Exact solution of a nonlinear eigenvalue problem in one dimension

Abstract

An exact solution of the eigenvalue problem {-d²/dx²+Δ[1-a₀ρ(x)]}ψ_n(x)=E_n ψ_n(x) with ρ(x)=∑_n=1^N|ψ_n(x)|² and with periodic boundary condition is presented. The solution gives rise to a density wave with [const-ρ(x)] proportional to sn²[(x/λ)|m] for suitable values of the parameters λ and m. The solution rests upon some remarkable properties of the solutions of Lamé's equation.