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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
Posted:
July 26, 2002
MathSciNet review:
1948113
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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), no. 2,
451–453. MR 1036981
(92f:34077), http://dx.doi.org/10.1090/S0002-9939-1991-1036981-X
- 2.
Mark
S. Ashbaugh and Rafael
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), http://dx.doi.org/10.1090/S0002-9939-1987-0875408-4
- 3.
Mark
S. Ashbaugh and Rafael
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)
- 4.
Mark
S. Ashbaugh and Rafael
Benguria, 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), no. 2, 419–424. MR 942630
(89f:81028), http://dx.doi.org/10.1090/S0002-9939-1989-0942630-X
- 5.
Mark
S. Ashbaugh and Rafael
D. Benguria, Eigenvalue ratios for Sturm-Liouville operators,
J. Differential Equations 103 (1993), no. 1,
205–219. MR 1218744
(94c:34125), http://dx.doi.org/10.1006/jdeq.1993.1047
- 6.
R.
D. Gentry and D.
O. Banks, Bounds for functions of eigenvalues of vibrating
systems, J. Math. Anal. Appl. 51 (1975),
100–128. MR 0372312
(51 #8528)
- 7.
Min-Jei
Huang, On the eigenvalue ratio for vibrating
strings, Proc. Amer. Math. Soc.
127 (1999), no. 6,
1805–1813. MR 1621941
(99i:34119), http://dx.doi.org/10.1090/S0002-9939-99-05015-7
- 8.
Yu
Ling Huang and C.
K. Law, Eigenvalue ratios for the regular
Sturm-Liouville system, Proc. Amer. Math.
Soc. 124 (1996), no. 5, 1427–1436. MR 1328351
(96g:34044), http://dx.doi.org/10.1090/S0002-9939-96-03396-5
- 9.
Joseph
B. Keller, The minimum ratio of two eigenvalues, SIAM J. Appl.
Math. 31 (1976), no. 3, 485–491. MR 0422751
(54 #10737)
- 10.
Richard
Lavine, The eigenvalue gap for one-dimensional
convex potentials, Proc. Amer. Math. Soc.
121 (1994), no. 3,
815–821. MR 1185270
(94i:35144), http://dx.doi.org/10.1090/S0002-9939-1994-1185270-4
- 11.
T.
J. Mahar and B.
E. Willner, An extremal eigenvalue problem, Comm. Pure Appl.
Math. 29 (1976), no. 5, 517–529. MR 0425244
(54 #13201)
- 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
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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:
http://dx.doi.org/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
Article copyright:
© Copyright 2002 American Mathematical Society
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