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Ionization of Coulomb Systems in \({\mathbb{R}^3}\) by Time Periodic Forcings of Arbitrary Size

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Abstract

We analyze the long time behavior of solutions of the Schrödinger equation \({i\psi_t=(-\Delta-b/r+V(t,x))\psi}\), \({x\in\mathbb{R}^3}\), r =  |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) =  V(t +  2π/ω, x) with zero time average.

We show that, for any V(t, x) of the form \({2\Omega(r) \sin (\omega t-\theta)}\), with Ω(r) nonzero on its support, Floquet bound states do not exist. This implies that the system ionizes, i.e. \({P(t, K) = \int_K|\psi(t,x)|^2dx\to 0}\) as t → ∞ for any compact set \({K\subset\mathbb{R}^3}\). Furthermore, if the initial state is compactly supported and has only finitely many spherical harmonic modes, then P(t, K) decays like \({t^{-5/3}}\) as t → ∞.

To prove these statements, we develop a rigorous WKB theory for infinite systems of ordinary differential equations.

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Authors and Affiliations

  1. Department of Mathematics, Ohio State University, 231 West 18th Ave., Columbus, OH, 43210, USA

    O. Costin & S. Tanveer

  2. Department of Mathematics, Rutgers University, 110 Frelinghaysen Rd., Piscataway, NJ, 08854, USA

    J. L. Lebowitz

  3. Department of Physics, Rutgers University, 136 Frelinghaysen Rd., Piscataway, NJ, 08854, USA

    J. L. Lebowitz

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  1. O. Costin
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  2. J. L. Lebowitz
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  3. S. Tanveer
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Correspondence to O. Costin.

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Communicated by M. Aizenman

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Costin, O., Lebowitz, J.L. & Tanveer, S. Ionization of Coulomb Systems in \({\mathbb{R}^3}\) by Time Periodic Forcings of Arbitrary Size. Commun. Math. Phys. 296, 681–738 (2010). https://doi.org/10.1007/s00220-010-1023-x

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