Manifold Decompositions and Indices of Schrödinger Operators

The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold M divided into components Ω1 and Ω2 by a separating hypersurface Σ. A homotopy argument relates the spectrum of a second-order elliptic operator on M to its Dirichlet and Neumann spectra on Ω1 and Ω2, with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on Σ. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant’s nodal domain theorem is given, with an explicit formula for the nodal deficiency.