The ODE/IM Correspondence

Abstract

This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

102 pages, 27 figures. v2: added references

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• PT-Symmetric Quantum Mechanics
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• How Algebraic Bethe Ansatz works for integrable model
• Introduction to PT-Symmetric Quantum Theory
• Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation
• Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation
• Schrödinger operators with complex potential but real spectrum
• Supersymmetry and the spontaneous breakdown of PT symmetry
• PT -symmetric harmonic oscillators
• Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
• Dilogarithm identities
• Unified Approach to Thermodynamic Bethe Ansatz and Finite Size Corrections for Lattice Models and Field Theories
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Integrable Quantum Field Theories in Finite Volume: Excited State Energies
• Quasi-exactly solvable quartic potential
• SUSY Quantum Mechanics with Complex Superpotentials and Real Energy Spectra
• sl(2, C) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues
• Spectral determinants for Schroedinger equation and Q-operators of Conformal Field Theory
• Excited State Destri - De Vega Equation for Sine-Gordon and Restricted Sine-Gordon Models
• A PT-Invariant Potential With Complex QES Eigenvalues
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• A family of complex potentials with real spectrum
• A new PT symmetric complex Hamiltonian with a real spectra
• Dynkin TBA's
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Non linear integral equation and excited--states scaling functions in the sine-Gordon model
• Pseudo-Hermiticity of Hamiltonians under imaginary shift of the co-ordinate : real spectrum of complex potentials
• The Importance of being Odd
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• g-function flow in perturbed boundary conformal field theories
• Bethe Equations "on the Wrong Side of Equator"
• Excited states in some simple perturbed conformal field theories
• Complex Calogero model with real energies
• Integrable Model of Boundary Interaction: The Paperclip
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Complex Square Well --- A New Exactly Solvable Quantum Mechanical Model
• A New Class of PT-symmetric Hamiltonians with Real Spectra
• Differential equations and integrable models: the SU(3) case
• Exponential Type Complex and non-Hermitian Potentials in PT-Symmetric Quantum Mechanics
• On nonequilibrium states in QFT model with boundary interaction
• Generating Converging Bounds to the (Complex) Discrete States of the $P^2+i{X}^3+i\alpha X$ Hamiltonian
• Beyond the WKB approximation in PT-symmetric quantum mechanics
• Functional relations in Stokes multipliers - Fun with $x^6+\alpha x^2$ potential-
• Non-Hermitian SUSY Hydrogen-like Hamiltonians with real spectra
• PT symmetry of a conditionally exactly solvable potential
• Integrable Circular Brane Model and Coulomb Charging at Large Conduction
• Bound and scattering states of extended Calogero model with an additional PT invariant interaction
• Notes on parafermionic QFT's with boundary interaction
• A Q-Operator Identity for the Correlation Functions of the Infinite XXZ Spin-Chain
• Numerical Simulations of PT-Symmetric Quantum Field Theories
• New Quasi-Exactly Solvable Sextic Polynomial Potentials
• Geometrical Loci and CFTs via the Virasoro Symmetry of the mKdV-SG hierarchy: an excursus
• Analytical results for the Coqblin-Schrieffer model with generalized magnetic fields
• On the Real Spectra of Calogero Model with Complex Coupling
• Boundary RG Flow Associated with the AKNS Soliton Hierarchy
• Higher-level eigenvalues of Q-operators and Schroedinger equation
• Generating Converging Eigenenergy Bounds for the Discrete States of the -ix^3 Non-Hermitian Potential
• The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions
• Paperclip at $\theta =\pi$
• Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz
• Detecting Broken PT-Symmetry
• Ordinary Differential Equations and Integrable Models
• Non-Hermitian Quantum Theory and its Holomorphic Representation: Introduction and Applications
• Quantization of soliton systems and Langlands duality
• Stokes Multipliers, Spectral Determinants and T-Q relations
• Real roots of Random Polynomials: Universality close to accumulation points
• An eigenvalue problem related to the non-linear sigma-model: analytical and numerical results
• Solving Baxter's TQ-equation via representation theory
• Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues
• PT invariant Non-Hermitian Potentials with Real QES Eigenvalues
• Non-Hermitian Quantum Theory and its Holomorphic Representation: Introduction and Some Applications

Cited by in corpus (182)

• Non-Hermitian Physics
• Pseudo-Hermitian Representation of Quantum Mechanics
• Non-Hermitian Topology and Exceptional-Point Geometries
• Observation of PT phase transition in a simple mechanical system
• Thermodynamic Bubble Ansatz
• Y-system for Scattering Amplitudes
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• The $\text{T}\bar{\text{T}}$ perturbation and its geometric interpretation
• T-systems and Y-systems in integrable systems
• Observation of Fast Evolution in Parity-Time-Symmetric System
• Mathematical and physical aspects of complex symmetric operators
• Defects and Quantum Seiberg-Witten Geometry
• Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods
• PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues
• Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Operators from mirror curves and the quantum dilogarithm
• Bethe Ansatz and Ordinary Differential Equation Correspondence for Degenerate Gaudin Models
• TBA equations and resurgent Quantum Mechanics
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• PT-symmetric quantum field theory in D dimensions
• Complex Correspondence Principle
• Scattering theory using smeared non-Hermitian potentials
• Ordinary versus PT-symmetric ${\varphi }^3$ quantum field theory
• Simulating broken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian systems by weak measurement
• Exact WKB and abelianization for the $T_3$ equation
• PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability
• Perturbed Defects and T-Systems in Conformal Field Theory
• A Solvable Deformation of Quantum Mechanics
• Exact WKB methods in $SU(2)$ $N_f=1$
• Momentum-independent reflectionless transmission in the non-Hermitian time-reversal symmetric system
• On the determinant representations of Gaudin models' scalar products and form factors
• Hermitian dynamics in a class of pseudo-Hermitian networks
• TBA-like integral equations from quantized mirror curves
• Chaotic systems in complex phase space
• Quantum catastrophes: a case study
• Probability-preserving evolution in a non-Hermitian two-band model
• Exactly Solvable Quasi-hermitian Transverse Ising Model
• Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model
• QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains
• Level statistics of a pseudo-Hermitian Dicke model
• Inner products in integrable Richardson-Gaudin models
• Fluctuations and skewness of the current in the partially asymmetric exclusion process
• Geometric aspects of the ODE/IM correspondence
• Liouville reflection operator, affine Yangian and Bethe ansatz
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• ODE/IM correspondence for the Fateev model
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• Lectures on Integrable Structures in Quantum Field Theory and Massive ODE/IM Correspondence
• Scaling behavior and phase diagram of a PT-symmetric non-Hermitian Bose-Hubbard system
• Probability Density in the Complex Plane
• Integrable Kondo problems
• TBA Equations and Quantization Conditions
• PT-Symmetric Representations of Fermionic Algebras
• Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
• Admissible perturbations and false instabilities in PT-symmetric quantum systems
• Q-operators for the open Heisenberg spin chain
• Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms
• The Bethe Ansatz and the Tzitzéica-Bullough-Dodd equation
• Six-point gluon scattering amplitudes from Z_4-symmetric integrable model
• TBA equations for the Schrödinger equation with a regular singularity
• PT Symmetry in Classical and Quantum Statistical Mechanics
• Quantum Phase Transition in a Pseudo-hermitian Dicke model
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Towards perturbative renormalization of ${\varphi }^2(i\varphi {)}^{\epsilon }$ quantum field theory
• Behavior of eigenvalues in a region of broken-PT symmetry
• Three solvable matrix models of a quantum catastrophe
• Discrete quantum square well of the first kind
• Characterizing the Bulk-Boundary Correspondence of one-dimensional non-Hermitian interacting systems by edge entanglement entropy
• Zeta-regularisation for exact-WKB resolution of a general 1D Schrödinger equation
• WKB periods for higher order ODE and TBA equations
• Twisting singular solutions of Bethe's equations
• Opers for higher states of quantum KdV models
• Does the complex deformation of the Riemann equation exhibit shocks?
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• PT symmetry breaking and exceptional points for a class of inhomogeneous complex potentials
• On the monodromy of the deformed cubic oscillator
• Periodic orbits for classical particles having complex energy
• Classical Particle in a Complex Elliptic Potential
• Conduction bands in classical periodic potentials
• Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians
• PT symmetry in relativistic quantum mechanics
• Polynomial Relations for q-Characters via the ODE/IM Correspondence
• Integrable boundary interaction in 3D target space: the "pillow-brane" model
• Functional Equations and Separation of Variables for Exact g-Function
• Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding
• Quantum inner-product metrics via recurrent solution of Dieudonne equation
• Non linear pseudo-bosons versus hidden Hermiticity
• PT phase transition in multidimensional quantum systems
• ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• Quasi-exactly solvable quartic: elementary integrals and asymptotics
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
• Quantum periods for $\mathcal{N}=2$ $SU(2)$ SQCD around the superconformal point
• Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation
• Quantum Big Bang without fine-tuning in a toy-model
• Opers for higher states of the quantum Boussinesq model
• Asymptotically vanishing PT-symmetric potentials and negative-mass Schroedinger equations
• Extending PT symmetry from Heisenberg algebra to E2 algebra
• Recursion relation for instanton counting for $SU(2)$ $\mathcal{N}=2$ SYM in NS limit of $\Omega$ background
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• N-site-lattice analogues of $V(x)=i{x}^3$
• Identification of observables in quantum toboggans
• A variational method for integrability-breaking Richardson-Gaudin models
• Anomalous real spectra of non-Hermitian quantum graphs in strong-coupling regime
• Hilbert space inner products for PT-symmetric Su-Schrieffer-Heeger models
• Non-adiabatic transitions in a non-symmetric optical lattice
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• Schwinger-Dyson equations and mass generation for an axion theory with a PT symmetric Yukawa fermion interaction
• Asymptotic analysis of noisy fitness maximization, applied to metabolism and growth
• Negative-energy PT-symmetric Hamiltonians
• On solutions of the Bethe Ansatz for the Quantum KdV model
• Topology-controlled spectra of imaginary cubic oscillators in the large-L approach
• Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions
• PT-symmetric quantum systems with positive P
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra
• Extended systems of Baxter Q-functions and fused flags I: simply-laced case
• KPZ fluctuations in finite volume
• $Q\tilde{Q}$-systems for twisted quantum affine algebras
• Exact WKB Analysis and TBA Equations for the Stark Effect
• ODE/IM correspondence and supersymmetric affine Toda field equations
• Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators
• Partial differential equations from integrable vertex models
• The coupled-cluster approach to quantum many-body problem in a three-Hilbert-space reinterpretation
• Elementary functions in Thermodynamic Bethe Ansatz
• T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras
• Entire functions with two radially distributed values
• On the ODE/IM correspondence for minimal models
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Universal spectral behavior of $x^2(ix{)}^{\epsilon }$ potentials
• Theory Including Future Not Excluded -- Formulation of Complex Action Theory II --
• Transfer matrices of rational spin chains via novel BGG-type resolutions
• Hermitian--to--quasi-Hermitian quantum phase transitions
• A Monoidal Category for Perturbed Defects in Conformal Field Theory
• Quantum Wronskian approach to six-point gluon scattering amplitudes at strong coupling
• WKB analysis of the linear problem for modified affine Toda field equations
• Resonant-amplified and invisible Bragg scattering based on spin coalescing modes
• Calculation of low-lying energy levels in quantum mechanics
• Scattering off PT-symmetric upside-down potentials
• Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: a Levinson type theorem on complex domains, and applications
• Perpetual emulation threshold of PT-symmetric Hamiltonians
• From Fuchsian differential equations to integrable QFT
• Analytic approximation for eigenvalues of a class of $\mathcal{P}\mathcal{T}$ symmetric Hamiltonians
• CPT-symmetric discrete square well
• Some spectral equivalences between Schrodinger operators
• Complex WKB Analysis of a PT Symmetric Eigenvalue Problem
• PT-symmetric eigenvalues for homogeneous potentials
• Six-vertex model and non-linear differential equations I. Spectral problem
• Faster than Hermitian Time Evolution
• Liouville Theory, AdS${}_2$ String, and Three-Point Functions
• Exact WKB in all sectors I: Potentials with degenerate saddles
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Affine opers and conformal affine Toda
• The large-g observability of the low-lying energies in the strongly singular potentials $V(x)=x^2+g^2/x^6$ after their PT-symmetric regularization
• High accuracy energy formulas for the attractive two-site Bose-Hubbard model
• Ad hoc physical Hilbert spaces in Quantum Mechanics
• Geometry of PT-symmetric quantum mechanics
• A(2|1) spectral equivalences and nonlocal integrals of motion
• Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin-$\frac{1}{2}$ chain
• Solvable spectral problems from 2d CFT and N=2 gauge theories
• Planarizable Supersymmetric Quantum Toboggans
• Necessary condition for information transfer under simulated parity-time-symmetric evolution
• Exact-WKB Analysis of Two-level Floquet Systems
• q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors
• Spectral zeta functions of a 1D Schrödinger problem
• Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin
• Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian $p^2-g{x}^4+a/x^2$
• The ODE/IM Correspondence between $C(2{)}^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT
• PT-symmetric quantum models living in an auxiliary Pontryagin space
• Two-parametric PT-symmetric quartic family
• Gluon scattering amplitudes from gauge/string duality and integrability
• Pseudo-Boson Coherent and Fock States
• The dynamical problem for a non self-adjoint Hamiltonian
• A short review on TBA equation and scattering amplitude/Wilson loop duality
• Exact WKB in all sectors II: Potentials with non-degenerate saddles
• Cryptohermitian Hamiltonians on graphs. II. Hermitizations
• PT-symmetry and Integrability