595 research outputs found
Weakly regular Floquet Hamiltonians with pure point spectrum
We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on
the parameter omega. We assume that the spectrum of H is discrete, {h_m (m =
1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian
operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose
that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m -
h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show
that in that case there exist a suitable norm to measure the regularity of V,
denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if
epsilon
|Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point
spectrum for all omega in Omega_infinity.Comment: 35 pages, Latex with AmsAr
Tissue fusion over non-adhering surfaces
Tissue fusion eliminates physical voids in a tissue to form a continuous
structure and is central to many processes in development and repair. Fusion
events in vivo, particularly in embryonic development, often involve the
purse-string contraction of a pluricellular actomyosin cable at the free edge.
However in vitro, adhesion of the cells to their substrate favors a closure
mechanism mediated by lamellipodial protrusions, which has prevented a
systematic study of the purse-string mechanism. Here, we show that monolayers
can cover well-controlled mesoscopic non-adherent areas much larger than a cell
size by purse-string closure and that active epithelial fluctuations are
required for this process. We have formulated a simple stochastic model that
includes purse-string contractility, tissue fluctuations and effective friction
to qualitatively and quantitatively account for the dynamics of closure. Our
data suggest that, in vivo, tissue fusion adapts to the local environment by
coordinating lamellipodial protrusions and purse-string contractions
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
A constant of quantum motion in two dimensions in crossed magnetic and electric fields
We consider the quantum dynamics of a single particle in the plane under the
influence of a constant perpendicular magnetic and a crossed electric potential
field. For a class of smooth and small potentials we construct a non-trivial
invariant of motion. Do to so we proof that the Hamiltonian is unitarily
equivalent to an effective Hamiltonian which commutes with the observable of
kinetic energy.Comment: 18 pages, 2 figures; the title was changed and several typos
corrected; to appear in J. Phys. A: Math. Theor. 43 (2010
On the spectrum of a bent chain graph
We study Schr\"odinger operators on an infinite quantum graph of a chain form
which consists of identical rings connected at the touching points by
-couplings with a parameter . If the graph is "straight",
i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum
with all the gaps open whenever . We consider a "bending"
deformation of the chain consisting of changing one position at a single ring
and show that it gives rise to eigenvalues in the open spectral gaps. We
analyze dependence of these eigenvalues on the coupling and the
"bending angle" as well as resonances of the system coming from the bending. We
also discuss the behaviour of the eigenvalues and resonances at the edges of
the spectral bands.Comment: LaTeX, 23 pages with 7 figures; minor changes, references added; to
appear in J. Phys. A: Math. Theo
Pulse-driven quantum dynamics beyond the impulsive regime
We review various unitary time-dependent perturbation theories and compare
them formally and numerically. We show that the Kolmogorov-Arnold-Moser
technique performs better owing to both the superexponential character of
correction terms and the possibility to optimize the accuracy of a given level
of approximation which is explored in details here. As an illustration, we
consider a two-level system driven by short pulses beyond the sudden limit.Comment: 15 pages, 5 color figure
Scattering solutions in a network of thin fibers: small diameter asymptotics
Small diameter asymptotics is obtained for scattering solutions in a network
of thin fibers. The asymptotics is expressed in terms of solutions of related
problems on the limiting quantum graph. We calculate the Lagrangian gluing
conditions at vertices for the problems on the limiting graph. If the frequency
of the incident wave is above the bottom of the absolutely continuous spectrum,
the gluing conditions are formulated in terms of the scattering data for each
individual junction of the network
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