Friedrichs Extension and Min–Max Principle for Operators with a Gap

Abstract

Semibounded symmetric operators have a distinguished self-adjoint extension, the Friedrichs extension. The eigenvalues of the Friedrichs extension are given by a variational principle that involves only the domain of the symmetric operator. Although Dirac operators describing relativistic particles are not semibounded, the Dirac operator with Coulomb potential is known to have a distinguished extension. Similarly, for Dirac-type operators on manifolds with a boundary a distinguished self-adjoint extension is characterised by the Atiyah-Patodi-Singer boundary condition. In this paper we relate these extensions to a generalisation of the Friedrichs extension to the setting of operators satisfying a gap condition. In addition we prove, in the general setting, that the eigenvalues of this extension are also given by a variational principle that involves only the domain of the symmetric operator.for k ≥ 1 are the discrete spectrum of A F in the interval (−∞, sup k≥1 λ k ), counted with multiplicities d k := # {j ≥ 1 : λ j = λ k } 2010 Mathematics Subject Classification. 49R05, 49S05, 47B25, 81Q10.

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

“…The proof is by combining a new variational characterization of E 1 (Ω), inspired by min-max techniques for operators with gaps introduced in [16,22] and the classical proof of Szegő about the eigenvalues of membranes of fixed area [39] (see also the recent work [36] on operators with gaps).…”

Dirac–Coulomb operators with general charge distribution II. The lowest eigenvalue

“…In (12) and ( 13), D 0 Ψ, V μ Ψ and σ • ∇ϕ are understood in the sense of distributions. That one can characterize the domain of the distinguished self-adjoint extension in terms of a Sobolev space with weight for the upper spinor ϕ was realized first in [22,23] and later revisited in [20,52].…”

“…In other words, no atom with a weight larger than or equal to 1 is allowed in the measure μ. We also gave in [21] a characterization of the domain using a method introduced in [22,23] and recently revisited in [20,52]. In this paper, we will always work with the so-defined distinguished extension.…”

“…The third difficulty is that the lowest Dirac eigenvalue in the gap (−1, 1) is not given by a minimum like in (1). In fact it is given by a min-max formula [14,21,26,52]. Although this variational characterization can be used to show that λ 1 (D 0 − μ * |x| −1 ) is monotone in μ, it is not obviously concave and this prevents us from using a simple argument such as (2).…”

Mathematical questions about the computation of eigenvalues of Dirac operators with critical potentials in atomic and molecular physics

“…Here, F ± = Λ ± F , with F a dense subspace of the domain of H , such that the quadratic form 〈ψ, Aψ〉 is well-defined on F + ⊕ F − . See also the recent articles [23][24][25]. Based on a simple and very useful orthogonal decomposition proposed by Talman [16], it was proved by Dolbeault, Esteban and Séré in [22] for the case without magnetic field, and later in [26,27] by Dolbeault, Esteban and Loss for the case of an external constant magnetic field, that the above abstract result implies that for electrostatic potentials having at worst singularities of the Coulomb type, −ν/|x|, with 0 < ν ≤ 1, the eigenvalues of the operator H A + V can be found by the following simple and computable procedure: for functions ϕ ∈ L 2 (R 3 , C 2 ), consider the quadratic form…”