Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism

Abstract

The integral representations for the eigenfunctions of $N$ particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open $N$-particle solutions have essentially the same structure being written as a generalized Fourier transform over the eigenfunctions of the $N-1$ particle open Toda chain with the kernels satisfying to the Baxter equations of the second and first order respectively. In the latter case this leads to recurrent relations which result to representation of the Mellin-Barnes type for solutions of an open chain. As byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of $GL(N,\RR)$ group.

Latex+amssymb.sty, 14 pages

References in corpus (1)

• Integral representation for the eigenfunctions of quantum periodic Toda chain

Cited by in corpus (40)

• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
• Directed polymers and the quantum Toda lattice
• Tropical Combinatorics and Whittaker functions
• Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
• Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain
• Separation of variables for the quantum SL(2,R) spin chain
• From Quantum Groups to Liouville and Dilaton Quantum Gravity
• TBA for the Toda chain
• Point-to-line polymers and orthogonal Whittaker functions
• Baxter operator and Archimedean Hecke algebra
• Splitting of surface defect partition functions and integrable systems
• Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C) magnet
• Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory
• Noncompact SL(2,R) spin chain
• Quantum Spin Systems and Supersymmetric Gauge Theories, I
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Unitarity of the SoV transform for the Toda chain
• Whittaker functions and related stochastic processes
• Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit
• O'Connell's process as a vicious Brownian motion
• Survival probability of mutually killing Brownian motions and the O'Connell process
• Bispectrality for the quantum open Toda chain
• Transition function for the Toda chain model
• Aspects of the inverse problem for the Toda chain
• On a Class of Representations of Quantum Groups
• On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models
• Generalized Toda mechanics associated with classical Lie algebras and their reductions
• On Baxter Q-operators And Their Arithmetic Implications
• Random polymers via orthogonal Whittaker and symplectic Schur functions
• Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains
• Matrix Whittaker processes
• System of Complex Brownian Motions Associated with the O'Connell Process
• On a Gauss-Givental Representation of Quantum Toda Chain Wave Function
• Bispectral dual difference equations for the quantum Toda chain with boundary perturbations
• Unitarity of the SoV Transform for $\mathrm{SL}(2,\mathbb{C})$ Spin Chains
• Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions
• BC-type open $SL(2,\mathbb{C})$ spin chain
• Determinantal Martingales and Interacting Particle Systems
• The Gelfand-Tsetlin basis for infinite-dimensional representations of $g{l}_n(\mathbb{C})$