Scattering with Critically-Singular and $\delta$-Shell Potentials

Abstract

We consider a scattering problem for electric potentials that have a component which is critically singular in the sense of Lebesgue spaces, and a component given by a measure supported on a compact Lipschitz hypersurface. We study direct and inverse point-source scattering under the assumptions that the potentials are real-valued and compactly supported. To solve the direct scattering problem, we introduce two functional spaces—sort of Bourgain type spaces—that allow to refine the classical resolvent estimates of Agmon and Hörmander, and Kenig, Ruiz and Sogge. These spaces seem to be very useful to deal with the critically-singular and $\delta$-shell components of the potentials at the same time. Furthermore, these spaces and their corresponding resolvent estimates turn out to have a strong connection with the estimates for the conjugated Laplacian used in the context of the inverse Calderón problem. In fact, we derive the classical estimates by Sylvester and Uhlmann, and the more recent ones by Haberman and Tataru after some embedding properties of these new spaces. Regarding the inverse scattering problem, we prove uniqueness for the potentials from point-source scattering data at fix energy. To address the question of uniqueness we combine some of the most advanced techniques in the construction of complex geometrical optics solutions.

Appendix A. The Functional Analytical Framework

Here we prove the Propositions 2.5,  2.6 and 2.7 which describe some basic properties of the functional spaces $X_\lambda$ and $X_\lambda ^*$. As we see, these propositions will be immediately derived from some properties related to the spaces $Y_\lambda$, $Y_\lambda ^*$, $Z_\lambda$ and $Z_\lambda ^*$.

Lemma A.1

The pair $(Y_\lambda , \Vert \centerdot \Vert _{Y_\lambda })$is a Banach space and its dual is isomorphic to $(Y_\lambda ^*, \Vert \centerdot \Vert _{Y_\lambda ^*})$. The Schwartz class ${\mathcal {S}}(\mathbb {R}^d)$is dense in $Y_\lambda$and $Y_\lambda ^*$with their corresponding norms.

Lemma A.2

The pair $(Z_{\lambda ,p^\prime }, \Vert \centerdot \Vert _{Z_{\lambda ,p^\prime }})$is a reflexive Banach space and its dual is isomorphic to $(Z_{\lambda , p}^*, \Vert \centerdot \Vert _{Z_{\lambda , p}^*})$with p and $p^\prime$duals. The Schwartz class ${\mathcal {S}}(\mathbb {R}^d)$is dense in $Z_{\lambda ,p^\prime }$and $Z_{\lambda , p}^*$with their corresponding norms.

Note that $2^k \simeq \lambda ^{1/2}$ when $k \in I$, $m_\lambda (\xi )^{1/2} |\widehat{P_{<I} f}(\xi )| \simeq \lambda ^{1/2} |\widehat{P_{<I} f}(\xi )|$, and $m_\lambda (\xi )^{1/2} |\widehat{P_k f}(\xi )| \simeq 2^{k} |\widehat{P_k f}(\xi )|$ when $k > k_\lambda + 1$. Thus, the norms of the spaces $Y_\lambda$, $Y_\lambda ^*$, $Z_{\lambda ,p^\prime }$ and $Z_{\lambda , p}^*$ can be re-written similarly to the norms of non-homogeneous Besov spaces with different weights and norms on the critical scales $k \in I$. This remark is the key to justify that these spaces are Banach and ${\mathcal {S}}(\mathbb {R}^d)$ is dense with respect to the corresponding topologies. The duality also works because of the same principle —since the norms in the corresponding pieces are taken to be dual. To be more precise, note that $|\!\Vert \centerdot |\!\Vert _*$ is the dual norm of $|\!\Vert \centerdot |\!\Vert$ and not the other way around, while $\Vert \centerdot \Vert _{L^{q_d^\prime }}$ and $\Vert \centerdot \Vert _{L^{q_d}}$ are dual of each other. Hence, $Z_{\lambda ,p^\prime }$ is reflexive and $Y_\lambda$ is not.

Now, we show how to derive the Propositions 2.5, 2.6 and 2.7 stated in the Sect.  2. Start by the first of these three propositions. The density of ${\mathcal {S}}(\mathbb {R}^d)$ in $Y_\lambda$ and $Z_\lambda$ is explicitly stated in the Lemmas A.1 and A.2, in particular, the density also holds for $X_\lambda = Y_\lambda + Z_\lambda$ with its corresponding norm. This proves the Proposition 2.5. Now, we turn to the Proposition 2.6. Since $(Y_\lambda , \Vert \centerdot \Vert _{Y_\lambda })$ and $(Z_\lambda , \Vert \centerdot \Vert _{Z_\lambda })$ are Banach spaces and $Y_\lambda$ and $Z_\lambda$ are subspaces of ${\mathcal {S}}^\prime (\mathbb {R}^d)$, we have by Theorem 1.3 in [3] that $(X_\lambda , \Vert \centerdot \Vert _{X_\lambda } )$ is a Banach space. The identity (17) is a standard property of Banach spaces (see Corollary 1.4 in [4]). This concludes the proof of the Proposition 2.6. Finally, let us focus on the last of these three propositions. It is a well-known fact—since ${\mathcal {S}}(\mathbb {R}^d)$ is dense in $Y_\lambda$ and $Z_\lambda$—that $(X_\lambda ^*, \Vert \centerdot \Vert _{X_\lambda ^*} )$ is isomorphic to the space $Y_\lambda ^*\cap Z_\lambda ^*$ endowed with the norm $\max \{ \Vert \centerdot \Vert _{Y_\lambda ^*} , \Vert \centerdot \Vert _{Z_\lambda ^*} \}$ (see 2 in the section Exercises and Further Results for Chapter 3 of the book [3]). Note that this later space is actually isomorphic to the space described in the Proposition 2.7 endowed with the norm (18). To finish the proof of this proposition, it is enough to check the density of ${\mathcal {S}}(\mathbb {R}^d)$ in $X_\lambda ^*$ with its corresponding norm. Note that this holds because ${\mathcal {S}}(\mathbb {R}^d)$ is dense in $Y_\lambda ^*$ and $Z_\lambda ^*$, and the norm $\Vert \centerdot \Vert _{X_\lambda ^*}$ is equivalent to $\max \{ \Vert \centerdot \Vert _{Y_\lambda ^*} , \Vert \centerdot \Vert _{Z_\lambda ^*} \}$.