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On creating new essential spectrum by self-adjoint extension of gapped operators

Alessandro Michelangeli

Abstract

Given a densely defined and gapped symmetric operator with infinite deficiency index, it is shown how self-adjoint extensions admitting arbitrarily prescribed portions of the gap as essential spectrum are identified and constructed within a general extension scheme. The emergence of new spectrum in the gap by self-adjoint extension is a problem with a long history and recent deep understanding, and yet it remains topical in several recent applications. Whereas it is already an established fact that, in case of infinite deficiency index, any kind of spectrum inside the gap can be generated by a suitable self-adjoint extension, the present discussion has the virtue of showing the clean and simple operator-theoretic mechanism of emergence of such extensions.

On creating new essential spectrum by self-adjoint extension of gapped operators

Abstract

Given a densely defined and gapped symmetric operator with infinite deficiency index, it is shown how self-adjoint extensions admitting arbitrarily prescribed portions of the gap as essential spectrum are identified and constructed within a general extension scheme. The emergence of new spectrum in the gap by self-adjoint extension is a problem with a long history and recent deep understanding, and yet it remains topical in several recent applications. Whereas it is already an established fact that, in case of infinite deficiency index, any kind of spectrum inside the gap can be generated by a suitable self-adjoint extension, the present discussion has the virtue of showing the clean and simple operator-theoretic mechanism of emergence of such extensions.
Paper Structure (3 sections, 9 theorems, 73 equations)

This paper contains 3 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction and background
  2. Preparation: unital deficiency index
  3. General case of gapped operators with infinite deficiency index

Key Result

Theorem 2.1

Let $S$ be a densely defined symmetric operator in a complex Hilbert space $\mathcal{H}$, which admits a self-adjoint extension $S_\mathrm{D}$ that has everywhere defined bounded inverse on $\mathcal{H}$ -- equivalently, assume that $S$ is a densely defined gapped symmetric operator with zero in the

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 12 more