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On some Schrödinger and wave equations with time dependent potentials

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Abstract

The existence and uniqueness of solutions in the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate.

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References

  1. D'Ancona, P., Pierfelice, V.: On the wave equation with a large rough potential. Preprint

  2. D'Ancona, P., Pierfelice, V., Visciglia, N.: Some remarks on the Schrödinger equation with a potential in L r t L s x , to appear in Math. Ann. (2005)

  3. Constantin, P., Saut, J.-C.: Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1(2), 413–439 (1988)

    Article  MathSciNet  Google Scholar 

  4. Georgiev, V., Visciglia, N.: Decay estimates for the wave equation with potential. Comm. Partial Diff. Equations 28(7–8), 1325–1369 (2003)

    Google Scholar 

  5. Ginibre, J., Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys. 123, 535–573 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Goldberg, M., Schlag, W.: Dispersive estimates for Schrödinger operators in dimensions one and three. Comm. Math. Phys. 251(1), 157–178 (2004)

    Article  Google Scholar 

  7. Ionescu, A., Kenig, C.: Well-posedness and local smoothing of solutions of Schrödinger equations. Math. Res. Lett. 12(2–3), 193–205 (2005)

    Google Scholar 

  8. Jensen, A., Nakamura, S.: Lp mapping properties of functions of Schrödinger operators and their applications to scattering theory. J. Math. Soc. Japan 47(2), 253–273 (1995)

    Google Scholar 

  9. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998)

    MATH  MathSciNet  Google Scholar 

  10. Kenig, C., Ponce, G., Vega, L.: Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math. 134(3), 489–545 (1998)

    Article  MathSciNet  Google Scholar 

  11. Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993)

    MATH  MathSciNet  Google Scholar 

  12. Montgomery-Smith, S.J.: Time Decay for the Bounded Mean Oscillation of Solutions of the Schrödinger and Wave Equations. Duke Math. J. 91(2), 393–408 (1998)

    Article  MathSciNet  Google Scholar 

  13. Rodnianski, I, Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)

    Article  MathSciNet  Google Scholar 

  14. Reed, M., Simon, B.: Methods of modern mathematical physics, IV: Analysis of operators. Academic Press, 1978

  15. Ruiz, A., Vega, L.: On local regularity of Schrödinger equations. Internat. Math. Res. Notices no. 1, 13–27 (1993)

    Article  MATH  Google Scholar 

  16. Simon, B.: The bound states of weakly coupled Schrödinger operators in one and two dimensions. Ann. Phys. (N.Y.) 97, 279–288 (1976)

    Article  MATH  Google Scholar 

  17. Simon, B.: Schrödinger operators in the twentieth century. J. Math. Phys. 41(6), 3523–3555 (2000)

    Article  Google Scholar 

  18. Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55(3), 699–715 (1987)

    Article  Google Scholar 

  19. Strichartz, R.S.: Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tao, T.: Counterexamples to the n=3 Endpoint Strichartz estimate for the wave equation. preprint (1998)

  21. Tao, T.: Spherically averaged end point Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25(7–8), 1471–1485 (2000)

    Google Scholar 

  22. Vega, L.: Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc. 102(4), 874–878 (1988)

    Article  MathSciNet  Google Scholar 

  23. Yajima, K.: Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110, 415–426 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rose-Hulman Institute of Technology, 5500 Wabash Avenue, Terre Haute, IN, 47803

    Virginia Naibo

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045

    Atanas Stefanov

Authors
  1. Virginia Naibo
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  2. Atanas Stefanov
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Correspondence to Virginia Naibo.

Additional information

The second author supported in part by NSF-DMS 0300511 and the University of Kansas General Research Fund # 2301716.

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Naibo, V., Stefanov, A. On some Schrödinger and wave equations with time dependent potentials. Math. Ann. 334, 325–338 (2006). https://doi.org/10.1007/s00208-005-0720-9

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  • DOI: https://doi.org/10.1007/s00208-005-0720-9

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