52 research outputs found
Positive Measure Spectrum for Schroedinger Operators with Periodic Magnetic Fields
We study Schroedinger operators with periodic magnetic field in Euclidean
2-space, in the case of irrational magnetic flux. Positive measure Cantor
spectrum is generically expected in the presence of an electric potential. We
show that, even without electric potential, the spectrum has positive measure
if the magnetic field is a perturbation of a constant one.Comment: 17 page
An inviscid dyadic model of turbulence: the fixed point and Onsager's conjecture
Properties of an infinite system of nonlinearly coupled ordinary differential
equations are discussed. This system models some properties present in the
equations of motion for an inviscid fluid such as the skew symmetry and the
3-dimensional scaling of the quadratic nonlinearity. It is proved that the
system with forcing has a unique equilibrium and that every solution blows up
in finite time in -norm. Onsager's conjecture is confirmed for the
model system
On products of skew rotations
Let , be two time-independent Hamiltonians with one
degree of freedom and , be the one-parametric groups of
shifts along the orbits of Hamiltonian systems generated by , . In
some problems of population genetics there appear the transformations of the
plane having the form under some
conditions on , . We study in this paper asymptotical properties of
trajectories of .Comment: 13 pages, 10 figure
Space-Time Complexity in Hamiltonian Dynamics
New notions of the complexity function C(epsilon;t,s) and entropy function
S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov
exponents or systems that exhibit strong intermittent behavior with
``flights'', trappings, weak mixing, etc. The important part of the new notions
is the first appearance of epsilon-separation of initially close trajectories.
The complexity function is similar to the propagator p(t0,x0;t,x) with a
replacement of x by the natural lengths s of trajectories, and its introduction
does not assume of the space-time independence in the process of evolution of
the system. A special stress is done on the choice of variables and the
replacement t by eta=ln(t), s by xi=ln(s) makes it possible to consider
time-algebraic and space-algebraic complexity and some mixed cases. It is shown
that for typical cases the entropy function S(epsilon;xi,eta) possesses
invariants (alpha,beta) that describe the fractal dimensions of the space-time
structures of trajectories. The invariants (alpha,beta) can be linked to the
transport properties of the system, from one side, and to the Riemann
invariants for simple waves, from the other side. This analog provides a new
meaning for the transport exponent mu that can be considered as the speed of a
Riemann wave in the log-phase space of the log-space-time variables. Some other
applications of new notions are considered and numerical examples are
presented.Comment: 27 pages, 6 figure
A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces
A Wasserstein spaces is a metric space of sufficiently concentrated
probability measures over a general metric space. The main goal of this paper
is to estimate the largeness of Wasserstein spaces, in a sense to be precised.
In a first part, we generalize the Hausdorff dimension by defining a family of
bi-Lipschitz invariants, called critical parameters, that measure largeness for
infinite-dimensional metric spaces. Basic properties of these invariants are
given, and they are estimated for a naturel set of spaces generalizing the
usual Hilbert cube. In a second part, we estimate the value of these new
invariants in the case of some Wasserstein spaces, as well as the dynamical
complexity of push-forward maps. The lower bounds rely on several embedding
results; for example we provide bi-Lipschitz embeddings of all powers of any
space inside its Wasserstein space, with uniform bound and we prove that the
Wasserstein space of a d-manifold has "power-exponential" critical parameter
equal to d.Comment: v2 Largely expanded version, as reflected by the change of title; all
part I on generalized Hausdorff dimension is new, as well as the embedding of
Hilbert cubes into Wasserstein spaces. v3 modified according to the referee
final remarks ; to appear in Journal of Topology and Analysi
One-Dimensional Discrete Stark Hamiltonian and Resonance Scattering by Impurities
A one-dimensional discrete Stark Hamiltonian with a continuous electric field
is constructed by extension theory methods. In absence of the impurities the
model is proved to be exactly solvable, the spectrum is shown to be simple,
continuous, filling the real axis; the eigenfunctions, the resolvent and the
spectral measure are constructed explicitly. For this (unperturbed) system the
resonance spectrum is shown to be empty. The model considering impurity in a
single node is also constructed using the operator extension theory methods.
The spectral analysis is performed and the dispersion equation for the
resolvent singularities is obtained. The resonance spectrum is shown to contain
infinite discrete set of resonances. One-to-one correspondence of the
constructed Hamiltonian to some Lee-Friedrichs model is established.Comment: 20 pages, Latex, no figure
Isoperiodic deformations of the acoustic operator and periodic solutions of the Harry Dym equation
We consider the problem of describing the possible spectra of an acoustic
operator with a periodic finite-gap density. We construct flows on the moduli
space of algebraic Riemann surfaces that preserve the periods of the
corresponding operator. By a suitable extension of the phase space, these
equations can be written with quadratic irrationalities.Comment: 15 page
Phonon Localization in One-Dimensional Quasiperiodic Chains
Quasiperiodic long range order is intermediate between spatial periodicity
and disorder, and the excitations in 1D quasiperiodic systems are believed to
be transitional between extended and localized. These ideas are tested with a
numerical analysis of two incommensurate 1D elastic chains: Frenkel-Kontorova
(FK) and Lennard-Jones (LJ). The ground state configurations and the
eigenfrequencies and eigenfunctions for harmonic excitations are determined.
Aubry's "transition by breaking the analyticity" is observed in the ground
state of each model, but the behavior of the excitations is qualitatively
different. Phonon localization is observed for some modes in the LJ chain on
both sides of the transition. The localization phenomenon apparently is
decoupled from the distribution of eigenfrequencies since the spectrum changes
from continuous to Cantor-set-like when the interaction parameters are varied
to cross the analyticity--breaking transition. The eigenfunctions of the FK
chain satisfy the "quasi-Bloch" theorem below the transition, but not above it,
while only a subset of the eigenfunctions of the LJ chain satisfy the theorem.Comment: This is a revised version to appear in Physical Review B; includes
additional and necessary clarifications and comments. 7 pages; requires
revtex.sty v3.0, epsf.sty; includes 6 EPS figures. Postscript version also
available at
http://lifshitz.physics.wisc.edu/www/koltenbah/koltenbah_homepage.htm
Incommensurate Charge Density Waves in the adiabatic Hubbard-Holstein model
The adiabatic, Holstein-Hubbard model describes electrons on a chain with
step interacting with themselves (with coupling ) and with a classical
phonon field \f_x (with coupling \l). There is Peierls instability if the
electronic ground state energy F(\f) as a functional of \f_x has a minimum
which corresponds to a periodic function with period , where
is the Fermi momentum. We consider irrational so that
the CDW is {\it incommensurate} with the chain. We prove in a rigorous way in
the spinless case, when \l,U are small and {U\over\l} large, that a)when
the electronic interaction is attractive there is no Peierls instability
b)when the interaction is repulsive there is Peierls instability in the
sense that our convergent expansion for F(\f), truncated at the second order,
has a minimum which corresponds to an analytical and periodic
\f_x. Such a minimum is found solving an infinite set of coupled
self-consistent equations, one for each of the infinite Fourier modes of
\f_x.Comment: 16 pages, 1 picture. To appear Phys. Rev.
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