35 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
Operators of harmonic analysis in weighted spaces with non-standard growth
Last years there was increasing an interest to the so-called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues. (C) 2008 Elsevier Inc. All rights reserved.INTAS [06-1000017-8792]; Center CEMAT, Instituto Superior Tecnico, Lisbon, Portugalinfo:eu-repo/semantics/publishedVersio
One class of solutions of Volterra equations with regular singularity
The Volterra integral equation of the second order with a regular singularity is considered. Under the conditions that a kernel K(x,t) is a real matrix function of order n×n with continuous partial derivatives up to order N+1 inclusively and K(0,0) has complex eigenvalues ν±i μ (ν>0), it is shown that if ν>2|‖K|‖ C -N-1, then a given equation has two linearly independent solutions.Розглядається інтегральне рівняння Вольтерра другого роду з регулярного сипгулярністю. У припущенні, що ядро K(x,t) —дійсна матричпозпачна функція порядку n×n з неперервними частинними похідними до порядку N+1 включно, і K(0,0) має комплексні власні значення ν±i μ (ν>0). Показано, що коли ν>2|‖K|‖ C -N-1, тоді існують два лінійно незалежних розв'язки даного рівняння