Abstract:  As Arnol’d pointed out, Sturm’s 19th century theorem regarding the oscillation of solutions to a second-order selfadjoint ODE has a topological nature: it describes the rotation of a straight line in the phase space of the equation. The topological ingredient here is the Maslov index, a homotopy invariant which counts the (signed) intersections of a path of Lagrangian planes with a codimension-one subset of the set of all Lagrangian planes. In this talk, I’ll begin with a discussion of Sturm’s theorem, the main idea being that one can glean spectral information from the geometric structure of an eigenfunction. I'll then show how these ideas translate to study the eigenvalues of a class of Hamiltonian differential operators that I studied in my PhD, which do not enjoy the selfadjointness property of the operators that Sturm studied. In particular, by viewing the problem symplectically, one can use the Maslov index to give a lower bound for the number of positive real eigenvalues in terms of the Morse indices of two related selfadjoint operators, as well as a mysterious correction term. The Hamiltonian operators in focus here arise, for example, when determining the (spectral) stability of standing waves in NLS type equations; if time permits, I’ll go through some applications to such problems on both bounded and unbounded domains.

 

Part of this talk was joint work with Graham Cox, Yuri Latushkin, and Robby Marangell.