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Strong convergence of a resolution of the identity via canonical coherent states

Ryo Namiki

Abstract

A resolution of the identity due to canonical coherent states is often proven in the weak operator topology. However, such a resolution with an integral symbol is typically supposed to hold in the strong operator topology associated with the framework of the spectral theorem. We provide an elementary proof of the strong convergence for the resolution of the identity due to canonical coherent states starting with a mostly familiar setup. Further, we enjoy a different proof and show that the relevant uniform limit does not exist.

Strong convergence of a resolution of the identity via canonical coherent states

Abstract

A resolution of the identity due to canonical coherent states is often proven in the weak operator topology. However, such a resolution with an integral symbol is typically supposed to hold in the strong operator topology associated with the framework of the spectral theorem. We provide an elementary proof of the strong convergence for the resolution of the identity due to canonical coherent states starting with a mostly familiar setup. Further, we enjoy a different proof and show that the relevant uniform limit does not exist.
Paper Structure (11 sections, 9 theorems, 116 equations)

This paper contains 11 sections, 9 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Basic notions and an elementary proof of the strong convergence
  3. Review of Klauder's approach
  4. no uniform limit
  5. conclusion and remarks
  6. Bochner integral
  7. elementary integration
  8. detail of calculation
  9. action of bra
  10. order of integral and summation
  11. An Elementary Approach to Exchange the Order of Integral and Summation for $\cal H$-valued Integrals Associated with Power Series

Key Result

Theorem 1

Let be $\varphi \in {\cal H}$. For any $\epsilon >0$, there exists $R>0$ such that whenever $r \ge R$.

Theorems & Definitions (28)

  • Theorem 1
  • Remark 2
  • proof
  • Remark 3
  • Remark 4
  • Theorem 5
  • proof
  • Theorem 6
  • Remark 7
  • proof
  • ...and 18 more