6,374 research outputs found
Ionization for Three Dimensional Time-dependent Point Interactions
We study the time evolution of a three dimensional quantum particle under the
action of a time-dependent point interaction fixed at the origin. We assume
that the ``strength'' of the interaction (\alpha(t)) is a periodic function
with an arbitrary mean. Under very weak conditions on the Fourier coefficients
of (\alpha(t)), we prove that there is complete ionization as (t \to \infty),
starting from a bound state at time (t = 0). Moreover we prove also that, under
the same conditions, all the states of the system are scattering states.Comment: Some improvements and some references added, 26 pages, LaTe
Do Prices Coordinate Markets?
Walrasian equilibrium prices can be said to coordinate markets: They support
a welfare optimal allocation in which each buyer is buying bundle of goods that
is individually most preferred. However, this clean story has two caveats.
First, the prices alone are not sufficient to coordinate the market, and buyers
may need to select among their most preferred bundles in a coordinated way to
find a feasible allocation. Second, we don't in practice expect to encounter
exact equilibrium prices tailored to the market, but instead only approximate
prices, somehow encoding "distributional" information about the market. How
well do prices work to coordinate markets when tie-breaking is not coordinated,
and they encode only distributional information?
We answer this question. First, we provide a genericity condition such that
for buyers with Matroid Based Valuations, overdemand with respect to
equilibrium prices is at most 1, independent of the supply of goods, even when
tie-breaking is done in an uncoordinated fashion. Second, we provide
learning-theoretic results that show that such prices are robust to changing
the buyers in the market, so long as all buyers are sampled from the same
(unknown) distribution
Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions
In this paper we prove results on Birkhoff and Oseledets genericity along
certain curves in the space of affine lattices and in moduli spaces of
translation surfaces. We also prove applications of these results to dynamical
billiards, mathematical physics and number theory. In the space of affine
lattices , we prove that almost every
point on a curve with some non-degeneracy assumptions is Birkhoff generic for
the geodesic flow. This implies almost everywhere genericity for some curves in
the locus of branched covers of the torus inside the stratum
of translation surfaces. For these curves (and more in general curves which are
well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff
genericity) we also prove that almost every point is Oseledets generic for the
Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin.
As applications, we first consider a class of pseudo-integrable billiards,
billiards in ellipses with barriers, which was recently explored by Dragovic
and Radnovic, and prove that for almost every parameter, the billiard flow is
uniquely ergodic within the region of phase space in which it is trapped. We
then consider any periodic array of Eaton retroreflector lenses, placed on
vertices of a lattice, and prove that in almost every direction light rays are
each confined to a band of finite width. This generalizes a phenomenon recently
discovered by Fraczek and Schmoll which could so far only be proved for random
periodic configurations. Finally, a result on the gap distribution of
fractional parts of the sequence of square roots of positive integers, which
extends previous work by Elkies and McMullen, is also obtained.Comment: To appear in Journal of Modern Dynamic
Genericity in Topological Dynamics
We study genericity of dynamical properties in the space of homeomorphisms of
the Cantor set and in the space of subshifts of a suitably large shift space.
These rather different settings are related by a Glasner-King type
correspondence: genericity in one is equivalent to genericity in the other.
By applying symbolic techniques in the shift-space model we derive new
results about genericity of dynamical properties for transitive and totally
transitive homeomorphisms of the Cantor set. We show that the isomorphism class
of the universal odometer is generic in the space of transitive systems. On the
other hand, the space of totally transitive systems displays much more varied
dynamics. In particular, we show that in this space the isomorphism class of
every Cantor system without periodic points is dense, and the following
properties are generic: minimality, zero entropy, disjointness from a fixed
totally transitive system, weak mixing, strong mixing, and minimal self
joinings. The last two stand in striking contrast to the situation in the
measure-preserving category. We also prove a correspondence between genericity
of dynamical properties in the measure-preserving category and genericity of
systems supporting an invariant measure with the same property.Comment: 48 pages, to appear in Ergodic Theory Dynamical Systems. v2: revised
exposition, added proof that the universal odometer is generic among
transitive Cantor homeomorphism
Generic Morse-Smale property for the parabolic equation on the circle
In this paper, we show that, for scalar reaction-diffusion equations
on the circle , the Morse-Smale property is
generic with respect to the non-linearity . In \cite{CR}, Czaja and Rocha
have proved that any connecting orbit, which connects two hyperbolic periodic
orbits, is transverse and that there does not exist any homoclinic orbit,
connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown
that, generically with respect to the non-linearity , all the equilibria and
periodic orbits are hyperbolic. Here we complete these results by showing that
any connecting orbit between two hyperbolic equilibria with distinct Morse
indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is
automatically transverse. We also show that, generically with respect to ,
there does not exist any connection between equilibria with the same Morse
index. The above properties, together with the existence of a compact global
attractor and the Poincar\'e-Bendixson property, allow us to deduce that,
generically with respect to , the non-wandering set consists in a finite
number of hyperbolic equilibria and periodic orbits . The main tools in the
proofs include the lap number property, exponential dichotomies and the
Sard-Smale theorem. The proofs also require a careful analysis of the
asymptotic behavior of solutions of the linearized equations along the
connecting orbits
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