Wave functions of the toda chain with boundary interaction

Abstract

In this contribution, we give an integral representation of the wave functions of the quantum N -particle Toda chain with boundary interaction. In the case of the Toda chain with one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of Γ-functions. The wave function of the Toda chain with two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits separation of variables. The separated difference equations coincide with the Baxter equation.

The Baxter–Bazhanov–Stroganov model: separation of variables and the Baxter equation

“…Acting by both sides of this identity on Ψ ρn , fixing λ = λ n,k (i.e. at the zeroes of the eigenvalue polynomial of B n (λ)) and using the inverse of ( 70) with (67), we see that, very similar to A n (λ n,k ), also D n (λ n,k ) acts as a shift operator on Ψ ρn , compare (27):…”

Wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model

“…The integral (3.2) absolutely converges. The proof is identical to that of [9,10]. For instance, we can use an elegant estimate [10, (30)] by N. Iorgov and V.Shadura, which states that for fixed γ n,i = λ i ∈ ıR, i = 1, .…”

Aspects of the inverse problem for the Toda chain

“…Further, again in virtue of the symmetry, in doing so, one may also assume the ordering λ 1 < · · · < λ N . For such an ordering of both sets of variables, one has the identity [15] N k=1…”