Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field
HTML articles powered by AMS MathViewer

by Yaniv Almog, Bernard Helffer and Xing-Bin Pan PDF
Trans. Amer. Math. Soc. 365 (2013), 1183-1217 Request permission


We consider the linearization of the time-dependent Ginzburg-Landau system near the normal state. We assume that a constant magnetic field and an electric current are applied through the sample, which captures half of the plane, inducing thereby a linearly varying magnetic field. In the limit of small normal conductivity we prove that if the electric current is lower than some critical value, the normal state loses its stability. For currents stronger than this critical value, the normal state is stable. To obtain this stability result we analyze both the spectrum and the pseudo-spectrum of the linearized operator. The critical current tends, in this small conductivity limit, to another critical current which had been obtained for a reduced model which neglects magnetic field effects.
  • M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972.
  • J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279. MR 345551
  • Y. Almog, The stability of the normal state of superconductors in the presence of electric currents, SIAM J. Math. Anal. 40 (2008), no. 2, 824–850. MR 2438788, DOI 10.1137/070699755
  • Yaniv Almog, Bernard Helffer, and Xing-Bin Pan, Superconductivity near the normal state under the action of electric currents and induced magnetic fields in $\Bbb R^2$, Comm. Math. Phys. 300 (2010), no. 1, 147–184. MR 2725185, DOI 10.1007/s00220-010-1111-y
  • Patricia Bauman, Hala Jadallah, and Daniel Phillips, Classical solutions to the time-dependent Ginzburg-Landau equations for a bounded superconducting body in a vacuum, J. Math. Phys. 46 (2005), no. 9, 095104, 25. MR 2171207, DOI 10.1063/1.2012107
  • S. J. Chapman, S. D. Howison, and J. R. Ockendon, Macroscopic models for superconductivity, SIAM Rev. 34 (1992), no. 4, 529–560. MR 1193011, DOI 10.1137/1034114
  • J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251–270. MR 391792
  • E. B. Davies, Wild spectral behaviour of anharmonic oscillators, Bull. London Math. Soc. 32 (2000), no. 4, 432–438. MR 1760807, DOI 10.1112/S0024609300007050
  • A. Dolgert, T. Blum, A. Dorsey, and M. Fowler, Nucleation and growth of the superconducting phase in the presence of a current, Phys. Rev. B, 57 (1998), 5432-5443.
  • Qiang Du, Max D. Gunzburger, and Janet S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev. 34 (1992), no. 1, 54–81. MR 1156289, DOI 10.1137/1034003
  • Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
  • Søren Fournais and Bernard Helffer, Spectral methods in surface superconductivity, Progress in Nonlinear Differential Equations and their Applications, vol. 77, Birkhäuser Boston, Inc., Boston, MA, 2010. MR 2662319
  • B. Helffer, On pseudo-spectral problems related to a time dependent model in superconductivity with electric current, Confluentes Math., 3 (2) (2011), 237–251.
  • B. I. Ivlev and N. B. Kopnin, Electric currents and resistive states in thin superconductors, Advances in Physics, 33 (1984), 47-114.
  • Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
  • J. Rubinstein and P. Sternberg, Formation and stability of phase slip centers in nonuniform wires with currents, Physica C-Superconductivity and Its Applications, 468 (2008), 260-263.
  • J. Rubinstein, P. Sternberg, and Q. Ma, Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents, Phys. Rev. Lett., 99 (2007).
  • Jacob Rubinstein, Peter Sternberg, and Kevin Zumbrun, The resistive state in a superconducting wire: bifurcation from the normal state, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 117–158. MR 2564470, DOI 10.1007/s00205-008-0188-3
  • Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
  • A. G. Sivakov, A. M. Glukhov, A. N. Omelyanchouk, Y. Koval, P. Müller, and A. V. Ustinov, Josephson behavior of phase-slip lines in wide superconducting strips, Phys. Rev. Lett., 91 (2003), art. no. 267001.
  • M. Tinkham, Introduction to Superconductivity, McGraw-Hill, 1996.
  • D. Y. Vodolazov, F. M. Peeters, L. Piraux, S. Matefi-Tempfli, and S. Michotte, Current-voltage characteristics of quasi-one-dimensional superconductors: An s-shaped curve in the constant voltage regime, Phys. Rev. Lett., 91 (2003).
Similar Articles
Additional Information
  • Yaniv Almog
  • Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
  • Email:
  • Bernard Helffer
  • Affiliation: Laboratoire de Mathématiques, Université Paris-Sud 11 et CNRS, Bât 425, 91 405 Orsay Cedex, France
  • MR Author ID: 83860
  • Email:
  • Xing-Bin Pan
  • Affiliation: Department of Mathematics and Center for PDE, East China Normal University, Shanghai 200062, People’s Republic of China
  • Email:
  • Received by editor(s): July 19, 2010
  • Received by editor(s) in revised form: February 20, 2011
  • Published electronically: August 9, 2012
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1183-1217
  • MSC (2010): Primary 82D55, 35B25, 35B40, 35Q55
  • DOI:
  • MathSciNet review: 3003262