1,287 research outputs found
The Cauchy problems for Einstein metrics and parallel spinors
We show that in the analytic category, given a Riemannian metric on a
hypersurface and a symmetric tensor on , the metric
can be locally extended to a Riemannian Einstein metric on with second
fundamental form , provided that and satisfy the constraints on
imposed by the contracted Codazzi equations. We use this fact to study the
Cauchy problem for metrics with parallel spinors in the real analytic category
and give an affirmative answer to a question raised in B\"ar, Gauduchon,
Moroianu (2005). We also answer negatively the corresponding questions in the
smooth category.Comment: 28 pages; final versio
Surgery and the spinorial tau-invariant
We associate to a compact spin manifold M a real-valued invariant \tau(M) by
taking the supremum over all conformal classes over the infimum inside each
conformal class of the first positive Dirac eigenvalue, normalized to volume 1.
This invariant is a spinorial analogue of Schoen's -constant, also
known as the smooth Yamabe number. We prove that if N is obtained from M by
surgery of codimension at least 2, then with . Various topological conclusions
can be drawn, in particular that \tau is a spin-bordism invariant below
. Below , the values of cannot accumulate from
above when varied over all manifolds of a fixed dimension.Comment: to appear in CPD
The Dirac operator on generalized Taub-NUT spaces
We find sufficient conditions for the absence of harmonic spinors on
spin manifolds constructed as cone bundles over a compact K\"ahler base. These
conditions are fulfilled for certain perturbations of the Euclidean metric, and
also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a
conjecture of Vi\csinescu and the second author.Comment: Final version, 16 page
The Ammann-Beenker tilings revisited
This paper introduces two tiles whose tilings form a one-parameter family of
tilings which can all be seen as digitization of two-dimensional planes in the
four-dimensional Euclidean space. This family contains the Ammann-Beenker
tilings as the solution of a simple optimization problem.Comment: 7 pages, 4 figure
Regularity for eigenfunctions of Schr\"odinger operators
We prove a regularity result in weighted Sobolev spaces (or
Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator.
More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space
obtained by blowing up the set of singular points of the Coulomb type potential
V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N}
\frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u
in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution
sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0.
Our result extends to the case when b_j and c_{ij} are suitable bounded
functions on the blown-up space. In the single-electron, multi-nuclei case, we
obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy
Invertible Dirac operators and handle attachments on manifolds with boundary
For spin manifolds with boundary we consider Riemannian metrics which are
product near the boundary and are such that the corresponding Dirac operator is
invertible when half-infinite cylinders are attached at the boundary. The main
result of this paper is that these properties of a metric can be preserved when
the metric is extended over a handle of codimension at least two attached at
the boundary. Applications of this result include the construction of
non-isotopic metrics with invertible Dirac operator, and a concordance
existence and classification theorem.Comment: Accepted for publication in Journal of Topology and Analysi
A spinorial energy functional: critical points and gradient flow
On the universal bundle of unit spinors we study a natural energy functional
whose critical points, if dim M \geq 3, are precisely the pairs (g, {\phi})
consisting of a Ricci-flat Riemannian metric g together with a parallel
g-spinor {\phi}. We investigate the basic properties of this functional and
study its negative gradient flow, the so-called spinor flow. In particular, we
prove short-time existence and uniqueness for this flow.Comment: Small changes, final versio
Quantum Poincar\'e Recurrences
We show that quantum effects modify the decay rate of Poincar\'e recurrences
P(t) in classical chaotic systems with hierarchical structure of phase space.
The exponent p of the algebraic decay P(t) ~ 1/t^p is shown to have the
universal value p=1 due to tunneling and localization effects. Experimental
evidence of such decay should be observable in mesoscopic systems and cold
atoms.Comment: revtex, 4 pages, 4 figure
Phase Control of Nonadiabaticity-induced Quantum Chaos in An Optical Lattice
The qualitative nature (i.e. integrable vs. chaotic) of the translational
dynamics of a three-level atom in an optical lattice is shown to be
controllable by varying the relative laser phase of two standing wave lasers.
Control is explained in terms of the nonadiabatic transition between optical
potentials and the corresponding regular to chaotic transition in mixed
classical-quantum dynamics. The results are of interest to both areas of
coherent control and quantum chaos.Comment: 3 figures, 4 pages, to appear in Physical Review Letter
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