BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map Cox, Graham


Courant's nodal domain theorem provides a natural  generalization of Sturmâ Liouville theory to higher dimensions; however,  the result is in general not sharp. It was recently shown that the nodal  deficiency of an eigenfunction is encoded in the spectrum of the  Dirichlet-to-Neumann operators for the eigenfunction's positive and  negative nodal domains. While originally derived using symplectic  methods, this result can also be understood through the spectral flow  for a family of boundary conditions imposed on the nodal set. In this  talk I will describe this flow for a Schrödinger operator with separable  potential on a rectangular domain, and describe a mechanism by which low  energy eigenfunctions do or do not contribute to the nodal deficiency.  Operators on non-rectangular domains and quantum graphs will also be  discussed. This talk represents joint work with Gregory Berkolaiko (Texas A&M) and  Jeremy Marzuola (UNC Chapel Hill).

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