Asymptotic Completeness and S-Matrix for Singular Perturbations
aa r X i v : . [ m a t h - ph ] J a n Asymptotic Completeness and S-Matrix for Singular Perturbations
Andrea Mantile a , Andrea Posilicano b, ∗ a Laboratoire de Math´ematiques, Universit´e de Reims - FR3399 CNRS, Moulin de la Housse BP 1039, 51687 Reims, France b DiSAT - Sezione di Matematica, Universit`a dell’Insubria, Via Valleggio 11, I-22100 Como, Italy
Abstract
We give a criterion of asymptotic completeness and provide a representation of the scattering matrix forthe scattering couple ( A , A ), where A and A are semi-bounded self-adjoint operators in L ( M , B , m ) suchthat the set { u ∈ dom( A ) ∩ dom( A ) : A u = Au } is dense. No sort of trace-class condition on resolventdi ff erences is required. Applications to the case in which A corresponds to the free Laplacian in L ( R n )and A describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces aregiven. R´esum´e
On fournit un crit`ere pour la compl´etude asymptotique et une repr´esentation de la matrice de la di ff usionpour un syst`eme de di ff usion ( A , A ), ´etant A et A op´erateurs autoadjointes demi-born´es dans L ( M , B , m )tels que l’ensemble { u ∈ dom( A ) ∩ dom( A ) : A u = Au } est dense. Aucune condition de trace sur lesr´esolvantes est requise. On consid`ere des applications aux cas o`u A est le Laplacien libre dans L ( R n ) et A d´ecrit le Laplacien avec conditions au bord autoadjointes sur une hypersurface compacte et non r´eguli`ere. Keywords:
Abstract scattering theory, Scattering matrix, Self-adjoint extensions of symmetric operators
1. Introduction.
Let A : dom( A ) ⊆ H → H be a self-adjoint operator in the Hilbert space H . Another self-adjointoperator A : dom( A ) ⊆ H → H is said to be a singular perturbation of A if the set N : = { u ∈ dom( A ) ∩ dom( A ) : A u = Au } is dense in H (see e.g. [23], [34]); in typical situations A and A correspond to thesame di ff erential expression and they di ff er due to some boundary conditions imposed on a null subset.Since the subspace N is closed with respect to the graph norm of A , the linear operator S : = A | N ,obtained by restricting A to N , is a densely defined closed symmetric operator and A is one of its self-adjoint extensions. Therefore to find all singular perturbations of A it su ffi ces to pick out H -dense subspaces N ( dom( A ), closed with respect to the graph norm of A , and then to look for the self-adjoint extensionsof S = A | N : for any of such a self-adjoint extensions A , A one has dom( S ) = N ⊆ N = { u ∈ dom( A ) ∩ dom( A ) : A u = Au } and so N is dense. Since dom( A ) is a Hilbert space with respect to thescalar product h u , v i A : = h u , v i H + h A u , A v i H , and N is closed with respect to the corresponding norm, ∗ Corresponding author
Email addresses: [email protected] (Andrea Mantile), [email protected] (AndreaPosilicano)
Preprint submitted to Journal de Math´ematiques Pures et Appliqu´ees January 29, 2019 ne has dom( A ) = N ⊕ N ⊥ and so, without loss of generality, we can suppose that N = ker( τ ), where τ : dom( A ) → h is a bounded and surjective linear operator, h ( ≃ N ⊥ ) being an auxiliary Hilbert space, i.e. τ is a sort of a (abstract) trace map.In Section 2, building on [32], we provide, by a Kre˘ın-type resolvent formula (see Theorem 2.4), theset of singular perturbations of a given self-adjoint A in terms of certain families Λ = { Λ z } z ∈ Z Λ of boundedlinear maps Λ z : b → b ∗ , where b is a reflexive Banach space such that h ֒ → b is a continuous immersionand Z Λ is a not empty subset of the resolvent set of A . By an abstract Green-type formula, this entailsthe relation h u , A Λ v i H = h A u , v i H + h τ u , ̺ v i h , h ∗ , where ̺ is another h ∗ -valued (abstract) trace map; such arelation permits us to employ a variation (due to Schechter, see [40] and [41]) of the Cook-Kato-Kurodamethod to get existence and completeness of the wave operators for the scattering couple ( A , A Λ ) in termsof conditions about the map τ and the operator family Λ (see Theorem 2.8).In order to implement such conditions towards applications, in Section 3 we provide a Limiting Ab-sorption Principle (LAP for short) holding, under certain conditions (see hypotheses H1-H4 there), forself-adjoint operators of the kind A Λ defined in spaces of square integrable functions on arbitrary measurespaces ( M , B , m ). This permits, under some further hypotheses (see hypotheses H5 and H6 in Section 3),to obtain an abstract result about asymptotic completeness for the scattering couple ( A , A Λ ) (see Theorem3.9).In Section 4, under the same hypotheses H1-H6 and using both the Birman-Kato invariance principleand Birman-Yafaev general scheme in scattering theory (see [8], [45], [46]), we provide an explicit relation(see Corollary 4.3) between the Scattering Matrix S Λ λ associated to the scattering couple ( A , A Λ ) and thelimit operator Λ + λ : = lim ǫ ↓ Λ λ + i ǫ ; such a limit exists in B ( h , h ∗ ) by LAP (see Lemma 3.6).Self-adjoint realizations of the Laplacian operator with boundary or interface conditions on a closedand bounded hypersurface in R n can be interpreted as singular perturbations of the free Laplacian; hencethe scattering theory for these models naturally develops within the abstract scheme presented above. Thispoint is considered in the final Section 5, where we specialize to the case in which the self-adjoint operator A coincides with the free Laplacian in L ( R n ), i.e. A = ∆ : H ( R n ) ⊂ L ( R n ) → L ( R n ), where H ( R n )denotes the usual Sobolev space of order two. Supposing that to the abstract trace map τ : H ( R n ) → h corresponds a distribution with compact support, i.e ran( τ ∗ ) ⊆ H − comp ( R n ), and under a compactnesshypothesis on Λ , we can apply our results to a wide set of singular perturbations of the free Laplacian (seeTheorem 5.1). Moreover, in such a setting the operator limit Λ + λ appearing in the representation of theScattering Matrix S Λ λ exists in the more convenient (as regards applications) space B ( b , b ∗ ). In particular,we give applications to the case of scattering from Lipschitz bounded obstacles in R n both with Dirichlet(see Subsection 5.1) and Neumann (see Subsection 5.2) boundary conditions, to scattering for Schr¨odingeroperators ∆ α in L ( R n ) with δ -type potentials with unbounded strengths α supported on bounded d -setswith 0 < n − d < d = n −
1, finite unions of Lipschitz hypersurfaces whichmay intersect on subsets having zero ( n − d is not an integer, self-similar fractals), see Subsection 5.3, and to scattering for Schr¨odinger operators ∆ θ in L ( R n ) with δ ′ -typepotentials with strength θ − supported on Lipschitz hypersurfaces (see Subsection 5.4).Beside their interest in Quantum Mechanics, Laplace operators with boundary or interface conditions onhypersurfaces (in particular with semi-transparent boundary conditions corresponding to δ and δ ′ singularpotentials) provide relevant models for classical scattering from obstacles or non-homogeneous acousticmedia (see the recent paper [28]). Playing a central role in direct and inverse scattering problems, thescattering amplitude (strictly related to the far-field pattern used in wave scattering, see, e.g., [21, Chapter6]) easily derives from the S -matrix. Hence, our results yield to a rigorous definition and an explicit formulafor this map, in the regime where the obstacles boundary or the singularity surface of the acoustic densityhave low regularity; this represents an important by-product and a relevant perspective of our work.We conclude this introduction describing how our results extend and connect with previously known2nes. Since, by [33] (see also [35, Theorem 2.5]), the operators A Λ have an additive representation of thekind A Λ = A + T Λ , our abstract results extend existence and completeness of scattering provided in [16]and in [9] for − ∆ + µ , µ a signed measure (in fact Ford’s paper [16] was our main inspiration in writing thepresent work).The construction developed in this work can be easily recast into the language of boundary triple theory(see [11], [42, Section 14]), the maps G z playing the role of γ -fields and the maps Λ z being the inverses ofthe Weyl functions (see [34]); since we do not require any trace-class condition on resolvents di ff erences,our results can be regarded as extensions of the abstract results provided in [7, Section 3].In Section 5 we extend to the Lipschitz case the results, there provided for smooth hypersurfaces, ap-pearing in [27]; these already extended the results given in [7, Section 5]. In more detail, the expressions forthe scattering matrix we provide in (5.38) relative to Dirichlet obstacles and in (5.39) relative to Neumannobstacles, extend to any dimension and to Lipschitz obstacles the similar ones obtained for two-dimensionalobstacles with piecewise C boundary in [12, Theorems 5.3 and 5.6] and [13, Theorems 4.2 and 4.3]; sim-ilar formulae are also given, in a smooth two dimensional setting in [7, Subsections 5.2 and 5.3] and in asmooth n -dimensional setting in [27, Subsections 6.1 and 6.2].The construction of the operator ∆ α with semi-transparent boundary condition of δ -type provided inTheorem 5.9 extend, as regards the regularity of the boundary and / or the class of admissible strength func-tions, previous constructions given, for example, in [19], [16], [10], [32], [6], [26], [15]. Asymptoticcompleteness for the scattering couple ( ∆ , ∆ α ) provided in Theorem 5.9 extend results on existence andcompleteness given, in the case the boundary is smooth and the strength are bounded, in [6] and [26]. Theformula for the scattering matrix provided in (5.44) (respectively in (5.47)) extends to d -sets (respectivelyto Lipschitz hypersurfaces) the results given, in the case of a smooth hypersurface, in [27, Subsections 6.4and 7.4] and, in the case of a smooth 2- or 3-dimensional hypersurface, in [7, Subsection 5.4] (see also theformula provided in [17] for Schr¨odinger operators of the kind − ∆ + µ , µ a signed measure).The construction of the operator ∆ θ with semi-transparent boundary condition of δ ′ -type provided inTheorem 5.15 extend, as regards the regularity of the boundary and / or the class of admissible strengthfunctions, previous constructions given in [6], [26], [15]. Asymptotic completeness for the scattering cou-ple ( ∆ , ∆ θ ) provided in Theorem 5.15 extend results on existence and completeness given, whenever theboundary is smooth and θ is bounded, in [6] and [26]. The formula for the scattering matrix provided in(5.53) extend to Lipschitz hypersurfaces the results given, in the case of a smooth hypersurface, in [27,Subsections 6.5 and 7.5]. Acknowledgments.
During the preparation of this work, the authors profited of some stays at the CNRSInstitute Wolfgang Pauli of Vienna, which they gratefully acknowledge for the kind financial support. • k · k X denotes the norm on the complex Banach space X ; in case X is a Hilbert space, h· , ·i X denotes the(conjugate-linear w.r.t. the first argument) scalar product. • h· , ·i X ∗ , X denotes the duality (assumed to be conjugate-linear w.r.t. the first argument) between the dualcouple ( X ∗ , X ). • L ∗ : dom( L ∗ ) ⊆ Y ∗ → X ∗ denotes the dual of the densely defined linear operator L : dom( L ) ⊆ X → Y ; ina Hilbert spaces setting L ∗ denotes the adjoint operator. • ρ ( A ) and σ ( A ) denote the resolvent set and the spectrum of the self-adjoint operator A ; σ p ( A ), σ pp ( A ), σ ac ( A ), σ sc ( A ), σ ess ( A ), σ disc ( A ), denote the point, pure point, absolutely continuous, singular continuous,essential and discrete spectra. 3 B ( X , Y ), B ( X ) ≡ B ( X , X ), denote the Banach space of bounded linear operator on the Banach space X tothe Banach space Y ; k · k X , Y denotes the corresponding norm. • S ∞ ( X , Y ) denotes the space of compact operators on the Banach space X to the Banach space Y . • X ֒ → Y means that X ⊆ Y and for any u ∈ X there exists c > k u k Y ≤ c k u k X ; we say that X iscontinuously embedded into Y . • Given the measure space ( M , B , m ), L ( M , B , m ) ≡ L ( M ) denotes the corresponding Hilbert space ofmeasurable, square-integrable functions. • u | Γ denotes the restriction of the function u to the set Γ ; L | V denotes the restriction of the linear operator L to the subspace V . • u ξλ denotes the plane wave with direction ξ and wavenumber | λ | , i.e. u ξλ ( x ) = e i | λ | ξ · x . • P in / ex z and Q in / ex z denote the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators relative to thedomain Ω in / ex , where Ω in ≡ Ω and Ω ex : = R n \ Ω .
2. Singular perturbations of self-adjoint operators.
Given the self-adjoint operator A : dom( A ) ⊆ H → H in the Hilbert space H and the auxiliary Hilbert space h , let τ : dom( A ) → h be continuous (w.r.t. the graph norm in dom( A )) and surjective. We further assume that ker( τ ) is densein H . For notational convenience we do not identify h with its dual h ∗ ; however we use h ∗∗ ≡ h . Typically h ֒ → h ֒ → h ∗ with dense inclusions and the h - h ∗ duality is defined in terms of the scalar product of theintermediate Hilbert space h .For any z ∈ ρ ( A ) we define R z ∈ B ( H , dom( A )) by R z : = ( − A + z ) − and G z ∈ B ( h ∗ , H ) by G z : h ∗ → H , G z : = ( τ R z ) ∗ , i.e. h G z φ, u i H = h φ, τ ( − A + ¯ z ) − u i h ∗ , h φ ∈ h ∗ , u ∈ H . (2.1)Since ker( τ ) is dense in H , one has (see [32, Remark 2.9],ran( G z ) ∩ dom( A ) = { } . (2.2)However, by the resolvent identity, G z − G w = ( w − z ) R w G z (2.3)and so ran( G z − G w ) ⊂ dom( A ) . (2.4)Notice that by (2.3) there follows G ∗ z G w = G ∗ ¯ w G ¯ z . (2.5)Let us now suppose that there exist a reflexive Banach space b ⊇ h , h ֒ → b , a set C \ R ⊆ Z Λ ⊆ ρ ( A ), and afamily Λ of linear bounded maps Λ z ∈ B ( b , b ∗ ), z ∈ Z Λ , such that (see [32, equations (2) and (4)]) Λ ∗ z = Λ ¯ z , (2.6) Λ w − Λ z = ( z − w ) Λ w G ∗ ¯ w G z Λ z . (2.7)4 emark 2.1. In writing (2.7) we are implicitly using the continuous embeddings h ֒ → b and b ∗ ֒ → h ∗ ; suchembeddings also give k Λ z k h , h ∗ ≤ c k Λ z k b , b ∗ . Remark 2.2.
Notice that whenever Λ z has inverse M z : = Λ − z , then (2.6) and (2.7) are equivalent to M ∗ z = M ¯ z , M z − M w = ( z − w ) G ∗ ¯ w G z . (2.8) Remark 2.3.
Notice that the class of families Λ satisfying (2.6) and (2.7) is not void: it can be parametrizedby couples ( Π , Θ ), where Π : h → ran( Π ) is an orthogonal projection in the Hilbert space h and Θ :dom( Θ ) ⊆ ran( Π ) ∗ → ran( Π ) is self-adjoint, setting (see [32, Section 2], [26, Section 2]) b = h , Λ z = Π ∗ ( Θ − Π τ ( G z − ( G z ◦ + G ¯ z ◦ ) / Π ∗ ) − Π , z ◦ ∈ ρ ( A ) , (2.9) Z Λ = Z Π , Θ : = { z ∈ ρ ( A ) : Θ − Π τ ( G z − ( G z ◦ + G ¯ z ◦ ) / Π ∗ has a bounded inverse } . The set Z Π , Θ always contains C \ R (see the proof of [35, Theorem 2.1]; see also [32, Proposition 2.1]) andso it is not void. In concrete situations it could happen that it is better to work with di ff erent representationsand / or to choose a space b strictly larger than h ; then (2.6) and (2.7) have to be checked case by case. Theorem 2.4.
Let Λ satisfy (2.6) and (2.7) . Then the family of bounded linear maps R Λ z ∈ B ( H ) , z ∈ Z Λ ,defined by R Λ z : = R z + G z Λ z G ∗ ¯ z . (2.10) is the resolvent of a self-adjoint operator A Λ which is a self-adjoint extension of the closed symmetricoperator S : = A | ker( τ ) .Proof. We proceed as in the proof of [32, Theorem 2.1]. By (2.7), z R Λ z is a pseudo-resolvent, i.e. itsatisfies the resolvent identity (see [32, page 113]). Since, by (2.10), u ∈ ker( R Λ z ) gives R z u ∈ ran( G z ), onegets u = R Λ z is injective. Moreover, by (2.6) one gets ( R Λ z ) ∗ = R Λ ¯ z . Thus, by [43, Theorems4.10 and 4.19], R Λ z is the resolvent of a self-adjoint operator A Λ . Let us now fix z ∈ Z Λ . By (2.2) and bydom( A Λ ) = ran( R Λ z ) = { u = u z + G z Λ z τ u z , u z ∈ dom( A ) } , (2.11)one gets dom( A ) ∩ dom( A Λ ) = { u ∈ dom( A ) : Λ z τ u = } . (2.12)Thus, by ( − A Λ + z ) u = ( R Λ z ) − u = ( − A + z ) u z , (2.13)one gets A Λ | ker( τ ) = A | ker( τ ) = S . Remark 2.5.
By the above proof there follows that Theorem 2.4 holds true without requiring that Z Λ contains the whole C \ R : it su ffi ces to suppose that Z Λ ⊆ ρ ( A ) is a not empty set which is symmetric withrespect to the real axis. However, the former hypothesis is used in our successive treatments of ScatteringTheory and Limiting Absorption Principle. Remark 2.6.
By (2.12) and (2.13), one getsker( τ ) ⊆ { u ∈ dom( A ) ∩ dom( A Λ ) : Au = A Λ u } . Since ker( τ ) is dense by our hypothesis, the set { u ∈ dom( A ) ∩ dom( A Λ ) : A u = A Λ u } is dense as well andso A Λ is a singular perturbation of A (see [34]). 5 emark 2.7. By [35, Corollary 3.2] (see also [26, Theorem 2.1]), the representation (2.9) shows that anyself-adjoint extension of S is of the kind provided in Theorem 2.4.Now, in order to simplify the exposition and since such an hypothesis holds true in the applicationsfurther considered, we suppose that A has a spectral gap, i.e. ρ ( A ) ∩ R , ∅ . Then, we pick λ ◦ ∈ ρ ( A ) ∩ R and set G ◦ : = G λ ◦ . (2.14)Let S be the symmetric operator defined by S : = A | ker τ as in Theorem 2.4. By [34, Theorem 3.1] and[26, Lemma 2.3 and Remark 2.4], one has (compare with (2.11) and (2.13))dom( S ∗ ) = { u = u ◦ + G ◦ φ , u ◦ ∈ dom( A ) , φ ∈ h ∗ } , (2.15)( − S ∗ + λ ◦ ) u = ( − A + λ ◦ ) u ◦ , (2.16)(one can check that the definition of S ∗ is λ ◦ -independent) and, defining the bounded linear map ̺ : dom( S ∗ ) → h ∗ , ̺ u ≡ ̺ ( u ◦ + G ◦ φ ) : = φ , (2.17)the following Green’s type identity holds (see [34, Theorem 3.1], [26, Remark 2.4]): h u , S ∗ v i H − h S ∗ u , v i H = h τ u ◦ , ̺ v i h , h ∗ − h ̺ u , τ v ◦ i h ∗ , h , u , v ∈ dom( S ∗ ) . (2.18)Thus, in particular, since A ⊂ S ∗ , dom( A ) = ker( ̺ ) and A Λ ⊂ S ∗ , h u , A Λ v i H = h A u , v i H + h τ u , ̺ v i h , h ∗ , u ∈ dom( A ) , v ∈ dom( A Λ ) . (2.19)The identity (2.19) is our starting point for the following abstract result about scattering for the couple( A , A Λ ): Theorem 2.8.
Let A Λ be defined according to Theorem 2.4. Suppose that there exists an open subset Σ ⊆ R of full measure such that for any open and bounded I, I ⊂ Σ , sup ( λ,ǫ ) ∈ I × (0 , ǫ k G λ ± i ǫ k h ∗ , H < + ∞ , (2.20) and sup ( λ,ǫ ) ∈ I × (0 , k Λ λ ± i ǫ k h , h ∗ < + ∞ . (2.21) Then the strong limitsW ± ( A Λ , A ) : = s- lim t →±∞ e − itA Λ e itA P ac , W ± ( A , A Λ ) : = s- lim t →±∞ e − itA e itA Λ P Λ ac , exist everywhere in H and are complete, i.e. ran( W ± ( A Λ , A )) = H Λ ac , ran( W ± ( A , A Λ )) = H ac , W ± ( A Λ , A ) ∗ = W ± ( A , A Λ ) , where P ac and P Λ ac are the orthogonal projectors onto H ac and H Λ ac , the absolutely continuous subspacesrelative to A and A Λ respectively. roof. At first let us show that (2.20) and (2.21) implysup ( λ,ǫ ) ∈ I × (0 , (cid:16) ǫ k τ R λ ± i ǫ k H , h + ǫ k ̺ R Λ λ ± i ǫ k H , h ∗ (cid:17) < + ∞ . (2.22)By the definition of G z one has k τ R z k H , h = k G ∗ ¯ z k H , h = k G z k h ∗ , H . By (2.10) and (2.4), one has ̺ R Λ z = ̺ ( R z + G z Λ z G ∗ ¯ z ) = ̺ ( R z u + ( G z − G ◦ ) Λ z G ∗ ¯ z u + G ◦ Λ z G ∗ ¯ z ) = Λ z G ∗ ¯ z and so k ̺ R Λ z k H , h ∗ = k Λ z G ∗ ¯ z k H , h ∗ ≤ k Λ z k h , h ∗ k G ∗ ¯ z k H , h = k Λ z k h , h ∗ k G z k h ∗ , H . (2.23)Now we follow the same reasonings as in the proof of [41, Theorem 9.4.2] (see also [40, Section 3]). Atfirst let us notice that in our setting equation (2.19) agree with [41, equation (9.4.1)] whenever the operatorsthere denoted by A and B correspond to τ and ι − ρ respectively, where ι : h → h ∗ is the duality mappinggiven by the canonical isomorphism from h onto h ∗ ; therefore (2.22) corresponds to [41, estimate (9.4.8)].Thus (compare with the first lines of the proof of [41, Theorem 9.4.2]), [41, Lemma 9.3.3, Corollary 9.3.1and Lemma 9.3.2] give, for any u c ∈ H c and u Λ c ∈ H Λ c , Z + ∞−∞ (cid:16) k τ e − itA E ( I ) u c k h + k ̺ e − itA Λ E Λ ( I ) u Λ c k h ∗ (cid:17) dt < + ∞ , (2.24)where E , H c and E Λ , H Λ c denote the spectral measures and the continuous subspaces relative to A and A Λ respectively. According to [41, Theorem 9.4.1], (2.22) and (2.24) give E ( Σ ) u c ∈ M ± ( A Λ , A ) : = { u ∈ H : lim t →±∞ e − itA Λ e itA u exists } and E Λ ( Σ ) u Λ c ∈ M ± ( A , A Λ ) : = { u ∈ H : lim t →±∞ e − itA e itA Λ u exists } . Thus, by H ac ⊆ H c , H Λ ac ⊆ H Λ c , and, since Σ c has Lebesgue measure zero, by E ( Σ ) P ac = P ac , E Λ ( Σ ) P Λ ac = P Λ ac , both the wave operators W ± ( A Λ , A ) and W ± ( A , A Λ ) exist; this also gives completeness (see e.g. [36,Proposition 3, Section XI.3]).
3. The Limiting Absorption Principle and Asymptotic completeness.
Now we suppose that H = L ( M , B , m ) ≡ L ( M ). Given a measurable ϕ : M → (0 , + ∞ ), we define theweighted L -space L ϕ ( M , B , m ) ≡ L ϕ ( M ) : = { u : M → C measurable : ϕ u ∈ L ( M ) } . (3.1)From now on h· , ·i and k · k denote the scalar product and the corresponding norm on L ( M ); h· , ·i ϕ and k · k ϕ denote the scalar product and the corresponding norm on L ϕ ( M ). In the following we suppose ϕ ≥ m -a.e.;therefore L ϕ ( M ) ֒ → L ( M ) ֒ → L ϕ − ( M ) ≃ L ϕ ( M ) ∗ . Then we introduce the following hypotheses: 7H1) both A and A Λ are bounded from above and there exists c > z R z and z R Λ z are continuous on { z ∈ C : Re( z ) > c } to B ( L ϕ ( M ));(H2) A satisfies a Limiting Absorption Principle (LAP for short), i.e. there exists an open set Σ ⊆ R offull measure such that for all λ ∈ Σ the limits R , ± λ : = lim ǫ ↓ R λ ± i ǫ (3.2)exist in B ( L ϕ ( M ) , L ϕ − ( M )) and the maps z R , ± z , where R , ± z ≡ R z whenever z ∈ ρ ( A ), are continuouson Σ ∪ C ± to B ( L ϕ ( M ) , L ϕ − ( M ));(H3) for any compact set K ⊂ Σ there exists c K > λ ∈ K and for any u ∈ L ϕ ( M ) ∩ ker( R , + λ − R , − λ ) one has k R , ± λ u k ≤ c K k u k ϕ ; (3.3)(H4) there exist c > γ > k ∈ N such that for all λ > c ( R Λ λ ) k − ( R λ ) k ∈ S ∞ ( L ( M ) , L ϕ + γ ( M )) . (3.4)Then A Λ satisfies a Limiting Absorption Principle as well: Theorem 3.1.
Suppose hypotheses (H1)-(H4) hold. Then Σ Λ : = Σ ∩ σ p ( A Λ ) is a (possibly empty) discreteset and for all λ ∈ Σ \ Σ Λ the limits R Λ , ± λ : = lim ǫ ↓ R Λ λ ± i ǫ (3.5) exist in B ( L ϕ ( M ) , L ϕ − ( M )) ; the maps z R Λ , ± z , where R Λ , ± z ≡ R Λ z whenever z ∈ ρ ( A Λ ) , are continuous on ( Σ \ Σ Λ ) ∪ C ± to B ( L ϕ ( M ) , L ϕ − ( M )) Proof.
Hypotheses (H1)-(H4) permit us to use the abstract results contained in [39], following the sameargumentations provided in the proof of [27, Theorem 4.2]: hypotheses (T1) and (E1) in [39, page 175]correspond to our (H2), (H3) and (H4); then, by [39, Proposition 4.2], the latter imply hypotheses (LAP),(E) in [39, page 166] and hypothesis (T) in [39, page 168]. In our setting (LAP), (E) and (T) correspondrespectively to (H2), ( R Λ λ ) k − ( R λ ) k ∈ S ∞ ( L ϕ − ( M ) , L ϕ ( M )) , and a technical variant of (H3). According to [39, Theorem 3.5], the three hypotheses (LAP), (E) and (T),together with (H1) (i.e. hypothesis (OP) in [39, page 165]), give the thesis. Remark 3.2.
In order to get Theorem 3.1 one does not need to require Σ to be a set of full measure.However that hypothesis is needed for next Theorem 3.9.Since, as is well known, LAP implies absence of singular continuous spectrum (see e.g. [1, Theorem6.1], [27, Corollary 4.7]), by (H2) and Theorem 3.1, one gets Corollary 3.3.
Suppose hypotheses (H1)-(H4) hold. Then σ sc ( A ) = σ sc ( A Λ ) = ∅ . (3.6) Equivalently ( L ( M ) pp ) ⊥ = L ( M ) ac , ( L ( M ) Λ pp ) ⊥ = L ( M ) Λ ac , (3.7) where L ( M ) pp , ( L ( M ) ac , L ( M ) Λ pp , ( L ( M ) Λ ac denotes the pure point and absolutely continuous subspacesof L ( M ) with respect to A and A Λ . z ∈ ρ ( A ), G ∗ z = τ R z : L ϕ ( M ) → h is surjective;(H5.2) for any λ ∈ Σ , the limits ( G ± λ ) ∗ : = lim ǫ ↓ τ R λ ∓ i ǫ (3.8)exist in B ( L ϕ ( M ) , h ) and are surjective; moreover the maps z ( G ± z ) ∗ , where ( G ± z ) ∗ ≡ G ∗ z whenever z ∈ ρ ( A ), are continuous on Σ ∪ C ∓ to B ( L ϕ ( M ) , h ). Remark 3.4.
By hypothesis (H5) and by duality, for any λ ∈ Σ , the limits G ± λ : = lim ǫ ↓ ( τ R λ ∓ i ǫ ) ∗ (3.9)exist in B ( h ∗ , L ϕ − ( M )) and are injective; moreover the maps z G ± z , where G ± z ≡ G z whenever z ∈ ρ ( A ),are continuous on Σ ∪ C ± to B ( h ∗ , L ϕ − ( M )). Remark 3.5.
Here we recall the definition of reduced minimum modulus γ ( T ) of a linear operator T ∈ B ( X , Y ), T , γ ( T ) : = inf {k T u k Y : dist X ( u , ker( T )) = } . (3.10)By [20, Theorem 5.2, page 231], T has closed range if and only if γ ( T ) >
0. Moreover, see [3, Proposition1.1], ker( T ) = ker( T ) = ⇒ | γ ( T ) − γ ( T ) | ≤ k T − T k X , Y . (3.11)Then, by (3.2), (3.5) and by the resolvent formula (2.10), one gets the following Lemma 3.6.
Suppose hypotheses (H1)-(H5) hold. Then, for any open and bounded I, I ⊂ Σ \ Σ Λ , one has sup ( λ,ǫ ) ∈ I × (0 , k Λ λ ± i ǫ k h , h ∗ < + ∞ . (3.12) Moreover, for any λ ∈ Σ \ Σ Λ , the limits Λ ± λ : = lim ǫ ↓ Λ λ ± i ǫ . (3.13) exist in B ( h , h ∗ ) and R Λ , ± λ = R , ± λ + G ± λ Λ ± λ ( G ∓ λ ) ∗ . (3.14) Proof.
Since ( G ± z ) ∗ are surjective by hypotheses (H5), G ± z are injective and have closed range by the closedrange theorem. Hence, by Remark 3.5, γ ( G ± z ) >
0, and, since ker( G ± z ) = { } for any z ∈ Σ ∪ C ± , the maps z γ ( G ± z ) are continuous on Σ ∪ C ± to (0 , + ∞ ) by (3.11). Hence, by (3.10), for any open and bounded I , I ⊂ Σ , for any ( λ, ǫ ) ∈ I × (0 ,
1) and for any φ ∈ h ∗ , there exist γ ± I > k G λ ± i ǫ φ k ϕ − ≥ γ ± I k φ k h ∗ . Therefore, by (2.10) and (2.6), k R Λ λ ∓ i ǫ − R λ ∓ i ǫ k L ϕ , L ϕ − = k G λ ∓ i ǫ Λ λ ∓ i ǫ G ∗ λ ± i ǫ k L ϕ , L ϕ − ≥ γ ∓ I k Λ λ ∓ i ǫ G ∗ λ ± i ǫ k L ϕ , h ∗ = γ ∓ I k G λ ± i ǫ Λ λ ± i ǫ k h , L ϕ − ≥ γ ∓ I γ ± I k Λ λ ± i ǫ k h , h ∗ . B ( L ϕ ( R n ) , L ϕ − ( R n )) of the limitslim ǫ ↓ G λ ± i ǫ Λ λ ± i ǫ G ∗ λ ∓ i ǫ = lim ǫ ↓ G ± λ Λ λ ± i ǫ ( G ∓ λ ) ∗ = R Λ , ± λ − R , ± λ . (3.15)Then, proceeding in a similar way as above, one has k G ± λ ( Λ λ ± i ǫ − Λ λ ± i ǫ )( G ∓ λ ) ∗ k L ϕ , L ϕ − ≥ γ ( G ± λ ) k ( Λ λ ± i ǫ − Λ λ ± i ǫ )( G ∓ λ ) ∗ k L ϕ , h ∗ = γ ( G ± λ ) k G ∓ λ ( Λ λ ∓ i ǫ − Λ λ ∓ i ǫ ) k h , L ϕ − ≥ γ ( G ± λ ) γ ( G ∓ λ ) k Λ λ ∓ i ǫ − Λ λ ∓ i ǫ k h , h ∗ . This and (3.15) give the existence of the limits (3.13) and then the limit resolvent formulae (3.14).Our last hypothesis is the following:(H6) for any z ∈ ρ ( A ), G z ∈ B ( h ∗ , L ϕ ( M )). Remark 3.7.
By duality, hypothesis (H6) is equivalent to requiring that τ R z has a bounded extension on L ϕ − ( M ) to h for any z ∈ ρ ( A ). Remark 3.8.
By (2.10) and (H6), if R z ∈ B ( L ϕ ( M )) for any z ∈ C such that Re( z ) > c , then the sameis true for R Λ z . Thus the maps z R z and z R Λ z satisfy hypothesis (H1) (they are continuous sincepseudo-resolvents in B ( L ϕ ( M ))).Then the previous results lead to Theorem 3.9.
Suppose that the couple ( A , A Λ ) satisfies hypotheses (H1)-(H6). Then asymptotic complete-ness holds, i.e. the strong limitsW ± ( A Λ , A ) : = s- lim t →±∞ e − itA Λ e itA P ac , W ± ( A , A Λ ) : = s- lim t →±∞ e − itA e itA Λ P Λ ac , exist everywhere in L ( M ) , ran( W ± ( A Λ , A )) = ( L ( M ) Λ pp ) ⊥ , ran( W ± ( A , A Λ )) = ( L ( M ) pp ) ⊥ , W ± ( A Λ , A ) ∗ = W ± ( A , A Λ ) , where P ac and P Λ ac are the orthogonal projectors onto the absolutely continuous subspaces L ( M ) ac andL ( M ) Λ ac .Proof. By Theorem 2.8, to get completeness we need to show that (2.20) and (2.21) hold true. Then,asymptotic completeness is consequence of Corollary 3.3. The bound (2.21) is given in Lemma 3.6 and sowe just need to prove the bound (2.20). Let µ ∈ ρ ( A ) ∩ R ; by (2.3), (2.5) and (2.4), one has | Im( z ) | k G ¯ z φ k = (cid:12)(cid:12)(cid:12) Im( z ) h G ∗ ¯ z G ¯ z φ, φ i h , h ∗ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) h τ ( G ¯ z − G z ) φ, φ i h , h ∗ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) h τ ( G ¯ z − G µ ) φ, φ i h , h ∗ − h τ ( G z − G µ ) φ, φ i h ∗ , h (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) ( µ − z ) h G ∗ z G µ φ, φ i h , h ∗ − ( µ − ¯ z ) h G ∗ ¯ z G µ φ, φ i h , h ∗ (cid:1)(cid:12)(cid:12)(cid:12) ≤ | µ − z | (cid:0) k G ∗ z k L ϕ , h + k G ∗ ¯ z k L ϕ , h (cid:1) k G µ k h ∗ , L ϕ k φ k h ∗ . ǫ k G λ ± i ǫ k h ∗ , L ≤
12 ( | µ − λ | + ǫ ) k G µ k h ∗ , L ϕ (cid:0) k τ R λ + i ǫ k L ϕ , h + k τ R λ − i ǫ k L ϕ , h (cid:1) (3.16)and the bound (2.20) is consequence of hypotheses (H5.2) and (H6).
4. The Scattering Matrix.
According to Theorem 3.9, under hypotheses (H1)-(H6), the scattering operator S Λ : = W + ( A Λ , A ) ∗ W − ( A Λ , A )is a well defined unitary map. Given a direct integral representation of L ( M ) ac with respect to the spectralmeasure of the absolutely continuous component of A (see e.g. [4, Section 4.5.1] ), i.e. a unitary map F : L ( M ) ac → Z ⊕ σ ac ( A ) ( L ( M ) ac ) λ d η ( λ ) (4.1)which diagonalizes the absolutely continuous component of A , we define the scattering matrix S Λ λ : ( L ( M ) ac ) λ → ( L ( M ) ac ) λ by the relation (see e.g. [4, Section 9.6.2]) F S Λ F ∗ u λ = S Λ λ u λ . Now, following the same scheme as in [27, Remark 5.7], which uses the Birman-Kato invariance principleand the Birman-Yafaev general scheme in stationary scattering theory (see e.g. [8], [45], [46]), we providean explicit relation between S Λ λ and Λ + λ .Given µ ∈ ρ ( A ) ∩ ρ ( A Λ ), we consider the scattering couple ( R Λ µ , R µ ) and the strong limits W ± ( R Λ µ , R µ ) : = s- lim t →±∞ e − itR Λ µ e itR µ P µ ac , where P µ ac is the orthogonal projector onto the absolutely continuous subspace of R µ ; we prove below thatsuch limits exist everywhere in L ( M ). Let S µ Λ the corresponding scattering operator S µ Λ : = W + ( R Λ µ , R µ ) ∗ W − ( R Λ µ , R µ ) . Using the unitary operator F µ which diagonalizes the absolutely continuous component of R µ , i.e. ( F µ u ) λ : = λ ( F u ) µ − λ , λ , µ − λ ∈ σ ac ( A ), one defines the scattering matrix S Λ ,µλ : ( L ( M ) ac ) µ − λ → ( L ( M ) ac ) µ − λ corresponding to the scattering operator S µ Λ by the relation F µ S µ Λ ( F µ ) ∗ u µλ = S Λ ,µλ u µλ . Before stating the next results, let us notice the relations (cid:16) − R µ + z (cid:17) − = z + z R µ − z ! , (cid:16) − R Λ µ + z (cid:17) − = z + z R Λ µ − z ! , (4.2)11herefore, by (H2) and Theorem 3.1, the limits (cid:16) − R µ + ( λ ± i (cid:17) − : = lim ǫ ↓ (cid:16) − R µ + ( λ ± i ǫ ) (cid:17) − , λ , , µ − λ ∈ Σ , (4.3) (cid:16) − R Λ µ + ( λ ± i (cid:17) − : = lim ǫ ↓ (cid:16) − R Λ µ + ( λ ± i ǫ ) (cid:17) − , λ , , µ − λ ∈ Σ \ Σ Λ , (4.4)exist in B ( L ϕ ( M ) , L ϕ − ( M )). Theorem 4.1.
Suppose that the couple ( A , A Λ ) satisfies hypotheses (H1)-(H6). Then the strong limitsW ± ( R Λ µ , R µ ) : = s- lim t →±∞ e − itR Λ µ e itR µ P µ ac (4.5) exist everywhere in L ( M ) . Moreover, for any λ , such that µ − λ ∈ σ ac ( A ) ∩ ( Σ \ Σ Λ ) , one hasS Λ ,µλ = − π i L µλ Λ µ (cid:0) + G ∗ µ (cid:16) − R Λ µ + ( λ + i (cid:17) − G µ Λ µ (cid:1) ( L µλ ) ∗ , (4.6) where L µλ : h ∗ → ( L ( M ) ac ) µ − λ , L µλ φ : = λ ( F G µ φ ) µ − λ . (4.7) Proof.
By (2.10), one has R Λ µ − R µ = G µ Λ µ G ∗ µ and we can use [45, Theorem 4’, page 178] (notice thatthe maps there denoted by G and V corresponds to our G ∗ µ and Λ µ respectively). Let us check that thehypotheses there required are satisfied. Since G ∗ µ ∈ B ( L ( M ) , h ), the operator G ∗ µ is | R µ | / -bounded. By(4.2), (H2), Theorem 3.1 and (H6), the limitslim ǫ ↓ G ∗ µ ( − R µ + ( λ ± i ǫ )) − , lim ǫ ↓ G ∗ µ ( − R Λ µ + ( λ ± i ǫ )) − , lim ǫ ↓ G ∗ µ ( − R Λ µ + ( λ ± i ǫ )) − G µ exist. Therefore, to get the thesis we need to check the validity of the remaining hypothesis in [45, Theorem4’, page 178]: G ∗ µ is weakly- R µ smooth, i.e., by [45, Lemma 2, page 154],sup <ǫ< ǫ k G ∗ µ ( − R µ + ( λ ± i ǫ )) − k L , h ≤ c λ < + ∞ , a.e. λ . (4.8)By (4.2), this is consequence ofsup <δ< δ k G ∗ µ R µ − λ ± i δ k L , h ≤ C λ < + ∞ , a.e. λ . (4.9)By k G ∗ µ R z k L , h = k τ R µ R z k L , h = k τ R z R µ k L , h = k R µ ( τ R z ) ∗ k h ∗ , L ≤ k R µ k L , L k G ¯ z k h ∗ , L , (4.9) follows by (3.16), hypotheses (H5.2) and (H6). Thus, by [45, Theorem 4’, page 178], the limits(4.5) exist everywhere in L ( M ) and the corresponding scattering matrix is given by (4.6), where L µλ φ : = ( F µ G µ φ ) λ = λ ( F G µ φ ) µ − λ . 12 emma 4.2. For any z , such that µ − z ∈ ρ ( A ) one has Λ µ (cid:0) + G ∗ µ (cid:16) − R Λ µ + z (cid:17) − G µ Λ µ (cid:1) = Λ µ − z . Proof.
By (2.3), one has G µ + z R µ − z G µ = G µ − z . (4.10)By (4.2), (4.10) and (2.7), one obtains Λ µ + Λ µ G ∗ µ (cid:16) − R Λ µ + z (cid:17) − G µ Λ µ =Λ µ + z Λ µ G ∗ µ G µ + z R µ − z G µ ! Λ µ + G µ − z z Λ µ − z G ∗ µ − z G µ Λ µ !! =Λ µ + z Λ µ G ∗ µ G µ − z Λ µ + z Λ µ G ∗ µ G µ − z (cid:0) Λ µ − z − Λ µ (cid:1) =Λ µ + z Λ µ G ∗ µ G µ − z Λ µ − z = Λ µ − z . Corollary 4.3.
Suppose that the couple ( A , A Λ ) satisfies hypotheses (H1)-(H6). ThenS Λ λ = − π iL λ Λ + λ L ∗ λ , λ ∈ σ ac ( A ) ∩ ( Σ \ Σ Λ ) , where L λ : h ∗ → ( L ( M ) ac ) λ is the µ -independent linear operator defined byL λ φ : = ( µ − λ )( F G µ φ ) λ . Proof.
By Theorem 3.9, Theorem 4.1 and by Birman-Kato invariance principle (see e.g. [4, Section II.3.3]),one has W ± ( A Λ , A ) = W ± ( R Λ µ , R µ )and so S Λ = S µ Λ . Thus, since ( F µ u ) λ = λ ( F u ) µ − λ , one obtains (see also [45, Equation (14), Section 6, Chapter 2]) S Λ λ = S Λ ,µ ( − λ + µ ) − . (4.11)By Lemma 4.2, whenever z = λ ± i ǫ and µ − λ ∈ Σ \ Σ Λ , one gets, as ǫ ↓ Λ µ (cid:0) + G ∗ µ (cid:16) − R Λ µ + ( λ ± i (cid:17) − G µ Λ µ (cid:1) = Λ ± µ − λ . The proof is then concluded by Theorem 4.1, by (4.11) and by setting L λ : = L µ ( − λ + µ ) − . The operator L λ is µ -independent by invariance principle, let us provide a direct proof: given µ , µ , by (2.3) and by( F R µ u ) λ = ( − λ + µ ) − ( F u ) λ , one gets the identity (cid:0) L µ − λ + µ − − L µ − λ + µ − (cid:1) φ = ( µ − λ )( F G µ φ ) λ − ( µ − λ )( F G µ φ ) λ = ( F (( µ − µ ) G µ − ( λ − µ )( G µ − G µ )) φ ) λ = ( µ − µ )( F G µ φ ) λ − ( λ − µ )( µ − µ )( F R µ G µ φ ) λ = . . Applications. Here we take A = ∆ : H ( R n ) ⊂ L ( R n ) → L ( R n ) , where H s ( R n ), s ∈ R , denotes the usual scale of Sobolev spaces and where ∆ denotes the distributionalLaplacian, and τ : H ( R n ) → h bounded and surjective onto the Hilbert space h and such that ker( τ ) is L ( R n )-dense.In the following we use the scale of weighted Sobolev spaces H sw ( R n ), s ∈ R , w ∈ R . Here H w ( R n ) ≡ L w ( R n ) denotes the weighted L -space which corresponds, according to the notation in the Section 3, to thechoice ϕ ( x ) = (1 + | x | ) w / . Then the weighted Sobolev space H mw ( R n ), m ≥ L w ( R n ) having k -order distributional derivatives, 1 ≤ k ≤ m , belonging to L w ( R n ); H sw ( R n ), s > H sw ( R n ), s ∈ ( −∞ , H − s − w ( R n ) (see e.g. [14, Section 4.2]). Theorem 5.1.
Let ∆ Λ denote the self-adjoint extension of the symmetric operator S : = ∆ | ker( τ ) given inTheorem 2.4, corresponding to the family Λ = { Λ z } z ∈ Z Λ , Λ z ∈ B ( b , b ∗ ) , h ֒ → b , and to the choice A = ∆ .Suppose that:i) ∆ Λ is bounded from above;ii) there exists c Λ > such that the embedding ran( Λ λ ) ֒ → h ∗ is compact for any λ > c Λ ;iii) there exists χ ∈ C ∞ comp ( R n ) such that, for any u ∈ H ( R n ) , τ u = τ ( χ u ) (5.1) Then asymptotic completeness holds for the scattering couple ( ∆ , ∆ Λ ) , σ ac ( ∆ Λ ) = σ ess ( ∆ Λ ) = ( −∞ , , σ sc ( ∆ Λ ) = ∅ and the scattering matrix S Λ λ is given byS Λ λ = − π iL λ Λ + λ L ∗ λ , λ ∈ ( −∞ , \ σ − p ( ∆ Λ ) , (5.2) where σ − p ( ∆ Λ ) : = ( −∞ , ∩ σ p ( ∆ Λ ) is a possibly empty discrete set, Λ + λ : = lim ǫ ↓ Λ λ + i ǫ , the limit existing in B ( b , b ∗ ) ,L λ : b ∗ → L ( S n − ) , L λ φ ( ξ ) : = | λ | n − (2 π ) n h τ ( χ u ξλ ) , φ i b , b ∗ . (5.3) Here S n − denotes the (n-1)-dimensional unitary sphere in R n and u ξλ is the plane wave with direction ξ ∈ S n − and wavenumber | λ | , i.e. u ξλ ( x ) = e i | λ | ξ · x . roof. According to [37, Lemma 1, page 170], one has R z ∈ B ( L w ( R n )) ; this entails (see [27, relation(4.8)]) ( − ∆ + z ) − ≡ R z ∈ B ( L w ( R n ) , H w ( R n )) . (5.4)Thus ∆ satisfies hypothesis (H1). It is a well-known fact that LAP holds for the free Laplacian, i.e. ∆ satisfies hypothesis (H2) (see e.g. [1, Theorem 4.1], [22, Theorem 18.3]): for any λ < w > , R , ± λ = lim ǫ ↓ ( − ∆ + λ ± i ǫ ) − exist in B ( L w ( R n ) , H − w ( R n )) (5.5)and the maps z R , ± z : = R z , z ∈ C \ ( −∞ , R , ± λ , z = λ ∈ ( −∞ , C ± \{ } to B ( L w ( R n ) , H − w ( R n )). Hypothesis (H3) holds true by [5, Corollary 5.7(b)].Hypothesis (H5) holds true by (5.1) and (5.5). By (5.5), supp( τ ∗ φ ) ⊆ supp( χ ). Since G z φ is the convolutionof the kernel of R z with the distribution τ ∗ φ ∈ H − comp ( R n ), one obtains G z ∈ B ( h ∗ , L w ( R n )) for any w andhypothesis (H6) holds true. By Remark 3.8, if (5.1) holds then the map z R Λ z satisfies (H1). If theembedding ran( Λ λ ) ֒ → h ∗ is compact, then hypothesis (H4) holds true with k = G ∗ λ ∈ B ( L ( R n , h )) and by G λ ∈ B ( h ∗ , L w ( R n )) for any w . Therefore, by Theorem 3.9, asymptotic completenessholds for the scattering couple ( ∆ , ∆ Λ ) and σ ac ( ∆ Λ ) = σ ac ( ∆ ) = ( −∞ , σ sc ( ∆ Λ ) = ∅ .Moreover, since R Λ z − R z is compact by ii ) and (2.10), σ ess ( ∆ Λ ) = σ ess ( ∆ ) = ( −∞ , S Λ λ is provided by Corollary 4.3. By (5.1), the distribution τ ∗ φ ∈ H − ( R n ), φ ∈ h ∗ ,is compactly supported, supp( τ ∗ φ ) ⊆ supp( χ ). Setting v ξ ( x ) : = e i ξ · x (2 π ) n , its Fourier transform is given by d τ ∗ φ ( ξ ) = h v ξ , τ ∗ φ i H loc ( R n ) , H − comp ( R n ) = h χ v ξ , τ ∗ φ i H ( R n ) , H − ( R n ) = h τ ( χ v ξ ) , φ i h , h ∗ . The unitary map F : L ( R n ) → R ⊕ ( −∞ , L ( S n − ) d λ ≡ L (( −∞ , L ( S n − )) given by( F u ) λ ( ξ ) : = − / | λ | n − b u ( | λ | / ξ ) (5.6)diagonalizes A = ∆ . Therefore, by R µ ∈ B ( H − ( R n ) , L ( R n )) and (5.6), one gets( µ − λ )( F R µ τ ∗ φ ) λ ( ξ ) = − / | λ | n − d τ ∗ φ ( | λ | / ξ ) = − / | λ | n − h τ ( χ v | λ | / ξ ) , φ i h , h ∗ . This gives the operator L λ provided in Corollary 4.3 (notice that for notational convenience in (5.3) we use − L λ ).In order to conclude the proof we need to show that the limits Λ ± λ , which exist in B ( h , h ∗ ) by Lemma3.6, in fact exist in B ( b , b ∗ ). By (2.7), for any z ∈ C \ ( −∞ ,
0] one has Λ λ ± i ǫ = Λ z + ( z − ( λ ± i ǫ )) Λ λ ± i ǫ τ R λ ± i ǫ G z Λ z . (5.7)Thus, since Λ z ∈ B ( b , b ∗ ) ⊆ B ( b , h ∗ ) for any z ∈ C \ ( −∞ , τ R λ ± i ǫ converges in B ( L w ( R n ) , h ) by(5.1) and (5.5) and since G z ∈ B ( h ∗ , L w ( R n )), the existence in B ( h , h ∗ ) of the limits Λ ± λ entails the existenceof such limits in B ( b , h ∗ ). Then, by duality and (2.6), the limits exist in B ( h , b ∗ ) as well. Thus, using again(5.7) and repeating the same reasonings, at the end one gets the existence of the limits Λ ± λ in B ( b , b ∗ ).15 .1. Traces, layer operators and Dirichlet-to-Neumann maps.5.1.1. Trace maps and single-layer operators on d-sets. Here we begin recalling some results about d -sets which are needed below (see [24] and [44] for moredetails).A Borel set Γ ⊂ R n is called a d -set, 0 < d < n , if there exists a Borel measure µ in R n such thatsupp( µ ) = Γ and ∃ c ± > ∀ x ∈ Γ , ∀ r ∈ (0 , , c − r d ≤ µ ( B xr ∩ Γ ) ≤ c + r d , (5.8)where B xr is the ball in R n of radius r centered at the point x (see e.g. [44, Definition 3.1]). By [44, Theorem3.4], once Γ is a d -set, µ d Γ , the d -dimensional Hausdor ff measure restricted to Γ , always satisfies (5.8) andso Γ has Hausdor ff dimension d in the neighborhood of any of its points.Examples of d -sets are, whenever d is an integer number, finite unions of d -dimensional Lipschitzmanifolds which intersect on a set of zero d -dimensional Hausdor ff measure and, whenever d is not aninteger, self-similar fractals of Hausdor ff dimension d .Let γ Γ be the map defined by the restriction of u ∈ C ∞ comp ( R n ) along the set Γ : γ Γ u : = u | Γ . Then, by [24,Theorem 1, Chapter VII], such a map has a bounded and surjective extension to H s + n − d ( R n ) for any s > γ Γ : H s + n − d ( R n ) → B s , ( Γ ) . (5.9)Here the Hilbert space B s , ( Γ ) is a Besov-like space (see [24, Section 2, Chapter V] for the precise defini-tions). Notice that if Γ is a manifold of class C κ, , κ ≥
0, then B s , ( Γ ) = H s ( Γ ) for any s ≤ κ +
1, where H s ( Γ ) denotes the usual fractional Sobolev space on Γ (see e.g. [31, Chapter 3]). Moreover, in the case0 < s < B s , ( Γ ) can be defined (see [24, Section 1.1, chap. V]) as the set of φ ∈ L ( Γ , µ d Γ ) having finitenorm k φ k B s , ( Γ ) : = Z Γ | φ ( x ) | d µ d Γ ( x ) + Z { ( x , y ) ∈ Γ × Γ : | x − y | < } | φ ( x ) − φ ( y ) | | x − y | d + s d ( µ d Γ × µ d Γ )( x , y ) . Since such a norm coincides with the usual norm in H s ( Γ ) whenever Γ is a Lipschitz hypersurface, forsuccessive convenience we use the notation B s , ( Γ ) ≡ H s ( Γ ) whenever 0 < s <
1. We also use the followingnotations for the dual (with respect to the L ( Γ )-pairing) spaces: ( B s , ( Γ )) ∗ ≡ B − s , ( Γ ) and, whenever 0 < s <
1, ( H s ( Γ )) ∗ ≡ H − s ( Γ ).By [44, Proposition 20.5], one has, similarly to the regular case, Γ compact d -set = ⇒ the embedding B s , ( Γ ) ֒ → B s , ( Γ ), s > s , is compact. (5.10) Γ compact d -set = ⇒ the embedding B s , ( Γ ) ֒ → L dd − s ( Γ ), 0 < s < d , is compact. (5.11)In the following the resolvent R z ≡ ( − ∆ + z ) − , z ∈ C \ ( −∞ , B ( H s ( R n ) , H s + ( R n )), s ∈ R . Given s >
0, by the mapping properties (5.9) one gets, for the adjoint of the trace map,( γ Γ ) ∗ : B − s , ( Γ ) → H − s − n − d ( R n )and so we can define the bounded operator (the single-layer potential)SL z : = R z ( γ Γ ) ∗ : B − s , ( Γ ) → H − s − n − d ( R n ) . (5.12)Notice that SL z = G z whenever τ = γ Γ and s = − n − d . By resolvent identity one has (compare with (2.5))SL z = SL w + ( w − z ) R z SL w . (5.13)16f n − < d < n , by (5.9) and (5.12), one obtains the bounded operator γ Γ SL z : B − s , ( Γ ) → B − s − ( n − d )2 , ( Γ ) . Since the map z R z is analytic on C \ ( −∞ ,
0] to B ( H s ( R n ) , H s + ( R n )) for any s ∈ R , the maps z SL z and z γ Γ SL z are analytic as well, with values in B ( B − s , ( Γ ) , H − s − n − d ( R n )) and B ( B − s , ( Γ ) , B − s − ( n − d )2 , ( Γ ))respectively .By (5.5), duality and interpolation one gets R , ± λ ∈ B ( H − sw ( R n ) , H − s + − w ( R n )) , ≤ s ≤ . (5.14)Thus, since Γ is bounded, the limits SL ± λ : = R , ± λ ( γ Γ ) ∗ = lim ǫ ↓ ( γ Γ R λ ∓ i ǫ ) ∗ exist in B ( B − s , ( Γ ) , H − s − n − d − w ( R n )), 0 < s ≤ − n − d . Let Γ be the boundary of a bounded Lipschitz domain Ω ; we set Ω in ≡ Ω and Ω ex : = R n \ Ω . In thefollowing ∆ Ω in / ex denote the distributional Laplacians on Ω in / ex .The one-sided, zero and first order, trace operators γ , in / ex Γ and γ , in / ex Γ = ν Γ · γ , in / ex Γ ∇ ( ν Γ denoting theoutward normal vector at the boundary) defined on smooth functions in C ∞ comp ( Ω in / ex ) extend to boundedand surjective linear operators (see e.g. [31, Theorem 3.38]) γ , in / ex Γ ∈ B ( H s + / ( Ω in / ex ) , H s ( Γ )) , < s < . (5.15)and γ , in / ex Γ ∈ B ( H s + / ( Ω in / ex ) , H s ( Γ )) , < s < H s ( Ω in / ex ) and H s ( Γ )). Using thesemaps and setting H s ( R n \ Γ ) : = H s ( Ω in ) ⊕ H s ( Ω ex ), the two-sided bounded and surjective trace operators aredefined according to γ Γ : H s + / ( R n \ Γ ) → H s ( Γ ) , γ Γ ( u in ⊕ u ex ) : =
12 ( γ , in Γ u in + γ , ex Γ u ex ) , (5.17) γ Γ : H s + / ( R n \ Γ ) → H s ( Γ ) , γ Γ ( u in ⊕ u ex ) : =
12 ( γ , in Γ u in + γ , ex Γ u ex ) , (5.18)while the corresponding jumps are[ γ Γ ] : H s + / ( R n \ Γ ) → H s ( Γ ) , [ γ Γ ]( u in ⊕ u ex ) : = γ , in Γ u in − γ , ex Γ u ex , (5.19)[ γ Γ ] : H s + / ( R n \ Γ ) → H s ( Γ ) , [ γ Γ ]( u in ⊕ u ex ) : = γ , in Γ u in − γ , ex Γ u ex . (5.20)Let us notice that in the case u = u in ⊕ u ex ∈ H s + / ( R n ), 0 < s < γ Γ in (5.17) coincides with the mapdefined in (5.9) and so there is no ambiguity in our notations; this also entails that γ Γ remains surjectiveeven if restricted to H ( R n ). Similarly the map γ Γ is surjective onto H s ( Γ ) even if restricted to H s + / ( R n ).By [31, Lemma 4.3], the trace maps γ , in / ex Γ can be extended to the spaces H ∆ ( Ω in / ex ) : = { u in / ex ∈ H ( Ω in / ex ) : ∆ Ω in / ex u in / ex ∈ L ( Ω in / ex ) } :17 , in / ex Γ : H ∆ ( Ω in / ex ) → H − / ( Γ ) . This gives the analogous extensions of the maps γ Γ and [ γ Γ ] defined on H ∆ ( R n \ Γ ) : = H ∆ ( Ω in ) ⊕ H ∆ ( Ω ex )with values in H − / ( Γ ).By using a cut-o ff function χ ∈ C ∞ comp ( R n ) such that χ = Ω in , all the maps definedabove can be extended (and we use the same notation) to functions u such that χ u is in the right functionspace.The single-layer operator SL z has been already introduced in the previous subsection, see (5.12); herewe recall the definition of double-layer operator DL z , z ∈ C \ ( −∞ , γ Γ ) ∗ : H − s ( Γ ) → H − s − / ( R n )and by the resolvent R z ∈ B ( H s ( R n ) , H s + ( R n )), one defines the bounded operatorDL z : H − s ( Γ ) → H − s + / ( R n ) , DL z : = R z ( γ Γ ) ∗ , < s < . (5.21)Let us notice that DL z = G z whenever τ = γ Γ and s = . By resolvent identity one has (compare with (2.5))DL z = DL w + ( z − w ) R z DL w . (5.22)By the mapping properties of the layer operators, one gets (see [31, Theorem 6.11]) χ SL z ∈ B ( H − / ( Γ ) , H ( R n )) , χ DL z ∈ B ( H / ( Γ ) , H ( R n \ Γ )) , (5.23)for any χ ∈ C ∞ comp ( R n ); by ( − ( ∆ Ω in ⊕ ∆ Ω ex ) + z )SL z φ = ( − ( ∆ Ω in ⊕ ∆ Ω ex ) + z )DL z ϕ =
0, one gets χ SL z φ ∈ H ∆ ( R n \ Γ ), φ ∈ H / ( Γ ), and χ DL z ϕ ∈ H ∆ ( R n \ Γ ), ϕ ∈ H − / ( Γ ). Thus γ Γ SL z ∈ B ( H − / ( Γ ) , H / ( Γ )) , γ Γ DL z ∈ B ( H / ( Γ ) , H − / ( Γ )) . These mapping properties can be extended to a larger range of Sobolev spaces (see [31, Theorem 6.12 andsuccessive remarks]): χ SL z ∈ B ( H s − / ( Γ ) , H s + ( R n )) , χ DL z ∈ B ( H s + / ( Γ ) , H s + ( R n \ Γ )) , − / ≤ s ≤ / ,γ Γ SL z ∈ B ( H s − / ( Γ ) , H s + / ( Γ )) , γ Γ DL z ∈ B ( H s + / ( Γ ) , H s − / ( Γ )) , − / ≤ s ≤ / γ Γ ]SL z φ = , [ γ Γ ]SL z φ = − φ , [ γ Γ ]DL z ϕ = ϕ , [ γ Γ ]DL z ϕ = . (5.24)Since the map z R z is analytic on C \ ( −∞ ,
0] to B ( H s ( R n ) , H s + ( R n )), the maps z γ Γ SL z , z γ Γ DL z ,are analytic as well.By (5.14), since Γ is bounded, the limitsSL ± λ : = R , ± λ ( γ Γ ) ∗ = lim ǫ ↓ SL λ ± i ǫ , DL ± λ : = R , ± λ ( γ Γ ) ∗ = lim ǫ ↓ DL λ ± i ǫ exist in B ( B − s , ( Γ ) , H / − s − w ( R n )), 0 < s ≤ /
2, and B ( H − s ( Γ ) , H / − s − w ( R n )), 0 < s ≤ /
2, respectively.Moreover, by the identities (5.13),(5.22) and by SL z ∈ B ( B − / , ( Γ ) , L w ( R n )), DL z ∈ B ( H − / ( Γ ) , L w ( R n ))(see [27, relation (4.10)]) one hasSL ± λ = SL z + ( z − λ ) R , ± λ SL z , DL ± λ = DL z + ( z − λ ) R , ± λ DL z . (5.25)18his entails, for any − / ≤ s ≤ / χ SL ± λ ∈ B ( H s − / ( Γ ) , H s + ( R n )) , χ DL ± λ ∈ B ( H s + / ( Γ ) , H s + ( R n \ Γ )) , (5.26) γ Γ SL ± λ ∈ B ( H s − / ( Γ ) , H s + / ( Γ )) , γ Γ DL ± λ ∈ B ( H s + / ( Γ ) , H s − / ( Γ )) , (5.27)and, by (5.24) and (5.25),[ γ Γ ]SL ± λ φ = , [ γ Γ ]SL ± λ φ = − φ , [ γ Γ ]DL ± λ ϕ = ϕ , [ γ Γ ]DL ± λ ϕ = . (5.28)Since the maps z R , ± z are continuous on C ± \{ } to B ( L w ( R n ) , H − w ( R n )), the maps z γ Γ SL ± z : = γ Γ SL z , z ∈ C \ ( −∞ , γ Γ SL ± λ , z = λ ∈ ( −∞ , , z γ Γ DL ± z : = γ Γ DL z , z ∈ C \ ( −∞ , γ Γ DL ± λ , z = λ ∈ ( −∞ , , are continuous as well. Let Ω ⊂ R n be a bounded Lipschitz domain and let us consider the boundary value problems (here Ω in ≡ Ω and Ω ex : = R n \ Ω as in the previous subsection) ( − ∆ Ω in + z ) u z , in φ = , z ∈ ρ ( ∆ D Ω in ) γ , in Γ u z , in φ = φ ∈ H / ( Γ ) ( ( − ∆ Ω in + z ) v z , in ϕ = , z ∈ ρ ( ∆ N Ω in ) γ , in Γ v z , in ϕ = ϕ ∈ H − / ( Γ ) (5.29)and ( − ∆ Ω ex + z ) u z , ex φ = , z ∈ ρ ( ∆ D Ω ex ) γ , ex Γ u z , ex φ = φ ∈ H / ( Γ ) u z , ex φ radiating ( − ∆ Ω ex + z ) v z , ex ϕ = , z ∈ ρ ( ∆ N Ω ex ) γ , ex Γ v z , ex ϕ = ϕ ∈ H − / ( Γ ) v z , ex ϕ radiating (5.30)where ∆ D Ω in / ex and ∆ N Ω in / ex denote the Dirichlet and Neumann Laplacian respectively; we refer to [31, Def-inition 9.5] for the definition of radiating solutions in the exterior problem. By [31, Theorem 4.10(i)],the solutions u z , in φ and v z , in ϕ of (5.29) exist and are unique in H ∆ ( Ω in ); by [31, Theorem 9.11 and Exer-cise 9.5] the solutions u z , ex φ and v z , ex ϕ of (5.30) exist and are unique in H ∆ , loc ( Ω ex ) : = { u : u | Ω ex ∩ B ∈ H ∆ ( Ω ex ∩ B ) for any open ball B ⊃ Ω } . Therefore the Dirichlet-to-Neumann and Neumann-to-Dirichlet op-erators P in / ex z : H / ( Γ ) → H − / ( Γ ) , z ∈ ρ ( ∆ D Ω in ) , P in / ex z φ : = γ , in / ex Γ u z , in / ex φ , Q in / ex z : H − / ( Γ ) → H / ( Γ ) , z ∈ ρ ( ∆ N Ω in ) , Q in / ex z ϕ : = γ , in / ex Γ v z , in / ex ϕ are well-defined.Let ˜ φ ∈ H − / ( Γ ) and ˜ ϕ ∈ H / ( Γ ); the functions SL + z ˜ φ z | Ω in / ex and DL + z ˜ ϕ z | Ω in / ex solve (5.29) and (5.30)with φ = γ Γ SL + z ˜ φ and ϕ = γ Γ DL + z ˜ ϕ (they are radiating according to [27, Lemma 5.3]). By (5.24) and (5.28),( P ex z − P in z ) γ Γ SL + z ˜ φ = γ , ex Γ (SL + z ˜ φ | Ω ex ) − γ , in Γ (SL + z ˜ φ | Ω in ) = [ γ Γ ]SL + z ˜ φ = − ˜ φ , ( Q ex z − Q in z ) γ Γ DL + z ˜ ϕ = γ , ex Γ (DL + z ˜ ϕ | Ω ex ) − γ , in Γ (DL + z ˜ ϕ | Ω in ) = [ γ Γ ]DL + z ˜ ϕ = ˜ ϕ . γ Γ SL + z ) = ker( γ Γ DL + z ) = { } . (5.31)By (5.25), one has γ Γ SL ± λ = γ Γ SL z + ( z − λ ) γ Γ R , ± λ SL z , γ Γ DL ± λ = γ Γ DL z + ( z − λ ) γ Γ R , ± λ DL z . Since ran( γ Γ R , ± λ ) ⊆ B / , ( Γ ) and ran( γ Γ R , ± λ ) ⊆ H / ( Γ ), by the compact embeddings (5.10), one gets γ Γ SL ± λ − γ Γ SL z ∈ S ∞ ( H − / ( Γ ) , H / ( Γ ))and γ Γ DL ± λ − γ Γ DL z ∈ S ∞ ( H / ( Γ ) , H − / ( Γ )) . By [31, Theorems 7.6 and 7.8]), both γ Γ SL z and γ Γ DL z are Fredholm with zero index; therefore both γ Γ SL ± z and γ Γ DL ± z are Fredholm with zero index as well. Thus, by (5.31), both γ Γ SL + z and γ Γ DL + z have boundedinverses and ( γ Γ SL + z ) − = P in z − P ex z , z ∈ C + \ (cid:0) σ disc ( ∆ D Ω in ) ∪ σ disc ( ∆ D Ω ex ) (cid:1) ∪ (cid:0) C \ ( −∞ , (cid:1) , (5.32)( γ Γ DL + z ) − = Q ex z − Q in z , z ∈ C + \ (cid:0) σ disc ( ∆ N Ω in ) ∪ σ disc ( ∆ N Ω ex ) (cid:1) ∪ (cid:0) C \ ( −∞ , (cid:1) . (5.33)By the mapping properties of the layer operators, for all s ∈ [0 , /
2] the maps z P ex z − P in z and z Q ex z − Q in z are analytic on C \ ( −∞ ,
0] to B ( H s + / ( Γ ) , H s − / ( Γ )) and to B ( H s − / ( Γ ) , H s + / ( Γ )) respectively. Let ∆ ◦ Ω in / ex denote the Laplacian in L ( Ω in / ex ) with domain dom( ∆ ◦ Ω in / ex ) = C ∞ comp ( Ω in / ex ). It is immediateto check (see e.g. [25, Section 2.3]) that ( ∆ ◦ Ω in / ex ) ∗ = ∆ max Ω in / ex , where ∆ max Ω in / ex denotes the distributional Laplacianwith domain dom( ∆ max Ω in / ex ) = H ∆ ( Ω in / ex ) : = { u in / ex ∈ L ( Ω in / ex ) : ∆ Ω in / ex u in / ex ∈ L ( Ω in / ex ) } . Moreover ∆ ◦ Ω in / ex is closable with closure given by ∆ ◦ Ω in / ex = ∆ min Ω in / ex (see [25, Section 2.3]), where ∆ min Ω in / ex denotes the distributional Laplacian with domain dom( ∆ min Ω in / ex ) = H ( Ω in / ex ) and H ( Ω in / ex ) denotes as usualthe completion of C ∞ comp ( Ω in / ex ) with respect to the H -norm. Therefore( ∆ min Ω in / ex ) ∗ = ∆ max Ω in / ex . (5.34)Since H ( Ω in / ex ) = { u in / ex ∈ H ( Ω in / ex ) : γ , ex / in Γ u in / ex = γ , ex / in Γ u in / ex = } (see [29, Theorem 1]) and H ( R n ) = ( H ( Ω in ) ⊕ H ( Ω ex )) ∩ ker([ γ Γ ]) ∩ ker([ γ Γ ]) (see e.g. [2, Theorem3.5.1]), one has ∆ min Ω in ⊕ ∆ min Ω ex = ∆ | ker( τ ) , τ = γ Γ ⊕ γ Γ . (5.35)Notice that for a generic Lipschitz boundary ran( τ ) is strictly contained in B / , ( Γ ) ⊕ H / ( Γ ) (see [30,Corollary 7.11]), while ran( τ ) = H / ( Γ ) ⊕ H / ( Γ ) whenever Γ is of class C κ, , κ > (see [29, Theorem 2]).By Green’s formula (see e.g. [31, Theorem 4.4]), for any couple u in / ex , v in / ex in H ∆ ( Ω in / ex ) one has h ∆ Ω in / ex u in / ex , v in / ex i L ( Ω in / ex ) − h u in / ex , ∆ Ω in / ex v in / ex i L ( Ω in / ex ) = j in / ex h γ , in / ex Γ u in / ex , γ , in / ex Γ v in / ex i H / ( Γ ) , H − / ( Γ ) − j in / ex h γ , in / ex Γ u in / ex , γ , in / ex Γ v in / ex i H − / ( Γ ) , H / ( Γ ) , j in = − j ex = +
1. Therefore, for any couple u = u in ⊕ u ex , v = v in ⊕ v ex in H ∆ ( R n \ Γ ) ∩ ker([ γ Γ ]) = H ( R n ) ∩ H ∆ ( R n \ Γ ) one has h ( ∆ Ω in ⊕ ∆ Ω ex ) u , v i L ( R n ) − h u , ( ∆ Ω in ⊕ ∆ Ω ex ) v i L ( R n ) = h γ Γ u , [ γ Γ ] v i H / ( Γ ) , H − / ( Γ ) − h [ γ Γ ] u , γ Γ v i H − / ( Γ ) , H / ( Γ ) (5.36)and, for any couple u = u in ⊕ u ex , v = v in ⊕ v ex in H ∆ ( R n \ Γ ) ∩ ker([ γ Γ ]) one has h ( ∆ Ω in ⊕ ∆ Ω ex ) u , v i L ( R n ) − h u , ( ∆ Ω in ⊕ ∆ Ω ex ) v i L ( R n ) = h [ γ Γ ] u , γ Γ v i H / ( Γ ) , H − / ( Γ ) − h γ Γ u , [ γ Γ ] v i H − / ( Γ ) , H / ( Γ ) . (5.37) Here we apply Theorem 5.1 to a case in which τ = γ Γ , h = B / , ( Γ ), b = H / ( Γ ) and Γ is the Lipschitzboundary of a bounded open set Ω ⊂ R n .Let ∆ D Ω in / ex : dom( ∆ D Ω in / ex ) ⊂ L ( Ω in / ex ) → L ( Ω in / ex ) , ∆ D Ω in / ex u : = ∆ Ω in / ex u , dom( ∆ D Ω in / ex ) : = { u in / ex ∈ H ∆ ( Ω in / ex ) : γ , in / ex Γ u in / ex = } , be the Dirichlet Laplacian in L ( Ω in / ex ). By (5.36), ∆ D Ω in / ex is symmetric; in fact it is self-adjoint and theself-adjoint operator ∆ D Ω in ⊕ ∆ D Ω ex has an alternative representation: Lemma 5.2.
The family of linear bounded maps Λ D Λ Dz : = P ex z − P in z : H / ( Γ ) → H − / ( Γ ) , z ∈ C \ ( −∞ , , satisfies (2.6) - (2.7) and ∆ Λ D = ∆ D Ω in ⊕ ∆ D Ω ex . Proof.
At first notice that, by the definition (5.12) and by resolvent identity, the operator family M Dz = − γ Γ SL z , z ∈ C \ ( −∞ , Λ Dz = ( M Dz ) − and so it satisfies (2.6) and (2.7) byRemark 2.2.Let u ∈ dom( ∆ Λ D ), so that, by (2.11), u = u z + G z Λ Dz τ = u z + SL z ( P ex z − P in z ) γ Γ u z , u z ∈ H ( R n ). By(5.12), SL z ∈ B ( H − / ( Γ ) , H ( R n )) and so, since ( − ∆ Ω in / ex + z )SL z =
0, one has u ∈ H ( R n ) ∩ H ∆ ( R n \ Γ ) ⊂ H ∆ ( R n \ Γ ). Then, by H ( R n ) ⊂ ker([ γ Γ ]) and (5.32), one gets γ , in / ex Γ u =
0; therefore u ∈ dom( ∆ D Ω in ) ⊕ dom( ∆ D Ω ex ) and so dom( ∆ Λ D ) ⊆ dom( ∆ D Ω in ⊕ ∆ D Ω ex ).By Theorem 2.4, ∆ Λ D is a self-adjoint extension of ( ∆ | ker( γ Γ )) ⊃ ∆ min Ω in ⊕ ∆ min Ω ex . Thus ∆ Λ D ⊂ ( ∆ | ker( γ Γ )) ∗ ⊂ ( ∆ min Ω in ⊕ ∆ min Ω ex ) ∗ = ∆ max Ω in ⊕ ∆ max Ω ex . Since ∆ max Ω in / ex | dom( ∆ D Ω in / ex ) = ∆ D Ω in / ex , one gets ∆ Λ D ⊆ ∆ D Ω in ⊕ ∆ D Ω ex . Since ∆ Λ D isself-adjoint and ∆ D Ω in ⊕ ∆ D Ω ex is symmetric by (5.36), one obtains ∆ Λ D = ∆ D Ω in ⊕ ∆ D Ω ex .By Lemma 5.2 and by the compact embedding H − / ( Γ ) ֒ → B − / , ( Γ ), we can apply Theorem 5.1: Theorem 5.3.
Let Ω be a bounded open domain with Lipschitz boundary Γ . Then asymptotic completenessholds for the scattering couple ( ∆ , ∆ D Ω in ⊕ ∆ D Ω ex ) and the corresponding scattering matrix S D λ is given byS D λ = − π iL D λ ( P ex λ − P in λ )( L D λ ) ∗ , λ ∈ ( −∞ , \ ( σ disc ( ∆ D Ω in ) ∪ σ disc ( ∆ D Ω ex )) , (5.38)21 disc ( ∆ D Ω ex ) = ∅ whenever Ω ex is connected, whereL D λ : H − / ( Γ ) → L ( S n − ) , L D λ φ ( ξ ) : = | λ | n − (2 π ) n h u ξλ | Γ , φ i H / ( Γ ) , H − / ( Γ ) . Proof.
By taking the limit ǫ ↓ − Λ D λ + i ǫ γ Γ SL λ + i ǫ = = − γ Γ SL λ + i ǫ Λ D λ + i ǫ and by (5.32), onegets Λ D , + λ = − ( γ Γ SL + λ ) − = P ex λ − P in λ .Moreover σ − p ( ∆ D Ω in ⊕ ∆ D Ω ex ) = σ p ( ∆ D Ω in ) ∪ σ p ( ∆ D Ω ex ) = σ disc ( ∆ D Ω in ) ∪ σ disc ( ∆ D Ω ex ). Finally, σ disc ( ∆ D Ω ex ) = ∅ whenever Ω ex is connected by the unique continuation principle. Remark 5.4.
Formula (5.38) extends to n -dimensional bounded Lipschitz domains the one which hasbeen obtained, in the case of 2-dimensional bounded piecewise C domains, in [12, Theorems 5.3 and5.6]; similar formulae are also given, in a smooth 2-dimensional setting in [7, Subsection 5.2] and in asmooth n -dimensional setting in [27, Subsection 6.1]. Let us mention that as regards the alone asymptoticcompleteness result, the Lipschitz regularity condition on the boundary is not necessary, see [18]. Here we apply Theorem 5.1 to a case in which τ = γ Γ , h = H / ( Γ ), b = H − / ( Γ ) and Γ is the Lipschitzboundary of a bounded open set Ω ⊂ R n .Let ∆ N Ω in / ex : dom( ∆ N Ω in / ex ) ⊂ L ( Ω in / ex ) → L ( Ω in / ex ) , ∆ N Ω in / ex u : = ∆ u , dom( ∆ N Ω in / ex ) : = { u in / ex ∈ H ∆ ( Ω in / ex ) : γ , in / ex Γ u in / ex = } , be the Neumann Laplacian in L ( Ω in / ex ). By (5.37), ∆ N Ω in / ex is symmetric; in fact it is self-adjoint and theself-adjoint operator ∆ N Ω in ⊕ ∆ N Ω ex has an alternative representation: Lemma 5.5.
The family of linear bounded maps Λ N Λ Nz : = Q in z − Q ex z : H − / ( Γ ) → H / ( Γ ) , z ∈ C \ ( −∞ , , satisfies (2.6) - (2.7) and ∆ Λ N = ∆ N Ω in ⊕ ∆ N Ω ex . Proof.
At first notice that, by the definition (5.21) and by resolvent identity, the operator family M Nz = − γ Γ DL z , z ∈ C \ ( −∞ , Λ Nz = ( M Nz ) − satisfies (2.6) and (2.7) by Remark 2.2.Let u ∈ dom( ∆ Λ N ), so that, by (2.11), u = u z + G z Λ Nz τ u z = u z − DL z ( Q ex z − Q in z ) γ Γ u z , u z ∈ H ( R n ). By[26, Lemma 3.1], DL z ∈ B ( H / ( Γ ) , H ( Ω in / ex )) and so, since ( − ∆ Ω in / ex + z )DL z =
0, one has u ∈ H ∆ ( R n \ Γ ).Then, by H ( R n ) ⊂ (ker([ γ Γ ]) ∩ ker([ γ Γ ])), (5.24) and by (5.33), one gets γ , in / ex Γ u =
0; therefore u ∈ dom( ∆ N Ω in ) ⊕ dom( ∆ N Ω ex ) and so dom( ∆ Λ N ) ⊆ dom( ∆ N Ω in ⊕ ∆ N Ω ex ).By Theorem 2.4, ∆ Λ N is a self-adjoint extension of ( ∆ | ker γ Γ ) ⊃ ∆ min Ω in ⊕ ∆ min Ω ex . Thus ∆ Λ N ⊂ ( ∆ | ker γ Γ ) ∗ ⊂ ( ∆ min Ω in ⊕ ∆ min Ω ex ) ∗ = ∆ max Ω in ⊕ ∆ max Ω ex . Since ∆ max Ω in / ex | dom( ∆ N Ω in / ex ) = ∆ N Ω in / ex , one gets ∆ Λ N ⊆ ∆ N Ω in ⊕ ∆ N Ω ex . Since ∆ Λ N isself-adjoint and ∆ N Ω in ⊕ ∆ N Ω ex is symmetric by (5.37), one obtains ∆ Λ N = ∆ N Ω in ⊕ ∆ N Ω ex .By Lemma 5.5 and by the compact embedding H / ( Γ ) ֒ → H − / ( Γ ), we can apply Theorem 5.1:22 heorem 5.6. Let Ω be a bounded open domain with Lipschitz boundary Γ . Then asymptotic completenessholds for the scattering couple ( ∆ , ∆ N Ω in ⊕ ∆ N Ω ex ) and the corresponding scattering matrix S N λ is given byS N λ = − π iL N λ ( Q in λ − Q ex λ )( L N λ ) ∗ , λ ∈ ( −∞ , \ ( σ disc ( ∆ N Ω in ) ∪ σ disc ( ∆ N Ω ex )) , (5.39) σ disc ( ∆ N Ω ex ) = ∅ whenever Ω ex is connected, whereL N λ : H − / ( Γ ) → L ( S n − ) , L N λ ϕ ( ξ ) : = | λ | n − (2 π ) n h ν Γ ·∇ u ξλ | Γ , ϕ i H / ( Γ ) , H − / ( Γ ) . Proof.
By taking the limit ǫ ↓ − Λ N λ + i ǫ γ Γ DL λ + i ǫ = = − γ Γ DL λ + i ǫ Λ N λ + i ǫ and by (5.32), onegets Λ N , + λ = − ( γ Γ DL + λ ) − = Q in λ − Q ex λ .Moreover σ − p ( ∆ N Ω in ⊕ ∆ N Ω ex ) ∪{ } = σ p ( ∆ N Ω in ) ∪ σ p ( ∆ N Ω ex ) = σ disc ( ∆ N Ω in ) ∪ σ disc ( ∆ N Ω ex ). Finally, σ disc ( ∆ N Ω ex ) = ∅ whenever Ω ex is connected by the unique continuation principle. Remark 5.7.
Formula (5.39) extends to n -dimensional bounded Lipschitz domains the one which has beenobtained, in the case of 2-dimensional bounded piecewise C domains, in [13, Theorems 4.2 and 4.3];similar formulae are also given, in a smooth 2-dimensional setting in [7, Subsection 5.3] and in a smooth n -dimensional setting in [27, Subsection 6.2] δ -type on d-sets. Here we apply Theorem 5.1 to a case in which τ = γ Γ , h = B s d , ( Γ ), s d : = − ( n − d ) / b = H s ( Γ ),0 < s < s d −
1, and Γ ⊂ R n is a d -set with 0 < n − d < Lemma 5.8.
Let α ∈ B ( H s ( Γ ) , H − s ( Γ )) , α ∗ = α , < s < − n − d . Then there exists a finite set Σ α ⊂ (0 , + ∞ ) such that for all z ∈ C \ (( −∞ , ∪ Σ α ) one has ( + αγ Γ SL z ) − ∈ B ( H − s ( Γ )) . Moreover the operator family Λ α in B ( H s ( Γ ) , H − s ( Γ ))) given by Λ α z : = − ( + αγ Γ SL z ) − α , z ∈ C \ (( −∞ , ∪ Σ α ) , satisfies (2.6) and (2.7) .Proof. By Fourier transform, one has the following estimate holding for any z ∈ C \ ( −∞ ,
0] and for any realnumber s : k R z k H s ( R n ) , H s + t ( R n ) ≤ d − t z , ≤ t ≤ , where d z : = dist( z , ( −∞ , γ Γ and ( γ Γ ) ∗ , one gets k γ Γ R z ( γ Γ ) ∗ k B − s , ( Γ ) , B − s − ( n − d ) + t , ( Γ ) ≤ d − t z k ( γ Γ ) ∗ k B − s , ( Γ ) , B − s − n − d , ( Γ ) k γ Γ k B − s − n − d + t , ( Γ ) , B − s − ( n − d ) + t , ( Γ ) . Choosing t = s + n − d , such an inequality shows that if 0 < s + n − d < c α > k γ Γ SL z α k H s ( Γ ) , H s ( Γ ) is strictly smaller than one whenever Re( z ) > c α . Therefore( + γ Γ SL z α ) − ∈ B ( H s ( Γ )) whenever Re( z ) > c α .Let 0 < s + n − d <
1. By (5.10), the embedding B − s − ( n − d )2 , ( Γ ) ֒ → B s , ( Γ ) is compact and so, byran( γ Γ SL z ) ⊆ B − s − ( n − d )2 , ( Γ ), the map γ Γ SL z : H − s ( Γ ) → H s ( Γ ) is also compact; thus γ Γ SL z α : H s ( Γ ) → s ( Γ ) is compact as well. Since the map z γ Γ SL z α is analytic from z ∈ C \ ( −∞ ,
0] to B ( H s ( Γ )) andthe set of z ∈ C \ (( −∞ ,
0] such that ( + γ Γ SL z α ) − ∈ B ( H s ( Γ )) is not void, by analytic Fredholm theory(see e.g. [37, Theorem XIII.13]), ( + γ Γ SL z α ) − ∈ B ( H s ( Γ )) for any z ∈ C \ (( −∞ , ∪ Σ α ), where Σ α is a discrete set. By Theorem 2.4, Remark 2.5 and next Theorem 5.9 (see (5.41)), Σ α is contained in thespectrum of a self-adjoint operator and so Σ α ⊂ R ; hence Σ α ⊆ [0 , c α ] and so it is finite being discrete, i.e.without accumulation points.By α = α ∗ and the same arguments as in the proof of [28, Corollary 2.4], one obtains (2.6) and( + αγ Γ SL z ) − = (cid:0) ( + γ Γ SL ¯ z α ) − (cid:1) ∗ ∈ B ( H − s ( Γ )) . (5.40)Finally, by SL z = R z ( γ Γ ) ∗ and resolvent identity for R z , it results(1 + αγ Γ SL w ) − (1 + αγ Γ SL z ) = ( z − w ) αγ Γ R w SL z . This yields Λ α w − Λ α z = ( z − w ) Λ α w γ Γ R w SL z Λ α z i.e. relation (2.7).Taking λ ◦ >
0, in the following we use the shorthand notation SL ◦ ≡ SL λ ◦ . Theorem 5.9.
Let Γ be a d-set with n − < d < n and let α ∈ B ( H s ( Γ ) , H − s ( Γ )) , α ∗ = α , < s < − n − d .Then1) The family of bounded linear operatorsR α z : = R z − SL z ( + αγ Γ SL z ) − αγ Γ R z , z ∈ C \ (( −∞ , ∪ Σ α ) (5.41) is the resolvent of the bounded from above self-adjoint operator ∆ α in L ( R n ) defined, in a λ ◦ -independentway, by dom( ∆ α ) : = { u ∈ H − s − n − d ( R n ) : u + SL ◦ αγ Γ u ∈ H ( R n ) } , (5.42) ∆ α u : = ∆ u − ( γ Γ ) ∗ αγ Γ u . (5.43) σ ess ( ∆ α ) = σ ac ( ∆ α ) = ( −∞ , , σ disc ( ∆ α ) = Σ α is finite, σ sc ( ∆ α ) = ∅ , σ − p ( ∆ α ) : = ( −∞ , ∩ σ p ( ∆ α ) is atmost discrete and asymptotic completeness holds for the scattering couple ( ∆ , ∆ α ) .3) The inverse ( ran( α ) + αγ Γ SL ± λ ) − : ran( α ) → ran( α ) exists for any λ ∈ ( −∞ , \ σ − p ( ∆ α ) and the scatteringmatrix S αλ is given byS αλ = + π iL D λ ( ran( α ) + αγ Γ SL + λ ) − α ( L D λ ) ∗ , λ ∈ ( −∞ , \ σ − p ( ∆ α ) , (5.44) L D λ : H − s ( Γ ) → L ( S n − ) , L D λ φ ( ξ ) : = | λ | n − (2 π ) n h u ξλ | Γ , φ i H s ( Γ ) , H − s ( Γ ) . Proof.
By Lemma 5.8, we can apply Theorem 2.4 and ∆ α : = ∆ Λ α is a well defined self-adjoint operatorwith resolvent given by (2.10). By (2.10) and Lemma 5.8, one gets σ ( ∆ α ) ⊆ ( −∞ , sup Σ α ] and so ∆ α isbounded from above since Σ α is finite. By Lemma 5.8, ran( Λ α z ) ֒ → H − s ( Γ ) ֒ → B − + n − d , ( Γ ) and so ran( Λ α z )is compactly embedded in h ∗ = B − + n − d , ( Γ ) by (5.10). Since Γ is bounded, (5.1) hold true. Thereforehypotheses i)-iii) in Theorem 5.1 hold.By (5.41) and [37, Theorem XIII.13], z R α z has poles (and the coe ffi cients of the Laurent expansionare finite-rank operators) only at Σ α ; so, by [37, Lemma 1, page 108], σ disc ( ∆ α ) = Σ α .24he proofs of (5.42) and (5.43) are the same as the ones given (in the case Γ is Lipschitz) in the proofof [28, Theorem 2.5] and are not reproduced here.Considering the limit ǫ ↓ Λ αλ ± i ǫ = − ( + αγ Γ SL λ ± i ǫ ) − α , one gets ker( α ) ⊆ ker( Λ α, ± λ ).Considering the limit ǫ ↓ − ( + αγ Γ SL λ ± i ǫ ) Λ αλ ± i ǫ = α = − Λ αλ ± i ǫ ( + γ Γ SL λ ± i ǫ α ) , one gets − ( + αγ Γ SL ± λ ) Λ α, ± λ = α = − Λ α, ± λ ( + γ Γ SL ± λ α ) , and − ( ˜ α − + γ Γ SL ± λ ) Λ α, ± λ | ker( α ) ⊥ = ker( α ) ⊥ , − Λ α, ± λ ( ˜ α − + γ Γ SL ± λ ) | ran( α ) = ran( α ) , where ˜ α : ker( α ) ⊥ → ran( α ) is the bijective bounded linear operator ˜ α : = α | ker( α ) ⊥ . This shows thatran( Λ α, ± λ ) ⊆ ran( α ) and that ˜ α − + γ Γ SL ± λ : ran( α ) → ker( α ) ⊥ is invertible with inverse − Λ α, ± λ | ker( α ) ⊥ , i.e. Λ α, ± λ | ker( α ) ⊥ = − ( ˜ α − + γ Γ SL ± λ ) − : ker( α ) ⊥ → ran( α ) . By ( ˜ α − + γ Γ SL ± λ ) − ˜ α − = ( α ( ˜ α − + γ Γ SL ± λ )) − = ( ran( α ) + αγ Γ SL ± λ ) − one gets the existence of the inverse( ran( α ) + αγ Γ SL ± λ ) − : ran( α ) → ran( α )and the identity Λ α, ± λ = − ( ran( α ) + αγ Γ SL ± λ ) − α : H s ( Γ ) → H − s ( Γ ) . Remark 5.10.
The limit single-layer operator SL ± λ admits the representationSL ± λ φ ( x ) = i Z Γ ∓| λ | / π k x − y k ! n − H (1) n − ( ∓| λ | / k x − y k ) φ ( y ) d µ d Γ ( y )whenever φ ∈ L ( Γ ) and x < Γ , where H (1) n − denotes the Hankel function of first kind of order n − Remark 5.11.
A particular case of operator α ∈ B (( H s ( Γ ) , H − s ( Γ )), such that α = α ∗ is α ∈ M ( H s ( Γ ) , H − s ( Γ )), α real-valued, where M ( H s ( Γ ) , H s ( Γ )) denotes the set of Sobolev multipliers on H s ( Γ ) to H s ( Γ ) (here andin the following we use the same notation for a function and for the corresponding multiplication operator).By the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ α ¯ φψ d µ d Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k| α | / φ k L ( Γ ) k| α | / ψ k L ( Γ ) ≤ k| α | / k H s ( Γ ) , L ( Γ ) k φ k H s ( Γ ) k ψ k H s ( Γ ) , one has | α | / ∈ M ( H s ( Γ ) , L ( Γ )) = ⇒ α ∈ M ( H s ( Γ ) , H − s ( Γ )) . Then, by the embeddings (5.11) and H¨older’s inequality, one gets p ≥ s = ⇒ L p ( Γ ) ⊆ M ( H s ( Γ ) , H − s ( Γ )) . Thus we can define ∆ α for any real-valued α ∈ L p ( Γ ), p > − ( n − d ) .25n the case Γ in Theorem 5.44 is a ( n − Corollary 5.12.
Let Ω be an open bounded set with a Lipschitz boundary Γ and α ∈ B ( H s ( Γ ) , H − s ( Γ )) , α = α ∗ , < s < / . Then ( − ∆ α + z ) − = R z + SL z ( P ex z − P in z )( α − ( P ex z − P in z )) − αγ Γ R z , z ∈ C \ (cid:0) ( −∞ , ∪ Σ α (cid:1) , (5.45)dom( ∆ α ) = { u ∈ H / − s ( R n ) ∩ H ∆ ( R n \ Γ ) : αγ Γ u = [ γ Γ ] u } . (5.46) ∆ α u = ( ∆ Ω in ⊕ ∆ Ω ex ) u . Whenever λ ∈ ( −∞ , \ ( σ − p ( ∆ α ) ∪ σ disc ( ∆ D Ω in ) ∪ σ disc ( ∆ D Ω ex )) , the scattering matrix S αλ has the alternativerepresentation S αλ = − π iL D λ ( P ex λ − P in λ )( α − ( P ex λ − P in λ )) − α ( L D λ ) ∗ . (5.47) If Ω ex is connected then σ − p ( ∆ α ) = σ disc ( ∆ D Ω ex ) = ∅ .Proof. Relation (5.45) is consequence of (5.32): by( α − Λ Dz ) γ Γ SL z ( + αγ Γ SL z ) − = = γ Γ SL z ( + αγ Γ SL z ) − ( α − Λ Dz ) , one gets ( α − Λ Dz ) − = γ Γ SL z ( + αγ Γ SL z ) − and so Λ α z = Λ Dz ( α − Λ Dz ) − α = ( P ex z − P in z )( α − ( P ex z − P in z )) − α . By H ( R n ) ⊆ ker([ γ Γ ]) and by (5.24), one gets dom( ∆ α ) ⊆ D α , where D α : = { ψ ∈ H / − s ( R n ) ∩ H ∆ ( R n \ Γ ) : αγ Γ u = [ γ Γ ] u } . Thus ∆ α ⊆ ( ∆ Ω in ⊕ ∆ Ω ex ) | D α . Since ∆ α is self-adjoint and ( ∆ Ω in ⊕ ∆ Ω ex ) | D α is symmetric by (5.36), the twooperators coincide.By the same reasonings as in the proof of Theorem 5.9, one shows that ( α + Λ D , ± λ ) | ran( α ) is invertibleand that Λ α, ± λ = Λ D , ± λ ( α − Λ D , ± λ ) − α = ( P ex λ − P in λ )( α − ( P ex λ − P in λ )) − α . If Ω ex is connected, then σ p ( ∆ α ) ∩ ( −∞ , = ∅ by the unique continuation principle (see [27, Remark3.8]). Remark 5.13.
The conditions providing the self-adjoint operator ∆ α in Theorem 5.9 are weaker, as regardsthe regularity of the boundary and / or the class of admissible strength functions, than the ones assumed inprevious works, see, for example, [19], [16], [10], [32], [6], [26], [15]. Asymptotic completeness for thescattering couple ( ∆ , ∆ α ) provided in Theorem 5.9 extend results on existence and completeness given, inthe case the boundary is smooth and the strength are bounded, in [6] and [26]. The formula for the scatteringmatrix provided in (5.44) (respectively in (5.47)) extends to d -sets (respectively to Lipschitz hypersurfaces)the results given, in the case of a smooth hypersurface, in [27, Subsections 6.4 and 7.4] and, in the case of asmooth 2- or 3-dimensional hypersurface, in [7, Subsection 5.4] (see also the formula provided in [17] forSchr¨odinger operators of the kind − ∆ + µ , µ a signed measure).26 .5. The Laplace operator with semi-transparent boundary condition of δ ′ -type on Lipschitz hypersurfaces. Here we apply Theorem 5.1 to a case in which τ = γ Γ , h = H / ( Γ ), b = H − / ( Γ ) and Γ is the boundaryof a bounded Lipschitz set Ω ⊂ R n . Lemma 5.14.
Let θ ∈ B ( H s ( Γ ) , H − s ( Γ )) , θ ∗ = θ , < s < . Then there exists a discrete set Σ θ ⊂ (0 , + ∞ ) such that for all z ∈ C \ (( −∞ , ∪ Σ θ ) one has ( + θ ( Q in z − Q ex z )) − ∈ B ( H − / ( Γ )) . Moreover the operatorfamily Λ θ in B ( H − / ( Γ ) , H / ( Γ ))) given by Λ θ z : = ( Q in z − Q ex z )( + θ ( Q in z − Q ex z )) − , z ∈ C \ (( −∞ , ∪ Σ θ ) , satisfies (2.6) and (2.7) .Proof. By θ ( Q in z − Q ex z ) ∈ B ( H − / ( Γ ) , H − s ( Γ )) and by the compact embedding H − s ( Γ ) ֒ → H − / ( Γ ), onegets θ ( Q in z − Q ex z ) ∈ S ∞ ( H − / ( Γ )). Therefore, by the Fredholm alternative, + θ ( Q in z − Q ex z ) is invertible ifand only if its kernel K z is trivial. By Q ex z − Q in z = ( γ Γ DL z ) − , K z , { } if and only if there is ψ ∈ H / ( Γ ) \{ } such that θψ = γ Γ DL z ψ . (5.48)By the definition (5.21) and by resolvent identity, we have γ Γ DL z − ( γ Γ DL z ) ∗ = γ Γ DL z − γ Γ DL ¯ z = (¯ z − z )DL ∗ z DL z . (5.49)Since θ = θ ∗ , (5.48) and (5.49) entail, for any z ∈ C \ R ,0 = ( z − ¯ z ) k DL z ψ k L ( R n ) . Since DL ∗ z = γ Γ R z is surjective, DL z has closed range by the closed range theorem and so (see Remark 3.5)there exists c > k DL z ψ k L ( R n ) ≥ c k ψ k H / ( Γ ) . Thus K z = { } whenever z ∈ C \ R and + θ ( Q in z − Q ex z )has a bounded inverse for any z ∈ C \ R . Since the operator-valued map z + θ ( Q in z − Q ex z ) is analytic on C \ ( −∞ , + θ ( Q in z − Q ex z ) − ∈ B ( H − / ( Γ ))for any z ∈ C \ (( −∞ , ∪ Σ θ ), where Σ θ ⊂ (0 , + ∞ ) is a discrete set.Since Λ θ z = Λ Nz ( + θ Λ Nz ) − , (5.50)one has ( Λ θ z ) − = ( + θ Λ Nz )( Λ Nz ) − = θ + ( Λ Nz ) − (5.51)Thus Λ θ z satisfies (2.6) and (2.7) by Remark 2.2 and Lemma 5.5.Taking λ ◦ >
0, in the following we use the shorthand notation DL ◦ ≡ DL λ ◦ . Theorem 5.15.
Let Γ be the boundary of a bounded Lipschitz set Ω ⊂ R n and let θ ∈ B ( H s ( Γ ) , H − s ( Γ )) , θ ∗ = θ , < s < . Then1) The family of bounded linear operatorsR θ z : = R z + DL z ( Q in z − Q ex z )( + θ ( Q in z − Q ex z )) − γ Γ R z , z ∈ C \ (( −∞ , ∪ Σ θ ) (5.52) is the resolvent of the bounded from above self-adjoint operator ∆ θ in L ( R n ) given by dom( ∆ θ ) = { u ∈ H ∆ ( R n ) : [ γ Γ ] u = , γ Γ u = θ [ γ Γ ] u } , ∆ θ u = ( ∆ Ω in ⊕ ∆ Ω ex ) u . ) σ ess ( ∆ θ ) = σ ac ( ∆ θ ) = ( −∞ , , σ disc ( ∆ θ ) = Σ θ is finite, σ sc ( ∆ θ ) = ∅ , σ − p ( ∆ θ ) = ( −∞ , ∩ σ p ( ∆ θ ) is atmost discrete.3) Asymptotic completeness holds for the scattering couple ( ∆ , ∆ θ ) and, whenever λ ∈ ( −∞ , \ ( σ − p ( ∆ θ ) ∪ σ disc ( ∆ N Ω in ) ∪ σ disc ( ∆ N Ω ex )) , the scattering matrix S θλ is given byS θλ = − π iL N λ ( Q in λ − Q ex λ )( + θ ( Q in λ − Q ex λ )) − ( L N λ ) ∗ . (5.53) If Ω ex is connected, then σ − p ( ∆ θ ) = σ disc ( ∆ N Ω ex ) = ∅ .Proof. By (2.11), u belongs to dom( ∆ θ ) if and only if u = u z + DL z Λ θ z γ Γ u z . By [26, Lemma 3.1], DL z ∈ B ( H / ( Γ ) , H ( Ω in / ex )) and so, since ( − ( ∆ Ω in ⊕ ∆ Ω in ) + z )DL z =
0, one has u ∈ H ∆ ( R n \ Γ ). Then, by H ( R n ) ⊂ (ker([ γ Γ ]) ∩ ker([ γ Γ ])) and (5.24), one obtains [ γ Γ ] u = Λ θ z γ Γ u z and [ γ Γ ] u =
0. Moreover, by(5.51), ( Λ θ z ) − = θ − γ Γ DL z , and so ( θ − γ Γ DL z )[ γ Γ ] u = γ Γ u z ; thus γ Γ u = θ [ γ Γ ] u anddom( ∆ θ ) ⊆ D θ : = { u ∈ H ∆ ( R n \ Γ ) : [ γ Γ ] u = , γ Γ u = θ [ γ Γ ] u } . Therefore ∆ θ ⊆ ( ∆ Ω in ⊕ ∆ Ω ex ) | D θ . Since ∆ θ is self-adjoint and ( ∆ Ω in ⊕ ∆ Ω ex ) | D θ is symmetric by (5.37), thetwo operators coincide.By Green’s formula (see e.g. [31, Theorem 4.4]) and by Ehrling’s lemma (see e.g. [38, Theorem 7.30]),one has (here 0 < s < / B ⊃ Ω is an open ball) h− ∆ θ u , u i L ( R n ) = k∇ u k L ( Ω in ) + k∇ u k L ( Ω ex ) + h θ [ γ Γ ] u , [ γ Γ ] u i H − s ( Γ ) , H s ( Γ ) ≥k∇ u k L ( Ω in ) + k∇ u k L ( Ω ex ) − k θ k H s ( Γ ) , H − s ( Γ ) (cid:0) k γ in0 u in k H s ( Γ ) + k γ ex0 u ex k H s ( Γ ) (cid:1) ≥k∇ u k L ( Ω in ) + k∇ u k L ( Ω ex ) − c k θ k H s ( Γ ) , H − s ( Γ ) (cid:0) k u in k H s + / ( Ω in ) + k u ex k H s + / ( Ω ex ∩ B ) (cid:1) ≥k∇ u k L ( Ω in ) + k∇ u k L ( Ω ex ) − c k θ k H s ( Γ ) , H − s ( Γ ) (cid:16) ǫ (cid:0) k u in k H ( Ω in ) + k u ex k H ( Ω ex ∩ B ) (cid:1) + c ǫ k u k L ( B ) (cid:17) ≥ − κ ǫ k u k L ( R n ) and so ∆ θ is bounded from above.By Lemma 5.14 and by (5.10), ran( Λ α z ) = H / ( Γ ) is compactly embedded in H − / ( Γ ) . Since Γ isbounded, (5.1) hold true. Therefore hypotheses i)-iii) in Theorem 5.1 hold.By (5.52) and [37, Theorem XIII.13], z R θ z has poles (and the coe ffi cients of the Laurent expansionare finite-rank operators) only at Σ θ ; so, by [37, Lemma 1, page 108], σ disc ( ∆ θ ) = Σ θ . Since ∆ θ is boundedfrom below, Σ θ is finite.If Ω ex is connected, then σ p ( ∆ θ ) ∩ ( −∞ , = ∅ by the unique continuation principle (see [27, Remark3.8]).By taking the limit ǫ ↓ + θ Λ N λ ± i ǫ )( Λ N λ ± i ǫ ) − Λ θλ ± i ǫ = = ( Λ N λ ± i ǫ ) − Λ θλ ± i ǫ ( + θ Λ N λ ± i ǫ )and ( + θ Λ N λ ± i ǫ ) − = ( Λ N λ ± i ǫ ) − Λ θλ ± i ǫ , one gets the existence of the inverse operator ( + θ Λ N , ± λ ) − andlim ǫ ↓ ( + θ Λ N λ ± i ǫ ) − = ( + θ Λ N , ± λ ) − . Thus Λ θ, ± λ = lim ǫ ↓ Λ N λ ± i ǫ ( + θ Λ N λ ± i ǫ ) − = Λ N , ± λ ( + θ Λ N , ± λ ) − = ( Q in λ − Q ex λ ) ± ( + θ ( Q in λ − Q ex λ ) ± ) − . emark 5.16. By remark 5.11, we can define ∆ θ for any real-valued θ ∈ L p ( Γ ), p > Remark 5.17.
In the quantum mechanics oriented literature, the δ ′ -like boundary conditions are usuallyrepresented in terms of a di ff erent parameter: let us suppose that β is a real-valued function which is a.e.di ff erent from zero and such that θ : = β − ∈ L p ( Γ ), p >
2; then one gets the self-adjoint operator ∆ β withdomain dom( ∆ β ) = { u ∈ H ∆ ( R n ) : [ γ Γ ] u = , βγ Γ u = [ γ Γ ] u } . That extends the results contained in [6, Section 3.2], where ∆ β is defined in case β − ∈ L ∞ ( Γ ) and Γ isa smooth hypersurface (see also [26, Section 5.5]). In the case β , Γ β ( Γ ,one can define the corresponding function θ as θ : = χ β β − , where χ β is the characteristic function of Γ β .Whenever such a function θ belongs to L p ( Γ ), p >
2, one gets again a self-adjoint operator ∆ β , with domaincharacterized by the boundary conditionsdom( ∆ β ) = { u ∈ H ∆ ( R n ) : [ γ Γ ] u = , (1 − χ β ) γ Γ u = , βγ Γ u = [ γ Γ ] u } . Operators with such kind of boundary conditions have been constructed (in case β and θ belong to L ∞ ( Γ ))in [15] (see also [26, Section 6.5] for a di ff erent construction in the case Γ is smooth). Asymptotic com-pleteness for the scattering couple ( ∆ , ∆ θ ) provided in Theorem 5.15 extends results on existence and com-pleteness given, in the case the boundary is smooth and θ is bounded, in [6] and [26]. The formula for thescattering matrix provided in (5.53) extends to Lipschitz hypersurfaces the results given, in the case of asmooth hypersurface and bounded θ , in [27, Subsections 6.5 and 7.5]. References [1] S. Agmon: Spectral properties of Schr¨odinger operators and Scattering Theory.
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