Resolvent estimates for the Dirac operator in weighted Sobolev spaces

Article type: Research Article

Authors: Pladdy, Christopher

Affiliations: Zenith Electronics Corporation, Research and Development Division, 2000 Millbrook Drive, Lincolnshire, IL 60069, USA

Abstract: We consider the Dirac operator with a scalar short‐range potential. Estimates for the extended resolvents, R±(λ), of this operator considered as a bounded operator from ℒs2 into the weighted Sobolev space ℋ−s1 are discussed. The extended resolvents R0±(λ) of the free Dirac operator are shown to be O(λ) as |λ|→∞ in the uniform operator topology of B(ℒs2,ℋ−s1), and this is shown to be the optimum result. R±(λ) are shown to be R0±(λ)+O(1) as |λ|→∞ in B(ℒs2,ℋ−s1). R±(λ)−R0±(λ) are shown to converge strongly to zero in B(ℒs2,ℋ−s1). The method of proof uses results for the free Schrödinger operator resolvent and the free Dirac operator resolvent, both considered as operators from ℒs2 to ℋ−s1, and the estimates for the free Dirac and Schrödinger resolvents, both considered as operators from ℒs2 to ℒ−s2.

Keywords: Dirac operators, limiting absorption principle, resolvent estimates, Sobolev space, pseudodifferential operator

Journal: Asymptotic Analysis, vol. 31, no. 3-4, pp. 279-295, 2002