Since the succesfull dynamic lattice model unfortunately cannot solve the problems posed by the failures of the independent electron model we can no longer ignore the interaction between the elctrons; actually it was surprising anyway that a model which does not take into account the electromagnetic forces between the electrons could successfully describe many effects in real systems. This relies on screening effects in the Coulomb gas.
So we suppose an interaction potential between two electrons with numbers i and j at position and . Normally this should be the Coulomb potential
which is bounded below and , so gives a nice self-adjoint Hamiltonian
Furthermore since we did not take into account spins we have to fulfill the Pauli exclusion principle by considering our system not on the Hilbert space but restricting it to the fermionic Fock space which is just the subspace of anti-symmetric wave functions. In the case of infinitely many electrons (as it should be for the infinite crystal and for grand-canonical quantum statistics) this rises some interesting mathematical questions since one has to deal with a quantum field theory.