The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase

TL;DR

This paper rigorously extends the adiabatic theorem to certain non-Hermitian quantum systems with real eigenvalues using complex geometric phases, providing a solid mathematical foundation for their adiabatic behavior.

Contribution

It proves the validity of the adiabatic theorem for diagonalizable non-Hermitian systems with real eigenvalues, justifying the complex Berry phase concept.

Findings

Adiabatic theorem holds for specified non-Hermitian systems.

Provides a rigorous mathematical proof using complex geometric phase.

Justifies the definition of complex Berry phase in non-Hermitian quantum mechanics.

Abstract

The adiabatic theorem is one of the most interesting and significant theorems in quantum mechanics. However, the adiabatic theorem can fail for general non-Hermitian quantum systems. In this paper, by utilizing the complex geometric phase, the functional calculus for biorthogonal systems and the Gr\"{o}nwall inequality, we prove rigorously that the adiabatic theorem is still valid for diagonalizable non-Hermitian systems with real eigenvalues. The proof also justifies the definition of a complex Berry phase in non-Hermitian systems.