The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

Abstract

We study the Schrödinger operator L = −∆ + V on a star-shaped domain Ω in R d with Lipschitz boundary ∂Ω. The operator is equipped with quite general Dirichlet-or Robin-type boundary conditions induced by operators between H 1/2 (∂Ω) and H −1/2 (∂Ω), and the potential takes values in the set of symmetric N × N matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian Grassmannian of the subspace of H 1/2 (∂Ω) × H −1/2 (∂Ω) corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of L (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative eigenvalues of a bilinear form related to the boundary operator.

An index theorem for Schrödinger operators on metric graphs

“…L t0 ∩ K λt 0 = {0}. Since the map t → L t is contained in C 1 [α, β], Λ( d L 2 (∂Γ)) , by [CJLS,Lemma 3.8] there exists a small neighbourhood Σ t0 ⊂ (α, β) of t 0 and a family of operators R t so that the map (3.25) and…”

The Maslov and Morse indices for Schrödinger operators on [0,1]

“…We isolate the main technical steps of our perturbation analysis in the following lemma, for which the statement and proof have been adapted with only minor changes from [15]. Lemma 3.21.…”

“…The Maslov index has its origins in the work of V. P. Maslov [41] and subsequent development by V. I. Arnol'd [2]. It has now been studied extensively, both as a fundamental geometric quantity [6,17,22,44,46] and as a tool for counting the number of eigenvalues on specified intervals [7,9,12,13,14,15,19,21,30,31,33]. In this latter context, there has been a strong resurgence of interest following the analysis by Deng and Jones (i.e., [19]) for multidimensional domains.…”

“…Our aim in the current analysis is to rigorously develop a relationship between the Maslov index and the Morse index in the relatively simple setting of (1.1), and to take advantage of this setting to compute the Maslov index directly for example cases so that these properties can be illustrated and illuminated. Our approach is adapted from [15,19], As a starting point, we define what we will mean by a Lagrangian subspace.…”

A symplectic perspective on constrained eigenvalue problems

“…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.…”