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One Dimensional Point Interactions and the Resolvent Algebra -- Simple Remarks

Antonio Moscato

Abstract

This paper shows that the resolvent algebra $\mathcal{R}\left( \mathbb{R}^2,σ\right)$ can accommodate dynamics induced by self-adjoint Hamiltonians on $L^2\left( \mathbb{R} \right)$ describing a single non-relativistic spinless particle undergoing one up to countably many different fixed point interactions located on the real line.

One Dimensional Point Interactions and the Resolvent Algebra -- Simple Remarks

Abstract

This paper shows that the resolvent algebra can accommodate dynamics induced by self-adjoint Hamiltonians on describing a single non-relativistic spinless particle undergoing one up to countably many different fixed point interactions located on the real line.
Paper Structure (11 sections, 11 theorems, 49 equations)

This paper contains 11 sections, 11 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The Resolvent Algebra
  3. Dynamics on $\mathcal{R}\left(X, \sigma \right)$
  4. Point Interactions
  5. One Fixed-Center Point Interaction
  6. $\boxed{k=1}$
  7. $\boxed{k=n}$
  8. Many Fixed-Centers Point Interactions
  9. Finitely Many Fixed-Centers Point Interactions
  10. Countably Many Fixed-Centers Point Interactions
  11. Conclusions

Key Result

Proposition 2.1

Let $\left( X, \sigma \right)$ be a symplectic space and let $\mathcal{R}_0$ be as in definition Def2.1. $\blacksquare$

Theorems & Definitions (31)

  • Definition 2.1
  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.3
  • Remark 2.5
  • Proposition 2.2
  • ...and 21 more