Abstract
We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the ‘commutator equation’. In addition, a minor modification of their argument allows for more direct treatment of eigenvalue crossings. We also obtain simple, explicit conditions that yield information on the rate of convergence in the adiabatic limit.
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References
Avron, J. E. and Elgart, A.: Adiabatic theorem without a gap condition, Comm. Math. Phys. 203 (1999), 445–463.
Bornemann, F.: Homogenization in Time of Singularly Perturbed Mechanical Systems, Lecture Notes in Math. 1687, Springer, New York, 1998.
Teufel, S.: A space-adiabatic theorem without gap condition and the massless Nelson model, in preparation.
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Teufel, S. A Note on the Adiabatic Theorem Without Gap Condition. Letters in Mathematical Physics 58, 261–266 (2001). https://doi.org/10.1023/A:1014556511004
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DOI: https://doi.org/10.1023/A:1014556511004