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Equivalence of classical and quantum completeness for real principal type operators on the circle

Kouichi Taira

Abstract

In this article, we prove that the completeness of the Hamilton flow and essential self-adjointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators. The proof is based on the construction of eigenfunctions with non-real eigenvalues which is well-known in scattering theory. Moreover, the relationship between scattering theory and the essential self-adjointness is explained.

Equivalence of classical and quantum completeness for real principal type operators on the circle

Abstract

In this article, we prove that the completeness of the Hamilton flow and essential self-adjointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators. The proof is based on the construction of eigenfunctions with non-real eigenvalues which is well-known in scattering theory. Moreover, the relationship between scattering theory and the essential self-adjointness is explained.

Paper Structure

This paper contains 35 sections, 29 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Motivation and history
  3. Results on classical and quantum completeness
  4. Results on the spectral properties
  5. Relation to scattering theory
  6. Idea of the proof of Theorem \ref{['mainess']}
  7. Possible generalization to higher dimensional cases
  8. Notation
  9. Pseudodifferential operators
  10. Pseudodifferential operators
  11. Quantization
  12. Fundamental properties of pseudodifferential operators
  13. Wavefront set
  14. Lagrangian distributions
  15. Some criterions of the essential self-adjointness
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $m\geq 0$ and $P\in \Psi^m_{\mathrm{phg}}$, which is symmetric on $C^{\infty}(\mathbb{T})$ with respect to the inner product of $L^2=L^2(\mathbb{T},dx)$. Suppose that the principal symbol $p:=\sigma(P)$ is homogeneous of order $m$ and is real principal type in the sense of Definition pridef. The

Theorems & Definitions (79)

  • Theorem 1.1
  • Remark 1.2
  • Example 1
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Remark 2.2
  • ...and 69 more