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research
A second eigenvalue bound for the Dirichlet Schroedinger operator
Authors
B. Baumgartner
B. Baumgartner
+11Â more
C. Haile
E. Krahn
G. Talenti
Helmut Linde
J.M. Luttinger
L.E. Payne
L.E. Payne
M.S. Ashbaugh
M.S. Ashbaugh
M.S. Ashbaugh
Rafael D. Benguria
Publication date
9 November 2005
Publisher
'Springer Science and Business Media LLC'
Doi
Cite
View
on
arXiv
Abstract
Let
λ
i
(
Ω
,
V
)
\lambda_i(\Omega,V)
λ
i
​
(
Ω
,
V
)
be the
i
i
i
th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain
Ω
⊂
R
n
\Omega \subset \R^n
Ω
⊂
R
n
and with the positive potential
V
V
V
. Following the spirit of the Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential
V
⋆
V_\star
V
⋆
​
, we prove that
λ
2
(
Ω
,
V
)
≤
λ
2
(
S
1
,
V
⋆
)
\lambda_2(\Omega,V) \le \lambda_2(S_1,V_\star)
λ
2
​
(
Ω
,
V
)
≤
λ
2
​
(
S
1
​
,
V
⋆
​
)
. Here
S
1
S_1
S
1
​
denotes the ball, centered at the origin, that satisfies the condition
λ
1
(
Ω
,
V
)
=
λ
1
(
S
1
,
V
⋆
)
\lambda_1(\Omega,V) = \lambda_1(S_1,V_\star)
λ
1
​
(
Ω
,
V
)
=
λ
1
​
(
S
1
​
,
V
⋆
​
)
. Further we prove under the same convexity assumptions on a spherically symmetric potential
V
V
V
, that
λ
2
(
B
R
,
V
)
/
λ
1
(
B
R
,
V
)
\lambda_2(B_R, V) / \lambda_1(B_R, V)
λ
2
​
(
B
R
​
,
V
)
/
λ
1
​
(
B
R
​
,
V
)
decreases when the radius
R
R
R
of the ball
B
R
B_R
B
R
​
increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density
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Last time updated on 11/12/2019