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On the Explicit Semiclassical Limiting Eigenvalue (Resonance) Distribution for the Zeeman (Stark) Hydrogen Atom Hamiltonian

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Spectral Theory and Mathematical Physics

Part of the book series: Latin American Mathematics Series ((LAMSUFSC))

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Abstract

The purpose of this paper is two fold: (1) to give a review of semiclassical limiting eigenvalue or resonance distribution theorems for perturbations of the hydrogen atom Hamiltonian and (2) to give a new result on the explicit weak limit measure for the Stark hydrogen atom problem. For the second goal, we provide a detailed analysis on several relevant measures related to the Kepler problem and study both their SO(4) invariance and Hamiltonian flow invariance.

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References

  1. R. Abraham, J. Marsden, Foundations of Mechanics, 2nd edn. (AMS Chelsea Publishing, New York, 1987)

    Google Scholar 

  2. M. Avendaño Camacho, P.D. Hislop, C. Villegas-Blas, Semiclassical Szegö Limit of eigenvalue clusters for the hydrogen atom Zeeman Hamiltonian. Ann. Henri Poincaré 18(12), 3933–3973 (2017)

    Article  MathSciNet  Google Scholar 

  3. M. Avendaño Camacho, P.D. Hislop, C. Villegas-Blas, On a limiting eigenvalue distribution theorem for sub-clusters of the hydrogen atom in a constant magnetic field (2019, in preparation)

    Google Scholar 

  4. J. Avron, I. Herbst, B. Simon, Schrdinger operators with magnetic fields. I. General interactions. Duke Math. J. 45(4), 847–883 (1978)

    Article  Google Scholar 

  5. I.W. Herbst, Dilation analyticity in constant electric field. I. The two body problem. Commun. Math. Phys. 64(3), 279–298 (1979)

    Article  Google Scholar 

  6. G. Hernández-Dueñas, S. Pérez Esteva, A. Uribe, C. Villegas-Blas, Perturbations of the Landau Hamiltonian: asymptotics of eigenvalue clusters. Preprint, arXiv:1911.08989 (2019)

    Google Scholar 

  7. P.D. Hislop, C. Villegas-Blas, Semiclassical Szego limit of resonance clusters for the hydrogen atom Stark Hamiltonian. Asymptot. Anal. 79(1–2), 17–44 (2011)

    MathSciNet  MATH  Google Scholar 

  8. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Non Relativistic Theory), 3rd edn. revised and enlarged (Butterworth Heinemann, Oxford, 2000)

    Google Scholar 

  9. A. Martinez, An Introduction to Semiclassical and Microlocal Analysis (Springer, Berlin, 2002)

    Book  Google Scholar 

  10. D. Ojeda-Valencia, C. Villegas-Blas, On limiting eigenvalue distribution theorems in semiclassical analysis, in Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol. 224 (Birkhauser, Springer Basel AG, Basel, 2012), pp. 221–252

    Google Scholar 

  11. A. Pushnitski, G. Raikov, C. Villegas-Blas, Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian. Commun. Math. Phys. 320, 425–453 (2013)

    Article  MathSciNet  Google Scholar 

  12. L.E. Thomas, C. Villegas-Blas, Singular continuous limiting eigenvalue distributions for Schrödinger operators on a 2-sphere. J. Funct. Anal. 141(1), 249–273 (1996)

    Article  MathSciNet  Google Scholar 

  13. L.E. Thomas, C. Villegas-Blas, Asymptotics of Rydberg states for the hydrogen atom. Commun. Math. Phys. 187(3), 623–645 (1997)

    Article  MathSciNet  Google Scholar 

  14. A. Uribe, C. Villegas-Blas, Asymptotics of spectral clusters for a perturbation of a hydrogen atom. Commun. Math. Phys. 280(1), 123–144 (2008)

    Article  MathSciNet  Google Scholar 

  15. C. Villegas-Blas, The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. Thesis, Mathematics Department, University of Virginia, 1996

    Google Scholar 

  16. A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44(4), 883–892 (1977)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors want to thank Salvador Pérrez-Esteva and Oscar Chavez-Molina for discussions on the content of the paper. The authors want to thank the projects PAPIIT-UNAM IN105718 and CONACYT Ciencia Básica 283531 for partial financial support. C. Villegas-Blas thanks the organizers of the “Conference on Spectral Theory and Mathematical Physics, Santiago 2018” for their invitation and financial support to attend the conference through the Chilean project CONICYT REDI17056.

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Authors and Affiliations

  1. Universidad Autónoma de Baja California, Ensenada, Mexico

    Carlos Pérez-Estrada

  2. Universidad Nacional Autónoma de México, Instituto de Matemáticas, Unidad Cuernavaca, Cuernavaca, Mexico

    Carlos Villegas-Blas

  3. “Laboratorio Solomon Lefschetz”, Unidad Mixta Internacional del CNRS, Cuernavaca, Mexico

    Carlos Villegas-Blas

Authors
  1. Carlos Pérez-Estrada
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  2. Carlos Villegas-Blas
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Corresponding author

Correspondence to Carlos Villegas-Blas .

Editor information

Editors and Affiliations

  1. Departamento de Matemáticas y C. C., Universidad de Santiago de Chile, Santiago, Chile

    Pablo Miranda

  2. Institut de Mathématiques de Bordeaux, Talence, Gironde, France

    Nicolas Popoff

  3. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Georgi Raikov

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Pérez-Estrada, C., Villegas-Blas, C. (2020). On the Explicit Semiclassical Limiting Eigenvalue (Resonance) Distribution for the Zeeman (Stark) Hydrogen Atom Hamiltonian. In: Miranda, P., Popoff, N., Raikov, G. (eds) Spectral Theory and Mathematical Physics. Latin American Mathematics Series(). Springer, Cham. https://doi.org/10.1007/978-3-030-55556-6_11

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