The Duistermaat index and eigenvalue interlacing for self-adjoint extensions of a symmetric operator

Abstract

Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts each eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as Weyl interlacing, Cauchy interlacing, Dirichlet--Neumann bracketing and so on. We prove a sharp version of the interlacing inequalities for ``finite-dimensional perturbations in boundary conditions'', expressed as bounds on the spectral shift between two self-adjoint extensions of a fixed symmetric operator with finite and equal defect numbers. The bounds are given in terms of the Duistermaat index, a topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the self-adjoint extensions being compared, while the third corresponds to the Friedrichs extension, which acts as a reference point. Along the way several auxiliary results are established, including one-sided continuity properties of the Duistermaat triple index, smoothness of the (abstract) Cauchy data space without unique continuation-type assumptions, and a formula for the Morse index of an extension of a non-negative symmetric operator.