Exact manipulations to Bloch states of a particle in a double cosine potential

For a single particle held in a double cosine potential with suitable parameters it is demonstrated that the Schrödinger equation has two sets of exact and simple Bloch solutions which are associated with the different boundary conditions. The corresponding eigenenergies can be positive or negative that depend on the amplitude and wave-vector of the laser potential. The results supply possibility for exactly manipulating the quantum motional states of the particle by using the laser beams.