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Adiabatic theorem without a gap condition. (English) Zbl 0936.47047
Summary: We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite-dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.

MSC:
47N50 Applications of operator theory in the physical sciences
47A10 Spectrum, resolvent
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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