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Journées équations aux dérivées partielles

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Bernard Helffer
On spectral problems related to a time dependent model in superconductivity with electric current
Journées équations aux dérivées partielles (2009), Exp. No. 3, 16 p., doi: 10.5802/jedp.56
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Résumé - 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.


[Alm1] Y. Almog. The motion of vortices in superconductors under the action of electric currents. Talk in CRM 2008.
[Alm2] 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.  MR 2438788 |  Zbl 1165.82029
[AHP] Y. Almog, B. Helffer, and X. Pan. Superconductivity near the normal state under the action of electric currents and induced magnetic field in $\mathbb{R}^2$. Preprint September 2009, submitted.
[AHHP] Y. Almog, C. Han, B. Helffer, and X. Pan. Critical magnetic field and critical current of type II superconductors. In preparation.
[BJP] 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.  MR 2171207 |  Zbl 1111.35071
[Dav] B. Davies. Linear operators and their spectra. Cambridge Studies in Advanced Mathematics.  Zbl 1138.47001
[DSZ] N. Dencker, J. Sjöstrand, and M. Zworski. Pseudospectra of semi-classical (pseudo) differential operators. Comm. Pure Appl. Math. 57 (3) (2004), p. 384-415.  MR 2020109 |  Zbl 1054.35035
[EN] K.J. Engel and R. Nagel. One-parameter semi-groups for linear evolution equations. Springer Verlag.  Zbl 0952.47036
[FoHel] S. Fournais and B. Helffer. Spectral methods in Superconductivity. To appear in Birkhäuser.  MR 2662319
[GGN] I. Gallagher, T. Gallay and F. Nier. Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator, Preprint 2008. arXiv |  MR 2511908
[Han] C. Han. The eigenvalue problem for the Schrödinger operator with electromagnetic field. Master Thesis (2009).
[Hel1] B. Helffer. Semi-Classical Analysis for the Schrödinger Operator and Applications, no. 1336 in Lecture Notes in Mathematics (1988).  MR 960278 |  Zbl 0647.35002
[Hel2] B. Helffer. The Montgomery model revisited. To appear in Colloquium Mathematicum.  MR 2602157 |  Zbl pre05691830
[HelMo] 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. Cedram |  MR 949012 |  Zbl 0638.47047
[HelNi] 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).  MR 2130405 |  Zbl 1072.35006
[IvKo] B.I. Ivlev and N.B. Kopnin. Electric currents and resistive states in thin superconductors. Adv. in Physics 33 (1984), p. 47-114.
[Mart] J. Martinet. Sur les propriétés spectrales d’opérateurs non-autoadjoints provenant de la mécanique des fluides. Thèse de doctorat, Dec. 2009.
[Sjö1] J. Sjöstrand. Resolvent estimates for non-self-adjoint operators via semi-groups. arXiv
[Sjö2] J. Sjöstrand. Spectral properties for non self-adjoint differential operators. Provisory lecture notes for Evian 2009.
[Tr] L.N. Trefethen. Pseudospectra of linear operators. SIAM Rev. 39 (3) (1997), p. 383-406.  MR 1469941 |  Zbl 0896.15006
[TrEm] L.N. Trefethen and M. Embree. Spectra and pseudospectra. A course in three volumes (2004-version).  MR 2155029 |  Zbl 1085.15009
[Vil] C. Villani. Hypocoercivity. Memoirs of the AMS (2009).  MR 2562709 |  Zbl 1197.35004
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