：: 10.4171/JNCG/11-3-4

英文摘要：

We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah-Patodi-Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regular spectral triples $(\mathcal{A},\mathcal{H},\mathcal{D})$ for manifolds with boundary of arbitrary dimension, where $\mathcal{H}$ is the space of square integrable sections. Starting out from Dirac operators with APS-conditions, these triples are even in case of even dimensional manifolds; we show that the closure of $\mathcal{A}$ in $\mathscr{L}(\mathcal{H})$ coincides with the continuous functions on the manifold being constant on each connected component of the boundary.

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