Related papers: Separation of variables for quantum integrable sys…
The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic…
There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation…
The discrete equations of motion for the quantum mappings of KdV type are given in terms of the Sklyanin variables (which are also known as quantum separated variables). Both temporal (discrete-time) evolutions and spatial (along the…
Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded…
An elliptic version of quantum groups is proposed. It comes form the quantization of the Knizhnik-Zamolodchikov- Bernard equation on the torus. The relation with elliptic IRF models is explained.
We will study solvability of nonlinear second-order elliptic system of partial differential equations with nonlinear boundary conditions. We study the generalized Steklov Robin eigensystem (with possibly matrices weights) in which the…
In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the "eigenvalues" of the decision variable. Here, by making use of the FTvN systems framework introduced by…
A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the…
The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is…
Kaneko and Sakai recently observed that certain elliptic curves whose associated newforms (by the modularity theorem) are given by the eta-quotients can be characterized by a particular differential equation involving modular forms and…
In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $\delta$- and $\delta'$-interactions are studied. We show the self-adjointness of these…
Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{+*}(2,{\bf R})$ are integrated by separation of variables in…
We consider the open XYZ spin chain with boundary fields. We solve the model by the new Separation of Variables approach introduced in arXiv:1904.00852. In this framework, the transfer matrix eigenstates are obtained as a particular…
By means of a linear scaling of the variables we convert a singular bifurcation equation in $\R^n$ into an equivalent equation to which the classical implicit function theorem can be directly applied. This allows to deduce the existence of…
To help reveal the complete picture of linear kinetic drift modes, four independent numerical approaches, based on integral equation, Euler initial value simulation, Euler matrix eigenvalue solution and Lagrangian particle simulation,…
An operator-splitting finite element scheme for the time-dependent, high-dimensional radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite…
We consider the integrable XXZ model with the special open boundary conditions. We perform Quantum Group reduction of this model in roots of unity and use it for the definition Minimal Models of Interable lattice theory. It is shown that…
The integrable system is introduced based on the Poisson $ rs $-matrix structure. This is a generalization of the Gaudin magnet, and in SL(2) case isomorphic to the generalized Neumann model. The separation of variables is discussed for…
An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent…
The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analyzed. In non-integrable regimes we…