Communications in Mathematical Physics

, Volume 337, Issue 1, pp 191–224 | Cite as

Lack of Diamagnetism and the Little–Parks Effect

  • Søren FournaisEmail author
  • Mikael Persson Sundqvist


When a superconducting sample is submitted to a sufficiently strong external magnetic field, the superconductivity of the material is lost. In this paper we prove that this effect does not, in general, take place at a unique value of the external magnetic field strength. Indeed, for a sample in the shape of a narrow annulus the set of magnetic field strengths for which the sample is superconducting is not an interval. This is a rigorous justification of the Little–Parks effect. We also show that the same oscillation effect can happen for disc-shaped samples if the external magnetic field is non-uniform. In this case the oscillations can even occur repeatedly along arbitrarily large values of the Ginzburg–Landau parameter κ. The analysis is based on an understanding of the underlying spectral theory for a magnetic Schrödinger operator. It is shown that the ground state energy of such an operator is not in general a monotone function of the intensity of the field, even in the limit of strong fields.


Unit Disc Ground State Energy Trial State Landau Equation Lower Eigenvalue 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhusDenmark
  2. 2.Department of Mathematical SciencesLund UniversityLundSweden

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