21 research outputs found
On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum
We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum
of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha,
with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay
as n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuously
differentiable in the strong sense and such that the matrix entries with
respect to the spectral decomposition of H obey the estimate
|V(t)_{m,n}|0,
p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be
arbitrarily small provided p is sufficiently large and \epsilon is small
enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the
diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where
\sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the
Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was
discussed earlier in the literature by Howland
Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations
We prove that the quantum harmonic oscillator is stable under spatially
localized, time quasi-periodic perturbations on a set of Diophantine
frequencies of positive measure. This proves a conjecture raised by
Enss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodic
setting. The motivation of the present paper also comes from construction of
quasi-periodic solutions for the corresponding nonlinear equation