A Variational Approach to Spectral Shift and Spectral Minimal Partitions

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Eigenvalue optimization for a parametric family of self-adjoint operators is a mathematical abstraction of a wide variety of questions arising in applied science. For example, the potential energy surface in quantum chemistry is the landscape of the dependence of an eigenvalue on the atoms' positions within the molecule; in this landscape, electrons seek out the lowest point, forcing the molecule into the corresponding configuration. Study of pattern-formation in reaction-diffusion systems can similarly lead to the question of optimizing energy (eigenvalue) of a partition of the available space; this question is known as the ‘spectral minimal partition problem’. The common feature of these two examples is a very large number of parameters. The underlying idea of the present project is that when the number of parameters is sufficiently large, it is possible to obtain global information about the operator family from the local information about the eigenvalue behavior. A practically important consequence of this is the ability to certify an experimentally found local minimum as being globally optimal. Questions from this project will be used to mentor graduate and undergraduate researchers, and a WikiBook devoted to spectral analysis on metric graphs will be created and maintained.The specific mathematical question to be addressed in the project is the eigenvalue dependence on the boundary conditions, which is directly related to the location of the partition boundaries in the ‘spectral minimal partition problem’. On the operator-theoretic level, this will be expressed as variation of the self-adjoint extension of a fixed symmetric operator. The goal is to prove a link between the Morse index of the eigenvalue at a critical point and the spectral shift of the operator with respect to a reference operator. Since the Morse index describes local stability of the eigenvalue with respect to perturbations in the boundary conditions, this link will allow one to obtain global information in the form of the spectral shift. The knowledge of spectral shift will then lead to bounds on the energy landscape which, in turn, will certify the local minimum as being globally optimal. The tools used for establishing the link will include parametrization of self-adjoint extensions by the Lagrangian Grassmannian, the Krein-Naimark resolvent formula, Dirichlet-to-Neumann map and the Maslov index techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.