Isolated versus nonisolated periodic orbits in variants of the two-dimensional square and circular billiards

Square and circular infinite wells are among the simplest two-dimensional potentials which can completely solved in both classical and quantum mechanics. Using the methods of periodic orbit theory, we study several variants of these planar billiard systems which admit both singular isolated and continuous classes of non-isolated periodic orbits. (In this context, isolated orbits are defined as those which are not members of a continuous family of paths whose orbits are all of the same length.) Examples include (i) various "folded" versions of the standard infinite wells (i.e., potentials whose geometrical shapes or "footprints" can be obtained by repeated folding of the basic square and circular shapes) and (ii) a square well with an infinite-strength repulsive δ-function "core," which is a special case of a Sinai billiard. In each variant case considered, new isolated orbits are introduced and their connections to the changes in the quantum mechanical energy spectrum are explored. Finally, we also speculate about the connections between the periodic orbit structure of supersymmetric partner potentials, using the two-dimensional square well and it superpartner potential as a specific example.