Resolvent estimates for the Dirac operator in weighted Sobolev spaces

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 ℒ_s² 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(ℒ_s²,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(ℒ_s²,ℋ_−s¹). R^±(λ)−R₀^±(λ) are shown to converge strongly to zero in B(ℒ_s²,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 ℒ_s² to ℋ_−s¹, and the estimates for the free Dirac and Schrödinger resolvents, both considered as operators from ℒ_s² to ℒ_−s².