Separation of Variables. New Trends.

Abstract

The review is based on the author's papers since 1985 in which a new approach to the separation of variables (\SoV) has being developed. It is argued that \SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown that the standard construction of the action-angle variables from the poles of the Baker-Akhiezer function can be interpreted as a variant of \SoV, and moreover, for many particular models it has a direct quantum counterpart. The list of the models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinear Schrödinger equation, $SL(3)$-invariant magnetic chain. New results for the 3-particle quantum Calogero-Moser system are reported.

33 pages, harvmac, no figures

Cited by in corpus (275)

• Baryon Distribution Amplitudes in QCD
• Integral representations for correlation functions of the XXZ chain at finite temperature
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Quantum Geometry of Refined Topological Strings
• Logarithmic scaling in gauge/string correspondence
• Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
• Off-diagonal Bethe ansatz and exact solution of a topological spin ring
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Worldsheet Scattering in AdS_5 x S^5
• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Off-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fields
• Nonequilibrium Cooper pairing in the nonadiabatic regime
• Solution for the dynamics of the BCS and central spin problems
• Classical Conformal Blocks and Painleve VI
• Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions
• Quantum quench in a p + i p superfluid: Winding numbers and topological states far from equilibrium
• Superuniversality of superdiffusion
• Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and matrix elements of some quasi-local operators
• Evolution equations for quark-gluon distributions in multi-color QCD and open spin chains
• Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I
• On the Dynamics of Finite-Gap Solutions in Classical String Theory
• Quantum Spectral Curve and Structure Constants in N=4 SYM: Cusps in the Ladder Limit
• Few remarks on Baecklund transformations for many-body systems
• Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
• Quantization of Solitons and the Restricted Sine-Gordon Model
• A Symplectic Structure for String Theory on Integrable Backgrounds
• Separation of variables for the quantum SL(2,R) spin chain
• On the Yang-Baxter equation for the six-vertex model
• Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism
• Factorisation of Macdonald polynomials
• Separation of variables for integrable spin-boson models
• Spectral Duality Between Heisenberg Chain and Gaudin Model
• Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
• Algebraic Bethe Ansatz for a discrete-state BCS pairing model
• New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains
• Generalized r-matrix structure and algebro-geometric solution for integrable systems
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Separation of variables for the Ruijsenaars system
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• Noncompact Heisenberg spin magnets from high-energy QCD: II. Quantization conditions and energy spectrum
• Integrable dynamics of coupled Fermi-Bose condensates
• Exactly solvable magnet of conformal spins in four dimensions
• Exact quantization conditions for cluster integrable systems
• Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model
• The Ruijsenaars-Schneider Model in the Context of Seiberg-Witten Theory
• Three-point functions in the SU(2) sector at strong coupling
• Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)
• Bethe states of the XXZ spin-1/2 chain with arbitrary boundary fields
• Quantum Painleve-Calogero Correspondence
• The Baxter's Q-operator for the W-algebra $W_N$
• On quantum separation of variables
• Long-range SL(2) Baxter equation in N=4 super-Yang-Mills theory
• Integrability in dipole-deformed N=4 super Yang-Mills
• Quantum mechanics of null polygonal Wilson loops
• Normal and anomalous solitons in the theory of dynamical Cooper pairing
• Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
• The hexagon in the mirror: the three-point function in the SoV representation
• Quantum integrability in systems with finite number of levels
• Separation of variables and scalar products at any rank
• Wave functions and correlation functions for GKP strings from integrability
• Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT
• Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame
• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• Explicit integration of the Hénon-Heiles Hamiltonians
• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• Towards Baxter equation in supersymmetric Yang-Mills theories
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• The Sine-Gordon model revisited I
• Dynamical r-matrix for the elliptic Ruijsenaars-Schneider system
• The Hamiltonian Structure of the Second Painleve Hierarchy
• Set theoretic Yang-Baxter & reflection equations and quantum group symmetries
• Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors
• The Link between Integrability, Level Crossings, and Exact Solution in Quantum Models
• Integrable structures and duality in high-energy QCD
• On determinant representations of scalar products and form factors in the SoV approach: the XXX case
• Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries
• Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems
• On maximally superintegrable systems
• Solitons in high-energy QCD
• On Bethe vectors in ${\mathfrak{gl}}_3$-invariant integrable models
• Backlund transformations for many-body systems related to KdV
• Simultaneous separation for the Neumann and Chaplygin systems
• Separation of variables for the Dn type periodic Toda lattice
• Baxter Q-operator for graded SL(2|1) spin chain
• Exactly solvable single-trace four point correlators in $\chi$CFT${}_4$
• Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and N<4 SYM
• Hexagons and Correlators in the Fishnet Theory
• SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra
• Q-operator and factorised separation chain for Jack polynomials
• The quasi-bi-Hamiltonian formulation of the Lagrange top
• A noncommutative discrete potential KdV lift
• Quantum and Classical Integrable Systems
• Introduction to classical and quantum integrability
• Separation of variables for soliton equations via their binary constrained flows
• High Energy QCD: Stringy Picture from Hidden Integrability
• Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C) magnet
• Degenerate Poisson pencils on curves: new separability theory
• Natural coordinates for a class of Benenti systems
• Riemann surfaces, separation of variables and classical and quantum integrability
• Separation of variables for quantum integrable systems on elliptic curves
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• Quantum Spin Systems and Supersymmetric Gauge Theories, I
• Separation of variables and Bäcklund transformations for the symmetric Lagrange top
• Noncompact SL(2,R) spin chain
• On the form factors of local operators in the lattice sine-Gordon model
• Classification of Isomonodromy Problems on Elliptic Curves
• On a Poisson reduction for Gel'fand--Zakharevich manifolds
• Classical Integrability for Three-point Functions: Cognate Structure at Weak and Strong Couplings
• Stationary problems for equation of the KdV type and dynamical $r$-matrices.
• Sekiguchi-Debiard operators at infinity
• A new integral representation for the scalar products of Bethe states for the XXX spin chain
• Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
• Colour-Twist Operators I: Spectrum and Wave Functions
• Classification of Parameter-Dependent Quantum Integrable Models, Their Parameterization, Exact Solution, and Other Properties
• New Compact Construction of Eigenstates for Supersymmetric Spin Chains
• A review of the AdS/CFT Quantum Spectral Curve
• Gaudin Models and Bending Flows: a Geometrical Point of View
• Integrability of the eta-deformed Neumann-Rosochatius model
• XXZ scalar products and KP
• Change in the adiabatic invariant in a nonlinear two-mode model of Feshbach resonance passage
• Dual Baxter equations and quantization of Affine Jacobian
• Manin matrices and Talalaev's formula
• Exact ground state and elementary excitations of a topological spin chain
• Duality Relations for Overlaps of Integrable Boundary States in AdS/dCFT
• New variables of separation for particular case of the Kowalevski top
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II
• Separation of variables for A2 Ruijsenaars model and new integral representation for A2 Macdonald polynomials
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Explicit solution of the (quantum) elliptic Calogero-Sutherland model
• Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ spin-$\frac{1}{2}$ chain
• On Scalar Products in Higher Rank Quantum Separation of Variables
• Quantum transfer-matrices for the sausage model
• On the generalized integrable Chaplygin system
• Bi-Hamiltonian representation of Stäckel systems
• The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method
• Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model
• Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems
• An integrable hierarchy with a perturbed Henon-Heiles system
• On quantum separation of variables beyond fundamental representations
• Noncompact Heisenberg spin magnets from high-energy QCD: III. Quasiclassical approach
• Discrete Dubrovin Equations and Separation of Variables for Discrete Systems
• Integrability in heavy quark effective theory
• From bi-Hamiltonian geometry to separation of variables: stationary Harry-Dym and the KdV dressing chain
• Complete spectrum of quantum integrable lattice models associated to ${\mathcal{U}}_q(\widehat{g{l}_n})$ by separation of variables
• On the Quantum Inverse Problem for the Closed Toda Chain
• Spectral curves of the hyperelliptic Hitchin systems
• Exact solution of the XXX Gaudin model with the generic open boundaries
• On bi-hamiltonian geometry of the Lagrange top
• On reciprocal equivalence of Stäckel systems
• Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems
• Functional Equations and Separation of Variables for Exact g-Function
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Systematic classical continuum limits of integrable spin chains and emerging novel dualities
• A new representation for two- and three-point correlators of operators from sl(2) sector
• Yangian Ward Identities for Fishnet Four-Point Integrals
• Integrable systems on $so(4)$ related with XXX spin chains with boundaries
• From Stäckel systems to integrable hierarchies of PDE's: Benenti class of separation relations
• The Kowalewski top: a new Lax representation
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
• On Separation of Variables for Reflection Algebras
• Generalized Lenard Chains, Separation of Variables and Superintegrability
• Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions
• Baxter Equation for Quantum Discrete Boussinesq Equation
• A note on ${\mathfrak{gl}}_2$-invariant Bethe vectors
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Hexagonalization of Fishnet integrals I: mirror excitations
• Exact Quantum Solutions of Extraordinary N-body Problems
• The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy
• Hamiltonian Structure of PI Hierarchy
• Bäcklund Transformations as exact integrable time-discretizations for the trigonometric Gaudin model
• Third rank Killing tensors in general relativity. The (1+1)-dimensional case
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Source identity and kernel functions for Inozemtsev-type systems
• Simultaneous separation for the Kowalevski and Goryachev-Chaplygin gyrostats
• Correlation functions by Separation of Variables: the XXX spin chain
• Tri-hamiltonian vector fields, spectral curves and separation coordinates
• Dynamical boundary conditions for integrable lattices
• Some integrable systems of algebraic origin and separation of variables
• Quantized W-algebra of sl(2,1) and quantum parafermions of U_q(sl(2))
• Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems
• Backlund transformations for the elliptic Gaudin model and a Clebsch system
• Constructing N-soliton solution for the mKdV equation through constrained flows
• The symplectic structure of the spin Calogero model
• Gustafson integrals for $SL(2,\mathbb{C})$ spin magnet
• A Conjectured R-Matrix
• Families of quai-bi-Hamiltonian systems and separability
• Digital quantum simulation of the BCS model with a central-spin-like quantum processor
• Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy
• Limits of Gaudin Systems: Classical and Quantum Cases
• Exact solution of an su(n) spin torus
• Deriving N-soliton solutions via constrained flows
• Seiberg-Witten Geometry of Four-Dimensional $\mathcal{N}=2$ Quiver Gauge Theories
• Amplitudes in the N=4 SYM from Quantum Geometry of the Momentum Space
• Analytic Bethe Ansatz and Baxter equations for long-range psl(2|2) spin chain
• On the Darboux-Nijenhuis Variables for the Open Toda Lattice
• Spin Chains and Gustafson's Integrals
• Baxter equations and Deformation of Abelian Differentials
• Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain
• Supersymmetric quantum mechanics of the flux tube
• The symplectic structure of rational Lax pair systems
• Integrable quantum Stäckel systems
• Spectrum of the quantum integrable $D_2^{(2)}$ spin chain with generic boundary fields
• Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. I. Frobenius integrability
• Mellin$-$Barnes Transformation for Two-Loop Master-Diagram
• The integrable structure of nonrational conformal field theory
• D-branes and Orbit Average
• Can a spin chain relate combinatorics to number theory?
• Bispectral duality and separation of variables from surface defect transition
• Classical and Quantum Superintegrability of Stäckel Systems
• A new method to introduce additional separated variables for high-order binary constrained flows
• Bäcklund Transformations for the Trigonometric Gaudin Magnet
• Non-adiabatic dynamics in d+id-wave fermionic superfluids
• Separable quantizations of Stäckel systems
• Separation of Variables for a flux tube with an end
• Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains
• Closed-form modified Hamiltonians for integrable numerical integration schemes
• Non-Hamiltonian systems separable by Hamilton-Jacobi method
• Exact surface energy of the $D_2^{(1)}$ spin chain with generic non-diagonal boundary reflections
• A representation basis for the quantum integrable spin chain associated with the su(3) algebra
• Matone's relation of N=2 super Yang-Mills and spectrum of Toda chain
• Hyperelliptic Integrable Systems on K3 and Rational Surfaces
• Yang-Baxter Algebra for the n-Harmonic Oscillator Realisation of sp(2n,R)
• Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables
• Dimers for Type D Relativistic Toda Model
• On Separation of Variables for Integrable Equations of Soliton Type
• An integrable system on the moduli space of rational functions and its variants
• Symplectic and orthogonal Lie algebra technology for bosonic and fermionic oscillator models of integrable systems
• Bethe ansatz solution of the ${\tau }_2$-model with arbitrary boundary fields
• Ground state solutions of inhomogeneous Bethe equations
• Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero-Gaudin system
• Boundary dissipative spin chains with partial solvability inherited from system Hamiltonians
• The extended Lotka-Volterra lattice and affine Jacobi varieties of spectral curves
• Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces
• Exact solution of the quantum integrable model associated with the twisted $D_3^{(2)}$ algebra
• Continuum limit of the Volterra model, separation of variables and non standard realizations of the Virasoro Poisson bracket
• T-W relation and free energy of the Heisenberg chain at a finite temperature
• Elliptic Curves and a New Construction of Integrable Systems
• Flat minimal quantizations of Stackel systems and quantum separability
• Exact solution of the trigonometric SU(3) spin chain with generic off-diagonal boundary reflections
• Orbit Approach to Separation of Variables in sl(3)-Related Integrable Systems
• On three-dimensional quasi-Stäckel Hamiltonians
• A convenient basis for the Izergin-Korepin model
• Canonical spectral coordinates for the Calogero-Moser space associated with the cyclic quiver
• Quantum Bäcklund Transformations: some ideas and examples
• Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain
• Orbit Approach to Separation of Variables in sl(4)-Related Integrable Systems
• Reducible Abelian varieties and Lax matrices for Euler's problem of two fixed centres
• Separation of variables for a lattice integrable system and the inverse problem
• Quantum discrete Dubrovin equations
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Bethe ansatz inside Calogero-Sutherland models
• Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields
• Separation of variables for higher rank integrable models: review of recent progress
• Separation of variables for the A_3 elliptic Calogero-Moser system
• Baxter T-Q Equation for Shape Invariant Potentials. The Finite-Gap Potentials Case
• Quantisation of bending flows
• Spectrum of the transfer matrices of the spin chains associated with the $A_3^{(2)}$ Lie algebra
• Symplectic Structures for the Cubic Schrodinger equation in the periodic and scattering case
• Separation of Variables and Vacuum Structure of N=2 SUSY QCD
• Selected Topics in Classical Integrability
• Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry
• Bethe States of the integrable spin-s chain with generic open boundaries
• Separation of Variables, Quasi-Trigonometric $r$-Matrices and Generalized Gaudin Models
• Trihamiltonian extensions of separable systems in the plane
• An Analytic Formula for the A_2 Jack Polynomials
• On the solutions of the $Z_n$-Belavin model with arbitrary number of sites
• New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin Chains
• BC-type open $SL(2,\mathbb{C})$ spin chain
• Algebraic curves as a source of separable multi-Hamiltonian systems
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
• SU(3) magnet: finite-gap integration on the lowest genus curve
• Motion on the n-dimensional ellipsoid under the influence of a harmonic force revisited
• The matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice