scholarly journals ON SEPARATION OF VARIABLES AND COMPLETENESS OF THE BETHE ANSATZ FOR QUANTUM N GAUDIN MODEL

2009 ◽  
Vol 51 (A) ◽  
pp. 137-145 ◽  
Author(s):  
E. MUKHIN ◽  
V. TARASOV ◽  
A. VARCHENKO
Keyword(s):  
Differential Operator ◽  
Bethe Ansatz ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Polynomial Kernel ◽  
Simple Spectrum ◽  
Gaudin Model ◽  
Bethe Algebra ◽  
One To One ◽  
Evaluation Parameters

AbstractIn this paper, we discuss implications of the results obtained in [5]. It was shown there that eigenvectors of the Bethe algebra of the quantum N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [5] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.

scholarly journals On the Gaudin model of type G2

2019 ◽  
Vol 21 (03) ◽  
pp. 1850012
Author(s):  
Kang Lu ◽  
Evgeny Mukhin
Keyword(s):  
Tensor Product ◽  
Bethe Ansatz ◽  
Irreducible Module ◽  
The Self ◽  
Gaudin Model ◽  
Bijective Correspondence ◽  
Dual Spaces ◽  
Finite Dimensional ◽  
Vector Representations ◽  
Evaluation Parameters

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G2. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G2 are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

scholarly journals Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz

2002 ◽  
Vol 31 (9) ◽  
pp. 513-553 ◽  
Author(s):  
Stanislav Pakuliak ◽  
Sergei Sergeev
Keyword(s):  
Bethe Ansatz ◽  
Weyl Algebra ◽  
Separation Of Variables ◽  
Toda Chain ◽  
Special Choice ◽  
Finite Dimensional Representation ◽  
Classical Counterpart ◽  
Discrete Integrable System ◽  
Root Of Unity ◽  
Finite Dimensional

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

First Order Operators on Manifolds With a Group Action

1996 ◽  
Vol 48 (4) ◽  
pp. 758-776 ◽  
Author(s):  
H. D. Fegan ◽  
B. Steer
Keyword(s):  
Differential Operator ◽  
Group Action ◽  
Lie Group ◽  
Differential Operators ◽  
Order Differential Operator ◽  
Compact Lie Group ◽  
First Order ◽  
Rank 2 ◽  
Homogeneous Operators

AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

Semibounded Extensions of Singular Ordinary Differential Operators

1988 ◽  
Vol 31 (4) ◽  
pp. 432-438
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Boundary Conditions ◽  
Differential Operators ◽  
The Self ◽  
Limit Circle ◽  
Singular Differential Operator ◽  
Ordinary Differential Operators

AbstractThe self-adjoint extensions of the singular differential operator Ly = [(py’)’ + qy]/w, where p < 0, w > 0, q ≧ mw, are characterized under limit-circle conditions. It is shown that as long as the coefficients of certain boundary conditions define points which lie between two lines, the extension they help define has the same lower bound.

scholarly journals A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

1997 ◽  
Vol 145 ◽  
pp. 125-142
Author(s):  
Takeshi Mandai
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Partial Differential Operator ◽  
Negative Integer ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Real Analytic

Consider a partial differential operator(1.1) where K is a non-negative integer and aj,a are real-analytic in a neighborhood of (0, 0)

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