133 research outputs found
Invariant distributions and X-ray transform for Anosov flows
For Anosov flows preserving a smooth measure on a closed manifold
, we define a natural self-adjoint operator which maps into
the space of invariant distributions in and
whose kernel is made of coboundaries in . We
describe relations to Livsic theorem and recover regularity properties of
cohomological equations using this operator. For Anosov geodesic flows on the
unit tangent bundle of a compact manifold, we apply this
theory to study questions related to -ray transform on symmetric tensors on
: in particular we prove that injectivity implies surjectivity of X-ray
transform, and we show injectivity for surfaces.Comment: 30 pages, few corrections and new results (e.g. the image of is
dense among invariant distributions
Calderon inverse Problem with partial data on Riemann Surfaces
On a fixed smooth compact Riemann surface with boundary , we show
that for the Schr\"odinger operator with potential for some , the Dirichlet-to-Neumann map
measured on an open set determines
uniquely the potential . We also discuss briefly the corresponding
consequences for potential scattering at 0 frequency on Riemann surfaces with
asymptotically Euclidean or asymptotically hyperbolic ends.Comment: 27 pages. Corrections and modifications in the Complex Geometric
Optics solutions; regularity assumption strenghtened to $C^{1,\alpha}
Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, I
We analyze the resolvent of Schr\"odinger operators
with short range potential on asymptotically conic manifolds
(this setting includes asymptotically Euclidean manifolds) near .
We make the assumption that the dimension is greater or equal to 3 and that
has no null space and no resonance at 0. In particular, we show that the
Schwartz kernel of is a conormal polyhomogeneous distribution on a
desingularized version of . Using this, we show that the
Riesz transform of is bounded on for and that this range is
optimal if is not identically zero or if has more than one end. We also
analyze the case V=0 with one end. In a follow-up paper, we shall deal with the
same problem in the presence of zero modes and zero-resonances.Comment: 28 pages, 1 figur
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