In this paper, we prove the optimal lower bound λ2λ14 of the Sturm-Liouville problem −(p(x)y′)′ + q(x)y = λρ(x)y, with Dirichlet boundary conditions for single-well potential q and single-barrier function on [0,1]. In the case of symmetric coefficients, we establish the lower bound λnλ1n2 for single-well q and single-barrier function with λ1 > 0 and μ1 ≤ 0, where μ1 is the first eigenvalue of the Neumann boundary problem. All the above estimates are given for q ≤ 0 without assumptions on the monotonicity of q.

1.
M.
Ashbaugh
and
R.
Benguria
, “
Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials
,”
Commun. Math. Phys.
124
,
403
415
(
1989
).
2.
M.
Ashbaugh
and
R.
Benguria
, “
Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials
,”
Proc. Am. Math. Soc.
105
,
419
424
(
1989
).
3.
M.
Ashbaugh
and
R.
Benguria
, “
Eigenvalue ratios for Sturm’Liouville operators
,”
J. Differ. Equations
103
,
205
219
(
1993
).
4.
J.
Ben Amara
and
J.
Hedhly
, “
Eigenvalue ratios for Schrödinger operators with indefinite potentials
,”
Appl. Math. Lett.
76
,
96
102
(
2018
).
5.
J.
Ben Amara
and
J.
Hedhly
, “
Upper bound for the ratios of eigenvalues of Schrödinger operators with nonnegative single-barrier potentials
,”
Math. Nachr.
291
,
1926
1940
(
2018
).
6.
J.
Ben Amara
and
H.
Jihed
, “
Lower bound for the ratios of eigenvalues of Schrödinger with nonpositive single-barrier potentials
,”
Math. Methods Appl. Sci.
42
,
4480
4487
(
2019
).
7.
J.
Hedhly
, “
Eigenvalue ratios for vibrating string equations with single-well densities
,”
J. Differ. Equations
307
,
476
485
(
2022
).
8.
M.-J.
Huang
, “
On the eigenvalue ratio for vibrating strings
,”
Proc. Am. Math. Soc.
127
,
1805
1813
(
1999
).
9.
M.
Horváth
, “
On the first two eigenvalues of Sturm–Liouville operators
,”
Proc. Am. Math. Soc.
131
,
1215
1224
(
2002
).
10.
M.
Kiss
, “
Eigenvalue ratios of vibrating strings
,”
Acta Math. Hung.
110
,
253
259
(
2006
).
11.
C. K.
Law
and
Y. L.
Huang
, “
Eigenvalue ratios and eigenvalue gaps of Sturm–Liouville operators
,”
Proc. -R. Soc. Edinburgh, Sect. A: Math.
128
,
337
347
(
1998
).
12.
W.
Leighton
,
Ordinary Differential Equations
, 3rd ed. (
Wadsworth Publishing
,
Belmont, CA
,
1970
).
13.
B. M.
Levitan
and
I.
Sargsjan
, “
Introduction to spectral theory: Selfadjoint ordinary differential operators
,”
Transl. Math. Monogr.
39
(
1975
), see Chap. 1.
14.
W.
Magnus
and
S.
Winkler
,
Hill’s Equation
(
Wiley
,
New York
,
1966
) reprinted by Dover, New York 1979, AMS, 1975.
You do not currently have access to this content.