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.


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