17,135 research outputs found
Transmutations and spectral parameter power series in eigenvalue problems
We give an overview of recent developments in Sturm-Liouville theory
concerning operators of transmutation (transformation) and spectral parameter
power series (SPPS). The possibility to write down the dispersion
(characteristic) equations corresponding to a variety of spectral problems
related to Sturm-Liouville equations in an analytic form is an attractive
feature of the SPPS method. It is based on a computation of certain systems of
recursive integrals. Considered as families of functions these systems are
complete in the -space and result to be the images of the nonnegative
integer powers of the independent variable under the action of a corresponding
transmutation operator. This recently revealed property of the Delsarte
transmutations opens the way to apply the transmutation operator even when its
integral kernel is unknown and gives the possibility to obtain further
interesting properties concerning the Darboux transformed Schr\"{o}dinger
operators.
We introduce the systems of recursive integrals and the SPPS approach,
explain some of its applications to spectral problems with numerical
illustrations, give the definition and basic properties of transmutation
operators, introduce a parametrized family of transmutation operators, study
their mapping properties and construct the transmutation operators for Darboux
transformed Schr\"{o}dinger operators.Comment: 30 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1111.444
Multi-Parton Scattering Amplitudes via On-Shell Methods
We present an overview of recent developments, based on on-shell techniques,
in the calculation of multi-parton scattering amplitudes at one loop that are
relevant for phenomenological studies at hadron colliders. These new on-shell
methods make efficient use of the physical properties of the hard scattering,
such as unitarity and factorization.Comment: Invited review, Annual Review of Nuclear and Particle Science; v2:
references adde
The Hamiltonian Approach to Yang-Mills (2+1): An Expansion Scheme and Corrections to String Tension
We carry out further analysis of the Hamiltonian approach to Yang-Mills
theory in 2+1 dimensions which helps to place the calculation of the vacuum
wave function and the string tension in the context of a systematic expansion
scheme. The solution of the Schrodinger equation is carried out recursively.
The computation of correlators is re-expressed in terms of a two-dimensional
chiral boson theory. The effective action for this theory is calculated to
first order in our expansion scheme and to the fourth order in a kinematic
expansion parameter. The resulting corrections to the string tension are shown
to be very small, in the range -0.3% to -2.8%, moving our prediction closer to
the recent lattice estimates.Comment: 33 pages, 10 figure
On-Shell Methods in Perturbative QCD
We review on-shell methods for computing multi-parton scattering amplitudes
in perturbative QCD, utilizing their unitarity and factorization properties. We
focus on aspects which are useful for the construction of one-loop amplitudes
needed for phenomenological studies at the Large Hadron Collider.Comment: 49 pages, 15 figures. v2: minor typos correcte
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