Bound states in mildly curved layers

It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states, provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e., the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schrodinger operators in a planar hard-wall layer.