Abstract
We show that the non-embedded eigenvalues of the Dirac operator on the real line with complex mass and non-Hermitian potential V lie in the disjoint union of two disks, provided that the L 1-norm of V is bounded from above by the speed of light times the reduced Planck constant. The result is sharp; moreover, the analogous sharp result for the Schrödinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on V implies the absence of non-real eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.
Article PDF
Similar content being viewed by others
References
Abramov A.A., Aslanyan A., Davies E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34(1), 57–72 (2001)
Bruneau, V., Ouhabaz, E.M.: Lieb-Thirring estimates for non-self-adjoint Schrödinger operators. J. Math. Phys. 49(9), 093504, 10 (2008)
Cascaval R.C., Gesztesy F., Holden H., Latushkin Y.: Spectral analysis of Darboux transformations for the focusing NLS hierarchy. J. Anal. Math. 93, 139–197 (2004)
Cuenin, J.-C.: Block-diagonalization of operators with gaps, with applications to Dirac operators. Rev. Math. Phys. 24(8), 1250021, 31 (2012)
Cuenin, J.-C., Tretter, C.: Perturbation of spectra and resolvent estimates. In preparation (2013)
Davies, E.B., Nath, J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math. 148(1), 1–28 (2002). On the occasion of the 65th birthday of Professor Michael Eastham
Dunford, N., Schwartz, J.T.: Linear Operators. Part I. Wiley Classics Library. John Wiley & Sons Inc., New York, (1988). General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, (1987) Oxford Science Publications
Frank R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)
Frank R.L., Laptev A., Lieb E.H., Seiringer R.: Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006)
Gesztesy F., Latushkin Y., Mitrea M., Zinchenko M.: Nonselfadjoint operators, infinite determinants, and some applications. Russ. J. Math. Phys. 12(4), 443–471 (2005)
Gohberg I., Goldberg S., Kaashoek M.A.: Classes of linear Operators. Vol. I, Volume 49 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1990)
Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)
Kato T.: Perturbation Theory for Linear Operators. Die Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York (1966)
Kato T.: Holomorphic families of Dirac operators. Math. Z. 183(3), 399–406 (1983)
Langer, H., Tretter, C.: Diagonalization of certain block operator matrices and applications to Dirac operators. In: Operator theory and analysis (Amsterdam, 1997), volume 122 of Oper. Theory Adv. Appl., pp. 331–358. Birkhäuser, Basel (2001)
Laptev A., Safronov O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Comm. Math. Phys. 292(1), 29–54 (2009)
Reed M., Simon B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)
Safronov O.: Estimates for eigenvalues of the Schrödinger operator with a complex potential. Bull. Lond. Math. Soc. 42(3), 452–456 (2010)
Šeba P.: The complex scaling method for Dirac resonances. Lett. Math. Phys. 16(1), 51–59 (1988)
Sjöstrand J., Zworski M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137(3), 381–459 (2007)
Syroid I.-P.P.: Nonselfadjoint perturbation of the continuous spectrum of the Dirac operator. Ukrain. Mat. Zh. 35(1), 115–119, 137 (1983)
Syroid, I.-P.P.: The nonselfadjoint one-dimensional Dirac operator on the whole axis. Mat. Metody i Fiz.-Mekh. Polya 25, 3–7, 101 (1987)
Thaller B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)
Tretter C.: Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London (2008)
Weder R.A.: Spectral properties of the Dirac Hamiltonian. Ann. Soc. Sci. Bruxelles Sér. I 87, 341–355 (1973)
Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil II. Mathematische Leitfäden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart, 2003. Anwendungen. [Applications]
Zworski, M.: Quantum resonances and partial differential equations. In: Proceedings of the International Congress of Mathematicians, vol. III (Beijing, 2002), pp. 243–252, Beijing, 2002. Higher Ed. Press
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Derezinski.
Rights and permissions
About this article
Cite this article
Cuenin, JC., Laptev, A. & Tretter, C. Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line. Ann. Henri Poincaré 15, 707–736 (2014). https://doi.org/10.1007/s00023-013-0259-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0259-3