Resonance statistics in a microwave cavity with a thin antenna

ArticleinPhysics Letters A 228(3) · February 1997with 10 Reads 
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Abstract
We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.

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    • H J Stoeckmann
    • J Stein
    H.J. Stoeckmann, J.Stein, Phys. Rev. Lett. 64 (1990), 2215–2218.
    • J A Folk
    • S F Godijn
    • A G Huibers
    • C M Cronenwelt
    • C M Marcus
    • K Campman
    • A C Gossard
    J.A. Folk, S.F. Godijn, A.G. Huibers, C.M. Cronenwelt, C.M. Marcus, K. Campman, A.C. Gossard, Phys. Rev.Lett. 76 (1996), 1699–1702.
    • F Haake
    F. Haake et al., Phys. Rev. A44 (1991), R6161–6164.
    • A Kudroli
    • S Sridhar
    • A Pandey
    • R Ramaswamy
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