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Slowly changing potential problems in quantum mechanics: adiabatic theorems, ergodic theorems, and scattering. (English) Zbl 1348.81227

The time-dependent Schrödinger equation dependent on a small parameter \(\epsilon\) (a time factor) is considered. The procedure of multiscale time averaging reduces, under certain hypothesis, the original dynamics to a piecewise time-independent one plus a correction, depending on a small parameter which determines the time intervals where the averaging is done. This procedure is introduced in Sec. II and applied into details to the Landau-Zener-Majorana system in Sec. III. In Sec. IV, the procedure is employed to prove an ergodic-type Theorem and an adiabatic Theorem, stating that, under certain hypothesis, the solutions of the Schrödinger equation depending on the slow time \(\epsilon t\) are approximated, for \(\epsilon \rightarrow 0\) and over \(0<t\leq \frac 1\epsilon\), by the initial condition multiplied by \(e^{i\theta(t)}\) for a suitable real function \(\theta\). From these results, in Sec. V a scattering theory for the slow-changing systems is discussed, obtaining a bound for the energy and an estimate of the propagation.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q41 Time-dependent Schrödinger equations and Dirac equations
47A35 Ergodic theory of linear operators
81U05 \(2\)-body potential quantum scattering theory
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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References:

[1] Fishman, S.; Soffer, A., Multiscale time averaging, reloaded, SIAM J. Math. Anal., 46, 2, 1385-1405 (2014) · Zbl 1310.34051
[2] 2.L.Landau, “Zur Theorie der Energieubertragung. II. Physikalische Zeitschrift der Sowjetunion,” Sov. Phys.1, 89 (1932);L.Landau, Z. Phys. Sov.1932, 2. · Zbl 0005.27801
[3] Zener, C., Non-adiabatic crossing of energy levels, Proc. R. Soc. London, Ser. A, 137, 696-702 (1932) · JFM 58.1356.02
[4] Course of Theoretical Physics, 4 (1971)
[5] Berry, M., Superadiabatic tracking of quantum evolution, J. Phys. A: Math. Gen., 24, 3255 (1991)
[6] Berry, M., The adiabatic limit and the semiclassical limit, J. Phys. A: Math. Gen., 17, 1225 (1984)
[7] Cycon; Hans, L., Schrödinger Operators: With Application to Quantum Mechanics and Global Geometry (2009)
[8] Derezinski, J.; Gerard, C., Scattering Theory of Classical and Quantum N-particle Systems (1997) · Zbl 0899.47007
[9] Enss, V.; Veselic, K., Bound states and propagating states for time-dependent Hamiltonians, Ann. Inst. Henri Poincare, Sect. A, 39, 2, 159-191 (1983) · Zbl 0532.47007
[10] Sparber, C., Weakly nonlinear time-adiabatic theory (2014)
[11] Enss, V.; Graffi, S., Quantum scattering theory for two- and three-body systems with potentials of short and long range, Schrödinger Operators, 1159, 39-176 (1985)
[12] Hagedorn, G. A.; Joye, A., Elementary exponential error estimates for the adiabatic approximation, J. Math. Anal. Appl., 267, 235-246 (2002) · Zbl 1051.81010
[13] Fishman, S.; Kieran, M.; Eshel, B.-J., Zener tunneling in systems without level crossing, Phys. Rev. A, 42, 9, 5181 (1990)
[14] Avron, J. E.; Seiler, R.; Yaffe, G., Adiabatic theorems and applications to the quantum Hall effect, Commun. Math. Phys., 110, 33-49 (1987) · Zbl 0626.58033
[15] Reed, M.; Simon, B., Fourier Analysis, Self-Adjointness, 2 (1975) · Zbl 0308.47002
[16] Wittig, C., The Landau-Zener formula, J. Phys. Chem. B, 109, 8428-8430 (2005)
[17] Vitanov, N. V.; Garraway, B. M., Landau-Zener model: Effects of finite coupling duration, Phys. Rev. A, 53, 6, 4288 (1996)
[18] Vitanov, N. V., Transition times in the Landau-Zener model, Phys. Rev. A, 59, 2, 988 (1999)
[19] Soffer, A., Completeness of wave operators in relativistic quantum mechanics, Lett. Math. Phys., 8, 6, 517-527 (1984) · Zbl 0571.47010
[20] Soffer, A., Monotonic local decay estimates (2011)
[21] Sigal, I. M.; Soffer, A., The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. Math., 126, 1, 35-108 (1987) · Zbl 0646.47009
[22] 22.I. M.Sigal and A.Soffer, Local Decay and Propagation Estimates for Time Dependent and Time Independent Hamiltonians (Princeton University, 1988);See also, W.Hunziker, I. M.Sigal, and AvySoffer, “Minimal escape velocities,” Commun. Partial Differ. Equations24(11-12), 2279-2295 (1999).10.1080/03605309908821502
[23] Sigal, I. M.; Soffer, A., Asymptotic completeness for N ≤ 4 particle systems with the Coulomb-type interactions, Duke Math. J., 71, 1, 243-298 (1993) · Zbl 0853.70010
[24] Kato, T., On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Jpn., 5, 435-439 (1950)
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