 |
Abstract: The `drum problem'---finding the modes (eigenfunctions) of the Laplacian in a cavity---is a classical problem of mathematics and physics with a 150-year history, and a wealth of applications. Modern applications can involve complex geometries, and high frequencies, where the multiscale nature of the problem makes it computationally challenging. I will present a variant of the Method of Particular Solutions which overcomes a normalization problem inherent in the original method. Rigorous perturbation analysis then leads to tighter new inclusion bounds on cavity eigenvalues. Another variant, the `scaling method', finds clusters of modes simultaneously, thus can be orders of magnitude faster than the MPS or any other known methods for this problem. I will present applications to i) high-frequency asymptotics of chaotic modes (involving `quantum chaos' and 'scars'), and ii) modeling of dielectric micro-cavity lasers.
This talk will be accessible to graduate students.
|