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arXiv:math-ph/9812024 · Journal of Mathematical Physics

Adiabatic evolution for systems with infinitely many eigenvalue crossings

Alain Joye, F. Monti, S. Guérin, Hans Rudolf Jauslin

Published Nov 1, 1999• 6 citations
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Abstract

We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered projector, and some geometric hypothesis on the local behavior of the eigenvalues at the crossings.

DOI
10.1063/1.533039

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We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered projector, and some geometric hypothesis on the local behavior of the eigenvalues at the crossings.
We prove the adiabatic theorem for quantum evolution without the traditional gap condition. We show that the theorem holds essentially in all cases where it can be formulated. In particular, our result implies that the adiabatic theorem holds also for eigenvalues embedded in the continuous spectrum. If there is information on the Hölder continuity of the spectral measure, then one can also estimate the rate at which the adiabatic limit is approached. The adiabatic theorem of Quantum Mechanics describes the long time behavior of the solutions of an initial value problem where the Hamiltonian generating the evolution depends slowly in time. Traditionally, the theorem is stated for Hamiltonians that have an eigenvalue which is separated by a gap from the rest of the spectrum. Folk wisdom is that a gap condition is a sine qua non for the adiabatic theorem to hold. In particular, according to this folk wisdom, one does not expect a general adiabatic theorem for Hamiltonians that have an eigenvalue embedded in, say, the continuous spectrum. Our purpose is to show that this folk wisdom is wrong, and there is a general adiabatic theorem without a gap condition. All one really needs for the adiabatic theorem is a spectral projection for the Hamiltonian that depends smoothly on time.
The authors consider the evolution of a two-level system driven by a nonself-adjoint Hamiltonian H( in t) and treat the adiabatic limit in to 0. While adiabatic theorem-like results do not hold true in general for this case, they prove that they are still valid for the subspace corresponding to the eigenvalue having the largest imaginary part (least dissipative eigenvalue). The theory gives the full asymptotic expansion of the evolution restricted to this subspace. The first correction beyond Berry's phase is to their best knowledge given explicitly for the first time.
Coherent states are well adapted to understand adiabatic decoupling for multi-components systems like the Dirac operator or systems obtained by reduction in the Born-Oppenheimer approximation [58]. Hagedorn [130] has done an accurate analysis of this using a variant of coherent states. The new fact for systems is that now states are vectors valued in Cm\mathbb C^mCm for m≥2m\ge 2m≥2 and the eigenvalues of the matrix of the system may cross so that one cannot diagonalize it smoothly. In the case when the eigenvalues are not crossing one can construct an adiabatic modification of the spectral projectors on the modes and compute the time evolution of coherent states following the same strategy as in the scalar case. Here we shall consider only non crossing cases where we can prove an adiabatic decoupling between the modes. For the more difficult crossing case we refer to the literature [130] where non-adiabatic transitions can be computed [101, 129] for more recent results.