Adiabatic theorem for non-Hermitian quantum systems with non-degenerate real eigenvalues: A proof following Kato's approach

TL;DR

The paper extends the adiabatic theorem to time-dependent non-Hermitian quantum systems with non-degenerate real spectra by adapting Kato’s approach to a biorthogonal basis and incorporating a complex Berry phase. It shows that an initial instantaneous eigenstate evolves, in the slow-drive limit, into the corresponding instantaneous eigenstate up to a dynamic phase $e^{-iT\int_0^s \lambda_j(\sigma)\,d\sigma}$ and a (generally complex) Berry phase $e^{-{\int_0^s \langle \xi_j(\sigma)|\dot{v}_j(\sigma)\rangle\,d\sigma}}$, while establishing the uniform boundedness of the evolution operators $U_T(s)$ and $U_T^{-1}(s)$. The proof relies on a generalized off-diagonal construction, biorthogonal eigenvectors, and Grönwall-type bounds, avoiding a direct projection-based argument. This work broadens the applicability of adiabatic reasoning to non-Hermitian dynamics and clarifies the central role of the complex Berry phase in slow, non-Hermitian evolution.

Abstract

The adiabatic theorem is one of the most interesting and significant theorem in quantum mechanics. In 1950, T. Kato gave an elegant proof of this result [1]. However, the validation of adiabatic theorem for non-Hermitian quantum systems is unrevealed. In this paper, by following Kato' approach, we prove rigorously that the adiabatic theorem is still valid for non-Hermitian systems with non-degenerate real eigenvalues. Moreover, our proof utilizes the complex Berry phase, instead of the orthogonal projections used in Kato's work.