Vladimir Inozemtsev

I describe a finite-dimensional manifold which contains all meromorphic solutions to the many-particle elliptic Calogero – Moser problem at some fixed values of the coupling constant. These solutions can be selected by purely algebraic calculations as it was shown in the simplest case of three interacting particles.

We consider the problem of finding integrals of motion for quantum elliptic Calogero–Moser systems with arbitrary number of particles extended by introducing spinexchange interaction. By direct calculation, after making certain ansatz, we found first two integrals — quite probably, lowest nontrivial members of the whole commutative ring. This result might be considered as the first step in constructing this ring of the operators which commute with the Hamiltonian of the model.

The relation between the characteristics of the equilibrium configurations of the classical Calogero–Moser integrable systems and properties of the ground state of their quantum analogs is found. It is shown that under the condition of factorization of the wave function of these systems the coordinates of classical particles at equilibrium are zeroes of the polynomial solutions of the second-order linear differential equation. It turns out that, under these conditions, the dependence of classical and quantum minimal energies on the parameters of the interaction potential is the same.

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.

The eigenvectors of the Hamiltonian $\mathscr{H}_N$ of $N$-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero–Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.