Resonances and balls in obstacle scattering with Neumann boundary conditions

We consider scattering by an obstacle in $\Real^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give related results for obstacles which are disjoint unions of several balls of the same radius.


Introduction
The purpose of this is note is to show that for obstacle scattering in Ê d , d ≥ 3 odd, a ball is uniquely determined by its resonances for the Laplacian with Neumann boundary conditions. We actually show a somewhat stronger result: If O 1 and O 2 have the same (Neumann) resonances and O 1 is the disjoint union of m balls, each of radius ρ, then so is O 2 . The multiplicity can be defined via Set B(ρ) to be the closed ball of radius ρ centered at the origin.
We note that a simple scaling argument shows that if ρ 1 = ρ 2 then R B(ρ1) = R B(ρ2) so that B(ρ) is determined (up to translation) by its Neumann resonances.
Hassell and Zworski [7] proved that for the Dirichlet Laplacian in Ê 3 , if a connected set O has the same resonances as B(ρ), then O is a translate of B(ρ). It seems that their argument also works for Neumann boundary conditions, again in dimension 3 with the restriction that O is connected. That we can prove this result for all odd dimensions for Neumann boundary conditions follows partly from the fact that for the Neumann case the resonances determine the Partially supported by NSF grant DMS 0500267. determinant of the scattering matrix up to one unknown constant, while for the Dirichlet case there are two unknown constants-compare Lemma 2.1 to [7].
We shall actually prove the following theorem, from which Theorem 1.1 follows immediately. The proof of Theorem 1.2 uses heat coefficients. These are closely related to the singularity at t = 0 of the distribution This is rather informal. One way to make precise sense of (1) is as follows. With ρ ′ chosen sufficiently large that O ⊂ B(ρ ′ ), where ½ E is the characteristic function of the set E, compare [15]. The Poisson formula for resonances in odd dimensions is see [3,8,9,14], and [13] for an application to the existence of resonances. This shows that any singularities of the distribution u O (t) at nonzero time are determined by the resonances of ∆ Ê d \O .
When O is not connected, we expect that the distribution of (2) has singularities at nonzero times, see [10]. In particular, if O consists of two disjoint convex obstacles, the distribution (2) has a singularity at twice the distance between them (e.g. [6] or [ is the disjoint union of 2 balls of radius ρ a distance δ > 0 apart. If R O1 = R O2 , then O 2 is obtained from O 1 by a rigid motion.
This paper was inspired by [7], and we use some of the same notation. In For a large number of more general self-adjoint operators (for example, for the Laplacian with different boundary conditions), the resonances determine the determinant of the scattering matrix up to two unknown constants, e.g. [7].
Proof. It follows from [14] that We now use the expression of [11] for the scattering matrix. Choose ρ ′ > 0 so that O is contained in the ball of radius ρ ′ centered at the origin.
− denotes the transpose of χ3 − . Then the scattering matrix S O (λ) associated to P is given by (4) S O (λ) = I + A O (λ).
We fix some notation. Proof. As t ↓ 0, t n/2 a n .
The trace in (6) can be made precise in exactly the same manner that (1) is made precise in (2). Explicit expressions for the first few a n can be found in [4]. For Neumann boundary conditions in Ê d , we have where the α i are nonzero constants which depend on the dimension. Using Lemma 2.1, (6), j (e.g. [12,Corollary 2.10] for the case of Ê d \ O connected, or [5]) and the expressions for the a i above, we prove the lemma.
We are now ready for the proof of our theorem. , with equality if and only if κ 1,∂Oi = κ 2,∂Oi = · · · = κ d−1,∂Oi . Since equality holds for O 1 , the three equalities of (7) mean that H ∂O2 is a constant and the principal curvatures of ∂O 2 are all equal. Thus O 2 must be the disjointn union of l balls of fixed radius ρ ′ . Again using the first two equalities of (7), we see that we must have l = m and ρ ′ = ρ.
We remark that the proof of this theorem is a bit delicate, in the sense that it is important for us that in the expression for a 3 the coefficients of H 2 ∂O and d−1 1 κ 2 j,∂O have the same sign. However, if R O = R B(ρ) and we know a priori that O is a (d-dimensional) convex set, then we do not need to use the coefficient a 3 to prove that O is, up to translation, B(ρ). Instead, one can use first two equalities of (7) and the fact that in the Alexandrov-Fenchel inequality Vol(∂O) Vol(∂B(ρ)) equality holds if and only if O is a ball [2, Chapter 4, Section 9] or [1].