Overview
This work establishes a Payne-Weinberger type inequality for quantum dot Dirac operators. The inequality applies to operators defined on bounded and simply connected planar domains. It provides a sharp upper bound for the first positive eigenvalue of these operators. This upper bound is solely dependent on the isoperimetric deficit of the domain.
Approach
The proof of this inequality leverages a recently investigated connection with the $\overline\partial$-Robin Laplacian. By utilizing this connection, an analogous inequality was established for the first eigenvalue of the $\overline\partial$-Robin Laplacian. This, in turn, relied on the corresponding inequality previously known for the Robin Laplacian.
Findings
- A Payne-Weinberger type inequality was proven for quantum dot Dirac operators.
- This inequality is defined for operators on bounded and simply connected planar domains.
- The inequality provides a sharp upper bound for the first positive eigenvalue of these operators.
- The upper bound's dependence is exclusively on the isoperimetric deficit of the domain.