Asymptotic eigenfunctions of the “bouncing ball” type for the two-dimensional Schrödinger operator with a symmetric potential

We construct asymptotic eigenfunctions for the two-dimensional Schrödinger operator with a potential in the form of a well that is mirror-symmetric with respect to a line. These functions correspond to librations on this line between two focal points. According to the Maslov complex germ theory, the asymptotic eigenfunctions in the direction transverse to the line with respect to which the well is symmetric have the form of the appropriate Hermite–Gauss mode. We obtain a global Airy-function representation for the asymptotic eigenfunctions in the longitudinal direction.