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

16.06.2023, 13:15  –  Haus 9, Raum 2.22 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Gregory Berkolaiko

Abstract:
Eigenvalue interlacing is a tremendously useful tool in linear algebra
and spectral analysis.  In its simplest form, the interlacing
inequality states that a rank-one positive perturbation shifts the
eigenvalue up, but not further than the next unperturbed eigenvalue.
For different types of perturbations, this idea is known as the "Weyl
interlacing" (additive perturbations), "Cauchy interlacing" (for
principal submatrices of Hermitian matrices), "Dirichlet-Neumann
bracketing" and so on.

We discuss the extension of this idea to general "perturbations in
boundary conditions", encoded as interlacing between eigenvalues of
two self-adjoint extensions of a fixed symmetric operator with finite
(and equal) defect numbers.  In this context, even the terms such as
"positive perturbation" or "rank of the perturbation" are not
immediately clear.  It turns out that concise answers can be obtained
in terms of the Duistermaat index, an integer-valued topological
invariant describing the relative position of three Lagrangian planes
in a symplectic space.  Two of the Lagrangian planes describe the two
self-adjoint extensions being compared, while the third one
corresponds to the distinguished Friedrichs extension.

We will illustrate our general results with simple examples, avoiding
technicalities as much as possible and giving intuitive explanations
of the Duistermaat index, the rank and signature of the perturbation
in the self-adjoint extension, and the curious role of the third
extension (Friedrichs) appearing in the answers.

Based on a work in progress with Graham Cox, Yuri Latushkin and Selim Sukhtaiev.