8,355 research outputs found
Dispersive Estimates for higher dimensional Schr\"odinger Operators with threshold eigenvalues II: The even dimensional case
We investigate dispersive
estimates for the Schr\"odinger operator when there is an
eigenvalue at zero energy in even dimensions . In particular, we show
that if there is an eigenvalue at zero energy then there is a time dependent,
rank one operator satisfying for such that With
stronger decay conditions on the potential it is possible to generate an
operator-valued expansion for the evolution, taking the form \begin{align*}
e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}}
A_{-1}+|t|^{-\frac{n}{2}}A_0, \end{align*} with and mapping
to while maps weighted
spaces to weighted spaces. The leading-order terms and
are both finite rank, and vanish when certain orthogonality conditions
between the potential and the zero energy eigenfunctions are satisfied. We
show that under the same orthogonality conditions, the remaining
term also exists as a map from to
, hence satisfies the same dispersive
bounds as the free evolution despite the eigenvalue at zero.Comment: This article continues the work of "Dispersive Estimates for higher
dimensional Schr\"odinger Operators with threshold eigenvalues I: The odd
dimensional case" by the authors to the case of even dimensions. To appear in
J. Spectr. Theor
The boundedness of wave operators for Schr\"odinger Operators with threshold singularities
Let be a Schr\"odinger operator on with
real-valued potential for and let . If decays
sufficiently, the wave operators are known to be bounded on for all if zero is not an eigenvalue, and on if zero is
an eigenvalue. We show that these wave operators are also bounded on
by direct examination of the integral kernel of the leading
term. Furthermore, if for all
eigenfunctions , then the wave operators are bounded for . If, in addition , then the wave
operators are bounded for .Comment: Incorporated referee comments and updated references. To appear in
Adv. Mat
Dispersive Estimates for higher dimensional Schr\"odinger Operators with threshold eigenvalues I: The odd dimensional case
We investigate dispersive
estimates for the Schr\"odinger operator when there is an
eigenvalue at zero energy and is odd. In particular, we show that if
there is an eigenvalue at zero energy then there is a time dependent, rank one
operator satisfying for such that With
stronger decay conditions on the potential it is possible to generate an
operator-valued expansion for the evolution, taking the form with and
finite rank operators mapping to while maps weighted spaces to weighted spaces. The
leading order terms and vanish when certain orthogonality
conditions between the potential and the zero energy eigenfunctions are
satisfied. We show that under the same orthogonality conditions, the remaining
term also exists as a map from to
, hence satisfies the same dispersive
bounds as the free evolution despite the eigenvalue at zero.Comment: To appear in J. Funct. Ana
Anomalous Creation of Branes
In certain circumstances when two branes pass through each other a third
brane is produced stretching between them. We explain this phenomenon by the
use of chains of dualities and the inflow of charge that is required for the
absence of chiral gauge anomalies when pairs of D-branes intersect.Comment: 7 pages, two figure
D-branes in a plane-wave background
The D-branes of the maximally supersymmetric plane-wave background are
described.Comment: 6 pages; contribution to the proceedings of the 35th Symposium
Ahrenshoop, 2002; v2: minor correction
An SL(2,Z) anomaly in IIB supergravity and its F-theory interpretation
The SL(2,Z) duality transformations of type IIB supergravity are shown to be
anomalous in generic F-theory backgrounds due to the anomalous transformation
of the phase of the chiral fermion determinant. The anomaly is partially
cancelled provided the ten-dimensional type IIB theory lagrangian contains a
term that is a ten-form made out of the composite U(1) field strength and four
powers of the curvature. A residual anomaly remains uncancelled, and this
implies a certain topological restriction on consistent backgrounds of the
euclidean theory. A similar, but slightly stronger, restriction is also derived
from an explicit F-theory compactification on K3 x M8 (where M8 is an
eight-manifold with a nowhere vanishing chiral spinor) where the cancellation
of tadpoles for Ramond--Ramond fields is only possible if M8 has an Euler
character that is a positive multiple of 24. The interpretation of this
restriction in the dual heterotic theory on T2 x M8 is also given.Comment: Argument has been streamlined and references have been added. 18
pages, harvmac (b
Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions
In this paper we consider Dirac operators in , , with a
potential . Under mild decay and continuity assumptions on and some
spectral assumptions on the operator, we prove a limiting absorption principle
for the resolvent, which implies a family of Strichartz estimates for the
linear Dirac equation. For large potentials the dynamical estimates are not an
immediate corollary of the free case since the resolvent of the free Dirac
operator does not decay in operator norm on weighted spaces as the
frequency goes to infinity.Comment: Updated Corollary 1.3 with a slightly stronger statement. To appear
in Comm. Math. Phys. arXiv admin note: text overlap with arXiv:0705.054
Multiparticle one-loop amplitudes and S-duality in closed superstring theory
Explicit expressions for one-loop five supergraviton scattering amplitudes in
both type II superstring theories are determined by making use of the pure
spinor formalism. The type IIB amplitude can be expressed in terms of a
doubling of ten-dimensional super Yang--Mills tree amplitude, while the type
IIA amplitude has additional pieces that cannot be expressed in that manner. We
evaluate the coefficients of terms in the analytic part of the low energy
expansion of the amplitude, which correspond to a series of terms in an
effective action of the schematic form D^{2k}R^5 for 0\le k \le 5 (where R is
the Riemann curvature). Comparison with earlier analyses of the tree amplitudes
and of the four-particle one-loop amplitude leads to an interesting extension
of the action of SL(2,Z) S-duality on the moduli-dependent coefficients in the
type IIB theory. We also investigate closed-string five-particle amplitudes
that violate conservation of the U(1) R-symmetry charge -- processes that are
forbidden in supergravity. The coefficients of their low energy expansion are
shown to agree with S-duality systematics. A less detailed analysis is also
given of the six-point function, resulting in the vanishing of the analytic
parts of the R^6 and D^4 R^6 interactions in the ten-dimensional effective
action, but not in lower dimensions.Comment: 62 pages, Mathematica notebook on integral expansion included in
submission. v2: minor modifications, references added, matches published
versio
Dispersive estimates for four dimensional Schr\"{o}dinger and wave equations with obstructions at zero energy
We investigate dispersive
estimates for the Schr\"odinger operator when there are
obstructions, a resonance or an eigenvalue, at zero energy. In particular, we
show that if there is a resonance or an eigenvalue at zero energy then there is
a time dependent, finite rank operator satisfying for such that
We also show that the operator if there is an eigenvalue but no
resonance at zero energy. We then develop analogous dispersive estimates for
the solution operator to the four dimensional wave equation with potential.Comment: 32 page
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