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 $2 and infinity, up to logarithmic losses for$6<p\leq 8$. 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