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Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps

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Abstract

We consider a smooth operator-valued functionH(t,δ) that has two isolated non-degenerate eigenvaluesE A (t,δ) andE B (t,δ) for δ>0. We assume these eigenvalues are bounded away from the rest of the spectrum ofH(t,δ), but have an avoided crossing with one another with a closest approach that isO(δ) as δ tends to zero. Under these circumstances, we study the small ε limit for the adiabatic Schrödinger equation

$$i\varepsilon \frac{{\partial \psi }}{{\partial t}} = H(t,\varepsilon ^{1/2} )\psi .$$

We prove that the Landau-Zener formula correctly describes the coupling between the adiabatic states associated with the eigenvaluesE A (t,δ) andE B (t,δ) as the system propagates through the avoided crossing.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, 24061-0123, Blacksburg, Virginia, USA

    George A. Hagedorn

  2. Center for Transport Theory and Mathematical Physics, Virginia Polytechnic Institute and State University, 24061-0123, Blacksburg, Virginia, USA

    George A. Hagedorn

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  1. George A. Hagedorn
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Communicated by T. Spencer

Supported in part by the National Foundation under Grant number DMS-8801360

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Hagedorn, G.A. Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps. Commun.Math. Phys. 136, 433–449 (1991). https://doi.org/10.1007/BF02099068

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  • DOI: https://doi.org/10.1007/BF02099068

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