I have been given the Hamiltonian of system (particle in a ring),


H=∑0N[−V1(|gn⟩⟨gn+1|+|gn+1⟩⟨gn|)−V2(|gn⟩⟨gn+2|+|gn+2⟩⟨gn |)]It's expressed in the basis
⟨m|gn⟩=δmn

How can I re-express this in the basis, |hm⟩:hm(n)=e2πinm/N where, n=1,2,3... N-1 and show that it is diagonal?