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Stability of Coulomb systems with magnetic fields. I: The one-electron atom. (English) Zbl 0595.35098
Authors’ summary: ”The ground state energy of an atom in the presence of an external magnetic field B (with the electron spin-field interaction included) can be arbitrarily negative when B is arbitrarily large. We inquire whether stability can be restored by adding the self energy of the field, \(\int B^ 2\). For a hydrogenic like atom we prove that there is a critical nuclear charge, \(z_ c\), such that the atom is stable for \(z<z_ c\) and unstable for \(z>z_ c.''\)
Reviewer: A.Martynyuk

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35B35 Stability in context of PDEs
81T60 Supersymmetric field theories in quantum mechanics
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References:
[1] Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields: III. Atoms in homogeneous magnetic field. Commun. Math. Phys.79, 529-572 (1981) · Zbl 0464.35086
[2] Remark 3 in Brezis, H., Lieb, E.H.: Minimum action solution of some vector field equations. Commun. Math. Phys.96, 97-113 (1984) · Zbl 0579.35025
[3] Daubechies, I., Lieb, E.H.: One-electron relativistic molecules with Coulomb interaction. Commun. Math. Phys.90, 497-510 (1983) · Zbl 0946.81522
[4] Kato, T.: Schrödinger operators with singular potentials. Israel J. Math.13, 135-148 (1972) · Zbl 0246.35025
[5] Lieb, E.H.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math.74, 441-448 (1983) · Zbl 0538.35058
[6] Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields: II. The many-electron atom and the one-electron molecule. Commun. Math. Phys.104, 271-282 (1986) · Zbl 0607.35082
[7] Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamlitonian and their relation to Sobolev inequalities. In: Studies in mathematical physics, essays in honor of Valentine Bargmann. Lieb, E.H., Simon, B., Wightman, A.S. (eds.) Princeton, NJ: Princeton University Press 1976 · Zbl 0342.35044
[8] Loss, M., Yau, H.T.: Stability of Coulomb systems with magnetic fields: III. Zero energy bound states of the Pauli operator. Commun. Math. Phys.104, 283-290 (1986) · Zbl 0607.35083
[9] Michel, F.C.: Theory of publsar magnetospheres. Rev. Mod. Phys.54, 1-66 (1982)
[10] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970 · Zbl 0207.13501
[11] Straumann, N.: General relativity and relativistic astrophysics. Berlin, Heidelberg, New York, Tokyo: Springer 1984
[12] Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A16, 1782-1785 (1977)
[13] Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields: I. General Interactions. Duke Math. J.45, 847-883 (1978) · Zbl 0399.35029
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