Scattering theory with both regular and singular perturbations

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(\Delta_B,\Delta)$, where $\Delta$ is the free self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $\Delta_B$ is a self-adjoint operator in a class of Laplacians with both a regular perturbation, given by a short-range potential, and a singular one describing boundary conditions (like Dirichlet, Neumann and semi-transparent $\delta$ and $\delta'$ ones) at the boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting absorption principle for $A_B$ and a Krein-like formula for the resolvent difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices) and the singular perturbations.


Introduction
The mathematical scattering theory for short-range potential is a well developed subject; the existence and completeness of the wave operators can be obtained by two essentially different approaches: the trace-class method and the smooth method (see, e.g., [22]). An important object defined in terms of the wave operators is the scattering operator and, even more important from the point of view of its physical applications, the scattering matrix, which is its reduction to a multiplication operator in the spectral representation of the self-adjoint free Laplacian.
The scattering problem for singular perturbations of self-adjoint operators, which is outside the original scope of these methods, is connected with scattering from obstacles with impenetrable or semi-transparent boundary conditions (see, e.g., [3], [4], [11]- [14]). On this side, a general scheme has been developed in [11] by combining the construction in [15] with an abstract version of the Limiting Absorption Principle (simply LAP in the following) due to W. Renger (see [18]) and a variant of the smooth method due to M. Schechter (see [19]). In particular, the results in [11] apply to obstacle scattering with a large class of interface conditions on Lipschitz hypersurfaces in any dimension. Let us recall that in [4] boundary triple theory and properties of the associated operator-valued Weyl functions were used to obtain a similar representation of the scattering matrix for singularly coupled self-adjoint extensions. It is worth to remark that, while the approach in [11] avoids any trace-class condition, these are needed in [4] and so the applications there are limited to the case of smooth obstacles in two dimensions.
The target of the present paper is to provide a general framework for the scattering with both potential type and singular perturbations. Since our concern is the scattering theory with respect to the free Laplacian, we regard the regular and the singular parts of the perturbation as a single object; this constitutes the main novelty of our approach. In particular, we give an abstract resolvent formula, generalizing the one in [15], which puts on an equal footing the two components of the perturbation. Such a representation is a key ingredient in the derivation of LAP which leads then to the main results of the first part: the asymptotic completeness and an explicit formula for the scattering matrix. These results rely on a certain number of assumptions whose validity is carefully analyzed in the second part where we consider the specific case of a short range potential plus a distributional term, supported on a closed surface and describing self-adjoint interface conditions. In this way, we obtain new representation formulae for the scattering matrix which are expected to be relevant in different physical applications involving wave propagation in inhomogeneous media with impenetrable or semi-transparent obstacles.
Here, in more details, the contents of the paper. In Section 2, following the scheme proposed in [15], we provide an abstract resolvent formula for a perturbations A B of the self-adjoint A by a linear combination of the adjoint of two bounded trace-like maps τ 1 : dom(A) → h 1 and τ 2 : dom(A) → h 2 ; while the kernel of τ 2 is required to be dense, so τ * 2 plays the role of a singular perturbations, no further hypothesis is required for τ 1 and in applications that allows τ * 1 to represent a regular perturbations by a short-range potential. In Subsection 2.3, by block operator matrices and the Schur complement, we re-write the obtained resolvent formula in terms of the resolvent of the operator corresponding to the non singular part of the perturbations; that plays an important role in the subsequent part regarding LAP and the scattering theory.
In Section 3, following the scheme proposed in [13] and further generalized in [11], at first we provide, under suitable hypothesis, a Limiting Absorption Principle for A B (see Theorem 3.1) and then an aymptotic completeness criterion for the scattering couple (A B , A) (see Theorem 3.5). Then, by a combination of LAP with stationary scattering theory in the Birman-Yafaev scheme and the invariance principle, we obtain a representation formula for the scattering matrix of the couple (A B , A) (see Theorem 3.11). Whenever A is the free Laplacian in L 2 (R 3 ), such a formula contains, as subcases, both the usual formula for the perturbation given by a short-range potential as given, e.g., in [22] and the formula for the case of a singular perturbation describing self-adjoint boundary conditions on a hypersurface as given in [11].
In Section 4, in order to apply our abstract results to the case in which A is the free 3D Laplacian and the regular part represents a perturbation by a potential, we give various regularity results for the boundary layer operators associated to ∆ + v, where v is a potential of Kato-Rellich type.
In Sections 5 and 6 we present various applications, where the free Laplacian is perturbed both by a regular term, given by a short range potential v decaying as |x| −κ(1+ǫ) , and by a singular one describing either separating boundary conditions (as Dirichlet and Neumann ones) or semitransparent (as δ and δ ′ type ones). In order to satisfy all our hypotheses, we need κ = 2. However, all our hypotheses but a single one (see Lemma 5.6) hold with κ = 1; we conjecture that the requirement κ = 2 is merely of technical nature and that our results are true for a short range potential decaying as |x| −(1+ǫ) . Finally, let us remark that whenever one is only interested in the construction of the operators and not in the scattering theory, then it is sufficient to assume that v is a Kato-Rellich potential (see Subsection 5.1).
Schrödinger operators with a Kato-Rellich potential plus a δ-like perturbation with a p-summable strength (p > 2) have been already considered in [14], while for a different construction with a bounded potential and a δ-or a δ ′ -like perturbation with bounded strength we refer to [3]. None of such references considered the scattering matrix (however, [14] provided a limiting absorption principle). Whenever the singular part of the perturbations is absent, our framework extends from compactly supported potentials in one dimension to short range potentials in three dimensions the kind of results provided in [5,Section 5].
Let us notice that, building on the results in [1] and [11], the abstract models introduced in Section 2 and the related scattering theory presented in Section 3 apply to perturbations of the Laplacian in R n , n ≥ 2, with a suitable short-range potential plus a singular term supported on a bounded hypersurface of co-dimension one.
1.1. Some notation and definition.
• · X denotes the norm on the complex Banach space X; in case X is a Hilbert space, ·, · X denotes the (conjugate-linear w.r.t. the first argument) scalar product.
• ·, · X * ,X denotes the duality (assumed to be conjugate-linear w.r.t. the first argument) between the dual couple (X * , X).
• L * : dom(L * ) ⊆ Y * → X * denotes the dual of the densely defined linear operator L : dom(L) ⊆ X → Y ; in a Hilbert spaces setting L * denotes the adjoint operator.
• B(X, Y ), B(X) ≡ B(X, X), denote the Banach space of bounded linear operator on the Banach space X to the Banach space Y ; · X,Y denotes the corresponding norm.
• S ∞ (X, Y ) denotes the space of compact operators on X to Y .
• X ֒→ Y means that X is continuously embedded into Y .
• H s (Ω) and H s (Ω ex ) denote the scales of Sobolev spaces.
where H s w (Ω ex ) denotes the weighted Sobolev space relative to the weight x w .
• γ in/ex 0 and γ in/ex 1 denote the interior/exterior Dirichlet and Neumann traces on the boundary Γ.
• SL z and DL z denote the single-and double-layer operators.
• D ⊂ R is said to be discrete in the open set E ⊃ D whenever the (possibly empty) set of its accumulations point is contained in R\E; D is said to be discrete whenever E = R. • Given x ≥ 0 and y ≥ 0, x y means that there exists c ≥ 0 such that x ≤ c y.
2. An abstract Kreȋn-type resolvent formula 2.1. The resolvent formula. Let A : dom(A) ⊆ H → H be a self-adjoint operator in the Hilbert space H. We denote by R z := (−A + z) −1 , z ∈ ̺(A), its resolvent; one has R z ∈ B(H, H A ), where H A is the Hilbert space given by dom(A) equipped with the scalar product be auxiliary Hilbert spaces with dense continuous embedding; we do not identify h k with its dual h * k (however, we use h k ≡ h * * k ) and we work with the h * k -h k duality ·, · h * k ,h k defined in terms of the scalar product of the intermediate Hilbert space h • k . The scalar product and hence the duality are supposed to be conjugate linear with respect to the first variable; notice that ϕ, φ h k ,h * k = φ, ϕ * h * k ,h k . Given the bounded linear maps we introduce the bounded operators We further suppose that there exist reflexive Banach spaces b k , k = 1, 2, with dense continuous is contained in the domain of definition of some (supposed to exist) (b 1 ⊕ b 2 )-valued extension of τ (which we denote by the same symbol) in such a way that Proof. By (2.4), one gets This entails, by the definitions (2.5) and (2.6), By the resolvent identity, there follows Now, we provide an additive representation of the self-adjoint A B in Theorem 2.1. Then

2.
3. An alternative resolvent formula. At first, let us notice that hypothesis (2.2), can be re-written as Regarding the well-posedness of (2.15), taking into account the definition of C B z , one has Then, by using the same kind of arguments as in the proofs of Lemma 2.5 and Theorem 2.6, one gets the following Theorem 2.11. Let A B be the self-adjoint operator in Theorem 2.9. Then, for any z ∈ ̺(A B ) ∩ ̺(A B 1 ), one has the representation

The Limiting Absorption Principle and the Scattering Matrix
Now, given the measure space From now on ·, · and · denote the scalar product and the corresponding norm on L 2 (M ); ·, · ϕ and · ϕ denote the scalar product and the corresponding norm on L 2 ϕ (M ). Then we introduce the following hypotheses: (H1) A B 1 is bounded from above and there exists a positive λ 1 ≥ sup σ(A B 1 ), such that R B 1 z ∈ B(L 2 ϕ (M )) for any z ∈ ̺(A B 1 ) such that Re(z) > λ 1 ; (H2) A B 1 satisfies a Limiting Absorption Principle (LAP for short), i.e. there exists a (eventually empty) closed set with zero Lebesgue measure e(A B 1 ) ⊂ R such that, for all λ ∈ R\e(A B 1 ), the limits We split next hypothesis (H4) in two separate points: (H4.1) A B is bounded from above; (H4.2) the embedding h 2 ֒→ b 2 is compact and there exists a positive λ 2 > sup σ(A B 1 ), such that G B 1 z ∈ B(h * 2 , L 2 ϕ 2+η (M )) for some η > 0 and for any z ∈ ̺(A B 1 ) such that Re(z) > λ 2 . Then, A B satisfies a Limiting Absorption Principle as well: Proof. We use [11, Theorem 3.1] (which builds on [18]). By (H1), (2.32) and (H4.2), R B 1 z and R B z are in B(L 2 ϕ (M )) and z → R B 1 z and z → R B z are continuous since pseudo-resolvents in B(L 2 ϕ (M )); A B is bounded from above by (H4.1). Therefore hypothesis (H1) in [11] holds true. Our hypotheses (H2) and (H3) coincides with the same ones in [11]. By (H4.2), the embedding b . Therefore hypothesis (H4) in [11] holds and the statement is a consequence of [11, Theorem 3.1]. Finally, (3.5) is an immediate consequence of Weyl's Theorem.
Let us now assume that (H5) the limits Then, by [11,Lemma 3.6], one gets the following: Moreover, for any λ ∈ R\e(A B ), the limits By the same reasoning as at the end of [11, proof of Theorem 5.1], one can improve the result regarding (3.8): . Before stating the next results, we recall the following: Definition 3.4. Given two self-adjoint operators A 1 and A 2 in the Hilbert space H, we say that completeness holds for the scattering couple (A 1 , A 2 ) whenever the strong limits where P ac k denotes the orthogonal projector onto the absolutely continuous subspace H ac k of A k . Furthermore, we say the asymptotic completeness holds for the scattering couple (A 1 , A 2 ) whenever, beside completeness, one has Proof. By (2.32) and by the same proof as in Lemma 2.4, one gets Then, by hypotheses (H1)-(H5) and by [ and denote by e ′ ess (A B 1 ) the set of accumulation points of e ess (A B 1 ). Since an open set minus a discrete subset is still open, one has where the I n 's are open intervals. Moreover, since I n ∩e ′ ess (A B 1 ) = ∅, then I n ∩e ess (A B 1 ) is discrete in I n and so I n \(I n ∩ e ess (A This gives By standard arguments (see e.g.

3.1.
A representation formula for the scattering matrix. According to Theorem 3.5, under the assumptions there stated, the scattering operator is a well defined unitary map. Let We define the scattering matrix λ u λ . Now, following the same scheme as in [11], which uses the Birman-Kato invariance principle and the Birman-Yafaev general scheme in stationary scattering theory, we provide an explicit relation between S B λ and Λ B, , we consider the scattering couple (R B µ , R µ ) and the strong limits where P µ ac is the orthogonal projector onto the absolutely continuous subspace of R µ ; we prove below that such limits exist everywhere in L 2 (M ). Let S µ B the corresponding scattering operator We introduce a further hypothesis (H7), which we split in four separate points: (H7.1) A is bounded from above and satisfies a Limiting Absorption Principle: there exists a (eventually empty) closed set e(A) ⊂ R of zero Lebesgue measure such that for all λ ∈ R\e(A) the limits Remark 3.7. By τ 2 G 1 z = τ 2 (τ 1 Rz) * = (τ 1 (τ 2 R z ) * ) * = (τ 1 G 2 z ) * , hypothesis (H7.4) entails the existence in B(b 2 , b * 1 ), for any λ ∈ R\e(A B 1 ), of the limits (3.16) Remark 3.8. Whenever one strengthens hypotheses (H7.2) as in (H5), then, by the same kind of proof that leads to the existence of the limit (3.8) (see [11,Lemma 3.6]), one gets the existence of the limits requested in hypotheses (H7.3).
Let us further notice that, whenever A is the free Laplacian in L 2 (R 3 ) and B 1 corresponds to a perturbation by a regular potential as in Section 5 below, then (3.34) gives the usual formula for the scattering matrix for a short-range potential (see, e.g., [22,Section 8]).

Kato-Rellich perturbations and their layers potentials
4.1. Potential perturbations. In this section we suppose that the real-valued potential v is of Kato-Rellich type, i.e., v ∈ L 2 (R 3 ) + L ∞ (R 3 ), equivalently, We use the same symbol v to denote both the potential function and the corresponding multiplication operator u → vu. Given Hence v ∈ B(H 2 (R 3 \Γ), L 2 (R 3 )). Then, for any u, v ∈ H 2 (R 3 \Γ), one has and so u → vu extends to a map in B(L 2 (R 3 ), H 2 (R 3 \Γ) * ). The proof is then concluded by interpolation.
In later proofs we will need the estimate provided in the following: Lemma 4.9. There exist c 1 > 0, c 2 > 0 such that, for any u ≡ u in ⊕ u ex ∈ H 1 (R 3 \Γ) and for any ε > 0, there holds Proof. By H 1 (Ω in/ex ) ֒→ H 3/4 (Ω in/ex ) ֒→ L 4 (Ω in/ex ), by the Gagliardo-Niremberg inequalities (see [6] for the interior case and [8] for the exterior one) and by Young's inequality Proof. By (4.18) and by the polarization identity, for any u and v in The proof is then concluded by taking u = R z u • , u • ∈ H −1 (R 3 ), and by where d z is the distance of z from [0, +∞).
In order to prove the jump relations of the double-layer operator relative to ∆ + v we need a technical result: Proof. At first let us notice that it suffices to show that the result holds for a single z ∈ C\(−∞, 0]. Indeed, by the resolvent identity In particular, we choose z such that ker(S z ) = {0} (see, e.g., Lemma (4.19) below).

Laplacians with regular and singular perturbations
Here we apply the abstract results in Section 2, presenting various examples were the self-adjoint operator A is the free Laplacian ∆ : All over this section we consider a Kato-Rellich potential v = v 2 + v ∞ of short-range type, i.e., We take , and, introducing the multiplication operator x by x u : x → (1 + |x| 2 ) 1/2 u(x), we define Further, we take either Hence, by what is recalled in Subsection 4.2, either G 2 z = SL z or G 2 z = DL z and either 2) holds. Notice that γ * 0 φ and γ * 1 φ, whenever φ ∈ L 2 (Γ), identify with the tempered distributions which act on a test function f respectively as where ν is the exterior normal to Γ. By a slight abuse of notation, in the following we set γ * 0 φ ≡ φδ Γ and φγ * 0 ≡ δ ′ Γ φ and so, either In this framework, given a couple of linear operators B 0 and B 2 as in (2.3) and such that the triple B = (B 0 , B 1 , B 2 ) satisfies the hypotheses in Theorem 2.1, equation (2.7) defines a selfadjoint operator ∆ B representing a Laplacian with a Kato-Rellich potential and a distributional one supported on Γ. Let us remark that, although τ 1 and B 1 depend on the index s, the operator ∆ B is s-independent whenever B 0 and B 2 are (see the next subsections). The choice s = 0 is a technical trick which we use to obtain LAP and a representation formula for the scattering couple (∆ B , ∆); whenever one is only interested in providing a resolvent formula for ∆ B , then the choice s = 0 is preferable. In particular, the resolvent formula for ∆ B holds in the setting s = 0 for any Kato-Rellich potential.
5.1. The Schrödinger operator. By our hypotheses on v, one has x 2s v ∈ L 2 (R 3 ) + L ∞ (R 3 ) and so, by Lemma 4.1, Considering the weight ϕ(x) = (1+ |x| 2 ) w/2 , w ∈ R, we use the notation L 2 denotes the corresponding scales of weighted Sobolev spaces. Since ) and, by duality, In particular, this gives Let v be as in (5.1), with κ = 1. Then, for s such that 0 ≤ 2s < 1 + ε and for z ∈ C sufficiently far away from (−∞, 0], Here we use the same kind of arguments as in the second part of the proof of Lemma 4.10. Thus we start from the resolvent identity By Lemma 4.10, such an equality holds in B(H 1 (R 3 \Γ) * ). By (4.9), By Lemma 5.1, Z B 1 = ∅ and by the relation . Therefore, Theorem 4.2 (see also Remark 4.3) yields The above relation shows that ∆ B 1 coincides with the Schrödinger operator ∆ + v provided by the Kato-Rellich theorem. This also shows that ∆ B 1 is s-independent. Nevertheless, the operator Λ B 1 z depends on the choice of s and the relations (5.9) and (5.10) with s = 0 are key objects in our analysis of LAP and scattering theory in the general case.

5.2.
Asymptotic completeness and scattering matrix. Before discussing the validity of our assumptions, we provide the following general results on the scattering couple (∆ B , ∆).
As in the previous subsections we use the weight ϕ(x) = (1+|x| 2 ) w/2 , w ∈ R; the notation for the corresponding weighted spaces are: L 2 w (R 3 ), H k w (R 3 ) and H k w (R 3 \Γ). From now on, the parameter s in the definitions (5.2) and (5.3) is restricted to the range (5.13) 1 < 2s < 1 + ε .