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On spectral stability for self-adjoint extensions

Mario Alberto Ruiz Caballero

Abstract

We prove that given a symmetric completely non-selfadjoint operator $B$ with finite deficiency indices $(n,n)$ on a Hilbert space and a boundary triplet $\left(\mathbb{C}^{n},Γ_{1},Γ_{2}\right)$ for $B^{*}$, the set of points in the spectrum of $A_{1}$ (the self-adjoint extension with domain $Ker\;Γ_{1}$) which are not eigenvalues of maximum multiplicity for any self-adjoint extension of $B$ disjoint of $A_{1}$, is a dense $\textit{G}_δ$ set in $σ(A_{1})$. Furthermore, a proof of a Malamud's theorem that generalizes a well-known result of the Aronszajn-Donoghue theory on the characterization of eigenvalues is offered.

On spectral stability for self-adjoint extensions

Abstract

We prove that given a symmetric completely non-selfadjoint operator with finite deficiency indices on a Hilbert space and a boundary triplet for , the set of points in the spectrum of (the self-adjoint extension with domain ) which are not eigenvalues of maximum multiplicity for any self-adjoint extension of disjoint of , is a dense set in . Furthermore, a proof of a Malamud's theorem that generalizes a well-known result of the Aronszajn-Donoghue theory on the characterization of eigenvalues is offered.
Paper Structure (6 sections, 17 theorems, 69 equations)

This paper contains 6 sections, 17 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Boundary Triplets Theory
  3. Nevanlinna-Herglotz matrices and matrix-valued measures
  4. Kats theorem and multiplicity function
  5. Aronszajn-Donoghue theory
  6. Consequences of Malamud's Theorem

Key Result

Theorem 1.1

Let $B$ denote a symmetric completely non-selfadjoint operator on a Hilbert space $\mathcal{H}$ with finite deficiency indices $(n,n)$ and $\Pi=\left( \mathbb{C}^{n},\Gamma_{1},\Gamma_{2}\right)$ be a boundary triplet for $B^{*}$. Then is dense $G_{\delta}$ in $\sigma(A_{1})$.

Theorems & Definitions (38)

  • Theorem 1.1: Forbidden energies-like
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Proposition 14.7, Sch and Proposition 1, MALAMUD
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5
  • ...and 28 more