5,281 research outputs found
A-priori bounds for the 1-d cubic NLS in negative Sobolev spaces
We consider the cubic Nonlinear Schrodinger Equation in one space dimension,
either focusing or defocusing. We prove that the solutions satisfy a-priori
local in time H^s bounds in terms of the H^s size of the initial data for s
greater than or equal to -1/6.Comment: 27 pages very minor misprints corrected (see formulas (4), (7)
Dispersive estimates for principally normal pseudodifferential operators
The aim of these notes is to describe some recent results concerning
dispersive estimates for principally normal pseudodifferential operators. The
main motivation for this comes from unique continuation problems. Such
estimates can be used to prove Carleman inequalities, which in turn yield
unique continuation results for various partial differential operators with
rough potentials.Comment: 72 page
Semiclassical Lp estimates
The purpose of this paper is to use semiclassical analysis to unify and
generalize Lp estimates on high energy eigenfunctions and spectral clusters. In
our approach these estimates do not depend on ellipticity and order, and apply
to operators which are selfadjoint only at the principal level. They are
estimates on weakly approximate solutions to semiclassical pseudodifferential
equations. The revision corrects an exponent in the main theorems.Comment: 33 pages, 1 figur
Sharp L^p bounds on spectral clusters for Lipschitz metrics
We establish L^p bounds on L^2 normalized spectral clusters for self-adjoint
elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we
obtain best possible bounds for all p between 6<p\leq 8$. In higher dimensions we obtain best
possible bounds for a limited range of p.Comment: 28 page
Subcritical Lp bounds on spectral clusters for Lipschitz metrics
We establish asymptotic bounds on the L^p norms of spectrally localized
functions in the case of two-dimensional Dirichlet forms with coefficients of
Lipschitz regularity. These bounds are new for the range p>6. A key step in the
proof is bounding the rate at which energy spreads for solutions to hyperbolic
equations with Lipschitz coefficients.Comment: 10 page
Observing the Berry phase in diffusive conductors: Necessary conditions for adiabaticity
In a recent preprint (cond-mat/9803170), van~Langen, Knops, Paasschens and
Beenakker attempt to re-analyze the proposal of Loss, Schoeller and Goldbart
(LSG) [Phys. Rev. B~48, 15218 (1993)] concerning Berry phase effects in the
magnetoconductance of diffusive systems. Van Langen et al. claim that the
adiabatic approximation for the Cooperon previously derived by LSG is not valid
in the adiabatic regime identified by LSG. It is shown that the claim of
van~Langen et al. is not correct, and that, on the contrary, the
magnetoconductance does exhibit the Berry phase effect within the LSG regime of
adiabaticity. The conclusion reached by van~Langen et al. is based on a
misinterpretation of field-induced dephasing effects, which can mask the Berry
phase (and any other phase coherent phenomena) for certain parameter values.Comment: 25 pages, 9 figure
Subcritical Lp bounds on spectral clusters for Lipschitz metrics
We establish asymptotic bounds on the L^p norms of spectrally localized
functions in the case of two-dimensional Dirichlet forms with coefficients of
Lipschitz regularity. These bounds are new for the range p>6. A key step in the
proof is bounding the rate at which energy spreads for solutions to hyperbolic
equations with Lipschitz coefficients.Comment: 10 page
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