×

Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value. (English) Zbl 1135.81030

The search for a proof of the stability of matter is very old and started with the work of Dyson and Lenard in 1967 by showing that Pauli’s exclusion principle for fermions is essential. Subsequently, Lieb and Thirring (1975) found a more satisfying proof using sophicated mathematical techniques. Lieb’s involvement with this problem now spans 33 years. Together with coworkers he looked at the stability of matter under the influence of intense magnetic and gravitational fields relevant to the behavior of massive stars and their possible collapse.
The present paper uses a pseudo-relativistic description of a quantum system of \(N\) electrons and \(K\) fixed nuclei avoiding the intricate Dirac theory. The given proof of the stability of relativistic matter including magnetic fields now goes further than previous proofs by permitting values for the nuclear charge \(Z\) all the way up to \(Z\alpha=2/\pi\) which is the well-known critical value when magnetic fields are absent.

MSC:

81V70 Many-body theory; quantum Hall effect
81Q99 General mathematical topics and methods in quantum theory
82-XX Statistical mechanics, structure of matter
81V45 Atomic physics
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Berezin F.A. (1972). Covariant and contravariant symbols of operators. (English transl.), Math. USSR. Izv. 6: 1117–1151 · Zbl 0259.47004 · doi:10.1070/IM1972v006n05ABEH001913
[2] Daubechies I. (1983). An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90: 511–520 · Zbl 0946.81521 · doi:10.1007/BF01216182
[3] Daubechies I. and Lieb E.H. (1983). One electron relativistic molecules with Coulomb interaction. Commun. Math. Phys. 90: 497–510 · Zbl 0946.81522 · doi:10.1007/BF01216181
[4] Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. http://arxiv.org/list/math.SP/0610593, 2006 · Zbl 1202.35146
[5] Herbst I.W. (1977). Spectral theory of the operator (p 2 + m 2)1/2e 2/r. Commun. Math. Phys. 53: 285–294 · Zbl 0375.35047 · doi:10.1007/BF01609852
[6] Kato T. (1976). Perturbation theory for linear operators. Springer, Berlin-Heidelberg-New York · Zbl 0342.47009
[7] Kovalenko V., Perelmuter M. and Semenov Ya. (1981). Schrödinger operators with \(L^{l/2}_{\mathrm w}(\mathbb {R}^{l/2})\) potentials J. Math. Phys. 22: 1033–1044 · Zbl 0463.47027 · doi:10.1063/1.525009
[8] Li P. and Yau S.-T. (1983). On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88: 309–318 · Zbl 0554.35029 · doi:10.1007/BF01213210
[9] Lieb E.H. (1990). The stability of matter: from atoms to stars. Bull. Amer. Math. Soc. (N.S.) 22: 1–49 · Zbl 0698.35135 · doi:10.1090/S0273-0979-1990-15831-8
[10] Lieb, E.H.: The stability of matter and quantum electrodynamics. In: Proceedings of the Heisenberg symposium (Munich, Dec. 2001), Fundamental physics–Heisenberg and beyond, G. Buschhorn, J. Wess, eds., Berlin-Heidelberg-New York: Springer 2004, pp. 53–68, A modified version appears in the Milan J. Math. 71, 199–217 (2003) · Zbl 1081.81566
[11] Lieb E.H. (1994). Flux phase of the half-filled band. Phys. Rev. Lett. 73: 2158–2161 · doi:10.1103/PhysRevLett.73.2158
[12] Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics 14, Providence, RI: Amer. Math. Soc., 2001 · Zbl 0966.26002
[13] Lieb E.H., Loss M. and Siedentop H. (1996). Stability of relativistic matter via Thomas-Fermi theory. Helv. Phys. Acta 69: 974–984 · Zbl 0866.47050
[14] Lieb E.H., Loss M. and Solovej J.P. (1995). Stability of matter in magnetic fields. Phys. Rev. Lett. 75: 985–989 · Zbl 1020.81957 · doi:10.1103/PhysRevLett.75.985
[15] Lieb E.H., Siedentop H. and Solovej J.P. (1997). Stability and instability of relativistic electrons in magnetic fields. J. Stat. Phys. 89: 37–59 · Zbl 0918.35121 · doi:10.1007/BF02770753
[16] Lieb E.H. and Yau H.-T. (1988). The stability and instability of relativistic matter. Commun. Math. Phys. 118: 177–213 · Zbl 0686.35099 · doi:10.1007/BF01218577
[17] Simon B. (1978). A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28: 377–385 · Zbl 0413.47029 · doi:10.1016/0022-1236(78)90094-0
[18] Simon B. (1979). Maximal and minimal Schrödinger forms. J. Operator Theory 1: 37–47 · Zbl 0446.35035
[19] Simon, B.: Trace ideals and their applications. Second edition, Mathematical Surveys and Monographs 120, Providence, RI: Amer. Math. Soc., 2005 · Zbl 1074.47001
[20] Weder R.A. (1975). Spectral analysis of pseudodifferential operators. J. Funct. Anal. 20: 319–337 · Zbl 0317.47035 · doi:10.1016/0022-1236(75)90038-5
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.