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Estimates on Green’s functions, localization and the quantum kicked rotor model. (English) Zbl 1213.82054
From the text: Consider the time-dependent Schrödinger equation $$\mathbb T=R/\mathbb Z$$,
$i\frac{\partial\Psi(t,x)}{\partial t}= a\frac{\partial^2\Psi(t,x)}{\partial x^2}+ ib\frac{\partial\Psi(t,x)}{\partial x}+ V(t,x)\Psi(t,x), \tag{1}$
where $$V$$ is 1-periodic in $$t$$ and $$x$$. More precisely, let
$V(t,x)= \kappa(\cos 2\pi x) \bigg(\sum_{n\in\mathbb Z}\delta(t-n)\bigg), \tag{2}$
corresponding to a periodic sequence of kicks. We then obtain the so-called quantum kicked rotor equation, introduced earlier as a model in quantum chaos (a quantum analogue of Chirikov’s standard map).
Thus $$i\Psi_t= H(t)\Psi$$ with time-dependent Hamiltonian
$H(t)=-aK^2+bL+V(x,t); \quad L=i\frac{\partial}{\partial x}$
and phase space $$L^2(\mathbb T)$$.
It has been conjectured by a number of authors that for typical parameter values for $$a,b$$, the wave function $$\Psi$$ satisfying equations (1), (2) will be almost periodic in time; hence $$\widehat{\psi(t)}$$ (The Fourier transform) remains localized. We proof this here, assuming the parameter $$\kappa$$ in (2) small. Thus we have:
Theorem. For $$\kappa$$ small and $$(a,b)$$ outside a set of small measure $$(\to 0$$ for $$\kappa\to 0)$$, the following holds. Let $$\Psi=\Psi(t,x)$$ solve (1), (2) and let $$\Psi_0= \Psi(0,x)$$ be a sufficiently smooth function on $$\mathbb T$$. Then $$\Psi$$ is an almost periodic function of time, say as an $$H^1(\mathbb T)$$-valued map. In particular $$\sup_t \|\Psi(t)\|_{H^1}<\infty$$.
For recent progress see J. Bourgain, M. Goldstein and W. Schlag [Commun. Math. Phys. 220, 583–621 (2001; Zbl 0994.82044); Acta Math. 188, No. 1, 41–86 (2002; Zbl 1022.47023)], J. Bourgain and M. Goldstein [Ann. Math. (2) 152, No. 3, 835–879 (2000; Zbl 1053.39035)] and J. Bourgain and S. Jitomirskaya [Invent. Math. 148, No. 3, 453–463 (2002; Zbl 1036.47019)].

##### MSC:
 82C10 Quantum dynamics and nonequilibrium statistical mechanics (general) 35A08 Fundamental solutions to PDEs 35Q40 PDEs in connection with quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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