320 research outputs found
Photon-photon scattering: a tutorial
Long-established results for the low-energy photon-photon scattering, gamma
gamma --> gamma gamma, have recently been questioned. We analyze that claim and
demonstrate that it is inconsistent with experience. We demonstrate that the
mistake originates from an erroneous manipulation of divergent integrals and
discuss the connection with another recent claim about the Higgs decay into two
photons. We show a simple way of correctly computing the low-energy gamma gamma
scattering.Comment: 9 pages, 2 figures; v2: typographical errors corrected in eqs 1, 20,
and 21; ref. [23] adde
Approximation Theory for Matrices
We review the theory of optimal polynomial and rational Chebyshev
approximations, and Zolotarev's formula for the sign function over the range
(\epsilon \leq |z| \leq1). We explain how rational approximations can be
applied to large sparse matrices efficiently by making use of partial fraction
expansions and multi-shift Krylov space solvers.Comment: 10 pages, 7 figure
On stationarity conditions for a certain periodic random process
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32356/1/0000428.pd
Some theorems on Cebysev approximation II
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33369/1/0000767.pd
Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
As an outgrowth of our investigation of non-regular spaces within the context
of quantum gravity and non-commutative geometry, we develop a graph Hilbert
space framework on arbitrary (infinite) graphs and use it to study spectral
properties of graph-Laplacians and graph-Dirac-operators. We define a spectral
triplet sharing most of the properties of what Connes calls a spectral triple.
With the help of this scheme we derive an explicit expression for the
Connes-distance function on general directed or undirected graphs. We derive a
series of apriori estimates and calculate it for a variety of examples of
graphs. As a possibly interesting aside, we show that the natural setting of
approaching such problems may be the framework of (non-)linear programming or
optimization. We compare our results (arrived at within our particular
framework) with the results of other authors and show that the seeming
differences depend on the use of different graph-geometries and/or Dirac
operators.Comment: 27 pages, Latex, comlementary to an earlier paper, general treatment
of directed and undirected graphs, in section 4 a series of general results
and estimates concerning the Connes Distance on graphs together with examples
and numerical estimate
Looking for magnetic monopoles at LHC with diphoton events
Magnetic monopoles have been a subject of interest since Dirac established
the relation between the existence of monopoles and charge quantization. The
intense experimental search carried thus far has not met with success. The
Large Hadron Collider is reaching energies never achieved before allowing the
search for exotic particles in the TeV mass range. In a continuing effort to
discover these rare particles we propose here other ways to detect them. We
study the observability of monopoles and monopolium, a monopole-antimonopole
bound state, at the Large Hadron Collider in the channel for
monopole masses in the range 500-1000 GeV. We conclude that LHC is an ideal
machine to discover monopoles with masses below 1 TeV at present running
energies and with 5 fb of integrated luminosity.Comment: This manuscript contains information appeared in Looking for magnetic
monopoles at LHC, arXiv:1104.0218 [hep-ph] and Monopolium detection at the
LHC.,arXiv:1107.3684 [hep-ph] by the same authors, rewritten for joint
publication in The European Physica Journal Plus. 26 pages, 22 figure
Kirchhoff's Rule for Quantum Wires
In this article we formulate and discuss one particle quantum scattering
theory on an arbitrary finite graph with open ends and where we define the
Hamiltonian to be (minus) the Laplace operator with general boundary conditions
at the vertices. This results in a scattering theory with channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy is explicitly given in
terms of the boundary conditions and the lengths of the internal lines. It is
shown to be unitary, which may be viewed as the quantum version of Kirchhoff's
law. We exhibit covariance and symmetry properties. It is symmetric if the
boundary conditions are real. Also there is a duality transformation on the set
of boundary conditions and the lengths of the internal lines such that the low
energy behaviour of one theory gives the high energy behaviour of the
transformed theory. Finally we provide a composition rule by which the on-shell
S-matrix of a graph is factorizable in terms of the S-matrices of its
subgraphs. All proofs only use known facts from the theory of self-adjoint
extensions, standard linear algebra, complex function theory and elementary
arguments from the theory of Hermitean symplectic forms.Comment: 40 page
Boundary relations and generalized resolvents of symmetric operators
The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint
exit space extensions of a, not necessarily densely defined, symmetric
operator, in terms of maximal dissipative (in \dC_+) holomorphic linear
relations on the parameter space (the so-called Nevanlinna families). The new
notion of a boundary relation makes it possible to interpret these parameter
families as Weyl families of boundary relations and to establish a simple
coupling method to construct the generalized resolvents from the given
parameter family. The general version of the coupling method is introduced and
the role of boundary relations and their Weyl families for the
Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page
Electromagnetic Dissociation as a Tool for Nuclear Structure and Astrophysics
Coulomb dissociation is an especially simple and important reaction
mechanism. Since the perturbation due to the electric field of the (target)
nucleus is exactly known, firm conclusions can be drawn from such measurements.
Electromagnetic matrixelements and astrophysical S-factors for radiative
capture processes can be extracted from experiments. We describe the basic
elements of the theory of nonrelativistic and relativistic electromagnetic
excitation with heavy ions. This is contrasted to electromagnetic excitation
with leptons (electrons), with their small electric charge and the absence of
strong interactions. We discuss various approaches to the study of higher order
electromagnetic effects and how these effects depend on the basic parameters of
the experiment. The dissociation of neutron halo nuclei is studied in a zero
range model using analytical methods. We also review ways how to treat nuclear
interactions, show their characteristics and how to avoid them (as far as
possible). We review the experimental results from a theoretical point of view.
Of special interest for nuclear structure physics is the appearence of low
lying electric dipole strength in neutron rich nuclei. Applications of Coulomb
dissociation to some selected radiative capture reactions relevant for nuclear
astrophysics are discussed. The Coulomb dissociation of 8B is relevant for the
solar neutrino problem. The potential of the method especially for future
investigations of (medium) heavy exotic nuclei for nuclear structure and
astrophysics is explored. We conclude that the Coulomb dissociation mechanism
is theoretically well understood, the potential difficulties are identified and
can be taken care of. Many interesting experiments have been done in this field
and many more are expected in the future.Comment: review article accepted for publication in "Prog. in Part. and Nucl.
Physics", 75 pages, 31 figure
Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions
© 2020 The Authors. Mathematische Nachrichten published by Wiley‐VCH Verlag GmbH & Co. KGaA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed
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