The $\bar{\mathrm{\partial }}$-Robin Laplacian

We study the family of operators $\{{\mathcal{R}}_a{\}}_{a\in [0,+\mathrm{\infty })}$ associated to the Robin-type problems in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ $$\{\begin{array}{ll}-\mathrm{\Delta }u=f& \text{in }\mathrm{\Omega },\\ 2\bar{\nu }{\mathrm{\partial }}_{\bar{z}}u+au=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ and their dependency on the boundary parameter $a$ as it moves along $[0,+\mathrm{\infty })$. 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.