89 research outputs found
Scattering theory for lattice operators in dimension
This paper analyzes the scattering theory for periodic tight-binding
Hamiltonians perturbed by a finite range impurity. The classical energy
gradient flow is used to construct a conjugate (or dilation) operator to the
unperturbed Hamiltonian. For dimension the wave operator is given by
an explicit formula in terms of this dilation operator, the free resolvent and
the perturbation. From this formula the scattering and time delay operators can
be read off. Using the index theorem approach, a Levinson theorem is proved
which also holds in presence of embedded eigenvalues and threshold
singularities.Comment: Minor errors and misprints corrected; new result on absense of
embedded eigenvalues for potential scattering; to appear in RM
Gauge Theoretic Invariants of, Dehn Surgeries on Knots
New methods for computing a variety of gauge theoretic invariants for
homology 3-spheres are developed. These invariants include the Chern-Simons
invariants, the spectral flow of the odd signature operator, and the rho
invariants of irreducible SU(2) representations. These quantities are
calculated for flat SU(2) connections on homology 3-spheres obtained by 1/k
Dehn surgery on (2,q) torus knots. The methods are then applied to compute the
SU(3) gauge theoretic Casson invariant (introduced in [H U Boden and C M
Herald, The SU(3) Casson invariant for integral homology 3--spheres, J. Diff.
Geom. 50 (1998) 147-206]) for Dehn surgeries on (2,q) torus knots for q=3,5,7
and 9.Comment: Version 3: minor corrections from version 2. Published by Geometry
and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper6.abs.htm
Nonrelativistic hydrogen type stability problems on nonparabolic 3-manifolds
We extend classical Euclidean stability theorems corresponding to the
nonrelativistic Hamiltonians of ions with one electron to the setting of non
parabolic Riemannian 3-manifolds.Comment: 20 pages; to appear in Annales Henri Poincar
On transversally elliptic operators and the quantization of manifolds with -structure
An -structure on a manifold is an endomorphism field
\phi\in\Gamma(M,\End(TM)) such that . Any -structure
determines an almost CR structure E_{1,0}\subset T_\C M given by the
-eigenbundle of . Using a compatible metric and connection
on , we construct an odd first-order differential operator ,
acting on sections of , whose principal symbol is of the
type considered in arXiv:0810.0338. In the special case of a CR-integrable
almost -structure, we show that when is the generalized
Tanaka-Webster connection of Lotta and Pastore, the operator is given by D
= \sqrt{2}(\dbbar+\dbbar^*), where \dbbar is the tangential Cauchy-Riemann
operator.
We then describe two "quantizations" of manifolds with -structure that
reduce to familiar methods in symplectic geometry in the case that is a
compatible almost complex structure, and to the contact quantization defined in
\cite{F4} when comes from a contact metric structure. The first is an
index-theoretic approach involving the operator ; for certain group actions
will be transversally elliptic, and using the results in arXiv:0810.0338,
we can give a Riemann-Roch type formula for its index. The second approach uses
an analogue of the polarized sections of a prequantum line bundle, with a CR
structure playing the role of a complex polarization.Comment: 31 page
Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes
We construct a class of solutions to the Cauchy problem of the Klein-Gordon
equation on any standard static spacetime. Specifically, we have constructed
solutions to the Cauchy problem based on any self-adjoint extension (satisfying
a technical condition: "acceptability") of (some variant of) the
Laplace-Beltrami operator defined on test functions in an -space of the
static hypersurface. The proof of the existence of this construction completes
and extends work originally done by Wald. Further results include the
uniqueness of these solutions, their support properties, the construction of
the space of solutions and the energy and symplectic form on this space, an
analysis of certain symmetries on the space of solutions and of various
examples of this method, including the construction of a non-bounded below
acceptable self-adjoint extension generating the dynamics
Once more on the Witten index of 3d supersymmetric YM-CS theory
The problem of counting the vacuum states in the supersymmetric 3d
Yang-Mills-Chern-Simons theory is reconsidered. We resolve the controversy
between its original calculation by Witten at large volumes and the calculation
based on the evaluation of the effective Lagrangian in the small volume limit.
We show that the latter calculation suffers from uncertainties associated with
the singularities in the moduli space of classical vacua where the
Born-Oppenheimer approximation breaks down. We also show that these
singularities can be accurately treated in the Hamiltonian Born-Oppenheimer
method, where one has to match carefully the effective wave functions on the
Abelian valley and the wave functions of reduced non-Abelian QM theory near the
singularities. This gives the same result as original Witten's calculation.Comment: 27 page
Comparative Study Between Solid State Welding and Radiant Energy Welding Processes for Joining Metallic Glassy Ribbons
Amorphous alloys have emerged as an important class of advanced materials that own a combination of properties, such as mechanical strength, hardness, high elasticity modulus and a very good corrosion resistance. Since the number of amorphous structures alloys increased in the last decades, ways of joining such materials were studied in order to produce complex structures or increase their size. Thus, if this kind of complex products are obtained, it will diversify their applicability in multiple and various domains. For this research two ways of joining amorphous ribbons has been studied: solid state welding and radiant energy welding. For the radiant energy welding process, it was selected electron beam welding (EBW) method and for the solid-state welding process, ultrasonic welding (UW) method was chosen. Seeing that these methods have found applicability in industries, a comparative study was done in order to see which one offers the best outcome. Recently, in the last years, such products were embedded in a polymer matrix, creating thus, composite materials that have improved mechanical properties. This raised curiosity for major industries, such as aero-space, medical and automotive. Amorphous ribbons from Ni-Fe-Cr-Si-B and Al-Ni-Nd-Co alloy families were welded by EBW method, and Cu-Zr-Al amorphous ribbons were welded by the UW method. Microstructure characterization has been performed by SEM, EDX, XRD and DSC analyses
General Spectral Flow Formula for Fixed Maximal Domain
We consider a continuous curve of linear elliptic formally self-adjoint
differential operators of first order with smooth coefficients over a compact
Riemannian manifold with boundary together with a continuous curve of global
elliptic boundary value problems. We express the spectral flow of the resulting
continuous family of (unbounded) self-adjoint Fredholm operators in terms of
the Maslov index of two related curves of Lagrangian spaces. One curve is given
by the varying domains, the other by the Cauchy data spaces. We provide
rigorous definitions of the underlying concepts of spectral theory and
symplectic analysis and give a full (and surprisingly short) proof of our
General Spectral Flow Formula for the case of fixed maximal domain. As a side
result, we establish local stability of weak inner unique continuation property
(UCP) and explain its role for parameter dependent spectral theory.Comment: 22 page
The Maslov index in weak symplectic functional analysis
We recall the Chernoff-Marsden definition of weak symplectic structure and
give a rigorous treatment of the functional analysis and geometry of weak
symplectic Banach spaces. We define the Maslov index of a continuous path of
Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces.
We derive basic properties of this Maslov index and emphasize the new features
appearing.Comment: 34 pages, 13 figures, 45 references, to appear in Ann Glob Anal Geom.
The final publication will be available at http://www.springerlink.com. arXiv
admin note: substantial text overlap with arXiv:math/040613
Fabrication and electrorotation of a novel epoxy based micromotor working in a uniform DC electric field
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