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 of the Schrödinger operator with a magnetic field in a bounded -dimensional domain, where curl vanishes non-degenerately, and 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.

**[A]**S. Agmon,*Lectures on exponential decay of solutions of second order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators*, Princeton University Press, 1982. MR**85f:35019****[BH]**C. Bolley and B. Helffer,*An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material*, Ann. Inst. Henri Poincaré, Physique Théorique,**58**(1993), 189-233. MR**94k:82120****[BPT]**P. Bauman, D. Phillips and Q. Tang,*Stable nucleation for the Ginzburg-Landau system with an applied magnetic field*, Arch. Rat. Mech. Anal.,**142**(1998), 1-43. MR**99g:58040****[BS]**A. Bernoff and P. Sternberg,*Onset of superconductivity in decreasing fields for general domains*, J. Math. Phys.**39**(1998), 1272-1284. MR**99a:82099****[C]**S. J. Chapman,*Nucleation of superconductivity in OAdecreasing fields*, European J. Appl. Math.,**5**(1994), part 1, 449-468; part 2, 468-494. MR**95m:82119****[CHO]**S. J. Chapman, S. D. Howison and J. R. Ockendon,*Macroscopic models for superconductivity,*SIAM Review,**34**(1992), 529-560. MR**94b:82037****[DFS]**M. del Pino, P. Felmer and P. Sternberg,*Boundary concentration for eigenvalue problems related to the onset of superconductivity*, Commun. Math. Phys.,**210**(2000), 413-446. MR**2001k:35231****[dG]**P. G. De Gennes,*Superconductivity of Metals and Alloys,*W. A. Benjamin, Inc., (1966).**[DGP]**Q. Du, M. Gunzburger and J. Peterson,*Analysis and approximation of the Ginzburg-Landau model of superconductivity,*SIAM Review,**34**(1992), 45-81. MR**93g:82109****[DH]**M. Dauge and B. Helffer,*Eigenvalues variation, I, Neumann problem for Sturm-Liouville operators*, J. Differential Equations,**104**(1993), 243-262. MR**94j:47097****[GL]**V. Ginzburg and L. Landau,*On the theory of superconductivity*, Zh. Eksper. Teoret. Fiz.**20**(1950), 1064-1082; English transl., L. D. Landau,*Collected Papers*, Gordon and Breach, New York, 1967, pp. 546-568. MR**38:5577****[GO]**M. Gunzburger and J. Ockendon,*Mathematical models in superconductivity*, SIAM News, November and December (1994).**[GP]**T. Giorgi and D. Phillips,*The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model*, SIAM Journal on Mathematical Analysis,**30**(1999), 341-359. MR**2000b:35235****[H1]**B. Helffer,*Semi-Classical Analysis for the Schrödinger Operator and Applications*, Lecture Notes in Mathematics, vol.1336, Springer-Verlag, 1988. MR**90c:81043****[H2]**B. Helffer,*Semiclassical analysis for the Schrödinger operator with magnetic wells (after R. Montgomery, B. Helffer-A. Mohamed)*, pp. 99-114, in: J. Rauch and B. Simon eds.,*Quasiclassical Methods*, The IMA Volumes in Mathematics and Its Applications, vol. 95, Springer, 1997. MR**98m:81034****[HMoh]**B. Helffer and A. Mohamed,*Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells*, J. Functional Analysis,**138**(1996), 40-81. MR**97h:35177****[HMor]**B. Helffer and A. Morame,*Magnetic bottles in connection with superconductivity*, J. Funct. Anal.**185**(2001), 604-680.**[HP]**B. Helffer and Xing-Bin Pan,*Upper critical field and location of surface nucleation of superconductivity*, Ann. L'Institut Henri Poincaré Analyse Non Linéaire, to appear.**[LP1]**Kening Lu and Xing-Bin Pan,*The first eigenvalue of Ginzburg-Landau operator*, in: Differential Equations and Applications, Bates et al. eds., International Press (1997), 215-226. MR**99j:35205****[LP2]**Kening Lu and Xing-Bin Pan,*Gauge invariant eigenvalue problems in**and in*, Trans. Amer. Math. Soc.,**352**(2000), 1247-1276. MR**2000j:35248****[LP3]**Kening Lu and Xing-Bin Pan,*Eigenvalue problems of Ginzburg-Landau operator in bounded domains*, J. Math. Phys.,**40**(1999), 2647-2670. MR**2001e:35167****[LP4]**Kening Lu and Xing-Bin Pan,*Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity*, Physica D,**127**(1-2) (1999), 73-104. MR**2000a:82075****[LP5]**Kening Lu and Xing-Bin Pan,*Surface nucleation of superconductivity in 3-dimension*, J. Differential Equations,**168**(2000), 386-452. MR**2002b:82069****[LP6]**Kening Lu and Xing-Bin Pan,*Surface nucleation of superconductivity*, Methods and Applications of Analysis**8**(2001), 279-300.**[M]**R. Montgomery,*Hearing the zero locus of a magnetic field*, Comm. Math. Phys.,**168**(1995), 651-675. MR**96e:81044****[P]**Xing-Bin Pan,*Upper critical field for superconductivity with edges and corners*, Calculus of Variations and PDE's (to appear).**[S]**Y. Sibuya,*Global theory of a second order linear ordinary differential equation with a polynomial coefficient*, North-Holland Mathematics Studies, Vol.**18**, North-Holland Publishing Co., Amsterdam-Oxford, 1975. MR**58:6561****[SdG]**D. Saint-James and P. G. De Gennes,*Onset of superconductivity in decreasing fields*, Physics Letters,**6**: (5) (1963), 306-308.**[SST]**D. Saint-James and G. Sarma and E.J. Thomas,*Type II Superconductivity*, Pergamon Press, Oxford, 1969.**[T]**M. Tinkham,*Introduction to Superconductivity*, McGraw-Hill Inc., New York, 1975.

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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