TABLE OF CONTENTS In this chapter we shall present the main ideas of the co-ordinate Bethe ansatz using as the basic model the simplest case of interacting spin- one- dimensional systems.
Now our goal is to find the eigenfunctions and eigenvalues of the Hamiltonian 
for the general case J z ? 0, J ? 0. This, by now well-known method, is due to H. Bethe who proposed it first for the Heisenberg spin chain, is called the Bethe's ansatz.
The total spin, as well as the z-projection of the total spin ? L j=1 S z j, commute with the Hamiltonian 
. This is why, we classify all states of the Hamiltonian by eigenvalues of the operator ? L j=1 S z j. It is convenient to choose the basis functions in the form
(here ? denotes the tensor product), such that the values x j in it determine M coordinates of sites with spins down (all other spins are directed up). [Naturally, we could choose the opposite basis with M spins up and L ? M spins down, and the results would be the same.] We suppose that 1 ? x 1 < x 2 < < x M ? L. Then the wave function can be written as
where a( x 1, , x M ) is the wave function in the co-ordinate representation. For...
