# How to derive the Sommerfeld radiation condition from the resolvent?

Suppose the resolvent $$R_0(\zeta)$$ for an operator $$P_0(\zeta)$$ is known to satisfy a limiting absorption principle at $$\lambda \in \Bbb R$$. For some perturbation $$P(\zeta)$$ of $$P_0(\zeta)$$ with $$V (\zeta)= P(\zeta) - P_0(\zeta)$$, then, if $$\lambda$$ is an embedded eigenvalue of $$P(\zeta)$$ with eigenfunction $$u$$, we can compute \begin{align} 0 &= R_0(\lambda + i \epsilon)P(\lambda)u= R_0(\lambda + i \epsilon) (P_0(\lambda + i\epsilon) + (P_0(\lambda) - P_0(\lambda + i\epsilon)) + V(\lambda))u \\ &\to u + R_0(\lambda + i 0)V(\lambda)u \Rightarrow u = - R_0(\lambda + i 0) V(\lambda) u \end{align} whenever $$P_0(\lambda + i \epsilon) - P_0(\lambda)$$ vanishes faster than $$R_0(\lambda + i \epsilon)$$ blows up. When $$P_0(\zeta) = - \Delta + \zeta^2$$, I can see that the asymptotic behavior of $$R_0(\lambda + i \epsilon)$$ at infinity implies the Sommerfeld radiation condition for $$u$$. This leads me to wonder whether we can, in general, derive these boundary constraints from knowledge of the asymptotics for the reference resolvent $$R_0$$ given any analytic family of operators $$P(\zeta)$$.

Is this the standard relation between the resolvent, the limiting absorption principle, and the Sommerfeld conditions mentioned in the Wikipedia article or is there a more canonical way to deduce the radiation conditions from knowledge of $$R_0$$?