Eigenvalues of Bethe vectors in the Gaudin model

According to the Feigin–Frenkel–Reshetikhin theorem, the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors can be found using the center of an affine vertex algebra at the critical level. We recently calculated explicit Harish-Chandra images of the generators of the center in all classical types. Combining these results leads to explicit formulas for the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors. The Harish-Chandra images can be interpreted as elements of classical W-algebras. By calculating classical limits of the corresponding screening operators, we elucidate a direct connection between the rings of q-characters and classical W-algebras.     

Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe Ansatz Conjecture

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two irreducible modules. We study sequences of polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight.

We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Piñeiro multiple orthogonal polynomial and others are given by Wronskian-type determinants of Jacobi-Piñeiro polynomials.

As a byproduct we describe a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.

     

Quadrics on Riemannian spaces of constant curvature. Separation of variables and connection with the Gaudin magnet

Integrable systems associated with separation of the variables in real Riemannian spaces of constant curvature are considered. An isomorphism between all such systems and the hyperbolic Gaudin magnet is established. This isomorphism is used in a classification of all coordinate systems that admit separation of the variables, the basis of which is the classification of the correspondingL operators of the Gaudin magnet.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 83–111, April, 1992.     

Gaudin subalgebras and wonderful models

Gaudin Hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin Hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini–Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes.     

The quantum Gaudin system

In this paper, we explicitly construct the quantum $\mathfrak{g}\mathfrak{l}_n $ Gaudin model for general n and for a general number N of particles. To this end, we construct a commutative family in $U(\mathfrak{g}\mathfrak{l}_n )^{ \otimes N} $ . When passing to the classical limit (which is the projection onto the associated graded algebra), our family gives the entire family of classical Gaudin Hamiltonians. The construction is based on the special limit of the Bethe subalgebra in the Yangian $Y(\mathfrak{g}\mathfrak{l}_n )$ .     

Quaternionic spherical wave functions

The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu‐type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of ‐hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation by herglotz wave functions

By a general argument, it is shown that Herglotz wave functions are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to the reduced wave equation in Ω. This is used to provide corresponding approximation results in global spaces (eg. in L2‐Sobolev‐spaces H^m(Ω)) and for boundary data. Copyright © 2004 John Wiley & Sons, Ltd.     

Semi-classical limit of wave functions

We study in one dimension the semi-classical limit of the exact eigenfunction of the Hamiltonian , for a potential being analytic, bounded below and . The main result of this paper is that, for any given with two turning points, the exact normalized eigenfunction converges to the classical probability density, and the momentum distribution converges to the classical momentum density in the sense of distribution, as and with fixed. In this paper we only consider the harmonic oscillator Hamiltonian. By studying the semi-classical limit of the Wigner's quasi-probability density and using the generating function of the Laguerre polynomials, we give a complete mathematical proof of the Correspondence Principle.

     

Asymptotic estimates of wave functions and the existence of wave operators

The existence of wave operators is proved for the case, where the unperturbed operator is the operator of multiplication by a smooth function in momentum space and the perturbation is an arbitrary operator satisfying a fall off condition near infinity or a weighted L_p-estimate in configuration space. Under somewhat more restrictive conditions the invariance principle is also proved.     

Generalized prolate spheroidal wave functions

The following equation $$(1 - x^2 )d^2 y/dx^2 + [(\beta - \alpha - (\alpha + \beta + 2)x]dy/dx + (\chi (c) - c^2 x^2 )y = 0$$ has been solved wherex(c) a separation constant is the characteristic value and is a function ofc. This solution is a generalization of spheroidal wave function into the series form ofP _n ^α;β (x),α andβ both separately are greater than ?1. The finite transform and its properties have been defined and a boundary value problem has been solved applying these tools.     

On a new form of Bethe ansatz equations and separation of variables in the $$\mathfrak{s}\mathfrak{l}_3 $$ Gaudin model

A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum Gaudin model is presented. Dedicated to V.I. Arnold on the occasion of his 70th birthday     

Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions

We consider the Heisenberg spin-1/2 XXZ magnet in the case where the anisotropy parameter tends to infinity (the so-called Ising limit). We find the temperature correlation function of a ferromagnetic string above the ground state. Our approach to calculating correlation functions is based on expressing the wave function in the considered limit in terms of Schur symmetric functions. We show that the asymptotic amplitude of the above correlation function at low temperatures is proportional to the squared number of strict plane partitions in a box.     

Hirota equation and Bethe ansatz

Recent analyses of classical integrable structures in quantum integrable models solved by various versions of the Bethe ansatz are reviewed. Similarities between elements of quantum and classical theories of integrable systems are discussed. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton theory. Functional relations for quantum transfer matrices become the classical Hirota bilinear difference equation; solving this classical equation gives all the basic results for the spectral properties of quantum systems. Vice versa, typical Bethe ansatz formulas under certain boundary conditions yield solutions of this classical equation. The Baxter T-Q relation and its generalizations arise as auxiliary linear problems for the Hirota equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 1, pp. 54–100, July, 1998.     

Light-cone wave functions of heavy baryons

We present a classification of the three-quark light-cone distribution amplitudes (LCDAs) for ground-state heavy baryons with the spin parities J^P = 1/2 ⁺ and J^P = 3/2 ⁺ in QCD in the heavy-quark symmetry limit and calculate several lowest moments of LCDAs based on QCD sum rules. We propose simple model functions for the heavy-baryon distribution amplitudes and analyze their dependence on the energy scale.     

Toeplitz operators on Herglotz wave functions

The space of Herglotz wave functions in R² consists of all the solutions of the Helmholtz equation that can be represented as the Fourier transform in R² of a measure supported in the circle and with density in L²(S¹). This space has a structure of a Hilbert space with reproducing kernel. The purpose of this article is to study Toeplitz operators with nonnegative radial symbols, defined on this space. We study the symbols defining bounded and compact Toeplitz operators as well as the Toeplitz operators belonging to the Schatten classes sp.