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Ionization for Three Dimensional Time-Dependent Point Interactions

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Abstract

We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the “strength” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t→∞, starting from a bound state at time t=0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states.

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References

  1. Albeverio, S.A., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. New York: Springer-Verlag, 1988

  2. Berezin, F.A., Faddeev, L.D.: A Remark on Schrödinger Equation with a Singular Potential. Sov. Math. Dokl. 2, 372–375 (1961)

    Google Scholar 

  3. Correggi, M., Dell’Antonio, G.F.: Rotating Singular Perturbations of the Laplacian. Ann. H. Poincaré 5, 773–808 (2004)

    MathSciNet  Google Scholar 

  4. Costin, O., Costin, R.D., Lebowitz, J.L., Rokhlenko, A.: Evolution of a Model Quantum System under Time Periodic Forcing: Conditions for Complete Ionization. Commun. Math. Phys. 221, 1–26 (2001)

    Article  Google Scholar 

  5. Costin, O., Lebowitz, J.L., Rokhlenko, A.: Decay versus Survival of a Localized State Subjected to Harmonic Forcing: Exact Results. J. Phys. A: Math. Gen. 35, 8943–8951 (2002)

    Article  Google Scholar 

  6. Costin, O., Lebowitz, J.L., Rokhlenko, A.: Exact Results for the Ionization of a Model Quantum System. J. Phys. A: Math. Gen. 33, 6311–6319 (2000)

    Article  Google Scholar 

  7. Costin, O., Costin, R.D., Lebowitz, J.L.: Transition to the Continuum of a Particle in Time-Periodic Potentials. In: Advances in Differential Equations and Mathematical Physics, Birmingham 2002, Contemp. Math. 327, Providence, RI: AMS, 2003, pp. 75–86

  8. Costin, O., Lebowitz, J.L., Rokhlenko, A.: Ionization of a Model Atom: Exact Results and Connection with Experiment. http://arxiv.org/abs/physics/9905038, 1999

  9. Dell’Antonio, G.F., Figari, R., Teta, A.: Schrödinger Equation with Moving Point Interactions in Three Dimensions. In: Stochastic Processes, Physics and Geometry: New Interplays, Leipzig 1999, CMS Conference Proceedings 28, Providence, RI: AMS, 2000, pp. 99–113

  10. Dell’Antonio, G.F.: Point Interactions. In: Mathematical Physics in Mathematics and Physics, Siena 2000, Fields Institute Communications 30, Providence, RI: AMS, 2001, pp. 139–150

  11. Enss, V., Veselic, K.: Bound States and Propagating States for Time-dependent Hamiltonians. Ann. Inst. H. Poincaré A 39, 159–191 (1983)

    Google Scholar 

  12. Figari, R.: Time Dependent and Non Linear Point Interactions. In: Proceedings of Mathematical Physics and Stochastic Analysis, Lisbon 1998, New York: World Scientific Publisher, 2000, pp. 184–197

  13. Graffi, S., Grecchi, V., Silverstone, H.J.: Resonances and Convergence of Perturbative Theory for N-body Atomic Systems in External AC-electric Field. Ann. Inst. H. Poincaré A 42, 215–234 (1985)

    CAS  Google Scholar 

  14. Howland, J.S.: Stationary Scattering Theory for Time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974)

    Article  Google Scholar 

  15. Howland, J.S.: Scattering Theory for Hamiltonians Periodic in Time. Indiana Univ. Math. J. 28(3), 471–494 (1979)

    Article  Google Scholar 

  16. Porter, D., Stirling, D.S.G.: Integral Equations. Cambridge: Cambridge University Press, 1990

  17. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol.I: Functional Analysis. San Diego: Academic Press, 1975

  18. Sayapova, M.R., Yafaev, D.R.: The Evolution Operator for Time-dependent Potentials of Zero Radius. Proc. Stek. Inst. Math. 2, 173–180 (1984)

    Google Scholar 

  19. Yafaev, D.R.: Scattering Theory for Time-dependent Zero-range Potentials. Ann. Inst. H. Poincaré A 40, 343–359 (1984)

    Google Scholar 

  20. Yajima, K., Kitada, H.: Bound States and Scattering States for Time Periodic Hamiltonians. Ann. Inst. H. Poincaré A 39, 145–157 (1983)

    Google Scholar 

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Authors and Affiliations

  1. International School for Advanced Studies, Trieste, Italy

    Michele Correggi

  2. Centro Linceo Interdisciplinare, Roma, Italy

    Gianfausto Dell’Antonio

  3. Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and Sezione INFN, Napoli, Italy

    Rodolfo Figari

  4. Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Napoli, Italy

    Andrea Mantile

Authors
  1. Michele Correggi
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  2. Gianfausto Dell’Antonio
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  3. Rodolfo Figari
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  4. Andrea Mantile
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Corresponding author

Correspondence to Michele Correggi.

Additional information

Communicated by B. Simon

On leave from Dipartimento di Matematica, Università di Roma, “La Sapienza”, Italy.

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Correggi, M., Dell’Antonio, G., Figari, R. et al. Ionization for Three Dimensional Time-Dependent Point Interactions. Commun. Math. Phys. 257, 169–192 (2005). https://doi.org/10.1007/s00220-005-1293-x

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  • DOI: https://doi.org/10.1007/s00220-005-1293-x

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