Manifold decompositions and indices of Schrödinger operators

Graham Cox, Christoper K. R. T. Jones, Jeremy L. Marzuola

Published 2015-06-24Version 1

We use the Maslov index to relate 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 two connected components, $\Omega_1$ and $\Omega_2$, by a separating hypersurface $\Sigma$. We use a homotopy argument to relate 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. We then use a homotopy argument to relate the Maslov index to 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, we give a new proof of Courant's nodal domain theorem, with an explicit formula for the nodal deficiency.