Perturbations of APS Boundary Conditions for Lorentzian Dirac Operators

TL;DR

This work addresses how far APS boundary conditions for a Lorentzian Dirac operator can be perturbed without losing Fredholmness. By developing Fredholm-criteria for perturbations of compact pairs of projections and translating these into the Lorentzian index framework, the authors establish a sharp constant $\\varepsilon=1$ for stability and a norm-difference condition via the Calkin norm. They introduce a graph-perturbation (graphfred) approach, showing that boundary data given by graphs over APS projections with $ig\,igig\|G_0igrac{C}igigig\\_C ig\|G_1ig riangledown_C<1$ preserve Fredholmness and keep the index aligned with the APS case, thereby extending Lorentzian index theory to a broader class of boundary conditions. The results yield explicit Fredholmness criteria and index formulas for perturbed Lorentzian boundary-value problems, providing a robust framework for analyzing Lorentzian Dirac operators beyond the canonical APS setup.

Abstract

We study how far APS boundary conditions for a Lorentzian Dirac operator may be perturbed without destroying Fredholmness of the Dirac operator. This is done by developing criteria under which the perturbation of a compact pair of projections is a Fredholm pair.

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Theorems & Definitions (23)

• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.6
• Definition 2.8
• Definition 2.9
• Remark 2.10
• Theorem 2.11
• Theorem 2.12
• Remark 2.13