Self-adjoint elliptic operators and manifold decompositions Part I: Low eigenmodes and stretching

This paper is the first of a three-part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds. In this article, we will show how the low eigensolutions of a self-adjoint elliptic operator over such a manifold can be studied by a splicing construction. This construction yields an approximated solution of the operator whenever we have two L²-solutions on both sides and a common limiting value of two extended L² -solutions. In Part II, the present analytic "Mayer-Vietoris" results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of Maslov indices of Lagrangians. In Part III after comparing infinite-and finite-dimensional Lagrangians and determinant line bundles and then introducing "canonical perturbations" of Lagrangian subvarieties of symplectic varieties, we will study invariants of 3-manifolds, including Casson's invariant.