Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

George D. Raikov

Annales de l'I.H.P. Physique théorique (1994)

  • Volume: 61, Issue: 2, page 163-188
  • ISSN: 0246-0211

How to cite

top

Raikov, George D.. "Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential." Annales de l'I.H.P. Physique théorique 61.2 (1994): 163-188. <http://eudml.org/doc/76652>.

@article{Raikov1994,
author = {Raikov, George D.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {discrete spectrum of the Schrödinger operator; asymptotic behaviour of the number of the eigenvalues},
language = {eng},
number = {2},
pages = {163-188},
publisher = {Gauthier-Villars},
title = {Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential},
url = {http://eudml.org/doc/76652},
volume = {61},
year = {1994},
}

TY - JOUR
AU - Raikov, George D.
TI - Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential
JO - Annales de l'I.H.P. Physique théorique
PY - 1994
PB - Gauthier-Villars
VL - 61
IS - 2
SP - 163
EP - 188
LA - eng
KW - discrete spectrum of the Schrödinger operator; asymptotic behaviour of the number of the eigenvalues
UR - http://eudml.org/doc/76652
ER -

References

top
  1. [Ale] A.B. Alekseev, On the spectral asymptotics of differential operators with polynomial occurrence of a small parameter, Probl. Mat. Fiz., Vol. 8, 1976, pp.3-15 (in Russian). MR510062
  2. [Av.Her.Sim 1] J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., Vol. 45, 1978, pp. 847-883. Zbl0399.35029MR518109
  3. [Av.Her.Sim 2] J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field, Commun. Math. Phys., Vol. 79, 1981, pp. 529-572. Zbl0464.35086MR623966
  4. [Bir] M.S. Birman, On the spectrum of singular boundary value problems, AMS Translations, (2), Vol. 53, 1966, pp. 23-80. Zbl0174.42502
  5. [Bir.Sol 1] M.S. Birman and M.Z. Solomjak, Estimates of the singular numbers of integral operators, Russian Math. Surveys, Vol. 32, 1977, pp. 15-89. Zbl0376.47023MR438186
  6. [Bir.Sol 2] M.S. Birman and M.Z. Solomjak, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, AMS Translations (2), Vol. 114, 1980. 
  7. [Bir.Sol 3] M.S. Birman and M.Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, D. REIDEL, Dordrecht, 1987. 
  8. [CdV] Y. Colin De Verdière, L'asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., Vol. 105, 1986, pp. 327-335. Zbl0612.35102MR849211
  9. [Com.Sch.Sei] J.M. Combes, R. Schrader and R. Seiler, Classical bounds and limits for energy distributions of Hamiltonian operators in electromagnetic fields, Ann. Physics, Vol. 111, 1978, pp. 1-18. MR489509
  10. [Cy.Fr.Ki.Sim] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger operators with Application to Quantum Mechanics and Global Geometry. Springer, Berlin, 1987. Zbl0619.47005MR883643
  11. [Dau.Rob] M. Dauge and D. Robert, Weyl's formula for a class of pseudodifferential operators of negative order on L2 (Rn), Lect. Notes Mah., Vol. 1256, 1987, pp. 91-122. Zbl0629.35098MR897775
  12. [Gre.Sze] U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, University of California Press, Berkeley-Los Angeles, 1958. Zbl0080.09501MR94840
  13. [Hel.Sjö 1] B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, II. Approximation champ magnétique faible: substitution de Peierls. In: Schrödinger operators, Lect. Notes Phys., Vol. 345, 1988, pp. 138-191. 
  14. [Hel.Sjö 2] B. Helffer and J. Sjöstrand, On the link between the Schrödinger operator with magnetic field and Harper's equation, In: Proceedings of Symp. Part. Diff. Equ., HOLZHAU, 1988. Teubner-Texte zur Mathematik, Vol. 112, 1989, pp. 139-156. Zbl0684.35074
  15. [Ivr 1] V.Ya. Ivrii, Estimations pour le nombre de valeurs propres négatives de l'opérateur de Schrödinger avec potentiels singuliers, C. R. Acad. Sci. Paris, Sér. I, Vol. 302, 1986, pp. 467-470. Zbl0611.35064MR838401
  16. [Ivr 2] V.Ya. Ivrii, Estimates of the number of negative eigenvalues of the Schrödinger operator with a strong magnetic field, Soviet Math. Dokl., Vol. 36, 1988, pp. 561-564. Zbl0661.35066MR936071
  17. [Ivr 3] V.Ya. Ivrii, Sharp spectral asymptotics for the two-dimensional Schrödinger operator with a strong magnetic field, Soviet Math. Dokl., Vol. 39, 1989, pp. 437-441. Zbl0714.35060MR1006532
  18. [Ivr 4] V.Ya. Ivrii, Semiclassical microlocal analysis and precise spectral asymptotics, Preprints 1-9, École Polytechnique, 1990-1992. 
  19. [Lei] H. Leinfelder, Gauge invariance of Schrödinger operators and related spectral properties, J. Opt. Theory, Vol. 9, 1983, pp. 163-179. Zbl0528.35024MR695945
  20. [Rai 1] G.D. Raikov, Spectral asymptotics for the Schrödinger operator with potential which steadies at infinity, Comm. Math. Phys., Vol. 124, 1989, pp. 665-685. Zbl0704.35110MR1014119
  21. [Rai 2] G.D. Raikov, Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Comm. P.D.E., Vol. 15, 1990, pp. 407-434. Zbl0739.35055MR1044429
  22. [Rai 3] G.D. Raikov, Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential, Letters Math. Phys., Vol. 21, 1991, pp. 41-49. Zbl0727.35102MR1088409
  23. [Rai 4] G.D. Raikov, Semiclassical and weak-magnetic-field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential, C. R. Acad. Sci. Bulg., Vol. 44, 1991, No. 1, pp.15-18. Zbl0739.35050MR1117317
  24. [Ree.Sim] M. Reed and B. Simon, Methods of Modem Mathematical Physics. IV. Analysis of Operators, Academic Press, London, 1978. Zbl0401.47001
  25. [Roz] G.V. Rozenbljum, An asymptotic of the negative discrete spectrum of the Schrödinger operator, Math. Notes U.S.S.R., Vol. 21, 1977, pp. 222-227. Zbl0399.35083MR447838
  26. [Sim] B. Simon, Functional Integration and Quantum Physics, Academic Press, London, 1979. Zbl0434.28013MR544188
  27. [Tam] H. Tamura, Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields, Osaka J. Math., Vol.25, 1988, pp. 633- 647. Zbl0731.35073MR969024

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.