Its solution can be expressed in terms of the resolvent Γ Γ of the integral kernel and the non-linear (repulsive) term ϕ ϕ ( x ) = − − β β ^ ^ ( x ) + λ λ ( 0 ) ∑ ∑ n ⟨ ⟨ − − β β ^ ^ ( x ) , ϕ ϕ n ⟩ ⟩ ϕ ϕ n λ λ n − − 1 + ( λ λ ( 0 ) ) 2 ∑ ∑ n ⟨ ⟨ − − β β ^ ^ ( x ) , ϕ ϕ n ⟩ ⟩ ϕ ϕ n λ λ n − − 1 ( λ λ n − − λ λ ( 0 ) ) − − 1 =