You can certainly shift your phase-space origin into the potential well (by a Galilean boost and translation). Then you know that the energy eigenstates will share the symmetry of the Hamiltonian, so they will be periodic on the tiles.

Now make an approximation of small deviations $p$ and $x$ away from the new origin, and you find that it is approximately the harmonic oscillator. Just take the harmonic oscillator wave-functions, make them periodic and normalized on the tile, and these are your approximate bound states. For the unbound states things will be a bit tricky, but I assume you could use a WKB quantization of the classical solutions to get a basic idea (again, the discrete symmetry on the tiles will help a lot).