# zbMATH — the first resource for mathematics

One-electron relativistic molecules with Coulomb interaction. (English) Zbl 0946.81522
Summary: As an approximation to a relativistic one-electron molecule, we study the operator $H=(-\Delta +m^2)^{1/2}-e^2 \sum ^K_{j=1}Z_j |x-R_j|^{-1}$ with $$Z_j \geq 0, e^{-2}=137.04$$. $$H$$ is bounded below if and only if $$e^2Z_j \leq 2/\pi$$ for all $$j$$. Assuming this condition, the system is unstable when $$e^2\sum Z_j >2/\pi$$ in the sense that $$E_0=\inf \text{spec} H\to -\infty$$ as the $$R_j\to 0$$ for all $$j$$. We prove that the nuclear Coulomb repulsion more than restores stability; namely $E_0+0.069e^2\sum _{i<j} Z_iZ_j|R_i -R_j|^{-1}\geq 0.$ We also show that $$E_0$$ is an increasing function of the internuclear distances $$|R_i-R_j|$$.

##### MSC:
 81V55 Molecular physics
Full Text:
##### References:
 [1] Dyson, F., Lenard, A.: Stability of matter. I. J. Math. Phys.8, 423-434 (1967) · Zbl 0948.81665 · doi:10.1063/1.1705209 [2] Lenard, A., Dyson, F.: Stability of matter. II. J. Math. Phys.9, 698-711 (1968) · Zbl 0948.81666 · doi:10.1063/1.1664631 [3] Lieb, E., Thirring, W.: A bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett.35, 687-689 (1975); Errata: Phys. Rev. Lett.35, 1116 (1975). See also Lieb, E.: The stability of matter. Rev. Mod. Phys.48, 553-569 (1976) · doi:10.1103/PhysRevLett.35.687 [4] Dyson, F.: Ground-state energy of a finite system of charged particles. J. Math. Phys.8, 1538-1545 (1967) · doi:10.1063/1.1705389 [5] Lieb, E.: TheN 5/3 law for bosons. Phys. Lett.A70, 71-73 (1979) [6] Weder, R.: Spectral analysis of pseudodifferential operators. J. Funct. Anal.20, 319-337 (1975) · Zbl 0317.47035 · doi:10.1016/0022-1236(75)90038-5 [7] Herbst, I.: Spectral theory of the operator (p 2+m 2)1/2?Ze 2/r. Commun. Math. Phys.53, 285-294 (1977); Errata: Commun. Math. Phys.55, 316 (1977) · Zbl 0375.35047 · doi:10.1007/BF01609852 [8] Kato, T.: Perturbation theory for linear operators. Berlin, New York: Springer 1966 (2nd edn. 1976) · Zbl 0148.12601 [9] Lieb, E., Simon, B.: Monotonicity of the electronic contribution to the Born-Oppenheimer energy. J. Phys.B11, L537-542 (1978) [10] Lieb, E.: Monotonicity of the molecular electronic energy in the nuclear coordinates. J. Phys.B15, L63-L66 (1982) [11] Daubechies, I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. (1983) · Zbl 0946.81521 [12] Lieb, E., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23, 22-116 (1977) · Zbl 0938.81568 · doi:10.1016/0001-8708(77)90108-6 [13] Kovalenko, V., Perelmuter, M., Semenov, Ya.: Schrödinger operators withL w 1/2 (? l ) potentials. J. Math. Phys.22, 1033-1044 (1981) · Zbl 0463.47027 · doi:10.1063/1.525009 [14] Brascamp, H., Lieb, E., Luttinger, M.: A general rearrangement inequality for multiple integrals. J. Funct. Anal.17, 227-237 (1974) · Zbl 0286.26005 · doi:10.1016/0022-1236(74)90013-5 [15] Lieb, E.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities (submitted) · Zbl 0527.42011 [16] Reed, M., Simon, B.: Methods of modern mathematical physics Vol. IV: Analysis of operators. New York: Academic Press 1978 · Zbl 0401.47001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.