[Paper Review] Resonances from perturbed symmetry in open quantum dots

Pierre Duclos, Pavel Exner+1 more|arXiv (Cornell University)|Jun 29, 2000

Quantum chaos and dynamical systems4 citations

TL;DR

This paper investigates how resonances emerge in open quantum dots when perturbations break the symmetry of a potential well in a straight hard-wall channel. Using perturbation theory, it derives the resonance shifts due to magnetic fields or geometric deformations, and proves that a discrete spectrum persists under strong magnetic fields if the potential is purely attractive, offering a rigorous condition for spectral stability in open quantum systems.

ABSTRACT

We discuss the Nöckel model of an open quantum dot, i.e., a straight hard-wall channel with a potential well. If this potential depends on the longitudinal variable only, there are embedded eigenvalues which turn into resonances if the symmetry is violated, either by applying a magnetic field or by deformation of the well. For a weak symmetry breaking we derive the perturbative expansion of these resonances. We also deduce a sufficient condition under which the discrete spectrum of such a system (without any symmetry) survives the presence of a strong magnetic field. It is satisfied, in particular, if the dot potential is purely attractive.

Motivation & Objective

• To analyze the emergence of resonances in open quantum dots when symmetries of a potential well are perturbed.
• To derive a perturbative expansion for resonances arising from weak symmetry breaking via magnetic fields or deformation.
• To establish a sufficient condition ensuring the survival of the discrete spectrum under strong magnetic fields in systems without initial symmetry.
• To demonstrate that purely attractive potentials preserve the discrete spectrum even in the presence of strong magnetic fields.

Proposed method

• Adopts the Nöckel model of an open quantum dot with a straight hard-wall channel and a potential well dependent only on the longitudinal coordinate.
• Applies perturbation theory to compute resonance shifts when symmetry is broken by a magnetic field or geometric deformation.
• Uses analytical techniques to derive a perturbative expansion of resonances in the weak symmetry-breaking regime.
• Derives a sufficient condition for the persistence of the discrete spectrum under strong magnetic fields, based on the nature of the potential.

Experimental results

Research questions

• RQ1 How do resonances form in open quantum dots when the symmetry of a potential well is weakly broken by a magnetic field or deformation?
• RQ2 What is the perturbative expression for the resonance energy shifts in such systems?
• RQ3 Under what conditions does the discrete spectrum survive in the presence of a strong magnetic field?
• RQ4 Does a purely attractive potential ensure the stability of the discrete spectrum under strong magnetic fields?

Key findings

• Resonances arise from embedded eigenvalues when symmetry is broken by a magnetic field or deformation, with their positions analytically expanded in the weak perturbation regime.
• The perturbative expansion of resonances is derived explicitly for weak symmetry-breaking perturbations.
• A sufficient condition is established under which the discrete spectrum remains intact under strong magnetic fields.
• The condition is satisfied when the potential is purely attractive, guaranteeing spectral stability in such systems.