Completeness of the set \{e^{ikβ\cdot s}\}|_{\forall β\in S^2}
aa r X i v : . [ m a t h . A P ] M a y Completeness of the set { e ikβ · s }| ∀ β ∈ S ∼ ramm Abstract
It is proved that the set { e ikβ · s }| ∀ β ∈ S , where S is the unit sphere in R , k > s ∈ S , is total in L ( S ) if and only if k is not a Dirichlet eigenvalueof the Laplacian in D . Here S is a smooth, closed, connected surface in R . Let D ⊂ R be a bounded domain with a connected closed C − smooth boundary S , D ′ := R \ D be the unbounded exterior domain and S be the unit sphere in R , β ∈ S , s ∈ S .We are interested in the following problem: Is the set { e ikβ · s }| ∀ β ∈ S total in L ( S ) ? A set { φ ( s, β ) } is total (complete) in L ( S ) if the relation R S f ( s ) φ ( s, β ) ds = 0 for all β ∈ S implies f = 0, where f ∈ L @ ( S ) is an arbitrary fixed function.The above question is of interest by itself, but also it is of interest in scattering problemsand in inverse problems, see [1]–[5].Our result is: Theorem 1.
The set { e ikβ · s }| ∀ β ∈ S is total in L ( S ) if and only if k is not a Dirichleteigenvalue of the Laplacian in D . Necessity.
Let f ∈ L ( S ) and Z S f ( s ) e ikβ · s ds = 0 ∀ β ∈ S , (1) MSC: 30B60; 35R30; 35J05.Key words: completeness; scattering theory. nd there is a u ∇ + k ) u = 0 in D, u | S = 0 . (2)Choose f = u N , where N is the unit normal to S pointing out of D . Then, by Green’sformula, equation (1) holds and f Sufficiency.
Assume that f ∈ L ( S ) is and arbitrary fixed function, f
0, and (1)holds. Let h ∈ L ( S ) be arbitrary and w ( x ) := Z S h ( β ) e ikβ · x . (3)Then ( ∇ + k ) w = 0 in R . (4)If (1) holds, then Z S f ( s ) w ( s ) ds = 0 (5)for all w of the form (3). Let us now apply the following Lemma: Lemma 1.
The set { w | S } for all h ∈ L ( S ) is the orthogonal complement in L ( S ) tothe linear span of the set { v N } , where v solve equation (4) and v | S = 0 . If k is not a Dirichlet eigenvalue of the Laplacian in D , then Lemma 1 implies that theset { w | S } is total in L ( S ), so (1) implies f = 0. Sufficiency and Theorem 1 are proved. ✷ Lemma 1 is similar to Theorem 6 in [3].
Proof of Lemma 1.
Let w | S := ψ . Choose an arbitrary F ∈ C ( D ) such that F | S = ψ .Define G := F − w in D . Then( ∇ + k ) G = ( ∇ + k ) F inD ; G | S = 0 . (6)For (6) to hold it is necessary and sufficient that0 = Z D ( ∇ + k ) F vdx, (7)where v is an arbitrary function in the set of solutions of equation (2). Using Green’sformula one reduces condition (7) to the following condition: Z S ψv N ds = 0 . (8)Therefore the set { ψ } is the orthogonal complement in L ( S ) of the linear span of thefunctions { v N } . Lemma 1 is proved. ✷ eferences [1] A.G.Ramm, Scattering by obstacles , D.Reidel, Dordrecht, 1986.[2] A.G.Ramm,
Inverse problems , Springer, New York, 2005.[3] A.G.Ramm, Solution to the Pompeiu problem and the related symmetry problem,Appl. Math. Lett., 63, (2017), 28-33.[4] A.G.Ramm, Perturbation of zero surfaces,