A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

Abstract

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of R 2 . Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegő type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences 1 supporting the existence of a Faber-Krahn type inequality.

Dirac operator spectrum in tubes and layers with a zigzag-type boundary

“…The corresponding problem for Dirac operators attracted attention only very recently, cf. [ 1 , 8 , 32 ], and many questions remain open. A particular class of problems concerns a confinement to unbounded regions of a nontrivial geometry.…”

Eigenvalue Curves for Generalized MIT Bag Models

“…The analogous question in the twodimensional framework -the optimization of the spectral gap for Dirac operators with infinite mass boundary conditions-is considered a hot open problem in spectral geometry [54]. More generally, the quest of geometrical upper and lower bounds for the spectral gap is a trending topic of research [5,24,30,56]. This quest is also addressed in the differential geometry literature for Dirac operators on spin manifolds, where sharp inequalities for spectral gaps in terms of geometric quantities are shown [2,4,12,13,35,45,46,47,53].…”