Bispectrality for the quantum open Toda chain (Q2855986)

The classical open \(n\)-particle Toda chain is an integrable system determined by the Hamiltonian NEWLINE\[NEWLINEH =\sum_{j=1}^n\frac12 X_j^2+\sum_{j=1}^{n-1} e^{x_{j+1}-x_j}NEWLINE\]NEWLINE and the canonical variables \(\{X_j,X_k\}=\{x_j,x_k\}=0\) with \(\{X_j,x_k\}=\delta_{jk}\) for \(j,k=1,\dots,n\). A closed chain is described by the periodicity requirement \(x_{n+1}=x_1\). There are obvious quantum analogs. In the papers [\textit{O. Babelon}, Lett. Math. Phys. 65, No. 3, 229--240 (2003; Zbl 1056.39028)] and [\textit{O. Babelon}, J. Phys. A, Math. Gen. 37, No. 2, 303--316 (2004; Zbl 1050.37038)], Babelon used Lax matrices, R-matrix theory and hyperelliptic spectral curves to construct dual canonical variables for the classical and quantum systems and to solve the inverse problem of expressing the original canonical variables in terms of the dual variables.NEWLINENEWLINEIn this paper, the author shows that Babelon's construction can be simplified substantially for open chains. The \(2\times2\) Lax matrix for the classical system is given by NEWLINE\[NEWLINEL(u)=\ell_n(u)\cdots\ell_2(u)\ell_1(u)=\left(\begin{matrix} A(u)&B(u)\\ C(u)&D(u)\end{matrix}\right), \text{ where } \ell_j=\left( \begin{matrix} u+X_j&-e^{x_j}\\e^{-x_j}&0\end{matrix}\right).NEWLINE\]NEWLINE A generating function for the \(n\) constants of motion is provided by the hyperelliptic spectral curve \(\det(v-L(u))=0\). The author points out that for the open system it is sufficient to work with the vector \((A(u),C(u))^\top\), where now the generating function for the constants of motion is \(A(u)=u^n+H_1u^{n-1}+\dots+ H_n\). In essence, one can consider this as the limit \(\epsilon\to 0\) of the quasiperiodic chain \(\det(v-L(u))=v^2-(A(u)+\epsilon D(u))v+\epsilon=0\) when the hyperelliptic spectral curve degenerates into a rational curve \(v=A(u)\). By working directly with the rational curve, the author can avoid the more complicated hyperelliptic constructions of Babelon. Also, the quantum analog follows quite directly from the classical construction once the rational curve is utilized.