The Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$
  Assuming a symmetric potential that approaches constant endstates with a sufficient asymptotic rate, we relate the Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index. 
Peter Howard, Yuri Latushkin, and Alim Sukhtayev
math.DS