Open Collections

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

Banff International Research Station for Mathematical Innovation and Discovery

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).

Mathematics; Quantum theory; Global analysis, analysis on manifolds; Mathematical physics

video/mp4

Author affiliation: Memorial University of Newfoundland

2019-03-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68672

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/