The Duistermaat index and eigenvalue interlacing for self-adjoint extensions of a symmetric operator
arXiv (Cornell University)(2023)
摘要
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.
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