We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for N = 1 in three dimensions. Based on these results, we formulate a conjecture on the weak-coupling asymptotic behaviour of such bound states.

Abstract. In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order α 5 log α −1 , where α denotes the fine structure constant, and prove rigorous bounds on the remainder term of the order o(α 5 log α −1 ). As a consequence, we prove that the binding energy is not a real analytic function of α, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of α, as conjectured in the recent literature.

We consider a Pauli-Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum P = 0.

We consider the Brown-Ravenhall model of a relativistic atom with N electrons and a nucleus of charge Z and prove the existence of an infinite number of discrete eigenvalues for N Z. As an intermediate result we prove a HVZ-type theorem for these systems.

We study a self-adjoint realization of a massless Dirac operator on a bounded connected domain Ω ⊂ R 2 which is frequently used to model graphene. In particular, we show that this operator is the limit, as M → ∞, of a Dirac operator defined on the whole plane, with a mass term of size M supported outside Ω.Equation (1) gives rise to a whole family of different boundary conditions (see Equation (2) below). In this present work we focus on one of these self-adjoint realizations, denoted by H ∞ , which corresponds to the so-called infinite mass boundary conditions. In the physics literature, the operator H ∞ has gained renewed interest due to its application to model quantum dots in graphene [6,7,15,11,14,12,2].Let H M be the Dirac operator defined on R 2 with a mass M on R 2 \ Ω, and 0 inside Ω. In [4] it was shown that certain plane-wave solutions of the eigenvalue equation H M ψ = Eψ, in the limit M → ∞, satisfy the same boundary conditions as the eigenfunctions of H ∞ . The main result of this work, Theorem 1, is the convergence, in the sense of spectral projections, of H M towards H ∞ .1.1. Definitions and main result. Let us introduce some notation used throughout this article. We denote by Ω ⊂ R 2 a bounded connected domain with boundary ∂Ω ∈ C 3 of length L > 0. We parametrize ∂Ω by the curve γ : [0, L] → ∂Ω in its arc-length, i.e., |γ ′ (s)| = 1. For a given self-adjoint operator H, we denote by σ(H) its spectrum, and by E I (H) its spectral projection on the set I ⊂ R. We use the symbols ·, · and (·, ·) to denote the scalar products in L 2 and C 2 , respectively.