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Estimates for negative eigenvalues of a random Schrödinger operator

Author(s): O. Safronov; B. Vainberg
Journal: Proc. Amer. Math. Soc. 136 (2008), 3921-3929.
MSC (2000): Primary 47F05
Posted: May 28, 2008
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References:

1.
Aizenman, M., Lieb, E.H.: On semi-classical bounds for eigenvalues of Schrödinger operators. Phys. Lett. 66A, 427-429 (1978). MR 598768 (81m:81020)

2.
Bourgain, J.: On random Schrödinger operators on $ {\mathbb{Z}}^2$. Discrete Contin. Dynam. Systems 8, no. 1, 1-15 (2002). MR 1877824 (2003f:47063)

3.
Bourgain, J: Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena, Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002 (V. D. Milman and G. Schechtman, Eds.), Lecture Notes in Mathematics, vol. 1807, Springer, Berlin, 2003, pp. 70-98. MR 2083389 (2006a:47058)

4.
Conlon, J.G.: A new proof of the Cwikel-Lieb-Rosenbljum bound. Rocky Mountain J. Math. 15, 117-122 (1985). MR 779256 (86j:35118)

5.
Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. (2) 106, 93-100 (1977). MR 0473576 (57:13242)

6.
Denisov, S.A.: Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not. 74, 3963-3982 (2004). MR 2103798 (2005h:35047)

7.
Fan, K.: Maximum properties and inequalities for the eigenvalues of completely continuous operators. Proc. Nat. Acad. Sci. USA 37, 760-766 (1951). MR 0045952 (13:661e)

8.
Glaser, V., Grosse, H., Martin, A.: Bounds on the number of eigenvalues of the Schrödinger operator. Commun. Math. Phys. 59, 197-212 (1978). MR 491613 (81a:35081)

9.
Helffer, B., Robert, D.: Riesz means of bounded states and semi-classical limit connected with a Lieb-Thirring conjecture I, II. I - Jour. Asymp. Anal. 3, 91-103 (1990); II - Ann. de l'Inst. H. Poincaré Phys. Théor. 53 (2), 139-147 (1990). MR 1061661 (91h:35241), MR 1079775 (91k:35183)

10.
Hundertmark, D.: On the number of bound states for Schrödinger operators with operator-valued potentials, Ark. Mat. 40, no. 1, 73-87 (2002). MR 1948887 (2003j:81058)

11.
Hundertmark, D., Lieb, E.H., Thomas, L.E.: A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator. Adv. Theor. Math. Phys. 2, 719-731 (1998). MR 1663336 (2000c:81062)

12.
Laptev, A., Weidl, T.: Sharp Lieb-Thirring inequalities in high dimensions, Acta Math. 184, no. 1, 87-111 (2000). MR 1756570 (2001c:35173)

13.
Lieb, E.H.: Bounds on the eigenvalues of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc. 82, 751-753 (1976). See also: The number of bound states of one body Schrödinger operators and the Weyl problem. Proc. Symp. Pure Math. 36, Amer. Math. Soc., 241-252 (1980). MR 0407909 (53:11679), MR 0573436 (82i:35134)

14.
Lieb, E.H.: On characteristic exponents in turbulence. Comm. in Math. Phys. 92, 473-480 (1984). MR 736404 (86c:35114)

15.
Lieb, E.H., Thirring, W.: 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).

16.
Rozenbljum, G.V.: Distribution of the discrete spectrum of singular differential operators. Dokl. Akad. Nauk SSSR 202, 1012-1015 (1972), Izv. VUZov, Matematika 1. 75-86 (1976). MR 0295148 (45:4216)

17.
Safronov, O.: Multi-dimensional Schrödinger operators with some negative spectrum. J. Funct. Anal. 238, no. 1, 327-339 (2006). MR 2253019 (2007i:35038)

18.
Safronov, O.: Multi-dimensional Schrödinger operators with no negative spectrum. Ann. Henri Poincaré 7, 4, 781-789 (2006). MR 2232372 (2007b:81078)

19.
Weidl, T.: On the Lieb-Thirring constants $ L_{\gamma,1}$ for $ \gamma \geq 1/2$. Comm. in Math. Phys. 178, 135-146 (1996). MR 1387945 (97c:81039)


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Additional Information:

O. Safronov
Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, North Carolina 28223
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
Email: bvainbe@uncc.edu

DOI: 10.1090/S0002-9939-08-09356-8
PII: S 0002-9939(08)09356-8
Keywords: Eigenvalue estimates, random Schr\"odinger operators
Received by editor(s): May 11, 2007,
Received by editor(s) in revised form: September 26, 2007
Posted: May 28, 2008
Communicated by: Mikhail Shubin
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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