Semiclassical asymptotics for open chaotic systems

The study of the quantum mechanical properties of open chaotic systems brings together the fields of fractal geometry and semiclassical asymptotics, giving rise to problems that are of fundamental mathematical importance and that are closely linked with scientific and technological developments concerning microelectronic, nanoelectronic and microlasing devices. Quantum mechanical properties at the semiclassical boundary with classical mechanics that were established decades ago for closed systems, from which particles cannot escape, are still far from being understood in open systems. For example, in closed systems there are formulae allowing one to count the number of quantum states (Weyl's law) and a detailed description of what the corresponding quantum wavefunctions look like. In open systems these remain outstanding unsolved problems. The long-term goal of our research programme is to develop a mathematically consistent semiclassical theory that will ultimately solve these problems.Scientifically and technologically this is a pressing issue. There are now a very considerable number of experiments starting to probe electronic and lasing devices on the mesoscopic and nanoscopic scale. Specifically, recent experimental developments have made it possible to construct clean mesoscopic and nanoscopic devices in which electronic transport is ballistic, i.e. not influenced by impurities. In these systems the corresponding classical dynamics has a controlling influence on the quantum properties and so semiclassical methods are essential. Another situation in which a semiclassical description is needed is in the theory of microlasers, which are currently the focus of considerable experimental and technological interest. Devices such as those just described are likely to form the basis for major new technologies. Yet we still lack many fundamental mathematical tools needed to probe them theoretically; tools that will be needed if they are to be computer-designed. One of the key reasons for this is that these systems are open, rather than closed. The central problem we face in developing a semiclassical theory of open quantum systems is that the classical motion in them, which must be used to form the skeleton of such a theory, is dominated by fractal structures. We need a fundamentally new mathematical approach. It is the goal of our research programme to understand how a semiclassical theory can be built on this fractal skeleton.