The proposal is to investigate the vacuum Casimir energy from a quantum field theory point of view and its connections to spectral theory and asymptotic properties of differential operators. Mathematicians and physicists are involved as PIs. They propose to work on questions about the consistency between local and global renormalized vacuum energies and the sign of the vacuum energy. The connection between the vacuum energy and periodic classical orbits via the semiclassical Maslov indices will be analyzed in particular by looking at the eigenvalue energy distributions. The PIs propose also to calculate the vacuum energies and spectral densities near boundary edges and corners: the spectral effects of such singular boundaries are also of mathematical interest in connection with generalized index theorems, which may include other explicit physical applications. The broader impact of the project stems partly from its interdisciplinary nature. Vacuum energy is relevant both to new nanotechnology devices and to cosmological issues ("dark energy"). Mathematically, the connection between periodic-orbit theory and vacuum energy and the implications of vacuum energy for spectral theory have barely been explored. Undergraduate and graduate student research assistants from both physics and mathematics will be recruited.
