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Spectral and scattering theory for a class of strongly oscillating potentials

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Abstract

Following a recent investigation by Pearson [23] on scattering theory for some class of oscillating potentials, we consider the Schrödinger operator inL 2(IRn) given by:H =-e −U∇·e 2U∇e −U +e −2U(F + (∇·Q)). HereU andF are real functions ofx, andQ is a IRn-valued function ofx, such that:

  1. (1)

    U is bounded, and the local singularities ofF andQ 2 are controlled in a suitable sense by the kinetic energy,

  2. (2)

    U, Q, andF tend to zero at infinity faster than ‖x‖−1. We defineH by a method of quadratic forms and derive the usual results of scattering theory, namely: the absolutely continuous spectrum is [0, ∞) and the singular continuous spectrum is empty, the wave operators exist and are asymptotically complete. This enlarges the class of already studied strongly oscillating potentials that give rise to the scattering and spectral properties mentioned above.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique et Hautes Energies, Université de Paris-Sud, F-91405, Orsay, France

    Monique Combescure

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  1. Monique Combescure
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Communicated by B. Simon

Laboratoire associé au Centre National de la Recherche Scientifique

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Combescure, M. Spectral and scattering theory for a class of strongly oscillating potentials. Commun.Math. Phys. 73, 43–62 (1980). https://doi.org/10.1007/BF01942693

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  • DOI: https://doi.org/10.1007/BF01942693

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