arXiv subjects
Publications and source records attributed to S. Kharchev.
In these notes we review the technique of Baxter Q-operators in the Ruijsenaars-Sutherland hyperbolic systems in the cases of one and two particles. Using these operators we show in particular that eigenfunctions of these systems admit two dual integral representations and prove their orthogonality and completeness.
In the previous paper we introduced a commuting family of Baxter Q-operators for the quantum Ruijsenaars hyperbolic system. In the present work we show that the wave functions of the quantum system found by M. Hallnäs and S. Ruijsenaars also diagonalize Baxter operators. Using this property we prove the conjectured duality relation for the wave function. As a corollary, we show that the wave function solves bispectral problems for pairs of dual Macdonald and Baxter operators. Besides, we prove the conjectured symmetry of the wave function with respect to spectral variables and obtain new integral representation for it.
We introduce Baxter Q-operators for the quantum Ruijsenaars hyperbolic system. We prove that they represent a commuting family of integral operators and also commute with Macdonald difference operators, which are gauge equivalent to the Ruijsenaars Hamiltonians of the quantum system. The proof of commutativity of the Baxter operators uses a hypergeometric identity on rational functions that generalize Ruijsenaars kernel identities.
We introduce and study a new family of commuting Baxter operators in the Ruijsenaars hyperbolic system, different from that considered by us earlier. Using a degeneration of Rains integral identity we verify the commutativity between the two families of Baxter operators and explore this fact for the proof of the coupling constant symmetry of the wave function. We also establish a connection between new Baxter operators and Noumi-Sano difference operators.
In the previous paper we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation we prove orthogonality and completeness relations for the wave functions or, equivalently, unitarity of the corresponding integral transform.
We present and prove hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model.
We obtain certain Mellin-Barnes integrals which present wave functions for $GL(n,\mathbb{R})$ hyperbolic Sutherland model with arbitrary positive coupling constant.
Recently we found Mellin-Barnes integrals, representing the wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model. In present paper, we establish bispectral properties of this wave function with respect to dual Ruijesenaars-Macdonald operators.
We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups.
We define the quasi-compact Higgs $G^{\mathbb C}$-bundles over singular curves introduced in our previous paper for the Lie group SL($N$). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of $G^{\mathbb C}$ at marked points of the curves. We demonstrate that in particular cases this construction leads to the classical integrable systems of Hitchin type. The examples of the systems are analogues of the classical Calogero-Sutherland systems related to a simple complex Lie group $G^{\mathbb C}$ with two types of interacting spin variables. These type models were introduced previously by Feher and Pusztai. We construct the Lax operators of the systems as the Higgs fields defined over a singular rational curve. We also construct hierarchy of independent integrals of motion. Then we pass to a fixed point set of real involution related to one of the complex structures on the moduli space of the Higgs bundles. We prove that the number of independent integrals of motion is equal to the half of dimension of the fixed point set. The latter is a phase space of a real completely integrable system. We construct the classical $r$-matrix depending on the spectral parameter on a real singular curve, and in this way prove the complete integrability of the system. We present three equivalent descriptions of the system and establish their equivalence.
We consider the classical Calogero-Sutherland system with two types of interacting spin variables. It can be reduced to the standard Calogero-Sutherland system, when one of the spin variables vanishes. We describe the model in the Hitchin approach and prove complete integrability of the system by constructing the Lax pair and the classical $r$-matrix with the spectral parameter on a singular curve.
It is shown that the Jacobi and Riemann identities of degree four for the multidimensional theta functions as well as the Weierstrass identities emerge as algebraic consequences of the fundamental multidimensional binary identities connecting the theta functions with Riemann matrices $τ$ and $2τ$.
This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $θ_1$, $θ_2$, $θ_3$, $θ_4$ of one complex variable, presented in a ready-to-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with modular parameters $τ$ and $2τ$.
We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of $U((\mathfrak{gl}(N))$ in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter $Q$-operator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.
This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and affine quantum groups at zero level $U_q(\hat{\mathfrak{g}})_{c=0}$ corresponding to an arbitrary finite-dimensional semisimple Lie algebra $\mathfrak{g}$. At the intermediate step we construct the embedding of the quantum groups into the algebra of the rational functions on the quantum multi-dimensional torus. The explicit parameterization of the quantum groups used in this paper turns out to be closely related to the parameterization of the moduli spaces of the monopoles. As a result the proposed constructions of the representations provide a quantization of the moduli spaces of the monopoles on $\RR^3$ and $\RR^2\times S^1$.
A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of $G$-monopoles defined as the components of the space of based maps of $\mathbb{P}^1$ into the generalized flag manifold $X=G/B$. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.
We describe new constructions of the infinite-dimensional representations of $U(\mathfrak{g})$ and $U_q(\mathfrak{g})$ for $\mathfrak{g}$ being $\mathfrak{gl}(N)$ and $\mathfrak{sl}(N)$. The application of these constructions to the quantum integrable theories of Toda type is discussed. With the help of these infinite-dimensional representations we manage to establish direct connection between group theoretical approach to the quantum integrability and Quantum Inverse Scattering Method based on the representation theory of Yangian and its generalizations. In the case of $U_q(\mathfrak{g})$ the considered representation is naturally supplied with the structure of $U_q(\mathfrak{g})\otimes U_{\tilde q}(\check{\mathfrak{g}})$-bimodule where $\check {\mathfrak{g}}$ is Langlands dual to $\mathfrak{g}$ and $\log q/2πi=- (\log{\tilde q}/2πi)^{-1}$. This bimodule structure is a manifestation of the Morita equivalence of the algebra and its dual.
Using the representation theory of $\frak{gl}(N,\RR)$, we express the wave function of the $GL(N,\RR)$ Toda chain, which two of us recently obtained by the Quantum Inverse Scattering Method, in terms of multiple integrals. The main tool is our generalization of the Gelfand-Zetlin method to the case of infinite-dimensional representations of $\frak{gl}(N,\RR)$. The interpretation of this generalized construction in terms of the coadjoint orbits is given and the connection with the Yangian $Y(\frak{gl}(N))$ is discussed. We also give the hyperbolic Sutherland model eigenfunctions expressed in terms of integrals in the Gelfand-Zetlin representation. Using the example of the open Toda chain, we discuss the connection between the Quantum Inverse Scattering Method and Representation Theory.