Mathematical Physics and Harmonic Analysis Seminar

Date: February 23, 2017

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Graham Cox, Memorial University, Newfoundland

Title: Manifold decompositions and indices of Schrödinger operators

Abstract: When finding the eigenvalues of a differential operator, it is often convenient to partition the spatial domain and then compute the spectrum on each component. This is useful when the operator has localized structure, such as a compactly supported defect superimposed over a known background. In this case the partitioning effectively decouples the defect and the background. One must then determine how the spectrum on the original domain is related to the spectra on the subdomains.

I will describe such spectral decompositions using the Maslov index, a generalized winding number for paths of Lagrangian subspaces. Using a homotopy argument, I will show that the Morse index of the original boundary value problem is given by the sum of the Morse indices on each subdomain plus a “coupling term” that depends on the Dirichlet-to-Neumann maps for the common boundary. An immediate corollary is a new proof of Courant's nodal domain theorem, with an explicit formula for the nodal deficiency. I will also discuss applications to periodic boundary conditions. This is joint with Christopher Jones and Jeremy Marzuola at UNC Chapel Hill.