The Maslov and Morse Indices for System Schrödinger Operators on R

Assuming a symmetric matrix-valued potential that approaches constant endstates with a sufficient asymptotic rate, we relate the Maslov and Morse indices for Schrödinger operators on R. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index. Our analysis is motivated, in part, by applications to stability of nonlinear waves, for which the Morse index of an associated linear operator typically determines stability. In a series of three examples, we illustrate the role of our result in such applications.