Abstract
We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to studying a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non-self-adjoint operators, which we develop in the Appendix. We also discuss some applications to the dispersive Helmholtz model in the quantum regime.
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Amrein, W.O., Boutet de Monvel, A., Georgescu, V.: C 0-groups, commutator methods and spectral theory of N-body Hamiltonians. Progress in Mathematics, Vol. 135, Basel: Birkhäuser Verlag, 1996
Arai M.M., Yamada O.: Essential self-adjointness and invariance of the essential spectrum for Dirac operators. Publ. Res. Inst. Math. Sci. 18, 973–985 (1982)
Astaburuaga M.A., Bourget O., Cortés V.H., Fernández C.: Floquet operators without singular continuous spectrum. J. Funct. Anal. 238(2), 489–517 (2006)
Berthier A., Georgescu V.: On the point spectrum of Dirac operators. J. Funct. Anal. 71(2), 309–338 (1987)
Benamou J.D., Castella F., Katsaounis T., Perthame B.: High frequency limit of the Helmholtz equations. Rev. Mat. Iberoam. 18(1), 187–209 (2002)
Benamou J.D., Lafitte O., Sentis R., Solliec I.: A geometrical optics-based numerical method for high frequency electromagnetic fields computations near fold caustics I. J. Comput. Appl. Math. 156(1), 93–125 (2003)
Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin: Springer-Verlag, 1976
Bony J.F., Häfner D.: Low frequency resolvent estimates for long range perturbations of the Euclidian Laplacian. Math. Res. Lett. 17(2), 301–306 (2010)
Bouclet, J.M.: Low frequency estimates for long range perturbations in divergence form. To appear in Candian Journal of Math.
Boussaid N.: Stable directions for small nonlinear Dirac standing waves. Commun. Math. Phys. 268(3), 757–817 (2006)
Boussaid N.: On the asymptotic stability of small nonlinear Dirac standing waves in a resonant case. SIAM J. Math. Anal. 40(4), 1621–1670 (2008)
Boutet de Monvel A., Kazantseva G., Măntoiu M.: Some anisotropic Schrödinger operators without singular spectrum. Helv. Phys. Acta 69, 13–25 (1996)
Boutet de Monvel, A., Măntoiu, M.: The method of the weakly conjugate operator. In: Lecture Notes in Physics, Vol. 488, Berlin-New York: Springer, 1997, pp. 204–226
Burq N., Planchon F., Stalker J.G., Tahvildar-Zadeh A.S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004)
D’Ancona P., Fanelli L.: Decay estimates for the wave and Dirac equations with a magnetic potential. Comm. Pure Appl. Math. 60(3), 357–392 (2007)
D’Ancona P., Fanelli L.: Strichartz and smoothing estimates of dispersive equations with magnetic potentials. Comm. Part. Diff. Eqs. 33(4–6), 1082–1112 (2008)
Dirac, P.: Principles of Quantum Mechanics, 4th ed. Oxford: Oxford University Press, 1982
Dereziński, J., Gérard, C.: Scattering theory of classical and quantum n-particle systems. Texts and Monographs in Physics. Berlin: Springer, 1997
Dereziński J., Skibsted E.: Quantum scattering at low energies. J. Funct. Anal. 257, 1828–1920 (2009)
Dereziński J., Skibsted E.: Scattering at zero energy for attractive homogeneous potentials. Ann. Henri Poincaré 10, 549–571 (2009)
Dolbeault J., Esteban M.J., Sere E.: On the eigenvalues of operators with gaps. Application to Dirac operators. J. Funct. Anal. 174(1), 208–226 (2000)
Escobedo M., Vega L.: A semilinear Dirac equation in H s(R 3) for s > 1. SIAM J. Math. Anal. 28(2), 338–362 (1997)
Esteban M.J., Loss M.: Self-adjointness for Dirac operators via Hardy-Dirac inequalities. J. Math. Phys. 48(11), 112107 (2007)
Faris W., Lavine R.: Commutators and selfadjointness of Hamiltonian operators. Commun. Math. Phys. 35, 39–48 (1974)
Fournais S., Skibsted E.: Zero energy asymptotics of the resolvent for a class of slowly decaying potentials. Math. Z. 248(3), 593–633 (2004)
Fröhlich J., Griesemer M., Sigal I.M.: Spectral theory for the standard model of non-relativisitc QED. Commun. Math. Phys. 283(3), 613–646 (2008)
Fröhlich, J., Griesemer, M., Sigal, I.M.: Spectral Renormalization Group and Local Decay in the Standard Model of the Non-relativistic Quantum Electrodynamics. http://arxiv.org/abs/0904.1014v1 [math-ph], 2009
Georgescu V., Gérard C.: On the virial theorem in quantum mechanics. Commun. Math. Phys. 208, 275–281 (1999)
Georgescu V., Golénia S.: Compact perturbations and stability of the essential spectrum of singular differential operators. J. Oper. Theory 59(1), 115–155 (2008)
Georgescu V., Măntoiu M.: On the spectral theory of singular Dirac type Hamiltonians. J. Operator Theory 46(2), 289–321 (2001)
Georgescu V., Gérard C., Møller J.: Commutators, C 0−semigroups and resolvent estimates. J. Funct. Anal. 216(2), 303–361 (2004)
Gérard C.: A proof of the abstract limiting absorption principle by energy estimates. J. Funct. Anal. 254, 2070–2704 (2008)
Gérard, C., Łaba, I.: Multiparticle quantum scattering in constant magnetic fields. Mathematical Surveys and Monographs 90. Providence, RI: Amer. Math. Soc., 2002
Golénia S., Jecko T.: A new look at Mourre’s commutator theory. Complex Anal. Oper. Theory 1(3), 399–422 (2007)
Golénia S., Moroianu S.: Spectral analysis of magnetic Laplacians on conformally cusp manifolds. Ann. Henri Poincaré 9(1), 131–179 (2008)
Guillarmou C., Hassell A.: Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds, Part I. Math. Ann. 341(4), 859–896 (2008)
Guillarmou C., Hassell A.: The resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. Part II. Ann. Inst. Fourier 59(4), 1553–1610 (2008)
Helffer, B., Sjöstrand, J.: Opérateurs de schrödinger avec champs magnétiques faibles et constants. Sé min. Équations Dériv. Partielles 1988–1989, Exp. No. 12, 11 p. Palaiseau: Ecole Polytechnique, 1989
Herbst I.W.: Spectral and scattering theory for Schrödinger operators with potentials independent of |x|. Am. J. Math. 113(3), 509–565 (1991)
Hunziker, W., Sigal, I.M., Soffer, A.: Minimal escape velocities, Comm. Part. Diff. Eqs. 24, 11, 2279–2295 (1999)
Iftimovici A., Măntoiu M.: Limiting absorption principle at critical values for the Dirac operator. Lett. Math. Phys. 49(3), 235–243 (1999)
Jackson J.D.: Classical electrodynamics. 3rd ed., John Wiley & Sons., New York, NY (1999)
Jensen A., Nenciu G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717–754 (2001)
Jensen, A., Nenciu, G. Erratum: “A unified approach to resolvent expansions at thresholds”. [Rev. Math. Phys. 13, no. 6, 717–754 (2001)], Rev. Math. Phys. 16, no. 5, 675–677 (2004)
Kalf H.: Essential self-adjointness of Dirac operators under an integral condition on the potential. Lett. Math. Phys. 44(3), 225–232 (1998)
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, New York: Springer-Verlag New York, Inc., 1966
Kato T., Yajima K.: Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1(4), 481–496 (1989)
Klaus M.: Dirac operators with several Coulomb singularities. Helv. Phys. Acta 53, 463–482 (1980)
Levitan, B.M., Otelbaev, M.: On conditions for selfadjointness of the Schrödinger and Dirac operators. Sov. Math. Dokl. 18, 1044–1048 (1978); translation from Dokl. Akad. Nauk SSSR 235, 768–771 (1977)
Landgren J., Rejtö P.: An application of the maximum principle to the study of essential self-adjointness of Dirac operators I. J. Math. Phys. l 20, 2204–2211 (1979)
Landgren J., Rejtö P., Klaus M.: An application of the maximum principle to the study of essential self-adjointness of Dirac operators II. J. Math. Phys. 21, 1210–1217 (1980)
Lawson, H.B., Michelsohn, M.L.: Spin geometry. Princeton Mathematical Series 38. Princeton, NJ: Princeton University Press., 1990
Machihara S., Nakamura M., Nakanishi K., Ozawa T.: Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219(1), 1–20 (2005)
Machihara S., Nakamura M., Ozawa T.: Small global solutions for nonlinear Dirac equations. Diff. Integral Eqs. 17(5-6), 623–636 (2004)
Mourre E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78, 519–567 (1981)
Nakamura S.: Low energy asymptotics for Schrödinger operators with slowly decreasing potentials. Commun. Math. Phys. 161(1), 63–76 (1994)
Nenciu G.: Eigenfunction expansions for Schrödinger and Dirac operators with singular potentials. Commun. Math. Phys. 42, 221–229 (1975)
Oguntuase, J.A.: On an inequality of Gronwall. J. Inequal. Pure Appl. Math. 2, no. 1, Article 9, 6 pp. (2001) (electronic)
Perry P., Sigal I., Simon B.: Spectral analysis of N-body Schrödinger operators. Ann. of Math. 114, 519–567 (1981)
Putnam C.R.: Commutator properties of Hilbert space operators and related topics. Springer Verlag, Berlin-Heidelberg-New York (1967)
Reed M., Simon B.: Methods of Modern Mathematical Physics: I–IV. Academic Press, New York-San Francisco-London (1979)
Richard S.: Some improvements in the method of the weakly conjugate operator. Lett. Math. Phys. 76(1), 27–36 (2006)
Richard S., Tiedrade Aldecoa R.: On the spectrum of magnetic Dirac operators with Coulomb-type perturbations. J. Funct. Anal. 250(2), 625–641 (2007)
Royer, J.: Limiting absorption principle for the dissipative Helmholtz equation, http://arxiv.org/abs/0905.0355v2 [math,AP], 2009
Thaller, B.: The Dirac equation, Texts and Monographs in Physics, Berlin: Springer-Verlag, 1992
Vasy, A., Wunsch, J.: Positive commutators at the bottom of the spectrum. http://arxiv.org/abs/0909.4583v2 [math.AP], 2009
Voronov B.L., Gitman D.M., Tyutin I.V.: The Dirac Hamiltonian with a superstrong Coulomb field. Teoret. Mat. Fiz. 150(1), 41–84 (2007)
Yafaev D.R.: The low energy scattering for slowly decreasing potentials. Commun. Math. Phys. 85(2), 177–196 (1982)
Yokoyama K.: Limiting absorption principle for Dirac operator with constant magnetic field and long-range potential. Osaka J. Math. 38(3), 649–666 (2001)
Wang, X.P.: Number of eigenvalues for a class of non-selfadjoint Schrödinger operators, Available at http://hal.archives-ouvertes.fr/docs/00/37/30/28/DDF/complex-eigenvalues.pdf, 2009
Wang X.P.: Time-decay of scattering solutions and classical trajectories. Annales de l’I.H.P., Sect. A 47(1), 25–37 (1987)
Wang X.P., Zhang P.: High-frequency limit of the Helmholtz equation with variable refraction index. J. Funct. Ana. 230, 116–168 (2006)
Xia J.: On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian. Trans. Amer. Math. Soc. 351, 1989–2023 (1999)
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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). https://doi.org/10.1007/s00220-010-1099-3
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DOI: https://doi.org/10.1007/s00220-010-1099-3