×

The complex scaling method for Dirac resonances. (English) Zbl 0659.47017

The complex scaling theory is successfully used in atomic and molecular physics. This method is generalized to the Dirac equation. It is shown that Dirac resonances are associated with nonreal eigenvalues of the scaled Dirac Hamiltonian, which describes a free particle with spin 1/2 and rest mass m in relativistic quantum mechanics. The perturbation theory for Dirac resonances is reduced to the standard Kato-Rellich theory.
Reviewer: S.D.Karokozov

MSC:

47A55 Perturbation theory of linear operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
Full Text: DOI

References:

[1] AguilarJ. and CombesJ. Commun. Math. Phys. 22, 269 (1971). · Zbl 0219.47011 · doi:10.1007/BF01877510
[2] BalslevE. and CombesJ., Commun. Math. Phys. 22, 286 (1971). · Zbl 0219.47005 · doi:10.1007/BF01877511
[3] Balslev, E., Resonance, resonance functions and spectral deformation, Proc. of the conference ’Resonances-Models and Phenomena’, Bielefeld, 9–14 April, 1984. Lecture Notes in Physics 211.
[4] SimonB., Commun. Math. Phys. 27, 1 (1972). · Zbl 0237.35025 · doi:10.1007/BF01649654
[5] SimonB., Ann. Math. 97, 247 (1973). · Zbl 0252.47009 · doi:10.2307/1970847
[6] JunkerB. J., Adv. Atom. Mol. Phys. 18, 207 (1982). · doi:10.1016/S0065-2199(08)60242-0
[7] ReinhardtE., Ann. Rev. Phys. Chem. 33, 223 (1982). · doi:10.1146/annurev.pc.33.100182.001255
[8] HoY. K., Phys. Rep. 99, 1 (1983). · doi:10.1016/0370-1573(83)90112-6
[9] RafelskiJ., FulchnerL. P., KleinA., Phys. Rep. 38, 236 (1978). · doi:10.1016/0370-1573(78)90116-3
[10] CowanR. et al., Phys. Rev. Lett. 54, 1761 (1985). · doi:10.1103/PhysRevLett.54.1761
[11] NATO Advanced Study Institute Series Vol. 80; Electrodynamics of Strong Fields, Plenum Press, New York, 1983.
[12] Weder, R. A., Spectral properties of the Dirac Hamiltonian, Preprint KUL, April 1973.
[13] WederR. A., J. Math. Phys. 15, 20 (1974). · doi:10.1063/1.1666495
[14] KatoT., Perturbation for Linear Operators, Springer-Verlag, Berlin, 1966, 1975.
[15] JoergensK., Perturbation of the Dirac operator; Lecture Notes in Mathematics 280, Springer-Verlag, Berlin, 1972.
[16] Kalf, H., Schmincke, U. W., Walter, J., and Wust, P., On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Lecture Notes in Mathematics 448. · Zbl 0311.47021
[17] BabbitD. and BalslevE., J. Funct. Anal. 18, 1 (1975). · Zbl 0304.47009 · doi:10.1016/0022-1236(75)90026-9
[18] SchechterM., J. Math. Anal. Appl. 13, 205 (1966). · Zbl 0147.12101 · doi:10.1016/0022-247X(66)90085-0
[19] GustafsonK., Michigan J. Math. 19, 71 (1972). · Zbl 0213.14301 · doi:10.1307/mmj/1029000800
[20] ReedM. and SimonB., Methods of Modern Mathematical Physics, IV, Academic Press, New York, 1978.
[21] Šeba, P., The Dirac eigenvalues near upper and lower continuum, Preprint JINR E2-86-808, Dubna, 1986. · Zbl 0638.70016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.