On spectral problems related to a time dependent model in superconductivity with electric current

Bernard Helffer[1]

  • [1] Mathématiques, Bat. 425, Université Paris-Sud 11, 91405 Orsay Cedex, UMR 8628 du CNRS

Journées Équations aux dérivées partielles (2009)

  • page 1-16
  • ISSN: 0752-0360

Abstract

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This lecture is mainly inspired by a paper of Y. Almog appearing last year at Siam J. Math. Anal. Our goal here is first to discuss in detail the simplest models which we think are enlightning for understanding the role of the pseudospectra in this question and secondly to present proofs which will have some general character and will for example apply in a more physical model, for which we have obtained recently results together with Y. Almog and X. Pan.

How to cite

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Helffer, Bernard. "On spectral problems related to a time dependent model in superconductivity with electric current." Journées Équations aux dérivées partielles (2009): 1-16. <http://eudml.org/doc/116374>.

@article{Helffer2009,
abstract = {This lecture is mainly inspired by a paper of Y. Almog appearing last year at Siam J. Math. Anal. Our goal here is first to discuss in detail the simplest models which we think are enlightning for understanding the role of the pseudospectra in this question and secondly to present proofs which will have some general character and will for example apply in a more physical model, for which we have obtained recently results together with Y. Almog and X. Pan.},
affiliation = {Mathématiques, Bat. 425, Université Paris-Sud 11, 91405 Orsay Cedex, UMR 8628 du CNRS},
author = {Helffer, Bernard},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-16},
publisher = {Groupement de recherche 2434 du CNRS},
title = {On spectral problems related to a time dependent model in superconductivity with electric current},
url = {http://eudml.org/doc/116374},
year = {2009},
}

TY - JOUR
AU - Helffer, Bernard
TI - On spectral problems related to a time dependent model in superconductivity with electric current
JO - Journées Équations aux dérivées partielles
DA - 2009/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 16
AB - This lecture is mainly inspired by a paper of Y. Almog appearing last year at Siam J. Math. Anal. Our goal here is first to discuss in detail the simplest models which we think are enlightning for understanding the role of the pseudospectra in this question and secondly to present proofs which will have some general character and will for example apply in a more physical model, for which we have obtained recently results together with Y. Almog and X. Pan.
LA - eng
UR - http://eudml.org/doc/116374
ER -

References

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  1. Y. Almog. The motion of vortices in superconductors under the action of electric currents. Talk in CRM 2008. 
  2. Y. Almog. The stability of the normal state of superconductors in the presence of electric currents. Siam J. Math. Anal. 40 (2) (2008), p. 824-850. Zbl1165.82029MR2438788
  3. Y. Almog, B. Helffer, and X. Pan. Superconductivity near the normal state under the action of electric currents and induced magnetic field in ℝ 2 . Preprint September 2009, submitted. Zbl1201.35179
  4. Y. Almog, C. Han, B. Helffer, and X. Pan. Critical magnetic field and critical current of type II superconductors. In preparation. 
  5. P. Bauman, H. Jadallah, and D. Phillips. Classical solutions to the time-dependent Ginzburg-Landau equations for a bounded superconducting body in a vacuum. J. Math. Phys., 46 (2005), p. 095104, 25. Zbl1111.35071MR2171207
  6. B. Davies. Linear operators and their spectra. Cambridge Studies in Advanced Mathematics. Zbl1138.47001
  7. N. Dencker, J. Sjöstrand, and M. Zworski. Pseudospectra of semi-classical (pseudo) differential operators. Comm. Pure Appl. Math. 57 (3) (2004), p. 384-415. Zbl1054.35035MR2020109
  8. K.J. Engel and R. Nagel. One-parameter semi-groups for linear evolution equations. Springer Verlag. Zbl0952.47036
  9. S. Fournais and B. Helffer. Spectral methods in Superconductivity. To appear in Birkhäuser. Zbl1256.35001MR2662319
  10. I. Gallagher, T. Gallay and F. Nier. Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator, Preprint 2008. Zbl1180.35383MR2511908
  11. C. Han. The eigenvalue problem for the Schrödinger operator with electromagnetic field. Master Thesis (2009). 
  12. B. Helffer. Semi-Classical Analysis for the Schrödinger Operator and Applications, no. 1336 in Lecture Notes in Mathematics (1988). Zbl0647.35002MR960278
  13. B. Helffer. The Montgomery model revisited. To appear in Colloquium Mathematicum. Zbl1207.34110MR2602157
  14. B. Helffer and A. Mohamed. Sur le spectre essentiel des opérateurs de Schrödinger avec champ magnétique, Annales de l’Institut Fourier 38 (2) (1988), p. 95-113. Zbl0638.47047MR949012
  15. B. Helffer and F. Nier. Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians. no. 1862 in Lecture Notes in Mathematics, Springer-Verlag (2004). Zbl1072.35006MR2130405
  16. B.I. Ivlev and N.B. Kopnin. Electric currents and resistive states in thin superconductors. Adv. in Physics 33 (1984), p. 47-114. 
  17. J. Martinet. Sur les propriétés spectrales d’opérateurs non-autoadjoints provenant de la mécanique des fluides. Thèse de doctorat, Dec. 2009. 
  18. J. Sjöstrand. Resolvent estimates for non-self-adjoint operators via semi-groups. http://arxiv.org/abs/0906.0094. 
  19. J. Sjöstrand. Spectral properties for non self-adjoint differential operators. Provisory lecture notes for Evian 2009. 
  20. L.N. Trefethen. Pseudospectra of linear operators. SIAM Rev. 39 (3) (1997), p. 383-406. Zbl0896.15006MR1469941
  21. L.N. Trefethen and M. Embree. Spectra and pseudospectra. A course in three volumes (2004-version). Zbl1085.15009MR2155029
  22. C. Villani. Hypocoercivity. Memoirs of the AMS (2009). Zbl1197.35004MR2562709

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