Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

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)

  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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Balslev E., Helffer B.: Limiting absorption principle and resonances for the Dirac operator. Adv. Adv. Math. 13, 186–215 (1992)

    MathSciNet  Article  MATH  Google 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)

    ADS  Article  MATH  Google 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)

    ADS  Article  MATH  Google Scholar 

  7. 7

    Berthier A., Georgescu V.: On the point spectrum of Dirac operators. J. Funct. Anal. 71(2), 309–338 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Bouclet J.-M., Tzvetkov N.: On global Strichartz estimates for non trapping metrics. J. Funct. Anal. 254(6), 1661–1682 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Boussaid N.: Stable directions for small nonlinear Dirac standing waves. Commun. Math. Phys. 268(3), 757–817 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Boussaid N., Comech A.: On spectral stability of the nonlinear Dirac equation. J. Funct. Anal. 271, 1462–1524 (2016)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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

    MathSciNet  MATH  Google Scholar 

  16. 16

    Christ M., Kiselev A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409–425 (2001)

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  27. 27

    Georgescu V., Mantoiu M.: On the spectral theory of singular Dirac type Hamiltonians. J. Oper. Theory 46(2), 289–321 (2001)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Ginibre J., Velo G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. 33

    Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34

    Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  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)

    ADS  MathSciNet  Article  MATH  Google 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)

    Article  Google Scholar 

  41. 41

    Stefanov A.: Strichartz estimates for the magnetic Schrödinger equation. Adv. Math. 210(1), 246–303 (2007)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to William R. Green.

Additional information

The first author was partially supported by NSF Grant DMS-1501041. The second author is supported by Simons Foundation Grant 281057. The third author is supported by Simons Foundation Grant 511825 and acknowledges the support of a Rose-Hulman summer professional development Grant.

Communicated by W. Schlag

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Burak Erdoğan, M., Goldberg, M. & Green, W.R. Limiting Absorption Principle and Strichartz Estimates for Dirac Operators in Two and Higher Dimensions. Commun. Math. Phys. 367, 241–263 (2019). https://doi.org/10.1007/s00220-018-3231-8

Download citation