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 $R_0$?



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