The Bethe Ansatz

Until now, we have considered our system as being defined on the whole real line. The disadvantage of this is that considering a finite particle number $N$ then only allows to access the zero-density case. As in many standard problems of quantum mechanics, in order to classify states, it is more reasonable to introduce a finite quantization length. We will thus put our system on an interval of length $L$ and impose for now simple periodic boundary conditions on the wavefunctions.