Exact Solution of a Nonlinear Eigenvalue Problem in One Dimension

An exact solution of the eigenvalue problem $\{-\frac{d^2}{d{x}^2}+\Delta [1-a_0\rho \mathrm{}\left(x\right)\left]\right\}{\psi }_n\mathrm{}\left(x\right)=E_n{\psi }_n\mathrm{}\left(x\right)\mathrm{}$ with $\rho \mathrm{}\left(x\right)={\Sigma }_{n=1\mathrm{}}^N{\left|{\psi }_n\mathrm{}\right(x\left)\right|}^2$ and with periodic boundary condition is presented. The solution gives rise to a density wave with [const-$\rho \mathrm{}\left(x\right)\mathrm{}$] proportional to $\mathrm{s}{\mathrm{n}}^2\left[\right(\frac{x}{\lambda }\left)\right|m]\mathrm{}$ for suitable values of the parameters $\lambda$ and $m$. The solution rests upon some remarkable properties of the solutions of Lamé's equation.