Counting spectrum via the Maslov index for one dimensional \(\theta\)-periodic Schrödinger operators (Q2832835)

Let \([a, b]\) be a finite interval and \(V\in L^{\infty }([a, b], \mathbb{R}^{n\times n})\) be symmetric a.e. For a parameter \(\theta \) the authors consider the Schrödinger operator \(H_{\theta }:=(-\partial _x^2)_{\theta }+V\) and its eigenvalues problem reformulated in terms of existence of \textit{conjugate points}. Denoting by \(N(r, \theta )\) the number of eigenvalues of \(H_{\theta }\) located below a fixed \(r\in \mathbb{R}\), the main result of this work states that \(N(r, \theta _2)-N(r, \theta _1)=\mathrm{Mas} (\gamma |_{\lambda =r})\) which is the Maslov index, i.e., the total number of conjugated points (counting their signs) for the part of the loop where \(\lambda =r\). Here \(\lambda \) is a spectral parameter while \(\gamma \) is a loop in the set of Lagrangian planes of \(\mathbb{R}^{16n}\).