Infinite potential well with a sinusoidal bottom
  We construct a tridiagonal matrix representation of the wave operator that maps the wave equation into a three-term recursion relation for the expansion coefficients of the wavefunction. Finding a solution of the recursion relation is equivalent to solving the original problem. Consequently, a larger class of solvable potentials is obtained. The usual diagonal representation constraint results in a reduction to the conventional class of solvable potentials. To exhibit the power of this approach, we give an exact solution for the infinite potential well with sinusoidal bottom. 
A. D. Alhaidari, H. Bahlouli