A Morse Index Theorem for Elliptic Operators on Bounded Domains

Abstract

Abstract. Given a selfadjoint, elliptic operator L, one would like to know how the spectrum changes as the spatial domain Ω ⊂ R d is deformed. For a family of domains {Ωt} t∈ [a,b] we prove that the Morse index of L on Ωa 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 Ωa 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 Ωa). 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.

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

“…In [3] the first author, Kuchment and Smilansky gave an explicit formula for the nodal deficiency as the Morse index of an energy functional defined on the space of equipartitions of Ω. More recently [6], the second two authors, with Jones, computed the nodal deficiency in terms of the spectra of Dirichlet-to-Neumann operators using Maslov index tools developed in [7,5]. In particular, for a simple eigenvalue λ k , it was shown that δ(φ k ) = Mor (Λ + ( ) + Λ − ( )) (2) for sufficiently small > 0, where Λ ± ( ) denote the Dirichlet-to-Neumann maps for the perturbed operator ∆ + (λ k + ), evaluated on the positive and negative nodal domains Ω ± = {±φ k > 0}, and Mor denotes the Morse index, or number of negative eigenvalues.…”

A symplectic perspective on constrained eigenvalue problems

“…Smale's result, which only applies to the Dirichlet problem, was originally proved by variational methods (see also [25]). A proof using the Maslov index was given in [9] for star-shaped domains, and in [7] for the general case.…”

“…In this paper we give a symplectic formulation of this problem, and use it to prove a constrained version of the celebrated Morse-Smale index theorem. We begin by reviewing the symplectic formulation of the unconstrained spectral problem, which first appeared in [9], and was elaborated on in [6,7]. Hypothesis 1.…”

“…Let L denote the selfadjoint operator corresponding to the bilinear form D in (7), with form domain X = H 1 0 (Ω) or X = H 2 (Ω) (for the Dirichlet and Neumann problems, respectively). The subspaces µ(λ) and β comprise a Fredholm pair for each value of λ, so the Maslov index of µ with respect to β is well defined, and satisfies Mas(µ(·); β) = −n(L), (10) where n(L) denotes the number of strictly negative eigenvalues (i.e.…”

“…In Section 3.4 we show that these conditions are always satisfied when L 2 c (Ω) ⊥ is a finite-dimensional subspace of H 1 (Ω). Now consider the bilinear form (7) restricted to X ∩ L 2 c (Ω), where X is the form domain of the unconstrained operator L. This defines a selfadjoint operator L c , with dense domain D(L c ) ⊂ L 2 c (Ω). This is the constrained operator whose spectrum we want to compute.…”