A Morse index theorem for elliptic operators on bounded domains

TL;DR

This paper establishes a Morse index theorem linking the change in spectral properties of elliptic operators on deforming domains to the Maslov index, extending previous results to more general domains and boundary conditions.

Contribution

It introduces a novel method to compute the spectral index change via the Maslov index for general domain deformations, broadening the scope of Morse index theorems.

Findings

Morse index difference equals Maslov index of boundary Lagrangian path

Applicable to general domain deformations beyond star-shaped regions

Provides computational approach using crossing forms

Abstract

Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $\Omega \subset \mathbb{R}^d$ is deformed. For a family of domains $\{\Omega_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $\Omega_a$ differs from the Morse index of $L$ on $\Omega_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $\Omega$. This is particularly useful when $\Omega_a$ is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on $\Omega_b$) and the "simplified" problem (on $\Omega_a$). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.