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A Morse index theorem for elliptic operators on bounded domains

Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $Ω\subset \mathbb{R}^d$ is deformed. For a family of domains $\{Ω_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $Ω_a$ differs from the Morse index of $L$ on $Ω_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $Ω$. This is particularly useful when $Ω_a$ 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 $Ω_b$) and the "simplified" problem (on $Ω_a$). 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.