Real Axis Asymptotics and Estimates of Hamiltonian Resolvent Kernels

Abstract

The three space Newtonian potential of a spherically symmetric charge distribution of finite total charge then has summability Fourier transform proportional to ${\left| {\vec{z}} \right|^{{ - 2}}}g(\left| {\vec{z}} \right|)$ over $\vec{z} \in {R_{3}}$ secondly, this g is assumed to have an even analytic extension from [0, + ∞) to the strip |g[s]| ≤ b for some b ∈ (0, + ∞) such that also there $\left( {\sup {{\left( {1 + \left| {R\left[ s \right]} \right|} \right)}^{\eta }}\left| {g\left( s \right)} \right|} \right) < + \infty$ for some $\eta \in \left( {0,1} \right]$. For such potentials, both the usual Schrodinger Hamiltonian and the free electron state second quantized Dirac Hamiltonian are Fourier transformed to momentum space, and then by spherical harmonics reduced from three dimensional to one dimensional radial integral operators H_q on (0,+∞). Then we obtain a Faddeev integral operator representation of the resolvent ${\left( {\lambda I - {H_{q}}} \right)^{{ - 1}}}$ the kernel being explicitly constructed by contour deforming (0,+∞) off the real axis to the side opposite λ. From this construction and the resulting estimates and asymptotics, we find ${\lambda _{0}} \in (0, + \infty ),$ H_q has no point spectrum in (λ_o, + ∞) and is absolutely continuous there, and $\left( {{{\left( {\lambda I - {H_{q}}} \right)}^{{ - 1}}}u,u} \right)$ is sub $\frac{1}{4}$ order Hölder continuous in λ up to both sides of the real axis for a dense linear manifold of u; some speculative ramifications of the latter for pseudoeigenvalues are discussed.