Hamiltonian spectral theory and the Maslov index

Abstract

This thesis explores the real spectral theory of Hamiltonian differential operators with the canonical symplectic structure using the Maslov index. Such operators arise, for example, in the stability analysis of standing waves in nonlinear Schrödinger (NLS) type equations. The Maslov index is a topological invariant which counts the signed crossings of a path in the Lagrangian Grassmannian with a codimension-one set. It has been widely used to study eigenvalue problems for differential operators, especially those that are selfadjoint. A straightforward application of the Maslov index in this context can be found in Sturm’s oscillation theorem relating the nodal count of an eigenfunction for a Sturm-Liouville operator to where in the sequence of eigenvalues the corresponding eigenvalue sits. Here, the differential operator N studied is not selfadjoint, but can be written in terms of two selfadjoint operators, L+ and L-. The analysis focuses on two cases. In Chapter 2, L+ and L- are Schrödinger operators on a compact interval with Dirichlet boundary conditions; in Chapter 3, they are fourth-order operators on the line. In both cases, the eigenvalue equations for N have a Hamiltonian structure, and induce flows on the Lagrangian Grassmannian. This affords the use of the Maslov index. Exploiting homotopy invariance yields a lower bound for the number of positive real eigenvalues, which includes a contribution to the Maslov index from a non-regular crossing. In chapter 2, the compactness of the domain facilitates the use of the eigenvalue curves, which represent the evolution of the eigenvalues as the domain is shrunk or expanded. Analysing their behaviour locally provides a geometric means of computing the Maslov index at the non-regular crossing. In chapter 3, non-regular crossings are handled via the partial signatures of higher-order crossing forms. Applications to the spectral stability of standing waves in NLS equations are given for both problems.