20.1. Non-relativistic gas Consider a one-dimensional gas of N non-relativistic bosons on an interval of length L, with two-body repulsive interaction given by a delta-function. The Hamiltonian of such a system is H = −

N



i=1

∂2

∂x2i

+2c

i>j

δ(xi −xj) c > 0.

a. Find the phase shift of the two-body scattering process and write the Bethe ansatz equations.

b. Analyse the solutions in the thermodynamical limit N→∞, L→∞, N/L = ρ, ρ

finite.