Eigenvalue bounds for quantum dot Dirac operators

Abstract

We exploit the connection between quantum dot Dirac operators and $\overline{\partial }$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating bounds (from above and below) from one to the other. As an application, we provide new upper and lower bounds for the eigenvalues of the quantum dot Dirac operators, which depend only on geometric quantities of the underlying domain. In particular, we obtain some Faber-Krahn type inequalities for convex thin domains.

arXiv comment20 pages, 3 figures. v2: added a Note before Section 1.1 discussing a proof of Dirac-Faber-Krahn that appeared during the peer review process of this work