Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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
MathSciNet review: 2425732
Retrieve article in: PDF

References | Similar articles | Additional information

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)


Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47F05

Retrieve articles in all Journals with MSC (2000): 47F05


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.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia