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.
Similar content being viewed by others
References
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)
Agmon S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151–218 (1975)
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)
Balslev E., Helffer B.: Limiting absorption principle and resonances for the Dirac operator. Adv. Adv. Math. 13, 186–215 (1992)
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)
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)
Berthier A., Georgescu V.: On the point spectrum of Dirac operators. J. Funct. Anal. 71(2), 309–338 (1987)
Bouclet J.-M., Tzvetkov N.: On global Strichartz estimates for non trapping metrics. J. Funct. Anal. 254(6), 1661–1682 (2008)
Boussaid N.: Stable directions for small nonlinear Dirac standing waves. Commun. Math. Phys. 268(3), 757–817 (2006)
Boussaid N., Comech A.: On spectral stability of the nonlinear Dirac equation. J. Funct. Anal. 271, 1462–1524 (2016)
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)
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)
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)
Cacciafesta F.: Virial identity and dispersive estimates for the n-dimensional Dirac equation. J. Math. Sci. Univ. Tokyo 18, 1–23 (2011)
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
Christ M., Kiselev A.: Maximal functions associated with filtrations. J. Funct. Anal. 179, 409–425 (2001)
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)
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)
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)
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)
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)
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)
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)
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
Fanelli L., Vega L.: Magnetic virial identities, weak dispersion and Strichartz inequalities. Math. Ann. 344(2), 249–278 (2009)
Fefferman, C.L.M.I. Weinstein: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326(1), 251–286 (2014)
Georgescu V., Mantoiu M.: On the spectral theory of singular Dirac type Hamiltonians. J. Oper. Theory 46(2), 289–321 (2001)
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)
Ginibre J., Velo G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)
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)
Hörmander L.: The Analysis of Linear Partial Differential Operators, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1985)
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)
Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)
Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)
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)
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)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978)
Rodnianski I., Schlag W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)
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
Roze S.N.: On the spectrum of the Dirac operator. Theor. Math. Phys. 2(3), 377–382 (1970)
Stefanov A.: Strichartz estimates for the magnetic Schrödinger equation. Adv. Math. 210(1), 246–303 (2007)
Thaller B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)
Vogelsang V.: Absolutely continuous spectrum of Dirac operators for long-range potentials. J. Funct. Anal. 76(1), 67–86 (1988)
Yamada O.: A remark on the limiting absorption method for Dirac operators. Proc. Jpn Acad. Ser. A Math. Sci. 69(7), 243–246 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by W. Schlag
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.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3231-8