Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian

 Access to full text

 Full (PDF)

1. S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations, 29 (1982), Princeton University Press Zbl0503.35001MR745286
2. P. Bauman, D. Phillips, Q. Tang, Stable nucleation for the Ginzburg-Landau system with an applied magnetic field, Arch. Rational Mech. Anal. 142 (1998), 1-43 Zbl0922.35157MR1629119
3. A. Bernoff, P. Sternberg, Onset of superconductivity in decreasing fields for general domains, J. Math. Phys. 39 (1998), 1272-1284 Zbl1056.82523MR1608449
4. C. Bolley, B. Helffer, An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material, Ann. Inst. H. Poincaré (Section Physique Théorique) 58 (1993), 169-233 Zbl0779.35104MR1217119
5. V. Bonnaillie, Analyse mathématique de la supraconductivité dans un domaine à coins : méthodes semi-classiques et numériques, (2003) 
6. V. Bonnaillie, On the fundamental state for a Schrödinger operator with magnetic fields in domains with corners, Asymptotic Anal. 41 (2005), 215-258 Zbl1067.35054MR2127997
7. V. Bonnaillie-Noël, M. Dauge, Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners, (2005) Zbl1134.81021
8. H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators, (1987), Springer Verlag, Berlin Zbl0619.47005
9. M. Dauge, B. Helffer, Eigenvalues variation I, Neumann problem for Sturm-Liouville operators, J. Differential Equations 104 (1993), 243-262 Zbl0784.34021MR1231468
10. M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the semi-classical limit, 268 (1999), Cambridge University Press Zbl0926.35002MR1735654
11. S. Fournais, B. Helffer, Energy asymptotics for type II superconductors, (2004) Zbl1160.82365MR2174430
12. V. V. Grušhin, Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols, Mat. Sb. (N.S.) 88 (1972), 504-521 Zbl0255.35022
13. B. Helffer, Introduction to the semiclassical analysis for the Schrödinger operator and applications, 1336 (1988), Springer Verlag Zbl0647.35002
14. B. Helffer, A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal. 138 (1996), 40-81 Zbl0851.58046MR1391630
15. B. Helffer, A. Morame, Magnetic bottles in connection with superconductivity, J. Funct. Anal. 185 (2001), 604-680 Zbl1078.81023MR1856278
16. B. Helffer, A. Morame, Magnetic bottles for the Neumann problem : curvature effect in the case of dimension 3 (General case), Ann. Sci. École Norm. Sup. 37 (2004), 105-170 Zbl1057.35061MR2050207
17. B. Helffer, X. Pan, Upper critical field and location of surface nucleation of superconductivity, Ann. Inst. H. Poincaré (Section Analyse non linéaire) 20 (2003), 145-181 Zbl1060.35132MR1958165
18. B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit I, Comm. Partial Differential Equations 9 (1984), 337-408 Zbl0546.35053MR740094
19. B. Helffer, J. Sjöstrand, Puits multiples en limite semi-classique II – Interaction moléculaire – Symétries – Perturbations, Ann. Inst. H. Poincaré (Section Physique théorique) 42 (1985), 127-212 Zbl0595.35031MR798695
20. B. Helffer, J. Sjöstrand, Puits multiples en limite semiclassique V – le cas des minipuits, Current topics in partial differential equations (1986), 133-186, Kinokuniya, Tokyo Zbl0628.35024
21. B. Helffer, J. Sjöstrand, Effet tunnel pour l’équation de Schrödinger avec champ magnétique, Ann. Scuola Norm. Sup. Pisa 14 (1987), 625-657 Zbl0699.35205MR963493
22. K. Lu, X-B. Pan, Eigenvalue problems of Ginzburg-Landau operator in bounded domains, J. Math. Phys. 40 (1999), 2647-2670 Zbl0943.35058MR1694223
23. K. Lu, X-B. Pan, Estimates of the upper critical field for the equations of superconductivity, Physica D 127 (1999), 73-104 Zbl0934.35174MR1678383
24. K. Lu, X-B. Pan, Gauge invariant eigenvalue problems on ${ℝ}^2$ and ${ℝ}_{+}^2$, Trans. Amer. Math. Soc. 352 (2000), 1247-1276 Zbl1053.35124MR1675206
25. K. Lu, X-B. Pan, Surface nucleation of superconductivity in $3$-dimension, J. of Differential Equations 168 (2000), 386-452 Zbl0972.35152MR1808455
26. M. del Pino, P.L. Felmer, P. Sternberg, Boundary concentration for eigenvalue problems related to the onset of superconductivity, Comm. Math. Phys. 210 (2000), 413-446 Zbl0982.35077MR1776839
27. M. Reed, B. Simon, Methods of modern Mathematical Physics, IV: Analysis of operators, (1978), Academic Press, New York Zbl0401.47001MR493421
28. D. Robert, Autour de l’approximation semi-classique, (1987), Birkhäuser, Boston Zbl0621.35001MR897108
29. D. Saint-James, G. Sarma, E.J. Thomas, Type II Superconductivity, (1969), Pergamon, Oxford 
30. B. Simon, Semi-classical analysis of low lying eigenvalues I, Ann. Inst. H. Poincaré (Section Physique Théorique) 38 (1983), 295-307 Zbl0526.35027MR708966
31. J. Sjöstrand, Operators of principal type with interior boundary conditions, Acta Math. 130 (1973), 1-51 Zbl0253.35076MR436226
32. P. Sternberg, On the Normal/Superconducting Phase Transition in the Presence of Large Magnetic Fields, Connectivity and Superconductivity M 62 (2000), 188-199, BergerJ.J. Zbl0976.82056
33. D.R. Tilley, J. Tilley, Superfluidity and superconductivity, (1990), Institute of Physics Publishing, Bristol and Philadelphia 
34. M. Tinkham, Introduction to Superconductivity, (1975), McGraw-Hill Inc, New York 

1. Soeren Fournais, Bernard Helffer, Strong diamagnetism for general domains and application
2. Søren Fournais, Le troisième champ critique en théorie de Ginzburg-Landau
3. Horia D. Cornean, Søren Fournais, Rupert L. Frank, Bernard Helffer, Sharp trace asymptotics for a class of $2D$-magnetic operators