Upper and lower bounds for an eigenvalue associated with a positive eigenvector

Journal of Mathematical Physics

Volumn 47, Issue 2, 2006, Pages

  Mouchet, Amaury ^a  

^a UNIVERSITÉ DE TOURS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33644619839     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2168124     Document Type: Article

References (36)

1. Avron, J., Herbst, I., and Simon, B., " The Zeeman effect revisted., " Phys. Lett. 62A (4), 214-216. (1977).
2. Avron, J., Herbst, I., and Simon, B., " Schrödinger operators with magnetic fields. I. General interactions., " Duke Math. J. 45 (4), 847-883 (1978).
3. Barnsley, M. F., " Lower bounds for quantum mechanical energy levels., " J. Phys. A 11 (1), 55-68 (1978).
4. Barnsley, M. F. and Duffin, R. J., " Bounds for the rth characteristic frequency of a beaded string or an electrical filter., " Proc. Natl. Acad. Sci. U.S.A. 77 (6), 3120-3124 (1980).
5. Barta, J., " Sur la vibration fondamentale d'une membrane., " C. R. Acad. Sci. 204 (7), 472-473 (1937) (in French).
6. Basdevant, J. L., Martin, A., and Richard, J. M., " Improved bounds on many-body Hamiltonians. I. Self-gravitating bosons., " Nucl. Phys. B 343 (1), 60-68 (1990).
7. Baumgartner, B., " A class of lower bounds for Hamiltonian operators., " J. Phys. A 12 (4), 459-467 (1979).
8. Bellissard, J. and Simon, B., " Cantor spectrum for the almost Mathieu equation., " J. Funct. Anal. 48, 408-419 (1982).
9. Benslama, A., Metatla, A., Bachkhaznadji, A., Zouzou, S. R., Krikeb, A., Basdevant, J. L., Richard, J. M., and Wu, T. T., " Optimized lower bound for four-body Hamiltonians., " Few-Body Syst. 24 (1), 39-54 (1998).
10. Berestycki, H. and Nirenberg, L., " The principal eigenvalue and maximum principle for second-order elliptic operators in general domains., " Commun. Pure Appl. Math. 47 (1), 47-92 (1994).
11. Bessa, G. P. and Montenegro, J. F., " An extension of Barta's theorem and geometric applications., " arXiv: math-ph/0308099 (2004).
12. Ceperley, D. M., " Fermion nodes., " J. Stat. Phys. 63 (5/6), 1237-1267 (1991).
13. Crandall, R. E. and Reno, M. H., " Ground state energy bounds for potentials ∥x∥ν, " J. Math. Phys. 0022-2488 10.1063/1.525207 23 (1), 64-70 (1982); Crandall, R. E. and Reno, M. H., J. Math. Phys. 23, 1737 (1982).
14. Crandall, R. E. and Reno, M. H., " Ground state energy bounds for potentials ∥x∥ν, " J. Math. Phys. 0022-2488 10.1063/1.525207 23 (1), 64-70 (1982); Crandall, R. E. and Reno, M. H., J. Math. Phys. 23, 1737 (1982).
15. Duffin, R. J., " Lower bounds for eigenvalues., " Phys. Rev. 71, 827-828 (1947).
16. Fernández, F. M., Ma, Q., and Tipping, R. H., " Eigenvalues fo the Schrödinger equation via the Riccati-Pad́ method., " Phys. Rev. A 40 (11), 6149-6153 (1989).
17. Fisher, M. E. and Ruelle, D., " The stability of many-particle systems., " J. Math. Phys. 7 (2), 260-270 (1966).
18. Handy, C. R. and Bessis, D., " Rapidly convergent lower bounds for the Schrödinger-equation ground-state energy., " Phys. Rev. Lett. 55 (9), 931-934 (1985).
19. Harper, P. G., " Single band motion of conduction electrons in a uniform magnetic field., " Proc. R. Soc. London, Ser. A 68, 879-892 (1955 a).
20. Harper, P. G., " The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals., " Proc. R. Soc. London, Ser. A 68, 874-878 (1955 b).
21. Harrell II, E. M., " Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part. " (unpublished).
22. Helffer, B., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., and Owen, M. P., " Nodal sets for groundstates of Schrödinger operators with zero magnetic fields in a non simply connected domain., " Commun. Math. Phys. 202, 629-649 (1999).
23. Hofstadter, D. R., " Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields., " Phys. Rev. B 14 (6), 2239-2249 (1976).
24. Lenard, A. and Dyson, F. J., " Stability of matter. I., " J. Math. Phys. 8 (3), 423-434 (1967).
25. Ĺvy-Leblond, J. M., " Nonsaturation of gravitational forces., " J. Math. Phys. 10 (5), 806-812 (1969).
26. Lieb, E. H., The Stability of Matter: From Atoms to Stars, 4th ed. (Springer-Verlag, Berlin, 2005).
27. Maslov, V. P. and Fedoriuk, M. V., Semi-Classical Approximation in Quantum Mechanics, Mathematical Physics and Applied Mathematics, Vol. 7 (D. Reidel, Dordrecht, 1981).
28. Mouchet, A., " Bounding the ground-state energy of a many-body system with the differential method., " Nucl. Phys. A 765, 319-341 (2006).
29. Mouchet, A., " A differential method for bounding the ground state energy., " J. Phys. A 38, 1039-1047 (2005).
30. Protter, M. H. and Weinberger, H. W., " On the spectrum of general second order operators., " Bull. Am. Math. Soc. 72, 251-255 (1966).
31. Reed, M. and Simon, B., Analysis of Operators, Methods of modern mathematical physics, Vol. 4 (Academic, New York, 1978a).
32. Reed, M. and Simon, B., Scattering theory, Methods of modern mathematical physics, Vol. 3 (Academic, New York, 1978b).
33. Saff, E. B. and Kuijlaars, A. B., " Distributing many points on a sphere., " Math. Intell. 19 (1), 5-11 (1997).
34. Schmutz, M., " The factorization method and ground state energy bounds., " Phys. Lett. 108A (4), 195-196 (1985).
35. Sigal, I. M., Lectures on large Coulomb systems, mathematical quantum theory. II. Schödinger operators, CRM Proceedings and Lecture Notes, Vol. 8 (Amer. Math. Soc., Providence, RI, 1995), pp. 73-107.
36. Thirring, W., Quantum Mechanics of Atoms and Molecules, A Course in mathematical physics, Vol. 3 (Springer-Verlag, New York, 1979).