The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mover accent="true"> <mml:mrow> <mml:mo>∂</mml:mo> </mml:mrow> <mml:mo>‾</mml:mo> </mml:mover> </mml:math> -Robin Laplacian

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.