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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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