A symplectic perspective on constrained eigenvalue problems

TL;DR

This paper extends the Maslov index framework to constrained eigenvalue problems, providing a new way to compute the Morse index and establishing a constrained Morse index theorem with applications to nonlinear Schrödinger equations.

Contribution

It introduces a novel application of the Maslov index to constrained eigenvalue problems and proves a constrained Morse index theorem.

Findings

The Maslov index can be used to compute the Morse index of constrained operators.

A constrained Morse index theorem is established, linking Morse index to conjugate points.

Application to nonlinear Schrödinger equation demonstrates practical relevance.

Abstract

The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, which designates the boundary condition, with the set of Cauchy data for the differential operator. We apply this methodology to constrained eigenvalue problems, in which the operator is restricted to a (not necessarily invariant) subspace. The Maslov index is defined and used to compute the Morse index of the constrained operator. We then prove a constrained Morse index theorem, which says that the Morse index of the constrained problem equals the number of constrained conjugate points, counted with multiplicity, and give an application to the nonlinear Schr\"odinger equation.