Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps. (English) Zbl 0723.35068

Summary: We consider a smooth operator-valued function H(t,\(\delta\)) that has two isolated non-degenerate eigenvalues \(E_{{\mathcal A}}(t,\delta)\) and \(E_{{\mathcal B}}(t,\delta)\) for \(\delta >0\). We assume these eigenvalues are bounded away from the rest of the spectrum of H(t,\(\delta\)), but have an avoided crossing with one another with a closest approach that is O(\(\delta\)) as \(\delta\) tends to zero. Under these circumstances, we study the small \(\epsilon\) limit for the adiabatic Schrödinger equation \(i\epsilon (\partial \psi /\partial t)=H(t,\epsilon^{1/2})\psi.\)
We prove that the Landau-Zener formula correctly describes the coupling between the adiabatic states associated with the eigenvalues \(E_{{\mathcal A}}(t,\delta)\) and \(E_{{\mathcal B}}(t,\delta)\) as the system propagates through the avoided crossing.


35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35P99 Spectral theory and eigenvalue problems for partial differential equations
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[1] Avron, J. E., Seiler, R., Yaffe, L. G.: Adiabatic theorems and applications to the quantum Hall Effect. Commun. Math. Phys.110, 33–49 (1987) · Zbl 0626.58033
[2] Berry, M. V.: Geometric amplitude factors in adiabatic quantum transitions, preprint
[3] Berry, M. V.: Histories of adiabatic quantum transitions, preprint
[4] Garrido, L. M.: Generalized adiabatic invariance. J. Math. Phys.5, 335–362 (1964)
[5] Gradshteyn, I. S., Ryzhik, I. M.: Tables of integrals, series, and products. New York: Academic Press 1980 · Zbl 0521.33001
[6] Hagedorn, G. A.: Adiabatic expansions near eigenvalue crossings. Ann. Phys.196, 278–296 (1989) · Zbl 0875.47002
[7] Jakšić, V., Segert, J.: Exponential approach to the adiabatic limit and the Landau-Zener formula, preprint · Zbl 0769.34006
[8] Landau, L. D.: Collected Papers of L. D. Landau, Edited and with an introduction by D. ter Haar. New York: Gordon and Breach 1967
[9] Lenard, A.: Adiabatic invariance to all orders. Ann. Phys.6, 261–276 (1959) · Zbl 0084.44403
[10] Nenciu, G.: Adiabatic theorem and spectral concentration. Commun. Math. Phys.82, 121–135 (1981) · Zbl 0493.47009
[11] Reed, M., Simon, B.: Methods of modern mathematical physics, Vol II. Fourier analysis, self-adjointness. New York: Academic Press 1975 · Zbl 0308.47002
[12] Sancho, S. J.:m th order adiabatic invariance. Proc. Phil. Soc. London.89, 1–5 (1966) · Zbl 0144.23501
[13] Simon, B.: Quantum mechanics for Hamiltonians defined as quadratic forms, Princeton, NJ: Princeton University Press 1971 · Zbl 0232.47053
[14] Yajima, K.: Unpublished notes, March 1988
[15] Yoshida, K.: Functional analysis. Berlin, Heidelberg, New York: Springer 1978
[16] Zener, C.: Non-adiabatic crossing of energy levels. Proc. R. Soc. Lond.137, 696–702 (1932) · Zbl 0005.18605
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