In this article we investigate spectral properties of the coupling H + V λ , where H = −iα · ∇ + mβ is the free Dirac operator in R 3 , m > 0 and V λ is an electrostatic shell potential (which depends on a parameter λ ∈ R) located on the boundary of a smooth domain in R 3 . Our main result is an isoperimetric-type inequality for the admissible range of λ's for which the coupling H + V λ generates pure point spectrum in (−m, m). That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman-Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible λ's, and we use this to relate the endpoints of the admissible range of λ's to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.2010 Mathematics Subject Classification. Primary 81Q10, Secondary 35Q40.

Spectral properties and the confinement phenomenon for the coupling H+V are studied, where H = −iα ·∇+mβ is the free Dirac operator in R 3 and V is a measure-valued potential. The potentials V under consideration are given in terms of surface measures on the boundary of bounded regular domains in R 3 .A criterion for the existence of point spectrum is given, with applications to electrostatic shell potentials. In the case of the sphere, an uncertainty principle is developed and its relation with some eigenvectors of the coupling is shown.Furthermore, a criterion for generating confinement is given. As an application, some known results about confinement on the sphere for electrostatic plus Lorentz scalar shell potentials are generalized to regular surfaces.

The Dirac operator, acting in three dimensions, is considered. Assuming that a large mass m ą 0 lies outside a smooth and bounded open set Ω Ă R 3 , it is proved that its spectrum is approximated by the one of the Dirac operator on Ω with the MIT bag boundary condition. The approximation, which is developed up to and error of order op1{? mq, is carried out by introducing tubular coordinates in a neighborhood of BΩ and analyzing the corresponding one dimensional optimization problems in the normal direction.2010 Mathematics Subject Classification. 35J60, 35Q75, 49J45, 49S05, 81Q10, 81V05, 35P15, 58C40.Key words and phrases. Dirac operator, relativistic particle in a box, MIT bag model, spectral theory.

Under certain hypothesis of smallness of the regular potential V, we prove that the Dirac operator in R 3 coupled with a suitable rescaling of V converges in the strong resolvent sense to the Hamiltonian coupled with a δ-shell potential supported on Σ, a bounded C 2 surface. Nevertheless, the coupling constant depends non-linearly on the potential V: the Klein's Paradox comes into play.

Abstract. For 1 ≤ n < d integers and ρ > 2, we prove that an n-dimensional AhlforsDavid regular measure µ in R d is uniformly n-rectifiable if and only if the ρ-variation for the Riesz transform with respect to µ is a bounded operator in L 2 (µ). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L 2 (µ) boundedness of the Riesz transforms to the uniform rectifiability of µ.