Vibrating quantum billiards on Riemannian manifolds

Quantum billiards provide an excellent forum for the analysis of quantum ch aos. Toward this end, we consider quantum billiards with time-varying surfa ces, which provide an important example of quantum chaos that does not requ ire the semiclassical ((h) over bar --> 0) or high quantum-number limits. W e analyze vibrating quantum billiards using the framework of Riemannian geo metry. First, we derive a theorem detailing necessary conditions for the ex istence of chaos in vibrating quantum billiards on Riemannian manifolds. Nu merical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freed om boundary vibrations and briefly discuss a generalization to billiards wi th two or more degrees-of-vibrations. The requisite conditions are direct c onsequences of the separability of the Helmholtz equation in a given orthog onal coordinate frame, and they arise from orthogonality relations satisfie d by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boun daries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semi-quantum chaos. Moreover, vibrating quant um billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.