On the Principle of Limiting Absorption for the Dirac Operator

Abstract

tion for elliptic Dirichlet problems in exterior domains, where the radiation conditions at infinity and the near-singularity properties of fundamental solutions were made use of in an essential manner. But recently Agmon j^l] proposed a new method without explicit recourse to these tools and involving a priori estimates, which is valid for elliptic operators in the

Decay estimates for the wave and Dirac equations with a magnetic potential

“…Agmon [1]); here we shall use a very precise version of the principle, due to Barcelo, Ruiz and Vega [4]. On the other hand, for the Dirac operator only a few results are available, which concern the case with a nonzero mass term (see [28], [42]). …”

“…In this section we will study the limiting absorption principle for the massless Dirac operator D; this property was studied by Yamada in [42] for the operator with mass. Moreover, as in the case of the magnetic Laplacian, we will extend this result to the perturbed operator D V = D + V (x), under a suitable assumption on the potential V .…”

“…As a byproduct of our method of proof, we obtain the limiting absorption principle for the perturbed Dirac operator under assumption (1.21) (see Section 3.2). The LAP had been proved earlier for the free Dirac equation by Yamada [42], and for the Dirac equation with potential (and with mass) in [28] under quite stronger assumptions.…”

Asymptotique de la phase de diffusion à haute énergie pour l'opérateur de Dirac

“…Pour 6 > 1, on a les propriétés suivantes ( [17], [19] Dans cet article, on fixe m=e==^==l,eton étudie le comportement asymptotique de s(\) quand X -> ±00. Cela revient à étudier celui de s^(^) = <s(--) pour ±/^ > 0 fixé h quand h \ 0.…”

Resolvent Estimates of the Dirac Operator

“…As a result, the existence of the extended resolvents R ± (λ) was assured (see Theorem 2.2 in section 2 below). The extended resolvents R ± (λ) play important roles in spectral and scattering theory for the operator H (see [13] and [14]). …”