A single electron is described by the (non-negative, essentially self-adjoint) Hamiltonian
for some large (compared with atomic distances) cube 
. Periodic boundary conditions are choosen in order to avoid any influence of the surface. This restriction (effectively to the compact 3-torus) of the elliptic operator 
guarantees that one has only to deal with a discrete spectrum.
To retain the desired "freedom" of the electrons in the mathematical description of the model its necessary to choose the (non-negative, essentially self-adjoint) Hamiltonian
which has absolutely continous spectrum 
, therefore describing "really free" particles, but facing with technical difficulties in counting states etc.
