The $\overline{\partial }$-Robin Laplacian

Abstract

We study the family of operators $\{{\mathcal{R}}_a{\}}_{a\in [0,+\infty )}$ associated to the Robin-type problems in a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ $$\{\begin{array}{ll}-\hat{I}”u=f& \text{in }\hat{I}\text{c◯},\\ 2\bar{\hat{I}}½{\partial }_{\bar{z}}u+au=0& \text{on }\partial \hat{I}\text{c◯},\end{array}$$ 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.

52 pages, 4 figures. v3: added Remark 1.11, Remark 2.15, and Figure 4