Mathematical Sciences: Spectral and Variational Problems of Mathematical Physics

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Professors Harrell and Loss will conduct research on spectral and variational analysis in several areas originating in theoretical physics. Three of their projects involve (i) direct and inverse spectral questions about Schroedinger and Laplace- Beltrami operators, (ii) proofs of inequalities using the method of competing symmetries, and (iii) continuous spectra and their gaps. The research program is a balance between pure and applied mathematics with a twofold purpose. First, Harrell and Loss wish to create formulae and theorems of real use to physicists, especially those working in quantum theory and statistical physics. Secondly, the investigation may only be distantly related to physical calculation, but can cast light on structural questions, such as what role symmetries and geometry play in the spectral and inverse spectral theory of Schroedinger operators and their generalizations.