The almost periodic gauge transform -- An abstract scheme with applications to Dirac Operators

TL;DR

This paper extends the gauge transform method to an abstract setting, enabling analysis of spectral properties of almost periodic and Dirac-type operators, with applications including asymptotic density of states and spectral band structure.

Contribution

It introduces a generalized gauge transform framework applicable to matrix-valued and Dirac operators, broadening the scope of spectral analysis tools.

Findings

Derived asymptotic expansions for density of states in almost periodic systems.

Proved the Bethe–Sommerfeld property for certain periodic Dirac systems.

Showed the spectrum contains semi-axes in specific Dirac operator cases.

Abstract

One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schr\"odinger operators is the method of gauge transform. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe--Sommerfeld property, that the spectrum contains a semi-axis -- or indeed two semi-axes in the case of operators that are not semi-bounded.