Schrödinger operator with magnetic field in domain with corners

Bonnaillie Noël, Virginie¹ 

¹ IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Journées équations aux dérivées partielles (2005), article no. 2, 12 p.

Résumé

We present here a simplified version of results obtained with F. Alouges, M. Dauge, B. Helffer and G. Vial (cf [4, 7, 9]). We analyze the Schrödinger operator with magnetic field in an infinite sector. This study allows to determine accurate approximation of the low-lying eigenpairs of the Schrödinger operator in domains with corners. We complete this analysis with numerical experiments.

Détail

DOI : 10.5802/jedp.15

Bibliographie

[1] Agmon, S. Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of $N$-body Schrödinger operators, vol. 29 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1982. | Zbl | MR

[2] Alouges, F., and Bonnaillie, V. Analyse numérique de la supraconductivité. C. R. Math. Acad. Sci. Paris 337, 8 (2003), 543–548. | Zbl | MR

[3] Bernoff, A., and Sternberg, P. Onset of superconductivity in decreasing fields for general domains. J. Math. Phys. 39, 3 (1998), 1272–1284. | Zbl | MR

[4] Bonnaillie, V. Analyse mathématique de la supraconductivité dans un domaine à coins; méthodes semi-classiques et numériques. Thèse de doctorat, Université Paris XI - Orsay, 2003.

[5] Bonnaillie, V. On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners. C. R. Math. Acad. Sci. Paris 336, 2 (2003), 135–140. | Zbl | MR

[6] Bonnaillie, V. Superconductivity in general domains. Prépublications d’Orsay 2004-09, 2004.

[7] Bonnaillie, V. On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners. Asymptot. Anal. 41, 3-4 (2005), 215–258. | Zbl | MR

[8] Bonnaillie Noël, V. A posteriori error estimator for the eigenvalue problem associated to the Schrödinger operator with magnetic field. Numer. Math. 99, 2 (2004), 325–348. | Zbl | MR

[9] Bonnaillie Noël, V., and Dauge, M. Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corner. In preparation, 2005. | MR

[10] Cycon, H. L., Froese, R. G., Kirsch, W., and Simon, B. Schrödinger operators with application to quantum mechanics and global geometry, study ed. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1987. | Zbl | MR

[11] Dauge, M., and Helffer, B. Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators. J. Differential Equations 104, 2 (1993), 243–262. | Zbl | MR

[12] Fournais, S., and Helffer, B. Accurate eigenvalue estimates for the magnetic neumann laplacian. To appear in Annales Inst. Fourier (2005). | Zbl | mathdoc-id

[13] Ginzburg, V., and Landau, L. On the theory of the superconductivity. Soviet. Phys. JETP 20 (1950), 1064–1082.

[14] Helffer, B. Semi-classical analysis for the Schrödinger operator and applications, vol. 1336 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. | Zbl | MR

[15] Helffer, B., and Mohamed, A. Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells. J. Funct. Anal. 138, 1 (1996), 40–81. | Zbl | MR

[16] Helffer, B., and Morame, A. Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185, 2 (2001), 604–680. | Zbl | MR

[17] Helffer, B., and Sjöstrand, J. Multiple wells in the semiclassical limit. I. Comm. Partial Differential Equations 9, 4 (1984), 337–408. | Zbl | MR

[18] Jadallah, H. T. The onset of superconductivity in a domain with a corner. J. Math. Phys. 42, 9 (2001), 4101–4121. | Zbl | MR

[19] Lu, K., and Pan, X.-B. Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity. Phys. D 127, 1-2 (1999), 73–104. | Zbl | MR

[20] Lu, K., and Pan, X.-B. Gauge invariant eigenvalue problems in ${\mathbf{R}}^2$ and in ${\mathbf{R}}_{+}^2$. Trans. Amer. Math. Soc. 352, 3 (2000), 1247–1276. | Zbl | MR

[21] Martin, D. http://perso.univ-rennes1.fr/daniel.martin/melina.

[22] Pan, X.-B. Upper critical field for superconductors with edges and corners. Calc. Var. Partial Differential Equations 14, 4 (2002), 447–482. | Zbl | MR

[23] Persson, A. Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8 (1960), 143–153. | Zbl | MR

[24] Simon, B. Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38, 3 (1983), 295–308. | Zbl | MR | mathdoc-id