Manifold decompositions and indices of Schr\"odinger operators

The Maslov index is used to compute the spectra of different boundary value problems for Schr\"{o}dinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into components $\Omega_1$ and $\Omega_2$ by a separating hypersurface $\Sigma$. A homotopy argument relates the spectrum of a second-order elliptic operator on $M$ to its Dirichlet and Neumann spectra on $\Omega_1$ and $\Omega_2$, 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 $\Sigma$. 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.