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A connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians in the context of shape optimization problems

Joaquim Duran, Albert Mas, Tomás Sanz-Perela

Abstract

This work addresses Faber-Krahn-type inequalities for quantum dot Dirac operators with nonnegative mass on bounded domains in $\mathbb{R}^2$. We show that this family of inequalities is equivalent to a family of Faber-Krahn-type inequalities for $\overline\partial$-Robin Laplacians. Thanks to this, we prove them in the case of simply connected domains for quantum dot boundary conditions asymptotically close to zigzag boundary conditions. Finally, we also study the case of negative mass.

A connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians in the context of shape optimization problems

Abstract

This work addresses Faber-Krahn-type inequalities for quantum dot Dirac operators with nonnegative mass on bounded domains in . We show that this family of inequalities is equivalent to a family of Faber-Krahn-type inequalities for -Robin Laplacians. Thanks to this, we prove them in the case of simply connected domains for quantum dot boundary conditions asymptotically close to zigzag boundary conditions. Finally, we also study the case of negative mass.

Paper Structure

This paper contains 16 sections, 16 theorems, 156 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Quantum dot Dirac operators
  3. The shape optimization problem
  4. Main results
  5. Equivalence of conjectures
  6. Asymptotic regimes
  7. Quantum dots with negative mass
  8. Equivalence of conjectures
  9. Connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians
  10. Properties of the eigenvalue curves
  11. Proof of the equivalence of conjectures
  12. Asymptotic regimes
  13. Embedding of a Hardy space into a Bergman space
  14. Quantum dots with negative mass
  15. Notation
  16. ...and 1 more sections

Key Result

Theorem 2.1

Assume that $m\geq0$. Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with $C^2$ boundary and let $D\subset\mathbb{R}^2$ be a disk with the same area as $\Omega$. The following hold:

Figures (6)

  • Figure 1: Eigenvalue curves of the quantum dot Dirac operators $\mathcal{D}_{\theta}$ for the disk $D_R$ of radius $R=2$ and with $m=1$. We have included two solid horizontal black lines to highlight the location of $\pm m$. The dashed vertical lines at $\theta=0$ and $\theta=\pi/2$ help to locate, respectively, the infinite mass and zigzag cases. The horizontal dashed lines represent the values $\pm \sqrt{\Lambda_{D_R}+m^2}$.
  • Figure 2: Eigenvalue curves of the $\overline\partial$-Robin Laplacians $\mathcal{R}_a$ for the disk $D_R$ of radius $R=2$. The horizontal dashed line represents the value $\Lambda_{D_R}$.
  • Figure 3: Plot of $a\mapsto \mu_\Omega(a)$ (left) and $\theta\mapsto\lambda_\Omega(\theta)$ (right), where $\Omega$ is a disk $D_R$ of radius $R=2$, and with $m=1$ (represented as a solid black horizontal line in the right picture). The horizontal dashed lines represent $\Lambda_{D_R}$ in the left and $\sqrt{\Lambda_{D_R}+m^2}$ in the right.
  • Figure 4: Eigenvalue curves of the quantum dot Dirac operators $\mathcal{D}_{\theta}(m)$ for the disk $D_R$ of radius $R=2$ and with $m=-1$. We have included two solid horizontal black lines to highlight the location of $\pm m$. The dashed vertical lines at $\theta=0$ and $\theta=\pi/2$ help to locate, respectively, the infinite mass and zigzag cases. The horizontal dashed lines represent the values $\pm \sqrt{\Lambda_{D_R}+m^2}$. The black dot illustrates the first crossing point in $(-\frac{\pi}{2},\frac{\pi}{2})$ between an eigenvalue curve of $\mathcal{D}_\theta(m)$ and the level set $|m|$, which is studied in \ref{['thm:shape_opt_neg_mass']}.
  • Figure 5: Schematic representation of the proof of $(i)$.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Conjecture 1.1
  • Remark 1.2
  • Conjecture 1.3
  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Theorem 2.7
  • ...and 26 more