Pólya's conjecture in the presence of a constant magnetic field

Journal of the European Mathematical Society

Volumn 11, Issue 6, 2009, Pages 1365-1383

  Frank, Rupert L ^a   Loss, Michael ^b   Weidl, Timo ^c  

^a PRINCETON UNIVERSITY   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^c UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Eigenvalue bounds; Laplace operator; Magnetic schr dinger operators; Polya's conjecture; Semi classical estimates

Indexed keywords

EID: 74349087923     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/184     Document Type: Article

References (27)

1. [AL] Aizenman, M., Lieb, E.: On semiclassical bounds for eigenvalues of Schrödinger operators. Phys. Lett. A 66, 427-429 (1978) MR 0598768
2. [B1] Berezin, F. A.: Covariant and contravariant symbols of operators. Math. USSR-Izv. 6 (1972), 1117-1151. Zbl 0259.47004 MR 0350504
3. [B2] Berezin, F. A.: Convex functions of operators. Math. USSR-Sb. 88, 269-277 (1972) Zbl 0271.47011 MR 0300121
4. [BS] Birman, M. S., Solomjak, M. Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel, Dordrecht (1987) Zbl 0744.47017 MR 1192782
5. [CV] Colin de Verdière, Y.: L'asymptotique de Weyl pour les bouteilles magnétiques. Comm. Math. Phys. 105, 327-335 (1986) Zbl 0612.35102 MR 0849211
6. [dB] de la Bretèche, R.: Preuve de la conjecture de Lieb-Thirring dans le cas des potentials quadratiques strictement convexes. Ann. Inst. H. Poincaré Phys. Théor. 70, 369-380 (1999) Zbl 0938.35148 MR 1686426
7. [ELV] Erdos, L., Loss, M., Vougalter, V.: Diamagnetic behavior of sums of Dirichlet eigenvalues. Ann. Inst. Fourier (Grenoble) 50, 891-907 (2000) Zbl 0957.35104 MR 1779898
8. [F] Fock, V.: Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld. Z. Physik 47, 446-448 (1928) JFM 54.0966.02
9. [FH] Frank, R. L., Hansson, A. M.: Eigenvalue estimates for the Aharonov-Bohm operator in a domain. In: Methods of Spectral Analysis in Mathematical Physics (Lund, 2006), J. Janas et al. (eds.), Oper. Theory Adv. Appl. 186, Birkhäuser, 115-137 (2009) Zbl pre005507104
10. [HR] Helffer, B., Robert, D.: Riesz means of bounded states and semi-classical limit connected with a Lieb-Thirring conjecture. II. Ann. Inst. H. Poincaré Phys. Théor. 53, 139-147 (1990) Zbl 0728.35078 MR 1079775
11. [L] Laptev, A.: Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal. 151, 531-545 (1997) Zbl 0892.35115 MR 1491551
12. [LS] Laptev, A., Safarov, Yu.: A generalization of the Berezin-Lieb inequality. Amer. Math. Soc. Transl. (2) 175, 69-79 (1996) Zbl 0868.47016 MR 1402917
13. [LW1] Laptev, A.,Weidl, T.: Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184, 87-111 (2000) Zbl 1142.35531 MR 1756570
14. [LW2] Laptev, A.,Weidl, T.: Recent results on Lieb-Thirring inequalities. In: Journées " Équations aux Dérivées Partielles" (La Chapelle sur Erdre, 2000), Exp. No.XX, Univ. Nantes, Nantes (2000) Zbl 1135.81337 MR 1775696
15. [LY] Li, P., Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88, 309-318 (1983) Zbl 0554.35029 MR 0701919
16. [L1] Lieb, E. H.: The classical limit of quantum spin systems. Comm. Math. Phys. 31, 327-340 (1973) Zbl 1125.82305 MR 0349181
17. [L2] Lieb, E. H.: The number of bound states of one-body Schrödinger operators and the Weyl problem. In: Geometry of the Laplace Operator (Hawaii, 1979), Proc. Sympos. Pure Math. 36, Amer. Math. Soc., 241-252 (1980) Zbl 0445.58029 MR 0573436
18. [L3] Lieb, E. H.: Flux phase of the half-filled band. Phys. Rev. Lett. 73, 2158-2161 (1994)
19. [LT] Lieb, E. H., Thirring,W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, Princeton Univ. Press, Princeton, NJ, 269-303 (1976) Zbl 0342.35044
20. [M] Métivier, G.: Valeurs propres de problèmes aux limites elliptiques irréguliers. Bull. Soc. Math. France Mém. 51-52, 125-229 (1977) Zbl 0401.35088 MR 0473578
21. [N] Nakamura, S.: A remark on the Dirichlet-Neumann decoupling and the integrated density of states. J. Funct. Anal. 179, 136-152 (2001) Zbl 0970.35084 MR 1807255
22. [P] Pólya, G.: On the eigenvalues of vibrating membranes. Proc. London Math. Soc. 11, 419-433 (1961) Zbl 0107.41805 MR 0129219
23. [RS] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol.4. Academic Press (1978) Zbl 0401.47001 MR 0493421
24. [R1] Rozenblyum, G. V.: On the eigenvalues of the first boundary value problem in unbounded domains. Math. USSR-Sb. 18, 235-248 (1972) Zbl 0267.35063 MR 0348295
25. [R2] Rozenblyum, G. V.: Domination of semigroups and estimates for eigenvalues. St. Petersburg Math. J. 12, 831-845 (2001) Zbl 1015.47023 MR 1812946
26. [S] Shubin, M. A.: Spectral theory and the index of elliptic operators with almost-periodic coefficients. Russian Math. Surveys 34, 109-158 (1979) Zbl 0448.47032 MR 0535710
27. [W] Weyl, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differential gleichungen (mit Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71, 441-479 (1911) JFM 43.0436.01 MR 1511670