## Estimates for negative eigenvalues of a random Schrödinger operator

HTML articles powered by AMS MathViewer

- by O. Safronov and B. Vainberg PDF
- Proc. Amer. Math. Soc.
**136**(2008), 3921-3929 Request permission

## References

- Michael Aizenman and Elliott H. Lieb,
*On semiclassical bounds for eigenvalues of Schrödinger operators*, Phys. Lett. A**66**(1978), no. 6, 427–429. MR**598768**, DOI 10.1016/0375-9601(78)90385-7 - Jean Bourgain,
*On random Schrödinger operators on $\Bbb Z^2$*, Discrete Contin. Dyn. Syst.**8**(2002), no. 1, 1–15. MR**1877824**, DOI 10.3934/dcds.2002.8.1 - J. Bourgain,
*Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 70–98. MR**2083389**, DOI 10.1007/978-3-540-36428-3_{7} - Joseph G. Conlon,
*A new proof of the Cwikel-Lieb-Rosenbljum bound*, Rocky Mountain J. Math.**15**(1985), no. 1, 117–122. MR**779256**, DOI 10.1216/RMJ-1985-15-1-117 - Michael Cwikel,
*Weak type estimates for singular values and the number of bound states of Schrödinger operators*, Ann. of Math. (2)**106**(1977), no. 1, 93–100. MR**473576**, DOI 10.2307/1971160 - Sergey A. Denisov,
*Absolutely continuous spectrum of multidimensional Schrödinger operator*, Int. Math. Res. Not.**74**(2004), 3963–3982. MR**2103798**, DOI 10.1155/S107379280414141X - Ky Fan,
*Maximum properties and inequalities for the eigenvalues of completely continuous operators*, Proc. Nat. Acad. Sci. U.S.A.**37**(1951), 760–766. MR**45952**, DOI 10.1073/pnas.37.11.760 - V. Glaser, H. Grosse, and A. Martin,
*Bounds on the number of eigenvalues of the Schrödinger operator*, Comm. Math. Phys.**59**(1978), no. 2, 197–212. MR**491613**, DOI 10.1007/BF01614249 - B. Helffer and D. Robert,
*Riesz means of bound states and semiclassical limit connected with a Lieb-Thirring’s conjecture*, Asymptotic Anal.**3**(1990), no. 2, 91–103. MR**1061661**, DOI 10.3233/ASY-1990-3201 - Dirk Hundertmark,
*On the number of bound states for Schrödinger operators with operator-valued potentials*, Ark. Mat.**40**(2002), no. 1, 73–87. MR**1948887**, DOI 10.1007/BF02384503 - Dirk Hundertmark, Elliott H. Lieb, and Lawrence E. Thomas,
*A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator*, Adv. Theor. Math. Phys.**2**(1998), no. 4, 719–731. MR**1663336**, DOI 10.4310/ATMP.1998.v2.n4.a2 - Ari Laptev and Timo Weidl,
*Sharp Lieb-Thirring inequalities in high dimensions*, Acta Math.**184**(2000), no. 1, 87–111. MR**1756570**, DOI 10.1007/BF02392782 - Elliott Lieb,
*Bounds on the eigenvalues of the Laplace and Schroedinger operators*, Bull. Amer. Math. Soc.**82**(1976), no. 5, 751–753. MR**407909**, DOI 10.1090/S0002-9904-1976-14149-3 - Elliott H. Lieb,
*On characteristic exponents in turbulence*, Comm. Math. Phys.**92**(1984), no. 4, 473–480. MR**736404**, DOI 10.1007/BF01215277
21 Lieb, E.H., Thirring, W.: - G. V. Rozenbljum,
*Distribution of the discrete spectrum of singular differential operators*, Dokl. Akad. Nauk SSSR**202**(1972), 1012–1015 (Russian). MR**0295148** - Oleg Safronov,
*Multi-dimensional Schrödinger operators with some negative spectrum*, J. Funct. Anal.**238**(2006), no. 1, 327–339. MR**2253019**, DOI 10.1016/j.jfa.2006.01.006 - Oleg Safronov,
*Multi-dimensional Schrödinger operators with no negative spectrum*, Ann. Henri Poincaré**7**(2006), no. 4, 781–789. MR**2232372**, DOI 10.1007/s00023-006-0268-6 - Timo Weidl,
*On the Lieb-Thirring constants $L_{\gamma ,1}$ for $\gamma \geq 1/2$*, Comm. Math. Phys.**178**(1996), no. 1, 135–146. MR**1387945**, DOI 10.1007/BF02104912

*Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities*. Studies in Math. Phys., Essays in Honor of Valentine Bargmann, Princeton, 269–303 (1976).

## Additional Information

**O. Safronov**- Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, North Carolina 28223
- MR Author ID: 607478
- Email: osafrono@uncc.edu
**B. Vainberg**- Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, 201 University City Boulevard, Charlotte, North Carolina 28223
- MR Author ID: 194146
- Email: bvainbe@uncc.edu
- Received by editor(s): May 11, 2007
- Received by editor(s) in revised form: September 26, 2007
- Published electronically: May 28, 2008
- Communicated by: Mikhail Shubin
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**136**(2008), 3921-3929 - MSC (2000): Primary 47F05
- DOI: https://doi.org/10.1090/S0002-9939-08-09356-8
- MathSciNet review: 2425732