Resolvent Convergence to Dirac Operators on Planar Domains

Abstract

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Ω. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Ω with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

“…In contrast to the 2D setting, was not directly investigated for general boundary parameters in 3D, as far as we know only the particular MIT bag operator is well studied. We emphasize the recent papers [ 2 , 56 ] for the analysis of general properties of the MIT bag operator and [ 3 , 9 , 24 , 54 , 62 ], where it is shown that the MIT bag boundary conditions and their 2D analogues can be interpreted as infinite mass boundary conditions (i.e., is surrounded by a medium with infinite mass). The strategy developed in [ 56 ] employing Calderón projections can also be used to study the self-adjointness of Dirac operators of the form ( 1.1 ).…”

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

“…Various mathematical studies have been undertaken, starting with a rigorous mathematical derivation of such Hamiltonians, see e.g. [19] for the effective Hamiltonian derivation or [3,8,30,38] for the justification of the so-called infinite mass boundary conditions. Many properties of such operators have been investigated as their self-adjointness in bounded domains with specified boundary conditions or coupled with the so-called δ-interactions, see [9,11].…”

Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

“…We focus first on Lorentz scalar potentials (aka mass potentials, see [42,Chapter 4.2]) which are defined in (4.5). Motivated by physics literature on quark confinement (MIT bag model) and on dynamics of electrons in graphene and other nanostructures, there is recently a large body of work in mathematical physics on Dirac operators on domains in R d ; see [3,[6][7][8]27,30,36,40], and the references therein. However, to the best of our knowledge, self-adjointness is obtained in all cases by imposing boundary conditions which encode additional forces acting on the system.…”

“…However, to the best of our knowledge, self-adjointness is obtained in all cases by imposing boundary conditions which encode additional forces acting on the system. Very recently is was proven (see [3,6,40]) that certain boundary conditions can be obtained via a limiting procedure starting from a Dirac operator on the full space R d and sending the value of the Lorentz scalar potential to infinity outside of . In the same spirit, one can also construct Dirac operators on manifolds embedded in R d , see [30].…”