On the Global Limiting Absorption Principle for Massless Dirac Operators

  • Alan Carey
  • Fritz Gesztesy
  • Jens Kaad
  • Galina Levitina
  • Roger Nichols
  • Denis Potapov
  • Fedor Sukochev


We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators \(H_0 = \alpha \cdot (-i \nabla )\) for all space dimensions \(n \in {{\mathbb {N}}}\), \(n \geqslant 2\). This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.


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  1. 1.
    Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Sc. Norm. Sup. Pisa Ser. 4 2, 151–218 (1975)MATHGoogle Scholar
  2. 2.
    Aiba, D.: Absence of zero resonances of massless Dirac operators. Hokkaido Math. J. 45, 263–270 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Amrein, W., Boutet de Monvel, A., Georgescu, V.: \(C_0\)-Groups, Commutator Methods and Spectral Theory of \(N\)-Body Hamiltonians. Progress in Mathematics, vol. 135. Birkhäuser, Basel (1996)CrossRefMATHGoogle Scholar
  4. 4.
    Balinsky, A., Evans, W.D.: Spectral Analysis of Relativistic Operators. Imperial College Press, London (2011)MATHGoogle Scholar
  5. 5.
    Balslev, E., Helffer, B.: Limiting absorption principle and resonances for the Dirac operator. Adv. Appl. Math. 13, 186–215 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bandara, L., Saratchandran, H.: Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics. J. Funct. Anal. 273, 3719–3758 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baumgärtel, H., Wollenberg, M.: Mathematical Scattering Theory. Akademie Verlag, Berlin (1983)CrossRefMATHGoogle Scholar
  8. 8.
    Ben-Artzi, M., Devinatz, A.: The limiting absorption principle for partial differential operators. Mem. Am. Math. Soc. 66(364), 1–70 (1987)MathSciNetMATHGoogle Scholar
  9. 9.
    Boussaid, N., Golénia, S.: Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies. Commun. Math. Phys. 299, 677–708 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boutet de Monvel-Berthier, A., Manda, D., Purice, R.: Limiting absorption principle for the Dirac operator. Ann. Inst. H. Poincaré 58, 413–431 (1993)MathSciNetMATHGoogle Scholar
  11. 11.
    Boutet de Monvel, A., Mantoiu, M.: The method of the weakly conjugate operator. In: Apagyi, B., Endrédi, G., Lévay, P. (eds.) Inverse and Algebraic Quantum Scattering Theory. Springer, Heidelberg (1997)Google Scholar
  12. 12.
    Carey, A., Gesztesy, F., Levitina, G., Nichols, R., Sukochev, F., and Zanin, D.: On the limiting absorption principle for massless Dirac operators (in preparation)Google Scholar
  13. 13.
    Carey, A., Gesztesy, F., Levitina, G., Potapov, D., Sukochev, F., Zanin, D.: On index theory for non-Fredholm operators: a \((1+1)\)-dimensional example. Math. Nachr. 289, 575–609 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Carey, A., Gesztesy, F., Levitina, G., Sukochev, F.: On the index of a non-Fredholm model operator. Oper. Matrices 10, 881–914 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Carey, A., Gesztesy, F., Grosse, H., Levitina, G., Potapov, D., Sukochev, F., Zanin, D.: Trace formulas for a class of non-Fredholm operators: a review. Rev. Math. Phys. 28(10), 1630002 (2016). (55 pages)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Carey, A., Gesztesy, F., Potapov, D., Sukochev, F., Tomilov, Y.: On the Witten index in terms of spectral shift functions. J. Anal. Math. 132, 1–61 (2017)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)MATHGoogle Scholar
  19. 19.
    Daude, T.: Scattering theory for massless Dirac fields with long-range potentials. J. Math. Pures Appl. 84, 615–665 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Erdoğan, M.B., Goldberg, M., Green, W.R.: Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions. arXiv:1706.05257
  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. 10, 507–531 (2008)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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, 719–757 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Faris, W.G.: Self-Adjoint Operators. Lecture Notes in Mathematics, vol. 433. Springer, Berlin (1975)MATHGoogle Scholar
  25. 25.
    Forsyth, I., Mesland, B., Rennie, A.: Dense domains, symmetric operators and spectral triples. N. Y. J. Math. 20, 1001–1020 (2014)MathSciNetMATHGoogle Scholar
  26. 26.
    Georgescu, V., Măntoiu, M.: On the spectral theory of singular Dirac type Hamiltonians. J. Oper. Theory 46, 289–321 (2001)MathSciNetMATHGoogle Scholar
  27. 27.
    Gérard, C.: A proof of the abstract limiting absorption principle by energy estimates. J. Funct. Anal. 254, 2707–2724 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Golénia, S., Jecko, T.: A new look at Mourre’s commutator theory. Complex Anal. Oper. Theory 1, 399–422 (2007)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Herbst, I.: Spectral theory of the operator \((p^2 + m^2)^{1/2} - Z e^2 /r\). Commun. Math. Phys. 53, 285–294 (1977)ADSCrossRefMATHGoogle Scholar
  30. 30.
    Higson, N., Roe, J.: Analytic \(K\)-Homology. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  31. 31.
    Iftimovici, A., Măntoiu, M.: Limiting absorption principle at critical values for the Dirac operator. Lett. Math. Phys. 49, 235–243 (1999)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kaad, J.: Differentiable absorption of Hilbert \(C^*\)-modules, connections, and lifts of unbounded operators. J. Noncommut. Geom. 11, 1037–1068 (2017)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kaad, J., Lesch, M.: A local global principle for regular operators in Hilbert \(C^*\)-modules. J. Funct. Anal. 262(10), 4540–4569 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1966)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kuroda, S.T.: An Introduction to Scattering Theory. Aarhus University Lecture Notes Series, No. 51 (1978)Google Scholar
  36. 36.
    Măntoiu, M., Pascu, M.: Global resolvent estimates for multiplication operators. J. Oper. Theory 36, 283–294 (1996)MathSciNetMATHGoogle Scholar
  37. 37.
    Mesland, B., Rennie, A.: Nonunital spectral triples and metric completeness in unbounded \(KK\)-theory. J. Funct. Anal. 271(9), 2460–2538 (2016)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Pladdy, C., Saitō, Y., Umeda, T.: Resolvent estimates for the Dirac operator. Analysis 15, 123–149 (1995)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Pladdy, C., Saitō, Y., Umeda, T.: Radiation condition for Dirac operators. J. Math. Kyoto Univ. 37(4), 567–584 (1998)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978)MATHGoogle Scholar
  41. 41.
    Richard, S.: Some improvements in the method of weakly conjugate operator. Lett. Math. Phys. 76, 27–36 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Ruzhansky, M., Sugimoto, M.: Structural resolvent estimates and derivative nonlinear Schrödinger equations. Commun. Math. Phys. 314, 281–304 (2012)ADSCrossRefMATHGoogle Scholar
  43. 43.
    Saitō, Y., Umeda, T.: The zero modes and zero resonances of massless Dirac operators. Hokkaido Math. J. 37, 363–388 (2008)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Vogelsang, V.: Absolutely continuous spectrum of Dirac operators for long-range potentials. J. Funct. Anal. 76, 67–86 (1988)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Yafaev, D.R.: Mathematical Scattering Theory. General Theory. Amer. Math. Soc, Providence, RI (1992)CrossRefMATHGoogle Scholar
  46. 46.
    Yafaev, D.R.: Mathematical Scattering Theory. Analytic Theory. Math. Surveys and Monographs, Vol. 158, Amer. Math. Soc., Providence, RI (2010)Google Scholar
  47. 47.
    Yamada, O.: On the principle of limiting absorption for the Dirac operator. Publ. RIMS, Kyoto Univ. 8, 557–577 (1972/73)Google Scholar
  48. 48.
    Yamada, O.: Eigenfunction expansions and scattering theory for Dirac operators. Publ. RIMS, Kyoto Univ. 11, 651–689 (1976)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Yamada, O.: A remark on the limiting absorption method for Dirac operators. Proc. Japan. Acad. Ser. A 69, 243–246 (1993)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alan Carey
    • 1
    • 2
  • Fritz Gesztesy
    • 3
  • Jens Kaad
    • 4
  • Galina Levitina
    • 5
  • Roger Nichols
    • 6
  • Denis Potapov
    • 5
  • Fedor Sukochev
    • 5
  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  3. 3.Department of MathematicsBaylor UniversityWacoUSA
  4. 4.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  5. 5.School of Mathematics and StatisticsUNSWKensingtonAustralia
  6. 6.Mathematics DepartmentThe University of Tennessee at ChattanoogaChattanoogaUSA

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