English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Eigenvalue interlacing is a useful tool in linear algebra and spectralanalysis. In its simplest form, the interlacing inequality states that arank-one positive perturbation shifts each eigenvalue up, but not further thanthe next unperturbed eigenvalue. For different types of perturbations, thisidea is known as Weyl interlacing, Cauchy interlacing, Dirichlet--Neumannbracketing and so on.  We prove a sharp version of the interlacing inequalities for``finite-dimensional perturbations in boundary conditions'', expressed asbounds on the spectral shift between two self-adjoint extensions of a fixedsymmetric operator with finite and equal defect numbers. The bounds are givenin terms of the Duistermaat index, a topological invariant describing therelative position of three Lagrangian planes in a symplectic space. Two of theLagrangian planes describe the self-adjoint extensions being compared, whilethe third corresponds to the Friedrichs extension, which acts as a referencepoint.  Along the way several auxiliary results are established, including one-sidedcontinuity properties of the Duistermaat triple index, smoothness of the(abstract) Cauchy data space without unique continuation-type assumptions, anda formula for the Morse index of an extension of a non-negative symmetricoperator.

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