Finite size spectrum, magnon interactions and magnetization of S=1 Heisenberg spin chains

We report on density matrix renormalization-group and analytical work on S=1 antiferromagnetic Heisenberg spin chains. We study the finite size behavior within the framework of the non-linear sigma model. The behavior of the gap $\sqrt{\Delta^2+v^2\sin^2{\frac{\pi}{L+2}}}$ is verified for the whole range of chain length L, starting from length as small as correlation length. We study the effect of magnon-magnon interactions on the finite size spectrum and on the magnetization curve close to the critical magnetic field.