Some estimates concerning the Zeeman effect

Wiesław Cupała

Studia Mathematica (1993)

  • Volume: 105, Issue: 1, page 13-23
  • ISSN: 0039-3223

Abstract

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The Itô integral calculus and analysis on nilpotent Lie grops are used to estimate the number of eigenvalues of the Schrödinger operator for a quantum system with a polynomial magnetic vector potential. An analogue of the Cwikel-Lieb-Rosenblum inequality is proved.

How to cite

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Cupała, Wiesław. "Some estimates concerning the Zeeman effect." Studia Mathematica 105.1 (1993): 13-23. <http://eudml.org/doc/215979>.

@article{Cupała1993,
abstract = {The Itô integral calculus and analysis on nilpotent Lie grops are used to estimate the number of eigenvalues of the Schrödinger operator for a quantum system with a polynomial magnetic vector potential. An analogue of the Cwikel-Lieb-Rosenblum inequality is proved.},
author = {Cupała, Wiesław},
journal = {Studia Mathematica},
keywords = {estimation of eigenvalues; Schrödinger operator; Schrödinger operators with polynomial potentials; Itô integral calculus; Cwikel-Lieb-Rosenblum inequality},
language = {eng},
number = {1},
pages = {13-23},
title = {Some estimates concerning the Zeeman effect},
url = {http://eudml.org/doc/215979},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Cupała, Wiesław
TI - Some estimates concerning the Zeeman effect
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 1
SP - 13
EP - 23
AB - The Itô integral calculus and analysis on nilpotent Lie grops are used to estimate the number of eigenvalues of the Schrödinger operator for a quantum system with a polynomial magnetic vector potential. An analogue of the Cwikel-Lieb-Rosenblum inequality is proved.
LA - eng
KW - estimation of eigenvalues; Schrödinger operator; Schrödinger operators with polynomial potentials; Itô integral calculus; Cwikel-Lieb-Rosenblum inequality
UR - http://eudml.org/doc/215979
ER -

References

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  1. [1] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York 1974. Zbl0278.60039
  2. [2] N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris 1971. Zbl0213.04103
  3. [3] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. 106 (1977), 93-100. Zbl0362.47006
  4. [4] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. Zbl0312.35026
  5. [5] E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, unpublished. Zbl0445.58029
  6. [6] K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216. Zbl0008.11301
  7. [7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Academic Press, 1978. Zbl0401.47001
  8. [8] G. W. Rosenblum, The distribution of the discrete spectrum of singular differential operators, Dokl. Akad. Nauk SSSR 202 (1972), 1012-1015 (in Russian). 
  9. [9] B. Simon, Schrödinger operators with singular magnetic vector potentials, Math. Z. 131 (1973), 361-370. Zbl0277.47006
  10. [10] B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979. Zbl0434.28013

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