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We prove that the resonance counting functions for Schrödinger operators HV = −∆+V on L2(Rd), for d ≥ 2 even, with generic, compactlysupported, realor 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 plurisubharmonic… (More)

We consider scattering by an obstacle in Rd, 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.

We describe the generic behavior of the resonance counting function for a Schrödinger operator with a bounded, compactlysupported 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 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 the asymptotics of certain short-range perturbations of ∆ can be recovered… (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)

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 relate the multiplicities of the poles of the continued scattering matrix to… (More)

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)

- Tanya J. Christiansen
- Asymptotic Analysis
- 2013

This paper is concerned with scattering by a smooth compact obstacle in R. 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 obstacles… (More)

The purpose of this paper is to give some refined results about the distribution of resonances in potential scattering. We use techniques and results from one and several complex variables, including properties of functions of completely regular growth. This enables us to find asymptotics for the distribution of resonances in sectors for certain potentials… (More)

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