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AUDIOVISUAL MATHEMATICS LIBRARY

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Spectral asymptotics of the one-particle density matrix for the Coulombic multi-particle systems

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Auteurs : ... (Auteur de la Conférence)
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Résumé : One-particle density matrix is the key object in the quantum-mechanical approximation schemes. In this talk I will give a short survey of recent regularity results with emphasis on sharp bounds for the eigenfunctions, and show how these bounds lead to the asymptotic formula for the eigenvalues of the one-particle density matrix. The argument is based on the results of M. Birman and M. Solomyak on spectral asymptotics for pseudo-differential operators with matrix-valued symbols.

Keywords : multi-particle Schrödinger operator; one-particle density matrix; eigenvalues; spectral asymptotics

Codes MSC :
35J10 - Schrödinger operator
 47G10 - Integral operators, See also {45P05}
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2985/Slides/A_Sobolev.pdf
    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 01/02/2024
    Date de captation : 15/01/2024
    Sous collection : Research talks
    arXiv category : Mathematical Physics ; Spectral Theory
    Domaine : Analysis and its Applications ; PDE ; Mathematical Physics
    Format : MP4 (.mp4) - HD
    Durée : 00:44:18
    Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-01-15_sobolev.mp4
Informations sur la Rencontre

Nom de la rencontre : Spectral Analysis for Quantum Hamiltonians / Analyse Spectrale pour des Hamiltoniens Quantiques
Dates : 15/01/2024 - 19/01/2024
Année de la rencontre : 2024
URL Congrès : https://conferences.cirm-math.fr/2985.html

Données de citation

DOI : 10.24350/CIRM.V.20127203
Citer cette vidéo: (2024). Spectral asymptotics of the one-particle density matrix for the Coulombic multi-particle systems. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20127203
URI : http://dx.doi.org/10.24350/CIRM.V.20127203

Voir aussi

Bibliographie

  • БИРМАН, Михаил Шлемович et СОЛОМЯК, Михаил Захарович. Асимптотика спектра слабо полярных интегральных операторов. Известия Российской академии наук. Серия математическая, 1970, vol. 34, no 5, p. 1142-1158. - https://doi.org/10.1070/IM1970v004n05ABEH000948

  • BIRMAN, M. Sh et SOLOMYAK, M. Z. Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. Vestnik Leningrad. Univ., 1977, vol. 13, no 3, p. 13-21. -

  • BIRMAN, M. Sh et SOLOMYAK, Mikhail Zakharovich. The asymptotics of the spectrum of pseudo-differential operators with anisotropic-homogeneous symbols. II, 1979, vol. 13, no 3, p. 5-10. -

  • BIRMAN, M. Sh. et SOLOMYAK, Mikhail Zakharovich. Spectral theory of self-adjoint operators in Hilbert space Mathematics and its Applications (Soviet Series), D.Reidel, 1987. Translated from the 1980 Russian original by S. Khrushchev and V. Peller. -

  • CIOSLOWSKI, Jerzy. Off-diagonal derivative discontinuities in the reduced density matrices of electronic systems. The Journal of Chemical Physics, 2020, vol. 153, no 15. - https://doi.org/10.1063/5.0023955

  • CIOSLOWSKI, Jerzy et PRĄTNICKI, Filip. Universalities among natural orbitals and occupation numbers pertaining to ground states of two electrons in central potentials. The Journal of Chemical Physics, 2019, vol. 151, no 18. - https://doi.org/10.1063/1.5123669

  • CIOSLOWSKI, Jerzy et STRASBURGER, Krzysztof. Angular-momentum extrapolations to the complete basis set limit: Why and when they work. Journal of Chemical Theory and Computation, 2021, vol. 17, no 6, p. 3403-3413. - https://doi.org/10.1021/acs.jctc.1c00202

  • COLEMAN, Albert John et YUKALOV, Vyacheslav I. Reduced density matrices: Coulson's challenge. Springer Science & Business Media, 2000. -

  • DAVIDSON, Ernest. Reduced density matrices in quantum chemistry. Academic Press, 1976. -

  • FOURNAIS, Søren et SØRENSEN, Thomas Østergaard. Pointwise estimates on derivatives of Coulombic wave functions and their electron densities. arXiv preprint arXiv:1803.03495, 2018. - https://doi.org/10.48550/arXiv.1803.03495

  • FOURNAIS, Søren, HOFFMANN-OSTENHOF, Maria, HOFFMANN-OSTENHOF, Thomas, et al. Analytic structure of many-body Coulombic wave functions. Communications in Mathematical Physics, 2009, vol. 289, p. 291-310. - http://dx.doi.org/10.1007/s00220-008-0664-5

  • FRIESECKE, G. On the infinitude of non–zero eigenvalues of the single–electron density matrix for atoms and molecules. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2003, vol. 459, no 2029, p. 47-52. - https://doi.org/10.1098/rspa.2002.1027

  • HATTIG, Christof, KLOPPER, Wim, KOHN, Andreas, et al. Explicitly correlated electrons in molecules. Chemical reviews, 2012, vol. 112, no 1, p. 4-74. - https://doi.org/10.1021/cr200168z

  • HEARNSHAW, Peter et SOBOLEV, Alexander V. Analyticity of the one-particle density matrix. arXiv preprint arXiv:2006.11785, 2020. - https://doi.org/10.1007/s00023-021-01120-6

  • KATO, Tosio. On the eigenfunctions of many‐particle systems in quantum mechanics. Communications on Pure and Applied Mathematics, 1957, vol. 10, no 2, p. 151-177. - https://doi.org/10.1002/cpa.3160100201

  • LEWIN, Mathieu, LIEB, Elliott H., et SEIRINGER, Robert. Universal functionals in density functional theory. In : Density Functional Theory: Modeling, Mathematical Analysis, Computational Methods, and Applications. Cham : Springer International Publishing, 2022. p. 115-182. - http://dx.doi.org/10.1007/978-3-031-22340-2_3

  • LIEB, Elliott H. et SEIRINGER, Robert. The stability of matter in quantum mechanics. Cambridge university press, 2010. -

  • SIMON, Barry. Methods of Modern Mathematical Physics: Fourier Analysis, Self-Adjointness. 1975. -

  • SIMON, B. Exponential decay of quantum wave functions. Online notes - http://www.math.caltech.edu/simon/Selecta/ExponentialDecay.pdf

  • SIMON, B. Simon's online Selecta - http://www.math.caltech.edu/simon/selecta.html

  • SOBOLEV, Alexander V. Eigenvalue estimates for the one-particle density matrix. arXiv e-prints, 2020, p. arXiv: 2008.10935. - https://doi.org/10.48550/arXiv.2008.10935


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