|
Eigenvalue estimates for magnetic Schrödinger operators in domains
Author(s):
Rupert
L.
Frank;
Ari
Laptev;
Stanislav
Molchanov
Journal:
Proc. Amer. Math. Soc.
136
(2008),
4245-4255.
MSC (2000):
Primary 35P15;
Secondary 35J10
Posted:
July 29, 2008
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Inequalities are derived for sums and quotients of eigenvalues of magnetic Schrödinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.
References:
-
- [A]
- M. S. Ashbaugh, The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H. C. Yang. Proc. Indian Acad. Sci. Math. Sci. 112 (2002), 3-30. MR 1894540 (2004c:35302)
- [AB1]
- M. S. Ashbaugh, R. D. Benguria, Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials. Proc. Amer. Math. Soc. 99 (1987), no. 3, 598-599. MR 875408 (88e:34039)
- [AB2]
- M. S. Ashbaugh, R. D. Benguria, Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials. Comm. Math. Phys. 124 (1989), no. 3, 403-415. MR 1012632 (91c:34114)
- [AB3]
- M. S. Ashbaugh, R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. Math. (2) 135 (1992), 601-628. MR 1166646 (93d:35105)
- [B]
- F. A. Berezin, Covariant and contravariant symbols of operators [Russian]. Math. USSR Izv. 6 (1972), 1117-1151 (1973). MR 0350504 (50:2996)
- [CY]
- Q.-M. Cheng, H. Yang, Bounds on eigenvalues of Dirichlet Laplacian. Math. Ann. 337 (2007), 159-175. MR 2262780 (2007k:35064)
- [C]
- G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators. Boll. Un. Mat. Ital. (6) 1-A (1982), 145-151. MR 652376 (83i:35140)
- [D]
- E. B. Davies, Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, 1989. MR 990239 (90e:35123)
- [HS]
- E. M. Harrell, J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Amer. Math. Soc. 349 (1997), no. 5, 1797-1809. MR 1401772 (97i:35129)
- [H]
- L. Hermi, Two new Weyl-type bounds for the Dirichlet Laplacian. Trans. Amer. Math. Soc. 360 (2008), 1539-1558. MR 2357704
- [L]
- A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal. 151 (1997), 531-545. MR 1491551 (99a:35027)
- [LW]
- A. Laptev, T. Weidl, Recent results on Lieb-Thirring inequalities. Journées ``Équations aux Dérivées Partielles'' (La Chapelle sur Erdre, 2000), Exp. No. XX, Univ. Nantes, Nantes, 2000. MR 1775696 (2001j:81064)
- [LP]
- M. Levitin, L. Parnovski, Commutators, spectral trace identities, and universal estimates for eigenvalues. J. Funct. Anal. 192 (2002), no. 2, 425-445. MR 1923409 (2003g:47040)
- [LY]
- P. Li, S.-T. Yau, On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88 (1983), 309-318. MR 701919 (84k:58225)
- [LL]
- E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001. MR 1817225 (2001i:00001)
- [RS]
- M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. MR 0493421 (58:12429c)
- [S]
- Yu. Safarov, Lower bounds for the generalized counting function. In: The Maz'ya anniversary collection, vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl. 110, Birkhäuser, Basel, 1999, 275-293. MR 1747899 (2001d:47036)
- [S1]
- B. Simon, Kato's inequality and the comparison of semi-groups. J. Funct. Anal. 32 (1979), 97-101. MR 533221 (80e:47036)
- [S2]
- B. Simon, Maximal and minimal Schrödinger forms. J. Operator Theory 1 (1979), no. 1, 37-47. MR 526289 (81m:35104)
- [Y]
- H. Yang, Estimates of the difference between consecutive eigenvalues, preprint, 1995 (revision of International Centre for Theoretical Physics preprint IC/91/60, Trieste, Italy, April 1991).
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
35P15,
35J10
Retrieve articles in all Journals with MSC
(2000):
35P15,
35J10
Additional Information:
Rupert
L.
Frank
Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Address at time of publication:
Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
Email:
rlfrank@math.princeton.edu
Ari
Laptev
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom - and - Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
Email:
a.laptev@imperial.ac.uk, laptev@math.kth.se
Stanislav
Molchanov
Affiliation:
Department of Mathematics, University of North Carolina, Charlotte, North Carolina 28223-0001
Email:
smolchan@uncc.edu
DOI:
10.1090/S0002-9939-08-09523-3
PII:
S 0002-9939(08)09523-3
Keywords:
Eigenvalue bounds,
semi-classical estimates,
Laplace operator,
magnetic Schr\"odinger operator
Received by editor(s):
May 29, 2007
Posted:
July 29, 2008
Communicated by:
Mikhail Shubin
Copyright of article:
Copyright
2008,
by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
|
Comments: webmaster@ams.org