On the Eigenfunctions of No-Pair Operators in Classical Magnetic Fields

Abstract

We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1, ∞) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue λ < 1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a < √ 1 − λ 2 , for λ ∈ [0, 1), and a < 1, for λ < 0. Mathematics Subject Classification (2000). Primary 81Q10; Secondary 47B25.

Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics. II. The no-pair operator

“…x , Ê) with fixed sign and satisfying |∇F | a, we have iy ∈ ̺(D A + iα · ∇F ), The bound (A.2) is essentially well-known. For instance, its proof given in [22] for classical vector potentials works for quantized ones as well. Next, we set for all ϕ ∈ D 0 .…”

Ground states of semi-relativistic Pauli-Fierz and no-pair Hamiltonians in QED at critical Coulomb coupling

“…More precisely, we shall prove a suitable bound on the spatial exponential localization of spectral subspaces corresponding to energies below the ionization threshold which applies to all γ γ c . This localization estimate is derived in Section 4 by adapting and extending ideas from [1,23,24]. We remark that by now we are able to improve the localization estimates of [24] thanks to some more recent results of [18] collected in Proposition 3.3.…”

Stability of impurities with Coulomb potential in graphene with homogeneous magnetic field

“…An error in the constant he obtained was corrected by Walter [23]. (7) The 3-dimensional magnetic case -for a rather big class of magnetic fields -was treated by Matte and Stockmeyer [14]. They showed that the critical constant is not lowered by an intricate resolvent method.…”

“…The generality of their result is paid for by the absence of an explicit lower bound on the energy. The bonus of our direct approach based on Lieb and Yau's [13] strategy in the variant found in [6] -compared to transfering the methods of [14] -is our result on the positivity of the energy. (8) The numerical value of the critical coupling constant is Z c ≈ 0.3780 which is compared with the expected critical coupling constantZ c of the existence of a distinguished self-adjoint extension of the non-magnetic Weyl operator W Z .…”