Once electron–hole densities exist, the generalized Wannier equation is not Hermitian anymore. As a result, the eigenvalue problem has both left- and right-handed eigenstates ϕ ϕ λ λ L ( k ) and ϕ ϕ λ λ R ( k ) , respectively. They are connected via the phase-space filling factor, i.e. ϕ ϕ λ λ L ( k ) = ϕ ϕ λ λ R ( k ) / ( 1 − − f k e − − f k h ) . The left- and right-handed eigenstates have the same eigen value E λ λ (that is real valued for the form shown) and they form a complete set of orthogonal solutions since