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Tunneling estimates for two-dimensional perturbed magnetic Dirac systems

Esteban Cárdenas, Benjamín Pavez, Edgardo Stockmeyer

Abstract

We prove tunneling estimates for two-dimensional Dirac systems which are localized in space due to the presence of a magnetic field. The Hamiltonian driving the motion admits the decomposition $H = H_0 + W$, where $H_0 $ is a rotationally symmetric magnetic Dirac operator and $W$ is a position-dependent matrix-valued potential satisfying certain smoothness condition in the angular variable. A consequence of our results are upper bounds for the growth in time of the expected size of the system and its total angular momentum.

Tunneling estimates for two-dimensional perturbed magnetic Dirac systems

Abstract

We prove tunneling estimates for two-dimensional Dirac systems which are localized in space due to the presence of a magnetic field. The Hamiltonian driving the motion admits the decomposition , where is a rotationally symmetric magnetic Dirac operator and is a position-dependent matrix-valued potential satisfying certain smoothness condition in the angular variable. A consequence of our results are upper bounds for the growth in time of the expected size of the system and its total angular momentum.
Paper Structure (32 sections, 27 theorems, 155 equations)

This paper contains 32 sections, 27 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. The model and main results
  3. The model
  4. Main result
  5. Consequences of the main result
  6. Preliminaries
  7. Unitary transformations and Hilbert spaces
  8. Essential self-adjointness
  9. The Pauli operator
  10. Analysis of the clasically allowed regions
  11. Localizing the ARs
  12. Partitions of unity
  13. Relative bounds for the perturbation
  14. Bounds for the potential
  15. The critical part
  16. ...and 17 more sections

Key Result

Theorem 2.8

Let $H$ be the Dirac Hamiltonian, satisfying Conditions condition 1 and condition 2, and let $E>0$ and $I=[-E,E]$. Then, there exist positive constants $\zeta_1$, $\zeta_2$, and $C_1<C_2$ such that

Theorems & Definitions (77)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • Remark 2.11
  • Corollary 1
  • proof
  • ...and 67 more