The project aims to investigate the analytic and spectral properties of some effective models commonly used in condensed matter physics, and their derivation from more fundamental models, in an appropriate scale limit.
We plan to investigate the derivation of tight-binding models in solids state physics via well-localized Wannier bases, and the topological obstruction to the existence of the latter; the spectral properties of Schrödinger and Dirac operators with non-Hermitian potentials; the self-adjointness and the spectral properties of zero-range N-body Hamiltonian operators; the topological properties of the minimizers of the Landau-de Gennes functional governing the phase transitions of liquid crystals.
We expect that the solution to these problems will require the interplay of different mathematical techniques, so the team involves experts in Calculus of Variations, Real Analysis, Spectral Theory and Mathematical Physics.
The project is highly innovative in view of the novel intertwining of different mathematical techniques, from Real Analysis, Functional Analysis, Spectral Theory, Calculus of Variations and Mathematical Physics. One goal of the project is to merge mathematical methods and ideas from all these disciplines.
The team includes some internationally known experts in the fields of expertise directly related to the goals. Therefore, we expect to be successful concerning most of the research goals outlined in the previous session. More specifically, we expect to reach goals (A2), (A3), (B1), (B2), (C1). Viceversa, goals (A1), (C2-4) and (D) are very challenging, so they should be regarded as high risk/high value tasks.