| Abstract: |
| When comparing the spectra of two self-adjoint operators, it is often useful to compare their eigenvalue counting functions. A well-known result in this direction for Schrodinger operators on metric graphs is known as Dirichlet-Neumann bracketing, which essentially states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacians differ by at most the size of the boundary. This can be seen as a manifestation of how local boundary conditions influence global spectral quantities.
The goal of this talk is to introduce a useful tool for such comparison results, known as the spectral flow, which is a topological invariant associated with one-parameter families of self-adjoint operators. In this setting, the parameter can be interpreted as a continuous change of boundary conditions or coupling along the graph. We show that for Schrodinger operators on metric graphs, the spectral flow can be effectively computed using the associated scattering matrices. We then present several applications, in the form of generalized nodal index theorems and eigenvalue interlacing results, which quantify how spectral data evolves under these changes.
The talk is based on joint work with Ram Band and Marina Prokhorova. |
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