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We prove that the resonance counting functions for Schrödinger operators H V = −∆ + V on L 2 (R d), for d ≥ 2 even, with generic, compactly-supported, real-or complex-valued potentials V , have the maximal order of growth d on each sheet Λm, m ∈ Z\{0}, of the logarithmic Riemann surface. We obtain this result by constructing, for each m ∈ Z\{0}, a… (More)

We describe the generic behavior of the resonance counting function for a Schrödinger operator with a bounded, compactly-supported real or complex valued potential in d ≥ 1 dimensions. This note contains a sketch of the proof of our main results [5, 6] that generically the order of growth of the resonance counting function is the maximal value d in the odd… (More)

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

Suppose that (X, g) is a conformally compact (n + 1)-dimensional manifold that is hyperbolic near infinity in the sense that the sectional curvatures of g are identically equal to minus one outside of a compact set K ⊂ X. We prove that the counting function for the resolvent resonances has maximal order of growth (n + 1) generically for such manifolds. This… (More)

We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time π on the boundary. Furthermore, it is shown that on R n the asymptotics of certain short-range perturbations of ∆ k can be… (More)

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions d ≥ 2. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto Λ, the Riemann surface of the logarithm function. We study the multiplicities of the poles of the continued scattering matrix on… (More)

This paper is concerned with scattering by a smooth compact obstacle in R d. The main results are that for certain classes of obstacles knowledge of the the scattering amplitude in only one or two pairs of incident and reflected directions suffices to recover the Taylor series of a boundary defining function for the obstacle at a point. Thus certain… (More)

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