A Morse Index Theorem for Elliptic Operators on Bounded Domains

We consider a second-order, selfadjoint elliptic operator L on a smooth one-parameter family of domains {Ωt}t∈[a,b] with no assumptions on the geometry of the Ωt’s. It is shown that the Morse index of L can be equated with the Maslov index of an appropriately defined path in a symplectic Hilbert space constructed on the boundary of Ωb. Our result is valid for a wide variety of boundary conditions, including (but not limited to) Dirichlet, Neumann and Robin. Specifically, the Maslov index of the path we define relates the Morse index of L on Ωb to the Morse index of L on Ωa. This is particularly useful when Ωa is a domain for which the spectrum is more readily understood, e.g. a region with very small volume. In other words, the Maslov index exactly computes the discrepancy between the Morse indices for the “original” problem (on Ωb) and the “simplified” problem (on Ωa). This generalizes previous results that were only available on star-shaped domains, or for Dirichlet boundary conditions. We then discuss how one can practically compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.