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The $\overline\partial$-Robin Laplacian

Joaquim Duran

Abstract

We study the family of operators $\{\mathcal{R}_a\}_{a\in [0,+\infty)}$ associated to the Robin-type problems in a bounded domain $Ω\subset\mathbb{R}^2$ $$ \begin{cases} -Δu = f & \text{in } Ω, \\ 2 \bar ν\partial_{\bar z} u + au = 0 & \text{on } \partialΩ, \end{cases} $$ and their dependency on the boundary parameter $a$ as it moves along $[0,+\infty)$. In this regard, we study the convergence of such operators in a resolvent sense. We also describe the eigenvalues of such operators and show some of their properties, both for all fixed $a$ and as functions of the parameter $a$. As shall be seen in more detail in arXiv:2507.18698, the eigenvalues of these operators characterize the positive eigenvalues of quantum dot Dirac operators.

The $\overline\partial$-Robin Laplacian

Abstract

We study the family of operators associated to the Robin-type problems in a bounded domain and their dependency on the boundary parameter as it moves along . In this regard, we study the convergence of such operators in a resolvent sense. We also describe the eigenvalues of such operators and show some of their properties, both for all fixed and as functions of the parameter . As shall be seen in more detail in arXiv:2507.18698, the eigenvalues of these operators characterize the positive eigenvalues of quantum dot Dirac operators.

Paper Structure

This paper contains 19 sections, 31 theorems, 252 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Qualitative behavior when the domain is a disk
  4. Main results
  5. Relation between the $\overline\partial$-Robin Laplacian and quantum dot Dirac operators
  6. Organization of the paper
  7. Preliminary results from complex analysis
  8. Maximal Wirtinger operators
  9. The Bergman space
  10. A Hilbert subspace of the Bergman space
  11. The $\overline\partial$-Robin Laplacian $\mathcal{R}_a$
  12. Hilbert space and sesquilinear form setting
  13. The associated self-adjoint operator
  14. Spectral properties of $\mathcal{R}_a$
  15. The family of operators $\{\mathcal{R}_a\}_{a>0}$
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every $a>0$, the operator is the unique self-adjoint operator such that, for every $f\in L^2(\Omega)$, there exists a unique function in its domain solving eq:BvPRodzinLaplacian in the weak sense. Moreover, for every $\lambda \in \mathbb{C}\setminus\mathbb{R}$, the resolvent $(\mathcal{R}_a-\lambda)^{-1}$ is a bounded operat

Figures (4)

  • Figure 1: Some eigenvalue curves $a\mapsto \mu(a)$ on $D_R$ for $R=3$.
  • Figure 2: Some eigenvalue curves $a\mapsto \mu(a)$ on $D_R$ for $R=3$. The dashed curves correspond to $a<0$. The continuous curves are the ones of \ref{['fig:EigenRodzinBall']}.
  • Figure 3: Some eigenvalue curves $a\mapsto \mu(a)$ on an annulus of inner radius $1$ and outer radius $\sqrt{10}$. The black pointed curves are not locally concave. The black dashed curve corresponds to the first eigenvalue curve on the disk of same area.
  • Figure 4: Plot of the radial part of the eigenfunctions \ref{['eq:AllEigenfunctionsDisk']} for $k=0$, $a=0.1$ (first column), $a=1$ (second column), $a=10$ (third column), $a=100$ (fourth column), and $j=0$ (first row), $j=1$ (second row), $j=2$ (third row). The eigenfunctions with angular sign $-$ are plotted with continuous red curves, and the ones with angular sign $+$ are plotted with dashed blue curves.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Conjecture 1.10
  • ...and 58 more