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Stability of the gapless pure point spectrum of self-adjoint operators

Paolo Facchi, Marilena Ligabò

Abstract

We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$ ensuring the global stability of the spectral nature of $T+V$.

Stability of the gapless pure point spectrum of self-adjoint operators

Abstract

We consider a self-adjoint operator on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of and on the bounded perturbation ensuring the global stability of the spectral nature of .
Paper Structure (19 sections, 11 theorems, 180 equations)

This paper contains 19 sections, 11 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Stability of the pure point spectrum of operators on $\ell^2(\mathbb{Z}^d)$
  3. Examples
  4. Road map of the proof of Theorem \ref{['thm:mainth2']}
  5. KAM iteration scheme
  6. First step
  7. Homological equation
  8. Iteration
  9. Strategy of the proof
  10. Notation and stability of small denominators
  11. Notation
  12. Stability of small denominators
  13. Proof of Theorem \ref{['thm:mainth2']}
  14. Part $(1)$. KAM scheme
  15. Base case, $\ell=0$
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $T(\varepsilon)=T+\varepsilon V$ be the family of operators defined on $D(T)$, $\forall\,\varepsilon\in\mathbb{R}$. Assume the validity of conditions (A.1)--(A.4). Then there exists $\varepsilon^*>0$ such that $\sigma(T(\varepsilon))$ is purely point and simple for all $\varepsilon\in\mathbb{R}

Theorems & Definitions (35)

  • Remark 1.1
  • Theorem 1.2: Stability of pure point spectrum
  • Remark 1.3
  • Definition 1.4
  • Theorem 2.1: Stability of pure point spectrum for operators on $\ell^2(\mathbb{Z}^d)$
  • Remark 2.2: Explicit bounds
  • Remark 2.3: Approximation functions
  • Corollary 2.4: Discrete Schrödinger operators
  • proof
  • Remark 2.5: Quasi-periodic Schrödinger operators
  • ...and 25 more