1,870 research outputs found
Surface Phonons and Other Localized Excitations
The diatomic linear chain of masses coupled by harmonic springs is a
textboook model for vibrational normal modes (phonons) in crystals. In addition
to propagating acoustic and optic branches, this model is known to support a
``gap mode'' localized at the surface, provided the atom at the surface has
light rather than heavy mass. An elementary argument is given which explains
this mode and provides values for the frequency and localization length. By
reinterpreting this mode in different ways, we obtain the frequency and
localization lengths for three other interesting modes: (1) the surface
vibrational mode of a light mass impurity at the surface of a monatomic chain;
(2) the localized vibrational mode of a stacking fault in a diatomic chain; and
(3) the localized vibrational mode of a light mass impurity in a monatomic
chain.Comment: 5 pages with 4 embedded postscript figures. This paper will appear in
the American Journal of Physic
Weak Chaos in a Quantum Kepler Problem
Transition from regular to chaotic dynamics in a crystal made of singular
scatterers can be reached by varying either sigma
or lambda. We map the problem to a localization problem, and find that in all
space dimensions the transition occurs at sigma=1, i.e., Coulomb potential has
marginal singularity. We study the critical line sigma=1 by means of a
renormalization group technique, and describe universality classes of this new
transition. An RG equation is written in the basis of states localized in
momentum space. The RG flow evolves the distribution of coupling parameters to
a universal stationary distribution. Analytic properties of the RG equation are
similar to that of Boltzmann kinetic equation: the RG dynamics has integrals of
motion and obeys an H-theorem. The RG results for sigma=1 are used to derive
scaling laws for transport and to calculate critical exponents.Comment: 28 pages, ReVTeX, 4 EPS figures, to appear in the I. M. Lifshitz
memorial volume of Physics Report
Small Deviation Probability via Chaining
We obtain several extensions of Talagrand's lower bound for the small
deviation probability using metric entropy. For Gaussian processes, our
investigations are focused on processes with sub-polynomial and, respectively,
exponential behaviour of covering numbers. The corresponding results are also
proved for non-Gaussian symmetric stable processes, both for the cases of
critically small and critically large entropy. The results extensively use the
classical chaining technique; at the same time they are meant to explore the
limits of this method.Comment: to appear in: Stochastic Processes and Their Application
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