Manifold decompositions and indices of Schr\\\"{o}dinger operators

Abstract

Abstract. The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold M divided into components Ω 1 and Ω 2 by a separating hypersurface Σ. A homotopy argument relates the spectrum of a second-order elliptic operator on M to its Dirichlet and Neumann spectra on Ω 1 and Ω 2 , with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on Σ. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, a new proof of Courant's nodal domain theorem is given, with an explicit formula for the nodal deficiency.

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.…”

“…Equations (2) and (3) remain valid for the Schrödinger operator L = −∆ + V with sufficiently regular potential V (for instance, V ∈ L ∞ (Ω)). While originally obtained as special cases of a general spectral decomposition formula, derived using symplectic methods in [6], these formulas can be obtained more directly from a spectral flow argument. Consider the family {L σ } of selfadjoint operators defined by L σ = −∆ + V with Dirichlet boundary conditions on ∂Ω and ∂u ∂ν + + ∂u ∂ν − + σu = 0 on the nodal set Γ = {x ∈ Ω : φ * (x) = 0}, where ν ± denote the outward unit normals to Ω ± .…”

A symplectic perspective on constrained eigenvalue problems

“…Proof. We follow the proof of Lemma 4 in [5]. Letting P β denote the orthogonal projection onto the boundary subspace β ⊂ H, and P ⊥ β = I − P β the complementary projection, it suffices to prove an estimate of the form…”

The Maslov index and the spectra of second order elliptic operators

“…It is of great interest in stability theory for multidimensional patterns for reaction-diffusion equations, see [KP, DJ]. In recent years the relation between the Morse index (the number of unstable eigenvalues) and the Maslov index has attracted much attention, see [CJLS,CJM1,CJM2,DJ,HS,HLS,JLM,JLS,LSS,PW]. These results can be viewed as a far reaching generalization of the classical Sturm Theorems for ODE's and systems of ODE's, cf.…”

“…Summarizing, one can say that the connection between self-adjoint extensions and Lagrangian planes resulted in various formulas relating the spectral flow and the Maslov index of paths of Lagrangian planes formed by strong traces of solutions to elliptic PDE's, see [BZ1,BZ2,BZ3]. In contrast, the Lagrangian planes considered in [DJ,CJLS,CJM1,CJM2] are formed by the weak traces of weak solutions to second order elliptic PDE's. The main contribution of this paper is in tying together all three topics discussed above.…”