A Variational Principle for Block Operator Matrices and its Application to the Angular Part of the Dirac Operator in Curved Spacetime

TL;DR

This paper develops a variational principle for block operator matrices with unbounded off-diagonal entries, providing bounds for the angular part of the Dirac operator in curved spacetime, and compares these bounds with existing numerical results.

Contribution

It introduces a new variational principle tailored for block operator matrices with unbounded off-diagonal entries, applied to the Dirac operator in Kerr-Newman spacetime.

Findings

Derived upper and lower bounds for the angular Dirac operator

Compared analytic bounds with numerical data from literature

Established a variational framework for complex block operator matrices

Abstract

The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above. In the last section, these analytic bounds are compared to numerical values from the literature.