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

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 Grönwall 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 for non-Hermitian systems, in both Abelian and non-Abelian cases.