Algebraic integrability of 
	    
	    PT
	    
	  -deformed Calogero models

Abstract We review some recents developments of the algebraic structures and spectral properties of non-Hermitian deformations of Calogero models. The behavior of such extensions is illustrated by the A 2 trigonometric and the D 3 angular Calogero models. Features like intertwining operators and conserved charges are discussed in terms of Dunkl operators. Hidden symmetries coming from the so-called algebraic integrability for integral values of the coupling are addressed together with a physical regularization of their action on the states by virtue of a PT -symmetry deformation.

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On String Integrability: A Journey through the Two-Dimensional Hidden Symmetries in the AdS/CFT Dualities

Advances in High Energy Physics ◽

10.1155/2010/471238 ◽

2010 ◽

Vol 2010 ◽

pp. 1-133 ◽

Cited By ~ 5

Author(s):

Valentina Giangreco Marotta Puletti

Keyword(s):

String Theory ◽

Two Dimensional ◽

Pure Spinor ◽

Hidden Symmetries ◽

Quantum Integrability ◽

Conserved Charges ◽

Phd Thesis

One of the main topics in the modern String Theory are the AdS/CFT dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, that is, the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality. We review the fundamental concepts and properties of integrability in two-dimensionalσ-models and in the AdS/CFT context. The first part is focused on theAdS5/CFT4duality, especially the classical and quantum integrability of the type IIB superstring onAdS5×S5which is discussed in both pure spinor and Green-Schwarz formulations. The second part is dedicated to theAdS4/CFT3duality with particular attention to the type IIA superstring onAdS4×ℂP3and its integrability. This review is based on the author's PhD thesis discussed at Uppsala University the 21st September 2009.

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Sonine Formulas and Intertwining Operators in Dunkl Theory

International Mathematics Research Notices ◽

10.1093/imrn/rnz313 ◽

2020 ◽

Cited By ~ 1

Author(s):

Margit Rösler ◽

Michael Voit

Keyword(s):

Root System ◽

Hypergeometric Functions ◽

Bessel Functions ◽

Jacobi Polynomials ◽

Necessary Conditions ◽

Integral Representations ◽

Intertwining Operators ◽

Dunkl Operators ◽

Intertwining Operator ◽

Type Integral

Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$, we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.

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INTEGRABILITY AND SYMMETRIC SPACES I: THE GROUP MANIFOLD

International Journal of Modern Physics A ◽

10.1142/s0217751x89000315 ◽

1989 ◽

Vol 04 (03) ◽

pp. 649-674 ◽

Cited By ~ 1

Author(s):

L. A. FERREIRA

Keyword(s):

Poisson Bracket ◽

Lie Group ◽

Symmetric Spaces ◽

Algebraic Structures ◽

Semisimple Lie Group ◽

Conserved Charges ◽

Group Manifold ◽

Coset Spaces

It is shown that any nonsingular Lagrangian describing the motion of a particle on a semisimple Lie group possesses a Fundamental Poisson bracket Relation (FPR) and consequently charges in involution. This property is independent of the dynamics of the model and can be derived in a quite simple and general way from the geometric and algebraic structures of the group manifold. The conditions a Hamiltonian has to satisfy in order those charges are to be conserved are discussed. These conditions lead to an algebra which plays an important role in the construction of conserved charges. In the second paper of the series, this work is extended to the coset spaces which are symmetric spaces.

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Existence of conserved quantities and their algebra in curved spacetime

International Journal of Modern Physics A ◽

10.1142/s0217751x20501626 ◽

2020 ◽

Vol 35 (26) ◽

pp. 2050162

Author(s):

Susobhan Mandal

Keyword(s):

General Relativity ◽

Dynamical Systems ◽

Important Task ◽

Conserved Quantities ◽

Curved Spacetime ◽

Hidden Symmetries ◽

Conserved Charges ◽

Geodesic Equations ◽

Spacetime Manifold ◽

Asymptotic Observer

In general relativity, finding out the geodesics of a given spacetime manifold is an important task because it determines which classical processes are dynamically forbidden. Conserved quantities play an important role in solving the geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system gives information about the hidden symmetries of that system since conserved quantities are deeply connected with the symmetries of the system. These are very important in their own right. Conserved quantities are also useful to capture certain features of spacetime manifold for an asymptotic observer. In this article, we show the existence of these conserved charges and their algebra in a generic curved spacetime for a class of dynamical systems with the Hamiltonians quadratic and linear in momentum and spin.

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Model of multicomponent narrow-band periodical non-stationary random signal

Information extraction and processing ◽

10.15407/vidbir2020.48.017 ◽

2020 ◽

Vol 2020 (48) ◽

pp. 17-24

Author(s):

I.M. Javorskyj ◽

◽

R.M. Yuzefovych ◽

P.R. Kurapov ◽

◽

...

Keyword(s):

Time Series ◽

Real Time ◽

Hilbert Transform ◽

Spectral Properties ◽

Narrow Band ◽

Analytic Signal ◽

Random Signal ◽

Hilbert Transformation ◽

The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

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