Perfectly matched layers for the stationary Schrödinger equation in a periodic structure

We construct a perfectly matched absorbing layer for stationary Schrödinger equation with analytic slowly decaying potential in a periodic structure. We prove the unique solvability of the problem with perfectly matched layer of finite length and show that solution to this problem approximates a solution to the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity.