scholarly journals Quantum integrable systems and differential Galois theory

1997 ◽  
Vol 2 (1) ◽  
pp. 31-56 ◽  
Author(s):  
A. Braverman ◽  
P. Etingof ◽  
D. Gaitsgory
Keyword(s):  
Integrable Systems ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Quantum Integrable Systems

scholarly journals Differential Galois theory and Darboux transformations for Integrable Systems

2017 ◽  
Vol 115 ◽  
pp. 75-88
Author(s):  
Sonia Jiménez ◽  
Juan J. Morales-Ruiz ◽  
Raquel Sánchez-Cauce ◽  
María-Angeles Zurro
Keyword(s):  
Integrable Systems ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Darboux Transformations

scholarly journals DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY

2009 ◽  
Vol 06 (08) ◽  
pp. 1357-1390 ◽  
Author(s):  
ANDRZEJ J. MACIEJEWSKI ◽  
MARIA PRZYBYLSKA
Keyword(s):  
Galois Group ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Homogeneous Potentials ◽  
Particular Solution

This paper is an overview of our works that are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realize the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

scholarly journals Separation of variables in AdS/CFT: functional approach for the fishnet CFT

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Andrea Cavaglià ◽  
Nikolay Gromov ◽  
Fedor Levkovich-Maslyuk
Keyword(s):  
Integrable Systems ◽  
Density Operator ◽  
Numerical Evaluation ◽  
Lagrangian Density ◽  
Separation Of Variables ◽  
Functional Approach ◽  
Quantum Integrable Systems ◽  
Arbitrary State ◽  
Nontrivial Coupling ◽  
The One

Abstract The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in $$ \mathcal{N} $$ N = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of $$ \mathcal{N} $$ N = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the $$ \mathcal{N} $$ N = 4 SYM case, as we speculate in the last part of the article.

scholarly journals ALGEBRAIC ASPECT AND CONSTRUCTION OF LAX OPERATORS IN QUANTUM INTEGRABLE SYSTEMS

1995 ◽  
Vol 10 (40) ◽  
pp. 3113-3117 ◽  
Author(s):  
B. BASU-MALLICK ◽  
ANJAN KUNDU
Keyword(s):  
Integrable Systems ◽  
Integrable Models ◽  
Quantum Integrable Systems ◽  
Algebraic Construction ◽  
Algebraic Aspect ◽  
Quantum Integrable Models

An algebraic construction which is more general and closely connected with that of Faddeev,1 along with its application for generating different classes of quantum integrable models is summarized to complement the recent results of Ref. 1.

scholarly journals Differential Galois Theory and Lie Symmetries

2015 ◽  
Author(s):  
David Blázquez-Sanz ◽  
◽  
Juan J. Morales-Ruiz ◽  
Jacques-Arthur Weil ◽  
◽  
...  
Keyword(s):  
Galois Theory ◽  
Lie Symmetries ◽  
Differential Galois Theory
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