In addition to the the theory of the Sommerfeld model we just have to add the lattice-periodic potential to the free Hamiltonian. By a lattice 
we mean a discrete abelian subgroup with three linearly independent generators. The potential 
should commute with the action of the lattice.
On a finite crystal we're therefore left with the (bounded below, essentially self-adjoint) Hamiltonian
for some large (compared with atomic distances) domain C which is compatible with the lattice action, i. e. which is divisible by some fundamental domain. This restriction (effectively to the compact 3-torus) of the elliptic operator 
guarantees that again one has only to deal with a discrete spectrum.
On the infinite crystal we have to consider the (bounded below, essentially self-adjoint) Hamiltonian
which is a periodic Schrödinger operator on 
.
