## The simplified Fermi accelerator in classical and quantum mechanics.(English)Zbl 0839.60075

Summary: We review the simplified classical Fermi acceleration mechanism and construct a quantum counterpart by imposing time-dependent boundary conditions on solutions of the “free” Schrödinger equation at the unit interval. We find similiar dynamical features in the sense that limiting KAM curves, respectively purely singular quasienergy spectrum, exist(s) for sufficiently smooth “wall oscillations” (typically of $${\mathcal C}^2$$ type). In addition, we investigate quantum analogs to local approximations of the Fermi map both in its quasiperiodic and irregular phase space regions. In particular, we find pure point q.e. spectrum in the former case and conjecture that “random boundary conditions” are necessary to model a quantum analog to the chaotic regime of the classical accelerator.

### MSC:

 60J75 Jump processes (MSC2010) 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
Full Text:

### References:

 [1] E. Fermi,Phys. Rev. 15:1169 (1949). · Zbl 0032.09604 [2] I. Percival and D. Richards,Introduction to Dynamics (Cambridge University, Cambridge, 1982). · Zbl 0499.70027 [3] S. Ulam, InProceedings 4th Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, 1961), Vol. 3, p. 315. [4] A. J. Lichtenberg and M. A. Lieberman,Phys. Rev. A 5:1852 (1972). [5] A. J. Lichtenberg and M. A. Lieberman,Physica 1D:291 (1980). [6] G. Bolz and T. Krüger, in G. Bolz, Ph.d. Thesis, Universität Bielefeld (1991). [7] A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion (Springer, New York, 1983). · Zbl 0506.70016 [8] B. V. Chirikov,Phys. Rep. 52:263 (1979). [9] G. Karner,Lett. Math. Phys. 17:329 (1989). · Zbl 0698.35143 [10] J. S. Howland, Spectrum of the Fermi accelerator, preprint, University of Virginia (1992). [11] P. Šeba,Phys. Rev. A 41:2306 (1990). [12] G. Karner,J. Math. Anal. Appl. 164:206 (1992). · Zbl 0773.47038 [13] J. S. Howland,Ann. Inst. H. Poincaré A 50:309, 325 (1989). [14] G. Nenciu,Ann. Inst. H. Poincaré A 59:91 (1993). [15] N. Dunford and J. T. Schwartz,Linear Operators, Part III (Interscience, New York, 1963). [16] M. S. Birman and M. G. Krein,Dokl. Akad. Nauk SSSR 144:475 (1962). [17] J. M. Combes, Connections between quantum dynamics and spectral properties of time-evolution operators, inDifferential Equations with Applications to Mathematical Physics, W. F. Ames, E. M. Harrell II, and J. V. Herod, eds. (Academic Press, New York, 1993). [18] H. Jauslin, Stability and chaos in classical and quantum Hamiltonian systems, inProceedings II Granada Seminar on Computational Physics, P. Carrido and J. Marro, eds. (World Scientific, Singapore, 1993). [19] G. Karner, The dynamics of the quantum standard map, preprint, University of Virginia (1993). [20] M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. I (Academic Press, New York, 1978). · Zbl 0401.47001 [21] G. Karner,Nonlinearity 7:623 (1994). · Zbl 0838.34057 [22] P. Ehrenfest,Z. Physik 45:455 (1927). [23] R. F. Fox and B. L. Lan,Phys. Rev. A 41:2952 (1990). [24] G. Karner, Quantum Fermi maps revisited, inProceedings of the IXth Bielefeld Encounters in Mathematics and Physics, S. Albeverio and L. Streit, eds. (World Scientific, Singapore, 1993). [25] J. S. Howland, Random perturbations of singular spectra, preprint, University of Virginia (1991). · Zbl 0736.47013 [26] S. Kotani,Contemp. Math. 50:277 (1986).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.