Deriving Sommerfeld radiation condition from limiting absorption principle

Question

For the Helmholtz equation $$ -(\Delta + k ^2) u = f, \label{1}\tag{1} $$ imposing the Sommerfeld radiation condition $$ \lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0 $$ on $u$ allows us to pick a unique solution for \eqref{1}. I have seen this condition derived by looking at the asymptotics in $r$ of the resolvent $R(k) = -(\Delta + k^2)^{-1}$ as $k$ tends to the real line, and I want to know if this is an approach that can be generalized to other operators. The Wikipedia page on the Sommerfeld radiation condition mentions there is a connection to the Limiting Absorption Principle.

I could not find any references, but for the Laplacian operator $P(k) = - \Delta - k ^2$ defined on $H^2(\Bbb R^m) \Subset L^2(\Bbb R^m)$, my best guess is as follows.

We know $$ \|u\|_{L^{2, -\sigma}} \lesssim \|P(k)u\|_{L^{2, \sigma}} \label{2}\tag{2} $$ for $\sigma > 1/2$, where $L^{2, \sigma}$ is given by integration against the measure $(1 +|x|^2)^{\sigma} \text d x$. From this, given any $f \in L^{2, \sigma}$, we have that $u_n = R(\lambda + i n^{-1}) f$ is bounded in $L^{2, -\sigma}$ for $\lambda \in \Bbb R$ and $n =1, \dots$, so the Banach-Alaoglu theorem furnishes a weak* limit $u \in L^{2, -\sigma}$ s.t. for all Schwartz-type test functions $\varphi$ we have $$ \langle u_n, \varphi\rangle \to \langle u, \varphi\rangle. $$ This implies that $u$ is a distributional solution to $P(\lambda) u = f$, so elliptic regularity results show $u \in H^{2, \text{loc}}$ and $e^{-r/n}u$ is in the domain of $P$. If $k = \lambda + i n ^{-1}$, we have $P(k) = P(\lambda) - 2i n ^{-1} \lambda + n ^{-2} $, so \begin{align} P(k) e ^{- r/n} u &= -(u \Delta e ^{- r/n} + 2 \nabla e ^{- r/n} \cdot \nabla u + e ^{- r/n} \Delta u + k ^2 e ^{- r/n} u) \\ &= -u \left( n ^{-2} e^{-r /n}-\frac{n ^{-1} e^{-r/n}}{r} \right) + 2 n ^{-1} e ^{- r/n} u_r + e ^{- r/n} P(k) u \\ &= \frac{e ^{-r/n}}{n r} u + 2 n ^{-1} e ^{- r/n} (u_r - i \lambda u) + e ^{- r/n} f , \end{align} hence \begin{align*} \|P(k) e ^{-r/n}u- e ^{-r/n}f\|_{L ^{2, \sigma}} &\lesssim n ^{-1}\| r ^{-1} e ^{-r/n} u\|_{L ^{2, \sigma}} + n ^{-1} \|e ^{-r/n}(u_r - i \lambda u)\|_{L ^{2, \sigma}} . \end{align*} Combining this with \eqref{2}, picking $\sigma$ slightly above $1/2$, and letting $n \to \infty$ yields: \begin{align*} \|u\|_{L ^{2, -\sigma}} \lesssim \|f\|_{L ^{2, \sigma}} + \lim_{n\to \infty} n ^{-1} \|e ^{-r/n}(u_r - i \lambda u)\|_{L ^{2, \sigma}}. \end{align*} This last term looks a lot like the Sommerfeld radiation condition. In fact, if the weak* limit that produced $u$ preserved this condition, the limit $u$ would be unique and the resolvent could be at least weakly extended to $\lambda$.

My question is whether this calculation is correct, and if it is, how much of it generalizes to other (possibly non-self adjoint) operators $P(k)$. Obviously the functions $e^{-r/n}$, as well as the pair of dual spaces $L^{2, \pm \sigma}$ would have to be replaced, but is there a general framework around this procedure?

Answer 1

The literature on this subject is indeed vast, so I'll just cite one recent paper that I'm familiar with that discusses the non-self adjoint case in a fair amount of generality: arXiv:1905.12587 [math.AP]. (If you only want to assume control on finitely many derivatives of the potential $V$, then I believe the argument still works as long as you have control on enough derivatives.) The upshot is that a version of the Sommerfeld radiation condition can be used to define function spaces between which $P(k)$ is Fredholm. If $P(k)$ is self-adjoint, then it is actually invertible. Otherwise, we can have some kernel and cokernel, but they are only finite-dimensional. The limiting absorption principle is only stated to apply in the self-adjoint case, but the Fredholm setup holds generally.

In the paper cited above, the Sommerfeld radiation condition is encoded in the definition of the Sobolev spaces used in the statement of the main theorem. In this case, the radiation condition is $(u_r- i k u) \in L^{2,1/2-\epsilon}$ for $\epsilon<1$, instead of the pointwise version stated in the question. (You can typically prove that various versions of the radiation condition are equivalent under weak hypotheses.) If you let $\tilde{u} = e^{- i k r}u$, then this is equivalent to $\tilde{u}_r \in L^{2,1/2-\epsilon}$, so a statement about the Sobolev regularity of $\tilde{u}$. For $\tilde{r}\in \mathbb{N}$, the $H_{\mathrm{b}}^{\tilde{r},s}$ spaces are defined by $H^{0,s}_{\mathrm{b}} = \langle r \rangle^{-s} L^2$ and $$H_{\mathrm{b}}^{\tilde{r},s} = \{u:Qu \in H^{0,s}_{\mathrm{b}} \text{ for }Q\in \operatorname{Diff}_{\mathrm{b}}^{\tilde{r}} \},$$ where $\operatorname{Diff}_{\mathrm{b}}^{\tilde{r}}$ is a module of order $\tilde{r}$ differential operators generated by $r\partial_r$ and $\partial_\theta$ away from the origin. Thus, we can impose the Sommerfeld radiation condition by requiring $\tilde{u} \in H_{\mathrm{b}}^{1,-1/2-\epsilon}$. This corresponds to the main theorem with $\tilde{r} = 1$ and $\ell = -1/2-\epsilon$.