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Transactions of the American Mathematical Society

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Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains


Authors: Xing-Bin Pan and Keng-Huat Kwek
Journal: Trans. Amer. Math. Soc. 354 (2002), 4201-4227
MSC (2000): Primary 35Q55, 81Q10, 82D55
DOI: https://doi.org/10.1090/S0002-9947-02-03033-7
Published electronically: May 15, 2002
MathSciNet review: 1926871
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Abstract: We establish an asymptotic estimate of the lowest eigenvalue $\mu (b\mathbf{F})$ of the Schrödinger operator $-\nabla _{b\mathbf{F}}^{2}$ with a magnetic field in a bounded $2$-dimensional domain, where curl $\mathbf{F}$ vanishes non-degenerately, and $b$is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.


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Additional Information

Xing-Bin Pan
Affiliation: Center for Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China; and Department of Mathematics, National University of Singapore, Singapore 119260
Email: matpanxb@nus.edu.sg

Keng-Huat Kwek
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260
Address at time of publication: The Logistics Institute—Asia Pacific National University of Singapore, Singapore 119260

DOI: https://doi.org/10.1090/S0002-9947-02-03033-7
Keywords: Schr\"{o}dinger operator with a magnetic field, eigenvalue, Ginzburg-Landau system, superconductivity, nucleation, upper critical field, Sturm-Liouville operator, Riccati type equation
Received by editor(s): July 17, 2000
Received by editor(s) in revised form: March 13, 2001
Published electronically: May 15, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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