By means of a q-boson-q-Toda correspondence pointed out by Duval and Pasquier, the n-particle hamiltonian of the periodic quantum relativistic Toda chain on Z(m+1) is mapped to the hamiltonian of a previously studied lattice discretization of the Lieb-Liniger model (which encodes the dynamics of m + 1 q-bosons on Z(n+1) in the center-of-mass frame). The map in question makes it possible to retrieve quantum integrals and an orthogonal eigenbasis of Bethe Ansatz wave functions given by Hall-Littlewood polynomials for the pertinent periodic q-difference Toda chain from the corresponding quantum integrals and eigenbasis for the lattice Lieb-Liniger model. This approach entails the spectrum of the periodic q-difference Toda chain in terms of the critical points of associated Yang-Yang type Morse functions and links the diagonalization via the algebraic Bethe Ansatz performed by Duval and Pasquier directly to the spectral analysis of the lattice Lieb-Liniger model.
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