252 research outputs found
The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2
The structure properties of multidimensional Delsarte transmutation operators
in parametirc functional spaces are studied by means of differential-geometric
tools. It is shown that kernels of the corresponding integral operator
expressions depend on the topological structure of related homological cycles
in the coordinate space. As a natural realization of the construction presented
we build pairs of Lax type commutive differential operator expressions related
via a Darboux-Backlund transformation having a lot of applications in solition
theory. Some results are also sketched concerning theory of Delsarte
transmutation operators for affine polynomial pencils of multidimensional
differential operators.Comment: 10 page
Transmuting CHY formulae
© The Author(s) 2019.The various formulations of scattering amplitudes presented in recent years have underlined a hidden unity among very different theories. The KLT and BCJ relations, together with the CHY formulation, connect the S-matrices of a wide range of theories: the transmutation operators, recently proposed by Cheung, Shen and Wen, provide an account for these similarities. In this note we use the transmutation operators to link the various CHY integrands at tree-level. Starting from gravity, we generate the integrands for YangMills, biadjoint scalar, Einstein-Maxwell, Yang-Mills scalar, Born-Infeld, Dirac-Born-Infeld, non-linear sigma model and special Galileon theories, as well as for their extensions. We also commence the study of the CHY-like formulae at loop level.Peer reviewe
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
Harmonic analysis associated with the Weinstein type operator on Rd
 We consider the Weinstein type operator ??;d on Rd. We build transmutation operators R? which turn out to be transmutation operator between ??;d and the Laplacian?d. Using this transmutation operators and its dual tR?, we develop a new commutative harmonic analysis on Rd corresponding to the operator ??;d
Harmonic analysis associated with the Weinstein type operator on Rd
 We consider the Weinstein type operator ??;d on Rd. We build transmutation operators R? which turn out to be transmutation operator between ??;d and the Laplacian?d. Using this transmutation operators and its dual tR?, we develop a new commutative harmonic analysis on Rd corresponding to the operator ??;d
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