On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator

Let $V(x)$, $x=(s_1,x_2,x_3)$, be a potential periodic in $x_1,x_2$ and exponentially decreasing as $|x_3|\to\infty$, and let $V_N(x)$ be the sum of shifts $V\bigl(x-(0,0,Nn_3)\bigr)$ over all integer $n_3$. We prove that the spectrum and eigenfunctions (not necessarily in the class $L^2$) of the Schrödinger operator with potential $V_N$, considered in a box, approximate the spectrum and eigenfunctions of the operator with potential $V$ and, for the negative part of the spectrum, the approximation converges exponentially in $N\to\infty$.