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Abstract

Propagators weakly associated to a family of Hamiltonians and the adiabatic theorem for the Landau Hamiltonian with a time-dependent Aharonov-Bohm flux. (English summary) J. Math. Phys. 46 (2005), no. 5, 053303, 26 pp. The adiabatic limit of the Laughlin-Halperin model for the quantum Hall effect (QHE) is at the heart of the mathematical physics literature on QHE. The authors discuss the existence of this adiabatic limit for the Hamiltonian corresponding to a constant magnetic field on R2 with an additional flux tube, without any electric potential. The magnetic flux of the tube is assumed to vary over the time interval τ. First of all, even the existence of the time evolution (propagator) is nontrivial: The Hamiltonian H(t) has a domain varying with t, and its time-derivative ˙ H(t) is not relatively bounded. The same is true in terms of the adiabatically rescaled time s, of course. The key idea for overcoming this difficulty goes back to [M. Born and V. Fock, Z. Phys. 51 (1928), 165–180; JFM 54.0994.03]: Replace H(s) by the adiabatic Hamiltonian Had(s): = H(s) + ı

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