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Communications in Mathematical Physics

, Volume 203, Issue 2, pp 421–444 | Cite as

Scattering Problem for Local Perturbations\newline of the Free Quantum Gas

  • Yu. G. Kondratiev
  • A. Yu. Konstantinov
  • M. Röckner
  • G. V. Shchepan'uk

Abstract:

Scattering theory for perturbations of the intrinsic Dirichlet (Laplace–Beltrami) operator H 0=−divΓΓ on L 2(Γ,π z ), i. e. the space of π z -square integrable functions on the configuration space Γ over ℝ d , is studied. Here π z denotes Poisson measure with intensity z. We show that for an arbitrary regular non-zero potential V the standard wave operators W ±(H 0,H 0+V) do not exist, and propose to consider Dirichlet operators of perturbed Poisson measures instead of potential perturbations of the Hamiltonian H 0. As case studies, cylindric smooth densities and finite volume Gibbs perturbations of the Poisson measure are considered. In these cases the existence of the corresponding wave operators is proved.

Keywords

Potential Versus Integrable Function Finite Volume Configuration Space Wave Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Yu. G. Kondratiev
    • 1
  • A. Yu. Konstantinov
    • 2
  • M. Röckner
    • 3
  • G. V. Shchepan'uk
    • 4
  1. 1.Bonn University, D-53013 Bonn, Germany. E-mail: kondratiev@wiener.iam.uni-bonn.de};¶BiBoS Research Center, Bielefeld University, D-33615 Bielefeld, GermanyDE
  2. 2.Department for Mathematics, Kyiv University, 64 Volodymyrs'ka St, 252033, Kyiv, Ukraine.¶E-mail: konst@faust.kiev.uaUA
  3. 3.Department for Mathematics, Bielefeld University, D-33615 Bielefeld, Germany.¶E-mail: roeckner@mathematik.uni-bielefeld.deDE
  4. 4.Institute for Mathematics of the Ukrainian National Science Academy, 3 Tereshchenkivs'ka St, MSP, Kyiv-4, 252601, Ukraine. E-mail: gena@imat.gluk.apc.orgUA

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