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Communications in Mathematical Physics

, Volume 367, Issue 1, pp 241–263 | Cite as

Limiting Absorption Principle and Strichartz Estimates for Dirac Operators in Two and Higher Dimensions

  • M. Burak Erdoğan
  • Michael Goldberg
  • William R. GreenEmail author
Article
  • 53 Downloads

Abstract

In this paper we consider Dirac operators in \({\mathbb{R}^n}\), \({n \ge 2}\), with a potential V. Under mild decay and continuity assumptions on V and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted L2 spaces as the frequency goes to infinity.

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC (1964)Google Scholar
  2. 2.
    Agmon S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151–218 (1975)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arai M., Yamada O.: Essential selfadjointness and invariance of the essential spectrum for Dirac operators. Publ. Res. Inst. Math. Sci. 18(3), 973–985 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balslev E., Helffer B.: Limiting absorption principle and resonances for the Dirac operator. Adv. Adv. Math. 13, 186–215 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bejenaru I., Herr S.: The cubic Dirac equation: small initial data in \({H^{1/2}(\mathbb{R}^3)}\). Commun. Math. Phys. 335, 43–82 (2015)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bejenaru I., Herr S.: The cubic Dirac equation: small initial data in \({H^{1/2}(\mathbb{R}^2)}\). Commun. Math. Phys. 343, 515–562 (2016)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Berthier A., Georgescu V.: On the point spectrum of Dirac operators. J. Funct. Anal. 71(2), 309–338 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bouclet J.-M., Tzvetkov N.: On global Strichartz estimates for non trapping metrics. J. Funct. Anal. 254(6), 1661–1682 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boussaid N.: Stable directions for small nonlinear Dirac standing waves. Commun. Math. Phys. 268(3), 757–817 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Boussaid N., Comech A.: On spectral stability of the nonlinear Dirac equation. J. Funct. Anal. 271, 1462–1524 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boussaid N., Comech A.: Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity. SIAM J. Math. Anal. 49, 2527–2572 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Boussaid N., D’Ancona P., Fanelli L.: Virial identiy and weak dispersion for the magnetic Dirac equation. J. Math. Pures Appl. 95, 137–150 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Boussaid N., Golenia S.: Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies. Commun. Math. Phys. 299(3), 677–708 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cacciafesta F.: Virial identity and dispersive estimates for the n-dimensional Dirac equation. J. Math. Sci. Univ. Tokyo 18, 1–23 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Carey A., Gesztesy F., Kaad J., Levitina G., Nichols R., Potapov D., Sukochev F.: On the global limiting absorption principle for massless Dirac operators. Ann. Henri Poincaré (2018)  https://doi.org/10.1007/s00023-018-0675-5 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Christ M., Kiselev A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409–425 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Comech A., Phan T., Stefanov A.: Asymptotic stability of solitary waves in generalized Gross-Neveu model. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 157–196 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D’Ancona P., Fanelli L.: Strichartz and smoothing estimates for dispersive equations with magnetic potentials. Commun. Partial Differ. Equ. 33(4–6), 1082–1112 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    D’Ancona P., Fanelli L.: Decay estimates for the wave and Dirac equations with a magnetic potential. Commun. Pure Appl. Math. 60(3), 357–392 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D’Ancona P., Fanelli L., Vega L., Visciglia N.: Endpoint Strichartz estimates for the magnetic Schrdinger equation. J. Funct. Anal. 258(10), 3227–3240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Erdoğan M.B., Goldberg M., Schlag W.: Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in \({\mathbb{R}^3}\). J. Eur. Math. Soc. (JEMS) 10(2), 507–531 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Erdoğan M.B., Goldberg M., Schlag W.: Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions. Forum Math. 21, 687–722 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Erdoğan M.B., Green W.R.: The Dirac equation in two dimensions: dispersive estimates and classification of threshold obstructions. Commun. Math. Phys. 352(2), 719–757 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Erdoğan, M.B., Green, W.R., Toprak, E.: Dispersive estimates for Dirac operators in dimension three with obstructions at threshold energies. Am. J. Math. ( to appear). arXiv:1609.05164
  25. 25.
    Fanelli L., Vega L.: Magnetic virial identities, weak dispersion and Strichartz inequalities. Math. Ann. 344(2), 249–278 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fefferman, C.L.M.I. Weinstein: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326(1), 251–286 (2014)Google Scholar
  27. 27.
    Georgescu V., Mantoiu M.: On the spectral theory of singular Dirac type Hamiltonians. J. Oper. Theory 46(2), 289–321 (2001)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Georgiev V., Stefanov A., Tarulli M.: Smoothing-Strichartz estimates for the Schrödinger equation with small magnetic potential. Discrete Contin. Dyn. Syst. 17(4), 771–786 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ginibre J., Velo G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Goldberg M., Schlag W.: A limiting absorption principle for the three-dimensional Schrödinger equation with L p potentials. Int. Math. Res. Not. 75, 4049–4071 (2004)CrossRefzbMATHGoogle Scholar
  31. 31.
    Hörmander L.: The Analysis of Linear Partial Differential Operators, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1985)Google Scholar
  32. 32.
    Kalf H., Yamada O.: Essential self-adjointness of n-dimensional Dirac operators with a variable mass term. J. Math. Phys. 42(6), 2667–2676 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)Google Scholar
  35. 35.
    Machihara S., Nakamura M., Nakanishi K., Ozawa T.: Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219, 1–20 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Marzuola J., Metcalfe J., Tataru D.: Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations. J. Funct. Anal. 255(6), 1497–1553 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978)Google Scholar
  38. 38.
    Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Rodnianski, I., Tao, T.: Effective limiting absorption principles, and applications. Commun. Math. Phys. 333, 1 (2015).  https://doi.org/10.1007/s00220-014-2177-8
  40. 40.
    Roze S.N.: On the spectrum of the Dirac operator. Theor. Math. Phys. 2(3), 377–382 (1970)CrossRefGoogle Scholar
  41. 41.
    Stefanov A.: Strichartz estimates for the magnetic Schrödinger equation. Adv. Math. 210(1), 246–303 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Thaller B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)Google Scholar
  43. 43.
    Vogelsang V.: Absolutely continuous spectrum of Dirac operators for long-range potentials. J. Funct. Anal. 76(1), 67–86 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yamada O.: A remark on the limiting absorption method for Dirac operators. Proc. Jpn Acad. Ser. A Math. Sci. 69(7), 243–246 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsUniversity of CincinnatiCincinnatiUSA
  3. 3.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA

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