On the first two eigenvalues of Sturm-Liouville operators

Author:
Miklós Horváth

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1215-1224

MSC (2000):
Primary 34L15, 34B25

DOI:
https://doi.org/10.1090/S0002-9939-02-06637-6

Published electronically:
July 26, 2002

MathSciNet review:
1948113

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Among the Schrödinger operators with single-well potentials defined on with transition point at , 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.

**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**

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, Műegyetem rkp. 3-9, Hungary

Email:
horvath@math.bme.hu

DOI:
https://doi.org/10.1090/S0002-9939-02-06637-6

Received by editor(s):
July 25, 2001

Received by editor(s) in revised form:
November 18, 2001

Published electronically:
July 26, 2002

Additional Notes:
Supported by the Hungarian Grant OTKA T032374

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society