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arXiv:2511.00968

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

Minyi Huang, Ray-Kuang Lee

quant-phmath-phSubmitted Nov 2, 2025• Announced Nov 4, 2025• 0 citations
v1 · Nov 2, 2025v2 · Mar 4, 2026v3 · Jun 28, 2026
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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.

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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.
We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their general linear GL(1,C)\text{GL}(1,\mathbb{C})GL(1,C) gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigenenergies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wave function but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian PT\mathcal{PT}PT-symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasiparticles.
Abstract A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.
With the growing interest in non-Hermitian quantum systems, the concepts of geometric phase such as Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase have been gradually extended from Hermitian to non-Hermitian quantum systems. However, extending the most general form, the nonadiabatic non-Abelian geometric phase, to non-Hermitian systems has remained an open problem. In this paper, we address this issue by proposing a formalism of the nonadiabatic non-Abelian geometric phase in non-Hermitian quantum systems. Our formalism is consistent with the previous extensions in non-Hermitian quantum systems, and can also naturally revert to the concepts of geometric phase in Hermitian quantum systems. It provides a general formulation of geometric phase for both Hermitian and non-Hermitian quantum systems.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians HHH are indeed real if they are invariant under the combined action of self-adjoint parity PPP and time reversal T.T.T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PTPTPT-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t)=ZH(t)exp(iMHt){U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)UH(t)=ZH(t)exp(iMHt) associated with the periodic pseudo-hermitian Hamiltonian H(t)=H(t+T)H(t)=H(t+T)H(t)=H(t+T). One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.