Mathematical Physics, Analysis and Geometry

, Volume 13, Issue 1, pp 83–103 | Cite as

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

  • Toufik Hmidi
  • Andrea Mantile
  • Francis Nier


The non autonomous Cauchy problem \(i\partial_{t}u=-\partial_{xx}^{2} u+\alpha(t) \delta_{0}u\) with u t = 0 = u 0 is considered in L 2 (ℝ) . The regularity assumptions for α are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.


Point interactions Solvable models in Quantum Mechanics Non-autonomous Cauchy problems 

Mathematics Subject Classifications (2000)

37B55 35B65 35B30 35Q45 


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  1. 1.
    Adami, R., Teta, A.: A class of nonlinear Schrödinger equations with concentrated nonlinearitie. J. Funct. Anal. 180, 148–175 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Albeverio, S., Gesztesy, F., Högh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn, with an appendix by P. Exner. AMS, Providence (2005)Google Scholar
  3. 3.
    Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. Ec. Norm. Super. (4), 14(2), 209–246 (1981)MATHMathSciNetGoogle Scholar
  4. 4.
    Chemin, J.-Y.: Perfect Incompressible Fluids. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications, 14. Clarendon, Oxford University Press, New York (1998)Google Scholar
  5. 5.
    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), 1–26 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Correggi, M., Dell’Antonio, G.F., Figari, R., Mantile, A.: Ionization for three dimensional time-dependent point interactions. Commun. Math. Phys. 257, 169–192 (2005)MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Correggi, M., Dell’Antonio, G.F.: Decay of a bound state under a time-periodic perturbation: a toy case. J. Phys., A, Math. Gen. 38, 4769–4781 (2005)MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Dell’Antonio, G.F., Figari, R., Teta, A.: A limit evolution problem for time-dependent point interactions. J. Funct. Anal. 142, 249–275 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fattorini, H.O., Kerber, A.: The Cauchy Problem. Cambridge University Press, Cambridge (1984)Google Scholar
  10. 10.
    Holmer, J.: The initial boundary value problem for the 1D nonlinear Schrödinger equation onthe half line. Differ. Integral Equ. 18(6), 647–668 (2005)MathSciNetGoogle Scholar
  11. 11.
    Kato, T.: Integration of the equation of evolution in a Banach spac. J. Math. Soc. Jpn. 5, 208–234 (1953)MATHCrossRefGoogle Scholar
  12. 12.
    Kato, T.: Linear evolution equations of ‘hyperbolic’ type. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 17, 241–258 (1970)MATHGoogle Scholar
  13. 13.
    Kisyński, J.: Sur les opérateurs de Green des problèmes de Cauchy abstraits. Stud. Math. 23, 285–328 (1964)MATHGoogle Scholar
  14. 14.
    Mantile, A.: Point interaction controls for the energy transfer in 3D quantum systems. Int. J. Control 81(5), 752–763 (2008)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Neidhardt, H., Zagrebnov, V.: Linear non-autonomous Cauchy problems and evolution semigroups. Adv. Differ. Equ. 14(3–4), 289–340 (2009)MATHMathSciNetGoogle Scholar
  16. 16.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 2. Academic, New York (1975)MATHGoogle Scholar
  17. 17.
    Sayapova, M., Yafaev, D.: The evolution operator for time-dependent potentials of zero radius. Trudy Mat. Inst. Steklov. 159, 167–174 (1983)MathSciNetGoogle Scholar
  18. 18.
    Simon, B.: Quantum Machanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton (1971)Google Scholar
  19. 19.
    Soffer, A., Weinstein, M.I.: Nonautonomous Hamiltonians. J. Stat. Phys. 93(1–2), 359–391 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Yafaev, D.: On “eigenfunctions” of a time-dependent Schrödinger equation (Russian). Teor. Mat. Fiz. 43(2), 228–239 (1980)MathSciNetGoogle Scholar
  21. 21.
    Yafaev, D.: Scattering theory for time-dependent zero-range potentials. Ann. IHP, Phys. Théor. 40 (1984)Google Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.IRMAR, UMR-CNRS 6625Université Rennes 1Rennes CedexFrance

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