374 research outputs found
Index theorems on manifolds with straight ends
We study Fredholm properties and index formulas for Dirac operators over
complete Riemannian manifolds with straight ends. An important class of
examples of such manifolds are complete Riemannian manifolds with pinched
negative sectional curvature and finite volume
Experimental Evidence for Quantum Interference and Vibrationally Induced Decoherence in Single-Molecule Junctions
We analyze quantum interference and decoherence effects in single-molecule
junctions both experimentally and theoretically by means of the mechanically
controlled break junction technique and density-functional theory. We consider
the case where interference is provided by overlapping quasi-degenerate states.
Decoherence mechanisms arising from the electronic-vibrational coupling
strongly affect the electrical current flowing through a single-molecule
contact and can be controlled by temperature variation. Our findings underline
the all-important relevance of vibrations for understanding charge transport
through molecular junctions.Comment: 5 pages, 4 figure
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
A theoretical investigation of low energy proton on hydrogen collisions
The Proton on Hydrogen collision problem is treated in the time-dependent formalism using a new self-consistent nuclear trajectory model in conjunction with a simple semi-classical approximation. In this method the nuclear trajectory is dependent on the time-evolution of the electronic wavefunction which is described by a basis of H₂⁺ eigenfunctions. The small-energy, large scattering angle region is well described in this way and agreement with available experimental data is obtained. Tile inclusion of the semi-classical approximation and the use of a larger molecular basis than hitherto employed allow these limits to be quite reasonably extended to describe the small angle and moderate energy region also. Results of charge exchange probabilities and differential-scattering cross-sections in the range 150-1000 e.v. (lab. energy of incident proton beam) are presented along with some inelastic calculations on excitation into the Hydrogen 2p±1 and 2S states. It is further shown that the inclusion of the Gerarde states (2Sσg, 3Dσg) in the basis set has a significant effect on the results obtained for collision energies of 700 e.v. and 1Kev.
A new numerical method is described which enables very rapid computation of all quantities required for the basis set, and leads to quick and simple integral calculations
Entropy of semiclassical measures for nonpositively curved surfaces
We study the asymptotic properties of eigenfunctions of the Laplacian in the
case of a compact Riemannian surface of nonpositive sectional curvature. We
show that the Kolmogorov-Sinai entropy of a semiclassical measure for the
geodesic flow is bounded from below by half of the Ruelle upper bound. We
follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus
on the main differences and refer the reader to (arXiv:0809.0230) for the
details of analogous lemmas.Comment: 20 pages. This note provides a detailed proof of a result announced
in appendix A of a previous work (arXiv:0809.0230, version 2
Manifolds with small Dirac eigenvalues are nilmanifolds
Consider the class of n-dimensional Riemannian spin manifolds with bounded
sectional curvatures and diameter, and almost non-negative scalar curvature.
Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 4. We show that if the square of
the Dirac operator on such a manifold has small eigenvalues, then the
manifold is diffeomorphic to a nilmanifold and has trivial spin structure.
Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a
non-trivial spin structure, then there exists a uniform lower bound on the r-th
eigenvalue of the square of the Dirac operator. If a manifold with almost
nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume
is not too small, then we show that the metric is close to a Ricci-flat metric
on M with a parallel spinor. In dimension 4 this implies that M is either a
torus or a K3-surface
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