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On the Maximal Excess Charge of the Chandrasekhar–Coulomb Hamiltonian in Two Dimension

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Abstract

We show that for the straightforward quantized relativistic Coulomb Hamiltonian of a two-dimensional atom – or the corresponding magnetic quantum dot – the maximal number of electrons does not exceed twice the nuclear charge. The result is then generalized to the presence of external magnetic fields and atomic Hamiltonians. This is based on the positivity of

$$|{\bf x}| T({\bf p}) + T({\bf p} ) |{\bf x}|$$

which – in two dimensions – is false for the non-relativistic case T(p) = p 2/2, but is proven in this paper for T(p) = |p|, i.e., the ultra-relativistic kinetic energy.

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References

  1. Egger R., De Martino A., Siedentop H., Stockmeyer E.: Multiparticle equations for interacting Dirac fermions in magnetically confined graphene quantum dots. J. Phys. A Math. Theor. 43(21), 215202 (2010)

    Article  ADS  Google Scholar 

  2. Flügge, S., Marschall, H.: Rechenmethoden der Quantentheorie. In: Heidelberger Taschenbücher, vol. 6, 3rd edn. Springer, Berlin (1965)

  3. Griesemer M., Siedentop H.: A minimax principle for the eigenvalues in spectral gaps. J. Lond. Math. Soc. (2) 60(2), 490–500 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Herbst I.W.: Spectral theory of the operator (p 2 + m 2)1/2−Ze 2/r. Comm. Math. Phys. 53, 285–294 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Könenberg, M., Matte, O., Stockmeyer, E.: Hydrogen-like atoms in relativistic QED. In: Siedentop, H. (eds.) Complex Quantum Systems: Analysis of Large Coulomb Systems. Lecture Notes Series, vol. 24. Institute for Mathematical Sciences, National University of Singapore. Singapore (2013)

  6. Lieb E.H.: Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29(6), 3018–3028 (1984)

    Article  ADS  Google Scholar 

  7. Lieb E.H., Yau H.-T.: The stability and instability of relativistic matter. Comm. Math. Phys. 118, 177–213 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Maier T., Siedentop H.: Stability of impurities with Coulomb potential in graphene with homogeneous magnetic field. J. Math. Phys. 53, 095207 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  9. Nam P.: New bounds on the maximum ionization of atoms. Commun. Math. Phys. 312, 427–445 (2012). doi:10.1007/s00220-012-1479-y

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Nam P., Portmann F., Solovej J.: Asymptotics for two-dimensional atoms. Ann. Henri Poincare 13, 333–362 (2012). doi:10.1007/s00023-011-0123-2

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Paananen T., Egger R.: Finite-size version of the excitonic instability in graphene quantum dots. Phys. Rev. B 84, 155456 (2011)

    Article  ADS  Google Scholar 

  12. Paananen T., Egger R., Siedentop H.: Signatures of Wigner molecule formation in interacting Dirac fermion quantum dots. Phys. Rev. B 83(8), 085409 (2011)

    Article  ADS  Google Scholar 

  13. Seiringer R.: On the maximal ionizaton of atoms in strong magnetic fields. J. Phys. A. Math. Gen. 34, 1943–1948 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Simon B.: Kato’s inequality and the comparison of semigroups. J. Funct. Anal. 32(1), 97–101 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333, München, Germany

    Michael Handrek & Heinz Siedentop

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  1. Michael Handrek
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  2. Heinz Siedentop
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Correspondence to Heinz Siedentop.

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Handrek, M., Siedentop, H. On the Maximal Excess Charge of the Chandrasekhar–Coulomb Hamiltonian in Two Dimension. Lett Math Phys 103, 843–849 (2013). https://doi.org/10.1007/s11005-013-0618-5

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  • DOI: https://doi.org/10.1007/s11005-013-0618-5

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