Manifold decompositions and indices of Schr\"odinger operators

TL;DR

This paper introduces a spectral decomposition formula for Schr"odinger operators on manifolds using the Maslov index, linking spectra to boundary conditions and providing new insights into nodal domain counts.

Contribution

It develops a novel approach to relate the spectra of Schr"odinger operators on divided manifolds to boundary data via the Maslov index, offering new proofs and formulas for nodal domain theorems.

Findings

Spectral decomposition formula involving Maslov index

Explicit relation between spectrum and boundary data

New proof and formula for Courant's nodal domain theorem

Abstract

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.