Block-Diagonalization of Operators with Gaps, with Applications to Dirac Operators

TL;DR

This paper develops new methods for block-diagonalizing Dirac operators with complex potentials, achieving exact diagonalization for high nuclear charges and improving approximation bounds, with broad applications in quantum physics.

Contribution

It introduces abstract theorems on spectral subspace perturbations that enable block-diagonalization of Dirac operators with unbounded, non-self-adjoint potentials, extending previous bounds.

Findings

Exact diagonalization up to Z=124 for Coulomb potentials.

Convergence of Douglas-Kroll-Heß approximation up to Z=62.

Improved bounds over previous results by H. Siedentop and E. Stockmeyer.

Abstract

We present new results on the block-diagonalization of Dirac operators on three-dimensional Euclidean space with unbounded potentials. Classes of admissible potentials include electromagnetic potentials with strong Coulomb singularities and more general matrix-valued potentials, even non-self-adjoint ones. For the Coulomb potential, we achieve an exact diagonalization up to nuclear charge Z=124 and prove the convergence of the Douglas-Kroll-He\ss\ approximation up to Z=62, thus improving the upper bounds Z=93 and Z=51, respectively, by H.\ Siedentop and E.\ Stockmeyer considerably. These results follow from abstract theorems on perturbations of spectral subspaces of operators with gaps, which are based on a method of H.\ Langer and C.\ Tretter and are also of independent interest.