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On trace identities and universal eigenvalue estimates for some partial differential operators
Author(s):
Evans
M.
Harrell II;
Joachim
Stubbe II
Journal:
Trans. Amer. Math. Soc.
349
(1997),
1797-1809.
MSC (1991):
Primary 35J10, 35J25, 58G25
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Abstract:
In this article, we prove and exploit a trace identity for the spectra of Schrödinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang.
References:
- [AsBe]
- Ashbaugh, Mark S., Benguria, Rafael D., A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. Math. 135 (1992), 601-628. MR 93d:35105
- [Da]
- Davies, E.B., Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 90e:35123
- [EaKa]
- Eastham, M.S.P., Kalf, H., Schrödinger-Type Operators with Continuous Spectra, Research Notes in Mathematics, vol. 65, Pitman, Boston, 1982. MR 84i:35107
- [Ha88]
- Harrell, Evans M., II, General Bounds for the Eigenvalues of Schrödinger Operators, Maximum Principles and Eigenvalue Problems in Partial Differential Equations, P.W. Schaefer, editor, Longman House, Essex, England, pp. 146-166, and New York, Wiley, 1988. MR 90d:35220
- [Ha93]
- Harrell, Evans M., II, Some geometric bounds on eigenvalue gaps, Communications in Partial Differential Equations 18 (1993), 179-198. MR 94c:35135
- [HaMi95]
- Harrell, Evans M., II and Michel, Patricia L., Commutator Bounds for Eigenvalues, with Applications to Spectral Geometry, Communications in Partial Differential Equations 19 (1994), 2037-2055; Erratum , vol. 20, 1995, pp. 1453. MR 95j:47028; CMP 95:13
- [HiPr80]
- Hile, G.N., M. H. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J. 29 (1980), 523-538. MR 82c:35052
- [Ho90]
- Hook, Stephen M., Domain-independent upper bounds for eigenvalues of elliptic operators, Trans. Amer. Math. Soc. 318 (1990), 615-642. MR 90h:35075
- [PaPoWe56]
- Payne, L.E., Pólya, G., Weinberger, H.F., On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289-298. MR 18:905c
- [QuRo77]
- Quigg, C., Rossner, J. L., Quantum mechanics with applications to quarkonium, Physics Reports 56 (1979), 167-235. MR 80m:81020
- [Ya91]
- Yang, Hongcang, An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of the Intl. Centre for Theoretical Physics,, 1991, Revised preprint, from Academia Sinica, 1995.
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Additional Information:
Evans
M.
Harrell
II
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email:
harrell@math.gatech.edu
Joachim
Stubbe
II
Affiliation:
Département de Physique Théorique, Université de Genève, Geneva, Switzerland
Email:
stubbe@cernvm.cern.ch
DOI:
10.1090/S0002-9947-97-01846-1
PII:
S 0002-9947(97)01846-1
Keywords:
Schr\"{o}dinger operator,
eigenvalue gap,
trace,
heat kernel,
partition function
Received by editor(s):
September 28, 1995
Additional Notes:
The first author was supported in part by US NSF Grant DMS 9211624.
Copyright of article:
Copyright
1997,
American Mathematical Society
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