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Non-relativistic limit of Dirac Hamiltonians with Aharonov-Bohm fields

Matteo Gallone, Alessandro Michelangeli, Diego Noja

TL;DR

The paper characterizes all self-adjoint realizations of Dirac- and Schrödinger-Aharonov-Bohm operators in two dimensions using Krein-Vishik-Birman-Grubb theory, revealing a block-wise structure by angular momentum. It then establishes a precise non-relativistic limit: after subtracting the rest energy and rescaling boundary data with $c$, the Dirac-AB family $H_ extα^{( extγ)}$ converges in norm-resolvent sense to a Schrödinger-AB family $S_ extα^{( extθ)}$, provided the extension parameters scale so that $ rac{2m c}{1+2 extα} ext γ(c) o ext θ$. This yields a clean mapping from the full relativistic operator family to the physically relevant $s$-wave, angular-momentum commuting non-relativistic AB operators with relativistic approximants, clarifying both squared operators and the role of boundary conditions. The results give explicit resolvent formulas and illuminate how scattering length is preserved along the limit, with complementary discussion of the positron sector and exceptional cases.$

Abstract

We characterise the families of self-adjoint Dirac and Schrödinger operators with Aharonov-Bohm magnetic field, and we exploit the non-relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short-scale boundary condition of self-adjointness. This ensures that the scattering length of the Aharonov-Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac-AB operators is mapped, in the non-relativistic limit, into the physically relevant sub-family of $s$-wave, angular-momentum-commuting, Schrö\-dinger-AB Hamiltonians with relativistic Dirac approximants.

Non-relativistic limit of Dirac Hamiltonians with Aharonov-Bohm fields

TL;DR

The paper characterizes all self-adjoint realizations of Dirac- and Schrödinger-Aharonov-Bohm operators in two dimensions using Krein-Vishik-Birman-Grubb theory, revealing a block-wise structure by angular momentum. It then establishes a precise non-relativistic limit: after subtracting the rest energy and rescaling boundary data with , the Dirac-AB family converges in norm-resolvent sense to a Schrödinger-AB family , provided the extension parameters scale so that . This yields a clean mapping from the full relativistic operator family to the physically relevant -wave, angular-momentum commuting non-relativistic AB operators with relativistic approximants, clarifying both squared operators and the role of boundary conditions. The results give explicit resolvent formulas and illuminate how scattering length is preserved along the limit, with complementary discussion of the positron sector and exceptional cases.$

Abstract

We characterise the families of self-adjoint Dirac and Schrödinger operators with Aharonov-Bohm magnetic field, and we exploit the non-relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short-scale boundary condition of self-adjointness. This ensures that the scattering length of the Aharonov-Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac-AB operators is mapped, in the non-relativistic limit, into the physically relevant sub-family of -wave, angular-momentum-commuting, Schrö\-dinger-AB Hamiltonians with relativistic Dirac approximants.

Paper Structure

This paper contains 17 sections, 30 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Relativistic setting
  3. Non-relativistic setting
  4. Non-relativistic limit
  5. Main Results and further remarks
  6. Notation
  7. Block-wise self-adjoint extensions of the Dirac-AB operator
  8. Operator closure and deficiency indices
  9. Characterisation of $\ker (\mathsf{h}_{\alpha,0}^*-mc^2+\lambda)$
  10. Distinguished self-adjoint extension
  11. Proof of Theorem \ref{['prop:Dirac']}
  12. Block-wise self-adjoint extensions of the Schrödinger-AB operator
  13. Proof of Theorem \ref{['thm:Main']}
  14. Elements of Kreı̆n-Višik-Birman-Grubb self-adjoint extension theory
  15. Canonical decomposition of the AB-operators
  16. ...and 2 more sections

Key Result

Theorem 1.5

Let $\alpha\in(-\frac{1}{2},\frac{1}{2})$, $m>0$, $c>0$, $\theta\in\mathbb{R}\cup\{\infty\}$, and let $\gamma\equiv\gamma(c)\in\mathbb{R}\cup\{\infty\}$ scale with $c$ according to Then there is a non-empty $c$-independent interval $J$ of the negative real line contained, eventually as $c\to +\infty$, in the infinite intersection $\bigcap_{c}\rho(H_{\alpha}^{(\gamma{(c)})}-mc^2)$ and also in $\rh

Theorems & Definitions (57)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 2.1
  • Proposition 2.2
  • ...and 47 more