Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

On the first two eigenvalues of Sturm-Liouville operators

Author(s): Miklós Horváth
Journal: Proc. Amer. Math. Soc. 131 (2003), 1215-1224.
MSC (2000): Primary 34L15, 34B25
Posted: July 26, 2002
MathSciNet review: 1948113
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Among the Schrödinger operators with single-well potentials defined on $(0,\pi)$ with transition point at $\frac\pi 2$ , the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric single-well potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.

References:

1.: S. Abramovich, The gap between the first two eigenvalues of a one-dimensional Schrödinger operator with symmetric potential, Proc. Amer. Math. Soc. 111 (1991), 451-453. MR 92f:34077
2.: M.S. Ashbaugh and R. Benguria, Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potential, Proc. Amer. Math. Soc. 99 (1987), 598-599. MR 88e:34039
3.: -, Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials, Commun. Math. Phys. 124 (1989), 403-415. MR 91c:34114
4.: -, Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials, Proc. Amer. Math. Soc. 105 (1989), 419-424. MR 89f:81028
5.: -, Eigenvalue ratios for Sturm-Liouville operators, J. Diff. Equations 103 (1993), 205-219. MR 94c:34125
6.: R.D. Gentry and D.O. Banks, Bound for functions of eigenvalues of vibrating systems, J. Math. Anal. Appl. 51 (1975), 100-128. MR 51:8528
7.: M.-J. Huang, On the eigenvalue ratio with vibrating strings, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1805-1813. MR 99i:34119
8.: Y.-L. Huang and C.K. Law, Eigenvalue ratios for the regular Sturm-Liouville system, Proc. Amer. Math. Soc. 124 (1996), 1427-1436. MR 96g:34044
9.: J.B. Keller, The minimum ratio of two eigenvalues, SIAM J. Appl. Math. 31 (1976), 485-491. MR 54:10737
10.: R. Lavine, The eigenvalue gap for one-dimensional convex potentials, Proc. Amer. Math. Soc. 121 (1994), 815-821. MR 94i:35144
11.: T.J. Mahar and B.E. Willner, An extremal eigenvalue problem, Comm. Pure Appl. Math. 29 (1976), 517-529. MR 54:13201

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34L15, 34B25

Retrieve articles in all Journals with MSC (2000): 34L15, 34B25

Additional Information:

Miklós Horváth
Affiliation: Department for Mathematical Analysis, Institute of Mathematics, Technical University of Budapest, H 1111 Budapest, Muegyetem rkp. 3-9, Hungary
Email: horvath@math.bme.hu

DOI: 10.1090/S0002-9939-02-06637-6
PII: S 0002-9939(02)06637-6
Received by editor(s): July 25, 2001
Received by editor(s) in revised form: November 18, 2001
Posted: July 26, 2002
Additional Notes: Supported by the Hungarian Grant OTKA T032374
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2002, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|