Related papers: Reflection operator and hypergeometry I: $SL(2, \m…
We consider noncompact open $SL(2, \mathbb{C})$ spin chain and construct eigenfunctions of $B$-element of monodromy matrix for the simplest case of the chain with one site. The reflection operator appearing in this construction can be used…
We diagonalize the $B$-element of monodromy matrix for noncompact open $SL(2,\mathbb{C})$ spin chain with boundary interaction. The monodromy matrix is defined in terms of $SL(2,\mathbb{C})$ $L$-operator and boundary $K$-matrix. The…
For the noncompact open ${\rm SL}(2,\mathbb{C})$ spin chain, the eigenfunctions of the special matrix element of monodromy matrix are constructed. The key ingredients of the whole construction are local Yang-Baxter $\mathcal{R}$-operators,…
We construct a class of representations of the quadratic R-matrix algebra, given by the reflection equation with the spectral parameter, in terms of certain ordinary difference operators. These operators turn out to act as parameter…
We construct a class of representations of the quadratic $R$-matrix algebra given by the reflection equation with the spectral parameter, $$ R{\,}(u-v)\,T^{(1)}(u)\,R{\,}(u+v)\,T^{(2)}(v)= T^{(2)}(v)\,R{\,}(u+v)\,T^{(1)}(u)\,R{\,}(u-v), $$…
We discuss the spectral decomposition of the hypergeometric differential operators on the line $\mathrm{Re}\, z=1/2$. Such operators arise in the problem of decomposition of tensor products of unitary representations of the universal…
Spin chain Hamiltonians can be written in terms of complex differential operators using the Bargmann representation of the Jordan-Schwinger map. In this case, the eigenfunctions are expressed as the product of orthonormal monomials of the…
In the previous two parts of this series of papers, we introduced and studied a large class of analytic difference operators admitting reflectionless eigenfunctions, focusing on algebraic and function-theoretic features in the first part,…
Eigenfunctions of the matrix elements of the monodromy matrix provide a convenient basis for studies of spin chain models. We present an iterative method for constructing the eigenfunctions in the case of the SL(2,C) spin chains. We derived…
In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional…
We obtain a reflection formula for the Gaussian hypergeometric function of real symmetric matrix argument. We also show that this result extends to the Gaussian hypergeometric function defined over the symmetric cones, and even to…
The Dirac-Dunkl operator on the 2-sphere associated to the $\mathbb{Z}_2^3$ reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
We construct a new representation for two- and three-point correlators of operators from sl(2) sector of planar N = 4 SYM. The spin and twist of operators are arbitrary. We start with the correlation function of light-ray operators and…
We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass…
We discuss the nearest neighbor distribution of the eigenvalues for hermitian generators in the Lie algebra of a semisimple complex Lie Group along a sequence of irreducible representations. After the basic definitions a limit theorem for…
We provide an explicit geometric algorithm involving only ruler and compass constructions in order to specify the specular reflection point on the surface of a reflecting sphere of radius $r$ given two focal points $A$ and $B$ lying outside…
A description of eigensubspaces of the cosine and sine operators is presented. The spectrum of each of these two operator consists of two eigenvalues (1,\,-1) and their eigensubspaces are infinite--dimensional. There are many possible bases…
Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are studied in the spirit of the treatment of the spherical harmonics by Stein and Weiss. As a special case, a simple self-contained proof of Laplace's integral for a Legendre…
In this work we study differential problems in which the reflection operator and the Hilbert transform are involved. We reduce these problems to ODEs in order to solve them. Also, we describe a general method for obtaining the Green's…