Abstract: We develop a general approach to study three-dimensional Schr¨odinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut locus in the Euclidean space. If the surface is asymptotically planar, we give an estimate on the location of the essential spectrum of the Schr¨odinger operator. Moreover, if the surface coincides up to a compact subset with a surface of revolution with strictly positive Gauss curvature, it is shown that the Schr¨odinger operator possesses an infinite number of discrete eigenvalues. This is joint work with David Krejèiøík.