Download PDF - Some estimates concerning the Zeeman effectAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Some estimates concerning the Zeeman effect

Authors 1

Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Wrocław, Branch, Kopernika 18, 51-617 Wrocław, Poland.

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.

Keywords

ENG
estimation of eigenvalues, Schrödinger operator

Bibliography

  1. L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York 1974.
  2. N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris 1971.
  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.
  4. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207.
  5. E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, unpublished.
  6. K. Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216.
  7. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Academic Press, 1978.
  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. B. Simon, Schrödinger operators with singular magnetic vector potentials, Math. Z. 131 (1973), 361-370.
  10. B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979.
Studia Mathematica
1993/105/1
Export to PBN, Opens in new tab
Pages:
13-23
Main language of publication
English
Received
1991-04-23
Accepted
1992-06-08
Published
1993
Exact and natural sciences