Related papers: Kernel functions and B\"acklund transformations fo…
We use the definition of the Calogero-Moser models as Hamiltonian reductions of geodesic motions on a group manifold to construct their $R$-matrices. In the Toda case, the analogous construction yields constant $R$-matrices. By contrast,…
We discuss elliptic quantum Calogero-Moser-Sutherland models, including their relativistic generalizations due to Ruijsenaars and van Diejen, and the relations of these models to classes of special functions developed and explored in recent…
Starting from the functional representation of the Ablowitz-Kaup-Newell-Segur (AKNS) and derivative nonlinear Schr\"odinger (DNLS) hierarchies and using the chains of the Miura-like transformations we derive a set of B\"acklund…
We consider the relativistic generalization of the quantum $A_{N-1}$ Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common…
The course of 5 lectures given at the seminar "Integrable Systems: from Classical to Quantum" (Universite de Montreal, Jul 26 -- Aug 6, 1999) contains a detailed comment on the recently discovered (Gaudin-Pasquier, 1992) connection between…
Explicit algebraic relations between the quantum integrals of the elliptic Calogero--Moser quantum problems related to the root systems ${\bf A_2}$ and ${\bf B_2}$ are found.
We derive the quantum analogue of a B\"acklund transformation for the quantised Ablowitz-Ladik chain, a space discretisation of the nonlinear Schr\"odinger equation. The quantisation of the Ablowitz-Ladik chain leads to the $q$-boson model.…
We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are…
Explicit algebraic relations between the quantum integrals of the elliptic Calogero-Moser quantum problems related to the root systems A_2 and B_2 are found.
For a planar domain $\Omega$, we consider the Dirichlet spaces with respect to a base point $\zeta\in\Omega$ and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note,…
Considering the kernel of an integral operator intertwining two realizations of the group of motions of the pseudo-Euclidian space, we derive two formulas for series containing Whittaker's functions or Weber's parabolic cylinder functions.…
In this paper we prove a sharp version of the Moser-Trudinger inequality for the Euler-Lagrange functional of a singular Toda system, motivated by the study of models in Chern-Simons theory. Our result extends those for the scalar case, as…
We introduce a classical integrable system associated with the torus-equivariant quantum $K$-theory of type C flag variety. We prove that its conserved quantities coincide with the generators of the defining ideal of the Borel presentation…
Ordinary and gl(n,R) generalized Toda systems as well as a related hierarchy are probed with respect to certain quantization characteristics. "Quantum" canonical and Poisson transformations are used to study quantizations of transformed…
Starting from the hyperoctahedral multivariate hypergeometric function of Heckman and Opdam (associated with the $BC_n$ root system), we arrive -- via partial confluent limits in the sense of Oshima and Shimeno -- at solutions of the…
We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the "equations of motion" on the defect point via the space-like and time-like description. We then exploit the structural…
The connection problem for isomonodromic tau functions on the one-punctured torus concerns the ratio between the tau function and its modular transform, associated to dual pants decompositions of the torus. In this paper, we study the…
Many integrable theories can be formulated universally in terms of Lie algebraic root systems. Well-studied are conformally invariant scalar field theories of Toda type and their massive versions, which can be expressed in terms of simple…
We consider the classical elliptic Calogero-Moser model. A set of canonical separated variables for this model has been constructed in Kuznetsov et al. However, the generating function of the separating canonical transform is known only for…
In our previous work \cite{LNS}, we constructed quasi-Casoratian solutions to the noncommutative $q$-difference two-dimensional Toda lattice ($q$-2DTL) equation by Darboux transformation, which we can prove produces the existing Casoratian…