Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the 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, R) group.

Exact quantization conditions for the relativistic Toda lattice

“…However, the four-dimensional NS free energy provides a powerful treatment of the spectrum in terms of convergent series, and the machinery of resurgent trans-series is not really needed. 4 As we have tried to emphasize in this paper, the 5d story is different, due to the poles in the NS free energy. Starting from the all-orders WKB result, one can definitely follow the route of trans-series.…”

“…Alternatively, one can write these quantization conditions in terms of integral equations of the TBA type [8]. These TBA equations are obtained by resumming the instanton expansion of the partition function (see [18,19] for a detailed derivation), and as shown in [20] they are equivalent to the quantization conditions in [1][2][3][4].…”

“…where L n (µ) is exactly the Lax operator of the non-relativistic Toda lattice (see for example [4]). In order to formulate the spectral problem for the quantum relativistic Toda lattice, we eliminate the motion of the center of mass in favor of N − 1 coordinates ζ 1 , · · · , ζ N −1 .…”

“…In the case of the Toda lattice, these conditions were obtained in [1][2][3][4], by finding appropriate solutions of the one-dimensional quantum Baxter equation associated to the integrable system. The Toda lattice has a relativistic generalization [5], and the techniques of separation of variables lead to a Baxter equation involving difference operators [6,7].…”

Relativistic Toda Chain with Boundary Interaction at Root of Unity

“…This progress was initiated by the paper [1] of Sklyanin who has proposed (using the R-matrix formalism) a recipe for a Separation of Variables in the case of quantum Toda chain where the algebraic Bethe ansatz fails. The next important step was done by Kharchev and Lebedev [2] who realized an iterative procedure of obtaining the eigenfunctions of the n-particle open Toda chain by some integral transformation from the eigenfunctions of the (n − 1)-particle open Toda chain. This iterative method was applied later to a relativistic Toda chain [3].…”