Uniqueness in potential scattering with reduced near field data
aa r X i v : . [ m a t h - ph ] F e b Uniqueness in potential scattering with reduced data
Evgeny Lakshtanov ∗ Boris Vainberg † Abstract
We consider inverse potential scattering problems where the source of the inci-dent waves is located on a smooth closed surface outside of the inhomogeneity ofthe media. The scattered waves are measured on the same surface at a fixed valueof the energy. We show that this data determines the bounded potential uniquely.
Key words: scattering data, inverse problem, backscattering, uniqueness.
Consider a potential scattering problem − ∆ u sc − λn ( x ) u sc = λ [ n ( x ) − u inc , x ∈ R d , λ = k > , (1)where the support of n ( x ) − O , n ( x ) is uniformly boundedin O , and the solution u sc satisfies the radiation condition: u sc = u sc ∞ ( k, θ ) e ikr r d − + O (cid:16) r − d +12 (cid:17) , θ = xr , r = | x | → ∞ . (2)Here u inc is an incident wave that satisfies the Helmholtz equation in R d \S where S isa set, where sources are distributed. We assume that S is a smooth surface that is aboundary of a bounded domain B located outside of O . To be more exact, u inc ( x ) = Z S e − ik | x − y | | x − y | ϕ ( y ) dS y , ϕ ∈ L ( S ) , x ∈ R d . (3) ∗ Department of Mathematics, Aveiro University, Aveiro 3810, Portugal. This work was supportedby Portuguese funds through the CIDMA - Center for Research and Development in Mathematics andApplications and the Portuguese Foundation for Science and Technology (“FCT–Fund¸c˜ao para a Ciˆenciae a Tecnologia”), within project UID/MAT/0416/2013 ([email protected]). † Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC 28223, USA.The work was partially supported by the NSF grant DMS-1008132 ([email protected]). R ) has been studied by Rakesh and Uhlmann [15].They assumed that the incident waves were emitted from points x varying in some sphere.They show the uniqueness for potentials with some restrictions on angular derivatives. In[9] we considered the scattering problem (1) when the incident waves were emitted fromsurface S and the receivers are also distributed over the same surface S , i.e., the followingdata are available: (cid:8) u sc | S : u inc emitted from S (cid:9) . (4)We have shown that data (4) allows one to determine the interior eigenvalues of thescatterer. In this article, we prove a uniqueness result. Namely, let us fix λ > B bounded by S . We show that data(4) for a fixed value of λ > n ( · ) uniquely. We also will assumethat λ is not an eigenvalue of the Dirichlet problem for the equation ( − ∆ − λn ( x )) u = 0in O . Since the support of n is bounded, the latter requirement can be enforced by aslight extension of O . Without loss of the generality, we can assume that the boundaryof O is infinitely smooth and the support of n ( x ) − O .Note also that the problem we consider is different from the problem of recovering ofthe potential from partial Cauchy data (see e.g. [3]). In the latter problem, it is assumedthat Cauchy data are available for all sufficiently regular solutions of the wave equation.The situation is different in the problem under consideration. Here only the fields on S are known that are produced by waves emitted from S . Acknowledgments.
The authors are thankful to Rakesh, Eemeli Bl˚asten, UweK¨ahler and Lassi P¨aiv¨arinta, David Colton and Armin Lechleiter for useful discussions.
From now on, for the sake of simplicity of notations, we assume that d = 3. Defineoperator L : L ( S ) → L ( ∂ O ) , L ∗ : L ( ∂ O ) → L ( S ) , ( L ϕ )( x ) = Z S e − ik | x − y | | x − y | ϕ ( y ) dS y , ( L ∗ µ )( x ) = Z ∂ O e ik | x − y | | x − y | µ ( y ) dS y , k = √ λ > . (5)2 emma 2.1. Suppose that λ > is not an eigenvalue of the negative Dirichlet Laplacianin either of the domains O or B (with the boundary S ). Then operators L , L ∗ have denseranges. Remark.
An outline of this proof can be found in [9]. Note also the integral kernelsof operators L , L ∗ are infinitely smooth, and the arguments below prove that their rangesare dense in any Sobolev space H s , s ≥ , not only in L . Proof.
Let us prove that the range of L is dense. Obviously, it is enough to showthat the kernel of the operator L ∗ is trivial. Assume that the opposite is true. Then thereexists µ ∈ L ( ∂ O ) such that µ u := Z ∂ O e ik | x − y | | x − y | µ ( y ) dS y , x ∈ R , k = √ λ > , which is defined on R and coincides with L ∗ µ on S , vanishes on S . Since( − ∆ − λ ) u = 0 , x / ∈ ∂ O , and λ is not an eigenvalue of the Dirichlet problem in B, u ≡ B . Then from theequation above it follows that u ≡ R \ O .If µ is continuous, the proof can be completed in a couple of lines using the potentialtheory. Indeed, u is continuous in R in this case. Thus u satisfies the Helmholtz equationand the homogeneous Dirichlet boundary condition in O . Since λ is not an eigenvalue, itfollows that u ≡ O , i.e., u ≡ R . The latter contradicts the fact that the jumpof the normal derivative of u on ∂ O is equal to − πµ µ ∈ L ( ∂ O ), then we approximate µ in L ( ∂ O ) by smooth functions µ n . Consider u n = Z ∂ O e ik | x − y | | x − y | µ n ( y ) dS y , x ∈ R . (6)If we restrict u n to ∂ O , then operator (6) becomes a pseudo-differential operator on ∂ O oforder −
1, and therefore u n | ∂ O has a limit in H / ( ∂ O ) as n → ∞ (as well as in H ( ∂ O )).Functions u n satisfy the Helmholtz equation outside of ∂ O , and they satisfy the radiationconditions. Thus the convergence of u n | ∂ O and standard a priori estimates in H for thesolutions of the Helmholtz equation imply that functions u n converge in H ( O ) and in H loc ( R \O ). Obviously, they converge to u ≡ H loc ( R \O ). Thus u n | ∂ O → H / ( ∂ O ) as n → ∞ . Hence u n converge in H ( O ) to a solution of the homogeneous Dirichlet problem. Since λ is not an eigenvalue of the Dirichlet problem in O , this implies that u n → H ( O ).Since µ n is smooth, the jump on ∂ O of the normal derivative of the potential u n defined by (6) is equal to − πµ n
0. On the other hand, the normal derivatives of weak(in H ) solutions of the Helmholtz equation are well defined, and from the weak (in H )3onvergence of u n to zero it follows that this jump (which is equal to µ n ) tends to zero in H − / ( ∂ O ). Since µ n approximates µ in L ( ∂ O ), it follows that µ = 0. This contradictsthe assumption made in the first lines of the proof. Thus the density of the range of theoperator L is proved. Similar arguments are valid for L ∗ . Definition . Consider the near-field operator F S = F S ( λ ) : L ( S ) → L ( S ) , F S ϕ = u sc | S , ϕ ∈ L ( S ) , where u sc is the solution of (1) with u inc given by (3).Note that formula (3) represents waves coming to S , while waves emitted from S have the different sign in the exponent. Thus F S ϕ is not the scattered wave produced bysources on S with the density ϕ . However, u sc | S = F S ϕ can be obtained (and measured)as a scattered field on S produced by some waves emitted from S . Namely, the followinglemma holds (see [9]). Lemma 2.2.
Suppose that λ > is not an eigenvalue of the negative Dirichlet Laplacianin either of the domains O or B . Then for each ϕ ∈ L ( S ) , one can construct a sequence ψ n ∈ L ( S ) of the source densities such that F S ϕ = lim n → u sc n | S , where the limit is takenin the space L ( S ) and u sc n is the solution of (1) with u inc ( x ) = Z S e ik | x − y | | x − y | ψ n ( y ) dS y , ψ n ∈ L ( S ) , x ∈ R d . One can determine the source densities ψ n without a priori knowledge of O except a valueof an ε > such that O is located inside of the ball | x | < /ε . Proof.
Consider a bounded domain e O that contains O and such that dist( B, e O ) > e O = ( R d \ B ε ) T {| x | < /ε } , where B ε is the ε -extension of B and ε > e O is infinitely smooth and λ is not an eigenvalue of the negative Dirichlet Laplacian in e O . From Lemma 2.1 it follows that the range of the operator( e L ϕ )( x ) = Z S e − ik | x − y | | x − y | ϕ ( y ) dS y , x ∈ ∂ e O , ϕ ∈ L ( S ) , is dense in H / ( ∂ e O ). Then the same is true for e L . Hence for every ϕ ∈ L ( S ), thereexists a sequence ψ n ∈ L ( S ) such that e L ψ n → e L ϕ in H / ( ∂ e O ). Below we considerfunctions e L ψ n , e L ϕ, L ψ n , L ϕ defined by the corresponding integrals for all x ∈ R . Thestandard a priory estimate (e.g., [10]) for the solution u = e L ψ n − e L ϕ of the Helmholtzequation in e O implies that k e L ψ n − e L ϕ k H ( e O ) ≤ C ( λ ) k e L ψ n − e L ϕ k H / ( ∂ e O ) → n → ∞ . O ⊂ e O , we have that kL ϕ − L ψ n k H / ( ∂ O ) → n → ∞ and kL ϕ − L ψ n k H ( O ) → n → ∞ . The statement of the lemma is an immediate consequence of the last two relations and apriory estimates (e.g., [10]) for the solutions of the problem (1) (with radiation conditionat infinity).
Theorem 2.3.
Consider two real-valued bounded potentials n and n and their backscat-tering far-field operators F S,i , i = 1 , . If λ = λ is not a Dirichlet eigenvalue for thedomain S , then the equality F S, ϕ = F S, ϕ, λ = λ , on a dense set { ϕ } in L ( S ) impliesthat k n − n k L ∞ = 0 . The following lemma will be needed to prove the theorem above.Denote by F ( λ ) , F out ( λ ) Dirichlet-to-Neumann maps for the Helmholtz equation inthe interior and exterior of O , respectively. The solutions are assumed to satisfy theradiation condition when F out is defined. Let F n be the Dirichlet-to-Neumann map forthe equation (∆ + λn ) u = 0 in O . The normal vector in all the cases is assumed tobe directed outside of O . Each of the Dirichlet-to-Neumann operators introduced aboveis a pseudo-differential operator of the first order and can be considered as a boundedoperator from a Sobolev space H s ( ∂ O ) into H s − ( ∂ O ) , s ∈ R . Lemma 2.4.
The near field operator F S has the following representation: F S = 14 π L ∗ ( F − F out )( F n − F out ) − ( F − F n ) L . (7) Remark.
These formulas are direct analogues of the formulas for the scatteringamplitude in the problem of scattering of the plane waves (see [8, Th.2.3] in the case ofthe transmission problem). The only difference is that a plane wave is defined by thedirection ω of the incident wave, and S is replaced by the unit sphere S = { ω : | ω | = 1 } in this case. The operators L , L ∗ are also slightly different in the case of the plane waves.In particular, L : L ( S ) → L ( ∂ O ) , L ϕ ( x ) = Z S e ikω · x ϕ ( ω ) dS ω . (8) Proof . Let us prove (7). Note that u inc | ∂ O = L ϕ . We will look for u sc outside of O in the form of the potential u sc = L ∗ µ with an unknown density µ , i.e., u sc = Z ∂ O e ik | x − y | | x − y | µ ( y ) dS y , x ∈ R \O . (9)5ore over, function µ must be chosen in such a way that u sc allows an extension in O that satisfies (1).Every solutions of the Schr¨odinger equation with a bounded potential belongs to H ( O ′ ) for any bounded domain O ′ . Therefore functions u sc , u inc and their normal deriva-tives are well defined on ∂ O . We reduce the scattering problem (1),(2) to the followingequation on ∂ O for unknown µ : F n ( u sc | ∂ O + u inc | ∂ O ) = F out ( u sc | ∂ O ) + F ( u inc | ∂ O ) . (10)This equation follows from the fact that u sc + u inc satisfies (1) in O , and u sc , u inc aresolutions of the Helmholtz equation in R \O and O , respectively.We note that operator F n is symmetric, and the imaginary part of the quadratic formof operator F out coincides with the total cross section, and therefore is positive (see [8,Lemma 2.1]). Thus, operator F n − F out is invertible, and equation (10) implies that u sc | ∂ O = ( F n − F out ) − ( F − F n )( u inc | ∂ O ) = ( F n − F out ) − ( F − F n ) L ϕ. (11)From (9) it follows that µ = π ( F − F out )( u sc | ∂ O )). It remains only to substitute (11)for u sc in the latter equation for µ and note that F S ϕ = u sc | S = L ∗ µ. Proof of Theorem 2.3.
We will reduce the statement of the theorem to the Gelfand-Calderon problem, which is solved in [11, Th.1] when d = 3 and in [2, Th.2.1] when d = 2.We preserve notations F , F out for the Dirichlet-to-Neumann maps for the Helmholtzequation in the interior and exterior of O , respectively, and we denote by F n , F n theDirichlet-to-Neumann maps for the Schr¨odinger equations in O with potentials λn and λn , respectively.Operators( F − F out )( F n i − F out ) − ( F − F n i ) : L ( ∂ O ) → L ( ∂ O ) , i = 1 , , (12)are bounded (and also compact). Indeed, each of the Dirichlet-to-Neumann operatorsintroduced above is a pseudo-differential operator of the first order (non-smoothness ofthe potential does not play any role here, since the support of the potential is strictly insideof the domain). Their full symbols were calculated in [7, Sect.3]. From this calculation itfollows that operator F − F out has order one, operator ( F n i − F out ) − has order −
1, anda couple of the first terms of the full symbol of operator F − F n i vanish, i.e., the latteroperator is compact. Thus (12) is compact.Assume that data (4) for n and n coincide on a dense set { ϕ } in L ( S ). Then fromLemma 2.1 it follows that operators (12) are equal. The first factor from the left in (12)is an invertible operator (see the justification of the transition from (10) to (11)). Hence,the equality of operators in (12) implies that( F n − F out ) − ( F − F n ) = ( F n − F out ) − ( F − F n )6s operators in L ( ∂ O ). Adding and subtracting F out in the right factors, we get( F n − F out ) − ( F − F out ) = ( F n − F out ) − ( F − F out )as operators in L ( ∂ O ). Hence, operators( F n − F out ) − , ( F n − F out ) − : H − ( ∂ O ) → L ( ∂ O ) , are equal, and therefore, F n − F out , F n − F out : L ( ∂ O ) → H − ( ∂ O )are equal. Thus F n ϕ = F n ϕ for every ϕ ∈ L ( ∂ O ).Now uniqueness follows from [2],[1] if d = 2 and [11] if d = 3. References [1] E. Bl˚asten, O.Y. Imanuvilov and M. Yamamoto, (2015),
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