Simple problem solvable with Bethe ansatz I want some exercise for my students. Is there any simple but still non-trivial problem which can be solved with Bethe ansatz? The Heisenberg model is still too heavy. 
 A: Here is one example. The book Quantum Inverse Scattering Method and Correlation Functions by Korepin, Bogoliubov and Izergin introduces the coordinate Bethe ansatz first for the 1d Bose gas (Chapter I.1), with wave function governed by the non-linear Schrödinger equation
 $$i \ \partial_t \Psi = -\partial_x^2 \Psi + 2\ c\ |\Psi|^2 \ \Psi \ ,$$
which is very closely related to the Lieb--Liniger model, with delta-function potential, and is also related to experiments. The two-body S-matrix is just rational in the rapidities, the Bethe have only real solutions, etc. The solutions and resulting thermodynamics was studied in detail by Yang and Yang.
A: How about a two-particle problem? See the model in this paper Tamm-Hubbard surface states in the continuum. The model is very simple but not so trivial. It is also a beautiful model as it can be solved exactly by Bethe ansatz---Bethe ansatz in the baby form. I think it can make a very good exercise in solid state physics. 
It is a variant of the model in this paper, which is also very simple, but is only semi-solvable in the sense that only half of the states conform to the Bethe ansatz. 
So you get at least two good exercises. 
