This article is initially inspired by a paper of Almog [2] on the effect of the electric current in a problem in superconductivity. Our goal here is to discuss in detail the simplest models which we think are enlightening for understanding the role of the pseudo-spectra in this question and to relate them to recent results obtained together with Almog and Pan. This paper is dedicated to Michelle Schatzman, who has shown her interest for these pseudo-spectral questions by giving a course in 2006 with the french title: "Une visite de la galerie des horreurs: pseudospectres de matrices de Toeplitz".
@article{CML_2011__3_2_237_0,
author = {Helffer, Bernard},
title = {On pseudo-spectral problems related to a time-dependent model in superconductivity with electric current},
journal = {Confluentes Mathematici},
pages = {237--251},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {2},
year = {2011},
doi = {10.1142/S1793744211000308},
language = {en},
url = {http://archive.numdam.org/articles/10.1142/S1793744211000308/}
}
TY - JOUR AU - Helffer, Bernard TI - On pseudo-spectral problems related to a time-dependent model in superconductivity with electric current JO - Confluentes Mathematici PY - 2011 SP - 237 EP - 251 VL - 3 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - http://archive.numdam.org/articles/10.1142/S1793744211000308/ DO - 10.1142/S1793744211000308 LA - en ID - CML_2011__3_2_237_0 ER -
%0 Journal Article %A Helffer, Bernard %T On pseudo-spectral problems related to a time-dependent model in superconductivity with electric current %J Confluentes Mathematici %D 2011 %P 237-251 %V 3 %N 2 %I World Scientific Publishing Co Pte Ltd %U http://archive.numdam.org/articles/10.1142/S1793744211000308/ %R 10.1142/S1793744211000308 %G en %F CML_2011__3_2_237_0
Helffer, Bernard. On pseudo-spectral problems related to a time-dependent model in superconductivity with electric current. Confluentes Mathematici, Tome 3 (2011) no. 2, pp. 237-251. doi : 10.1142/S1793744211000308. http://archive.numdam.org/articles/10.1142/S1793744211000308/
[1] Y. Almog, The motion of vortices in superconductors under the action of electric currents, talk at the CRM in Montreal (2008).
[2] Y. Almog, The stability of the normal state of superconductors in the presence of electric currents, Siam J. Math. Anal. 40 (2008) 824–850.
[3] Y. Almog, B. Helffer and X. Pan, Superconductivity near the normal state under the action of electric currents and induced magnetic field in R2 , Commun. Math. Phys. 300 (2010) 147–184.
[4] Y. Almog, B. Helffer and X. Pan, Superconductivity near the normal state in a half- plane under the action of a perpendicular electric current and an induced magnetic field, to appear in Trans. AMS.
[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) 095104.
[6] W. Bordeaux-Montrieux, Estimation de résolvante et construction de quasi-modes près du bord du pseudospectre, preprint 2010.
[7] L. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudo- spectra, J. Operator Theory 47 (2002) 413–429.
[8] B. Davies, Linear Operators and Their Spectra, Cambridge Studies in Advanced Mathematics (Cambridge Univ. Press, 2007).
[9] B. Davies, Wild spectral behaviour of anharmonic oscillators, Bull. London Math. Soc. 32 (2000) 432–438.
[10] E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. 70 (2004) 420–426.
[11] N. Dencker, J. Sjöstrand and M. Zworski, Pseudospectra of semiclassical (pseudo)differential operators, Comm. Pure Appl. Math. 57 (2004) 384–415.
[12] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer-Verlag, 2000).
[13] S. Fournais and B. Helffer, Spectral Methods in Surface Superconductivity, Progress in non linear PDE 77 (Birkhäuser, 2010).
[14] B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics, No. 1336 (Springer, 1988).
[15] B. Helffer, The montgomery model revisited, Colloq. Math. 118 (2010) 391–400.
[16] B. Helffer, On spectral problems related to a time dependent model in superconduc- tivity with electric current, Proc. Evian Conference (June, 2009), preprint.
[17] B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, No. 1862 (Springer- Verlag, 2004).
[18] B. Helffer and J. Sjöstrand, From resolvent bounds to semi-groups bounds, in [23].
[19] R. Henry, Analyse spectrale pour l’opérateur de Airy complexe et applications, Master Thesis, Université Paris-Sud (September, 2010).
[20] B. I. Ivlev and N. B. Kopnin, Electric currents and resistive states in thin supercon- ductors, Adv. Phys. 33 (1984) 47–114.
[21] J. Martinet, Sur les propriétés spectrales d’opérateurs non-autoadjoints provenant de la mécanique des fluides. Thèse de doctorat de l’Université Paris-Sud, December, 2009.
[22] J. Sjöstrand, Resolvent estimates for non-self-adjoint operators via semigroups, arXiv: 0906.0094.
[23] J. Sjöstrand, Spectral properties for non self-adjoint differential operators, Lecture notes for Evian, 2009, arXiv:1002.4844v1.
[24] L. N. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39 (1997) 383–406.
[25] L. N. Trefethen and M. Embree, Spectra and Pseudospectra, A course in three volumes (Princeton Univ. Press, 2005).
[26] C. Villani, Hypocoercivity, Memoirs of the AMS 202 (Amer. Math. Soc., 2009), No. 950.
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