Journal of Inequalities in Pure and
Applied Mathematics
LOWER BOUNDS FOR THE INFIMUM OF THE SPECTRUM
OF THE SCHRÖDINGER OPERATOR IN RN AND THE
SOBOLEV INEQUALITIES
volume 3, issue 4, article 63,
2002.
E.J.M. VELING
Delft University of Technology
Faculty of Civil Engineering and Geosciences
Section for Hydrology and Ecology
P.O. Box 5048,
NL-2600 GA Delft, The Netherlands.
EMail: Ed.Veling@CiTG.TUDelft.nl
Received 15 April, 2002;
accepted 27 May, 2002.
Communicated by: S.S. Dragomir
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
037-02
Quit
Abstract
This article is concerned with the infimum e1 of the spectrum of the Schrödinger
operator τ = −∆ + q in RN , N ≥ 1. It is assumed that q− = max(0, −q) ∈
Lp (RN ), where p ≥ 1 if N = 1, p > N/2 if N ≥ 2. The infimum e1 is estimated
in terms of the Lp -norm of q− and the infimum λN,θ of a functional ΛN,θ (ν) =
−1
1 N
k∇vkθ2 kvk1−θ
2 kvkr , with ν element of the Sobolev space H (R ), where θ =
N/(2p) and r = 2N/(N −2θ). The result is optimal. The constant λN,θ is known
explicitly for N = 1; for N ≥ 2, it is estimated by the optimal constant CN,s in
the Sobolev inequality, where s = 2θ = N/p. A combination of these results
gives an explicit lower bound for the infimum e1 of the spectrum. The results
improve and generalize those of Thirring [A Course in Mathematical Physics III.
Quantum Mechanics of Atoms and Molecules, Springer, New York 1981] and
Rosen [Phys. Rev. Lett., 49 (1982), 1885-1887] who considered the special
case N = 3. The infimum λN,θ of the functional ΛN,θ is calculated numerically
(for N = 2, 3, 4, 5, and 10) and compared with the lower bounds as found in this
article. Also, the results are compared with these by Nasibov [Soviet. Math.
Dokl., 40 (1990), 110-115].
2000 Mathematics Subject Classification: 26D10, 26D15, 47A30
Key words: Optimal lower bound, infimum spectrum Schrődinger operator, Sobolev
inequality
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
Contents
1
2
3
4
5
A
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
17
28
31
32
37
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.
Results
In this article we study the Schrödinger operator τ = −∆ + q on RN . The
real-valued potential q is such that q = q+ + q− , where
q+ = max(0, q) ∈ L2loc (RN ),
(1.1)
(1.2)
q− = max(0, −q) ∈ Lp (RN ),
N = 1: 1 ≤ p < ∞,
N ≥ 2: N/2 < p < ∞.
Associated with q is the closed hermitian form h,
Z
h(u, v) = (∇u, ∇v) +
(1.3)
quv̄dx,
u, v ∈ Q(h),
RN
(1.4)
1
N
Q(h) = H (R ) ∩ {u | u ∈ L2 (RN ),
q+1/2 ∈ L2 (RN )}.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
As will be shown in the course of the proof of Theorem 1.1, h is semibounded
below if the condition (1.2) is satisfied. Hence, we can define a unique selfadjoint operator H, such that Q(h) is its quadratic form (see [22, Theorem
VIII.15] or [26, Theorem 2.5.19]).
We remark that τ restricted to C0∞ (RN ) is essentially self-adjoint for the
following values of p :
(1.5)
p≥2
if N = 1, 2, 3;
p>2
if N = 4;
p ≥ N/2 if N ≥ 5;
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
see [21, Corollary, p. 199, with V1 = q+ , c = d = 0, V2 = q− ]. For N =
1, 2, 3 condition (1.5) imposes a restriction on the values of p allowed in (1.2).
Furthermore, D(H) = H02 (RN ) = H 2 (RN ) if q+ ∈ L∞ (RN ), p > N/2, N ≥
4; see [6, pp. 123, 246 (vi)].
It is our purpose to give a lower bound for the infimum of the spectrum of
H by estimating the Rayleigh quotient e1 = inf u∈D(H) h(u, u)/kuk22 . Since q+
enlarges e1 , it suffices to consider the Rayleigh quotient for the case q+ = 0.
Let ΛN,θ be the following functional on H 1 (RN ) :
(1.6)
k∇vkθ2 kvk1−θ
2
,
ΛN,θ (v) =
kvkr
r = 2N/(N − 2θ),
1
N
v ∈ H (R ),
where
E.J.M. Veling
0 < θ ≤ 1/2 if N = 1,
and
0 < θ < 1 if N ≥ 2.
Title Page
Let λN,θ be its infimum
Contents
λN,θ = inf ΛN,θ (v)|v ∈ H 1 (RN ), v 6= 0 .
(1.7)
It is possible to include the cases θ = 0, with λN,0 = ΛN,0 (v) = 1, and θ = 1,
provided N ≥ 2; see below. The functional ΛN,θ (v) is invariant for dilations in
the argument of v and for scaling of v.
We recall the following imbeddings
(1.8)
(1.9)
(1.10)
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
H 1 (R1 ) ,→ C 0,λ (R1 ), 0 < λ ≤ 1/2,
H 1 (R2 ) ,→ Ls (R2 ), 2 ≤ s < ∞,
1
N
s
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 49
N
H (R ) ,→ L (R ), 2 ≤ s ≤ 2N/(N − 2), N ≥ 3;
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
see [1, pp. 97, 98]. Here, C 0,λ (R1 ) is the space of bounded, uniformly continuous functions v on R1 with
sup
|v(x) − v(y)|/|x − y|λ < ∞.
x,y∈R1 , x6=y
Hence, u ∈ H 1 (R1 ) implies u ∈ L2 (R1 ) ∩ L∞ (R1 ) and, therefore, u ∈ Ls (R1 ),
2 ≤ s ≤ ∞. Thus, (1.8), (1.9), and (1.10) imply that there exist positive
constants K such that
(1.11)
2 ≤ s ≤ ∞ if N = 1,
2
2 1/2
[k∇vk2 + kvk2 ] /kvks ≥ K,
2 ≤ s < ∞ if N = 2,
2 ≤ s ≤ 2N/(N − 2) if N ≥ 3.
Returning to the functional ΛN,θ , we make for 0 < θ < 1 (0 < θ ≤ 1/2 if
N = 1) a dilation x = y, x, y ∈ RN , w(y) = v(x), such that
k∇wk22 /kwk22 = θ/(1 − θ).
ab ≤ aP /P + bQ /Q, a, b ≥ 0, 1 < P < ∞, 1/P + 1/Q = 1,
with equality if and only if aP = bQ , applied to Λ2N,θ (w) gives (P = 1/θ,
2θ
Q = 1/(1 − θ), a = ηk∇wk2θ
2 , b = kwk2 /η)
(1.13)
Λ2N,θ (w) ≤
E.J.M. Veling
Title Page
Contents
JJ
J
The inequality
(1.12)
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
θη 1/θ k∇wk22 + (1 − θ)η −1/(1−θ) kwk22
,
kwk2r
II
I
Go Back
Close
Quit
Page 6 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
for some number η > 0. Equality holds if and only if
η 1/θ k∇wk22 = η −1/(1−θ) kwk22 ,
i.e. η −1/(θ(1−θ)) = θ/(1 − θ).
In this case,
Λ2N,θ (w) = θθ (1 − θ)1−θ
(1.14)
k∇wk22 + kwk22
.
kwk2r
Since it is possible to perform this dilation for any v ∈ H 1 (RN ), and since θθ
(1 − θ)1−θ > 0 we conclude that λN,θ > 0 for 0 < θ < 1. The case N = 1,
θ = 1/2 (in that case r becomes undefined) is covered by the value s = ∞ in
(1.11). The cases θ = 1, N ≥ 2 are covered by a special form of the Sobolev
inequality
(1.15) k∇wks ≥ CN,s kwkt , t = sN/(N − s), 1 ≤ s < N, w ∈ H 1,s (RN ),
E.J.M. Veling
Title Page
Contents
where CN,s are the optimal constants and
1,s
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
JJ
J
N
(1.16) H (R )
= completion of {w | w ∈ C 1 (RN ), kuks1,s = kukss + k∇ukss < ∞}
II
I
Go Back
with respect to the norm k · k1,s .
Close
1
2
∞
2
If we take s = 2 we have λN,1 = CN,2 , N ≥ 3. Since H (R ) 6,→ L (R ),
it follows that λ2,1 = C2,2 = 0, i.e. K = 0 in (1.11). The numbers CN,s are
Quit
Page 7 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
known explicitly by the work of [2] and [25], see also [14]
(1.17) CN,s = N
(1.18)
1/s
(s−1)/s
N −s
s−1
1/N
N
N
× N ωN B
,N + 1 −
, 1 < s < N,
s
s
1/N
CN,1 = N ωN , N ≥ 2,
where ωN is the volume of the unit ball in RN :
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
(1.19)
(1.20)
ωN = π N/2 /Γ(1 + N/2),
B(a, b) = Γ(a)Γ(b)/Γ(a + b), a, b > 0,
and there is equality in (1.15) for functions of the form
(1.21)
1−N/s
wN,s (x1 , ..., xN ) = a + b|x|s/(s−1)
, a, b > 0, 1 < s < N.
Note that wN,s ∈
/ Ls (RN ) if s ≥ N 1/2 . For s = 1 there are no functions
such that there is equality, but by taking an approximating sequence {wi } ∈
H 1,1 (RN ) of the characteristic function of the unit ball, the bound CN,1 can be
approximated arbitrarily close. See further Lemma 2.1 for more information
about ΛN,θ and the explicit form for λ1,θ .
In Theorem 1.1 we give the lowest possible point of the spectrum of this
Schrödinger equation for all q− satisfying (1.2). Let us define the number
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
l(N, θ), where θ = N/(2p), as follows
(1.22)
R
k∇uk22 + RN qkuk22 dx
l(N, θ) =
inf
inf
kq− k−1/(1−θ)
.
p
2
p
N
1
N
kuk2
q− ∈L (R ) u∈H (R )
Theorem 1.1. Let q− ∈ Lp (RN ), 1 ≤ p < ∞ if N = 1, N/2 < p < ∞ if
N ≥ 2 (i.e. (1.2)). Then
−2/(1−θ)
(1.23)
l(N, θ) = −(1 − θ)θθ/(1−θ) λN,θ
, 0 < θ < 1/2 if N = 1,
0 < θ < 1 if N ≥ 2,
and explicitly for N = 1
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
l(1, θ)
(
E.J.M. Veling
(1.24)
−2θ )1/(1−θ)
1
1
,
, 0 < θ < 1/2,
= − (2θ)2θ (1 − 2θ)1−2θ B
2 2θ
(
−1 )2/(2p−1)
1
= − p−p (p − 1)p−1 B
,p
, 1 < p < ∞,
2
l(1, 1/2) = −1/4.
(1.25)
−2/(1−θ)
e1 ≥ − inf (1 − θ)θθ/(1−θ) λN,θ
θ∈Θ
Contents
JJ
J
II
I
Go Back
Remark 1.1. Of course, for any application of this method to find a lower
bound for e1 (the smallest eigenvalue) one can take the following infimum over
the allowed set Θ of θ-values (depending on q− ).
(1.26)
Title Page
1/(1−θ)
Close
Quit
Page 9 of 49
kq− kN/(2θ) .
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
Remark 1.2. Note that we do not include θ = 1 in the allowed θ-range, although for N ≥ 2 λN,1 is defined. It turns out that the method of the proof
does not work in this case; it gives however a criterion such that σd (H) = ∅
(i.e. there are no isolated eigenvalues), see the Remark 2.3 after the proof of
Theorem 1.1.
Remark 1.3. It is possible to allow the case p = ∞, i.e. θ = 0, then l(N, 0) =
−1. If q = −kq− k∞ this bound is achieved arbitrarily close by a sequence
of functions {ui } ∈ H 1 (RN ), where each ui is a smooth approximation of the
characteristic function of the i-ball in RN , because then the quotient
R
q|ui |2 dx
RN
k∇ui k22 /kui k22 → N ωN i−1 , i → ∞, and
kq− k−1
∞ = −1.
kui k22
Remark 1.4. Already Lieb and Thirring [15] characterize the infimum of the
spectrum with a number −(L1γ,N )1/γ (in their notation, γ = p − N/2), with
γ > max(0, 1 − N/2), and γ = 1/2, N = 1. Therefore,
−2/(1−θ)
(1.27)
(L1γ,N )1/γ γ=(1−θ)N/(2θ) = (1 − θ)θθ/(1−θ) λN,θ
.
L1γ,1
They give
for γ > 1/2 explicitly. Here, we also include the case N = 1,
γ = 1/2 (i.e. θ = 1/2, p = 1). However, the main reason of this article
is to show how one can give an explicit estimate for e1 by sharp estimates of
the numbers λN,θ , N ≥ 2, in terms of the numbers CN,s for some s = s(θ),
see Theorems 1.2 and 1.3. For a survey for other integral inequalities results
related to the infimum of the spectrum see [9] and [16].
Remark 1.5. The results for the ordinary differential case (N = 1, Ω = R) are
related to those for Ω = R+ with either a Dirichlet or a Neumann boundary
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
condition at x = 0 (respectively the operators T0 and Tπ/2 in the work of [8],
[27] and [10]). In those cases there holds 1 ≤ p ≤ ∞
R∞
ku0 k22 + 0 q|u|22 dx
(1.28)
inf
inf
kq− k−2p/(2p−1)
= l(1, 1/(2p)),
p
q− ∈Lp (R+ ) u∈D(T0 )
kuk22
(1.29)
R∞
ku0 k22 + 0 q|u|22 dx
inf
inf
kq− k−2p/(2p−1)
p
2
p
+
q− ∈L (R ) u∈D(Tπ/2 )
kuk2
2/(2p−1)
=2
l(1, 1/(2p)).
See for related work [3].
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Theorem 1.2. The following inequalities hold for N ≥ 2
(1.30)
Title Page
i) λN,θ > (λN,θ0 )α (λN,θ00 )1−α , 0 < α < 1,
00
00
θ = αθ0 + (1 − α)θ , θ0 6= θ ,
(1.31)
ii) λN,θ > (θCN,2θ )θ ,
(1.32)
iii) λN,θ > (θN CN,2θN )θ ,
θ
λN,θ > (θCN,2θ ) ,
(1.33)
iv) λN,θ > (CN,2 )θ ,
1/2 ≤ θ < 1,
0 < θ ≤ θN ,
θN ≤ θ < 1,
0 < θ < 1,
where CN,s is given by (1.17) and (1.18) and θN = θ(N ) ∈ (1/2, 1) is the
unique maximum of θCN,2θ , 1/2 ≤ θ ≤ 1. θN is given by θN = N/(2pN ) where
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
pN is the solution of M (N, p) = 0, with
N −p
N −p
(1.34)
M (N, p) = log
+
+ ψ(p) − ψ(N + 1 − p),
p−1
p(p − 1)
d
d
(1.35)
ψ(x) =
(log(Γ(x)) =
Γ(x) /Γ(x), x > 0.
dx
dx
It is now easy to combine both theorems in
Theorem 1.3. Under the conditions of Theorem 1.1 there holds
(
−(1 − θ)θθ/(1−θ) (θN CN,2θN )−2θ/(1−θ) , 0 < θ ≤ θN ,
(1.36) l(N, θ) >
−(1 − θ)θ−θ/(1−θ) (CN,2θ )−2θ/(1−θ) ,
θN ≤ θ < 1,
and also (generally less than optimal)
Proof. Equation (1.36) follows from (1.23) and (1.32); (1.37) follows from
00
(1.23), (1.30) (with θ = 0) and (1.31).
0
Remark 1.6. For N = 3, θ = 1 the result (1.37) reads explicitly
l(3, θ) > −(1 − θ)θθ/(1−θ) [31/2 2−2/3 π 2/3 ]−2θ/(1−θ) ,
E.J.M. Veling
Title Page
(1.37) l(N, θ) > −(1 − θ)θθ/(1−θ) (θ0 CN,2θ0 )−2θ/(1−θ) ,
0 < θ < 1, for any θ0 ≥ θ, 1/2 ≤ θ0 ≤ 1.
(1.38)
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
0 < θ < 1,
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 49
and this is the same result as [23, (14)].
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
Remark 1.7. [26, (3.5.30), and private communication by H. Grosse] gives the
following result for N = 3
(1.39) l(3, 3/(2p)) > −((p − 1)/p)2 (4π)−2/(2p−3)
(2p−2)/(2p−3)
2p − 3
× Γ
, 3/2 < p < ∞,
p−1
or in terms of θ,
(1.40) l(3, θ) > −(1 − 2θ/3)2 (4π)−2θ/(3−3θ)
(3−2θ)/(3−3θ)
6 − 6θ
, 0 < θ < 1.
× Γ
3 − 2θ
It can be proved that (1.38) is better than (1.40) for all 0 < θ < 1. For θ = 0 the
right-hand sides of both (1.38) and (1.40) give the correct value l(3, 0) = −1.
Remark 1.8. To show the superiority of (1.37) with θ0 < 1 against (1.37) with
θ0 = 1, i.e. (1.38), we evaluate the bound for l(3, 3/4) of (1.37) with θ = θ0 =
3/4. We find
(1.41)
l(3, 3/4) > −22 3−7 π −2 ' −1.8510−4 ,
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
while (1.38) gives
Close
l(3, 3/4) > −2−4 π −4 ' −6.4210−4 ,
and (1.40) gives
Quit
Page 13 of 49
l(3, 3/4) > −2−6 π −2 ' −15.8310−4 .
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
Based on our numerical calculations (see Section 3) we find l(3, 3/4) =
−1.75018010−4 . So the estimate (1.41) comes close to the actual value of l(3, 3/4).
Remark 1.9. The results in Theorems 1.1, 1.2, and 1.3 were announced in [28]
and [7, p. 337].
Remark 1.10. In the interesting paper [20] Nasibov has given a lower bound
(in his notation 1/k0 ) for λN,θ :
λN,θ =
(1.42)
1
1
>
,
k0
k0
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
with
E.J.M. Veling
1
(1.43) k0 = p
θθ (1 − θ)1−θ
N ωN B
N N (1 − θ)
,
2
2θ
θ/N
Title Page
× kB
p 1/p p0 −1/p0
kB (p) =
2π
2π
"
(1.44)
#N/2
,
2N
N + 2θ
1
1
+ 0 = 1.
p p
And, even better
(1.45)
λN,θ =
,
Contents
JJ
J
II
I
Go Back
Close
Quit
1
1
>
,
k0
k0
with
1
k0
>
1
,
k0
for
θ > N/4,
Page 14 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
with
(1.46) k0 =
1
kB
θ
θ (1 − θ)1−θ
N
N − 2θ
2
×kB
2N
N + 2θ
1/2
kG(|x|)k
N
N −2θ
,
G(|x|) = K N −2 (|x|)|x|−(N −2)/2 ,
(1.47)
2
with Kα the modified Bessel function of the second kind and order α. The
inequality (1.45) is only relevant for N = 2, 1/2 ≤ θ ≤ 1, and N = 3,
3/4 ≤ θ ≤ 1, since k0 < k0 , for N = 2, 0 < θ < 1/2, and N = 3, 0 < θ < 3/4,
and k0 = k0 , for N = 2, θ = 1/2, and N = 3, θ = 3/4.
The reader is advised to consult also the original paper (Dokl. Akad. Nauk
SSSR 307, No. 3, 538-542 (1989)) of [20] since there are a number of misprints
in the translated version. In Section 3 this lower bound will be compared with
(1.32). The function G reads
N = 2,
N = 3,
G(|x|) = K0 (|x|),
−1/2
G(|x|) = K 1 (|x|)|x|
2
r
=
π
exp(−|x|)/|x|,
2
so, one has to calculate the integrals in (1.46)
Z ∞
1−θ
1/(1−θ)
(1.48)
N = 2 : kG(|x|)k 1 =
K0
(r) 2πr dr
,
1−θ
0
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
(1.49)
N = 3 : kG(|x|)k 3
3−2θ
r Z ∞
(3−2θ)/3
π
3r
(3−4θ)/(3−2θ)
=
r
exp −
4π dr
.
2 0
3 − 2θ
For N = 3 the integral in (1.49) can be evaluated explicitly, while for N = 2,
i.e. (1.48), that is only possible for θ = 1/2:
N = 2 : kG(|x|)k2
Z ∞
1/2
2
= 2π
K0 (r) r dr
0
=
∞ 1/2 √
r2
2
2
K0 (r) − K1 (r) = π,
2π
2
0
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
N = 3 : kG(|x|)k 3
3−2θ
r
2−2θ (3−2θ)/3
π
3 − 2θ
6 − 6θ
(3−2θ)/3
=
(4π)
Γ
.
2
3
3 − 2θ
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
2.
Proofs
Firstly, we give more information on ΛN,θ in a lemma.
Lemma 2.1. The value λN,θ = inf v∈H 1 (RN ), v6=0 ΛN,θ (v) for the functional ΛN,θ (v)
defined in (6) is attained by radial symmetric monotonely decreasing positive
functions vN,θ (|x|) which satisfy, except for θ = 1/2, N = 1, the following
ordinary differential equation for 0 < θ < 1/2 if N = 1, and 0 < θ < 1 if
N ≥ 2,
d2
(N − 1) d
− 2v −
v − v|v|(N +2θ)/(N −2θ)−1 + v = 0, r = |x| > 0,
dr
r
dr
d
(2.1)
v(0) = 0, lim v(r) = 0,
r→∞
dr
and the value λN,θ is then given by
(2.2) λN,θ
E.J.M. Veling
Title Page
Z
θ/2
(N (1−θ)−2θ)/(2N )
= θ (1 − θ)
N ωN
∞
2
vN,θ
(r)rN −1 dr
θ/N
0
for 0 < θ < 1, N ≥ 2.
For N = 1 we have explicitly for x ≥ 0
v1,θ (x) = v1,θ (−x), 0 < θ ≤ 1/2,
−(1−2θ)/(2θ)
2θ
1/2
x
, 0 < θ < 1/2,
v1,θ (x) = (1 − 2θ) cosh
1 − 2θ
(2.4) v1,1/2 (x) = e−x ,
(2.3)
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
−θ −θ/2
(2.5) λ1,θ = 2 θ
(1 − θ)
−1/2
(2.6) λ1,N/(2p) = 2
(1−θ)/2
−(1−2θ)/2
(1 − 2θ)
θ
1 1
,
,
B
2 2θ
0 < θ < 1/2,
1/(2p)
1
(2p−1)/2
−(p−1)
,p
,
(2p − 1)
(p − 1)
B
2
1 < p < ∞,
λ1,1/2 = 1.
(2.7)
Proof. The case N = 1 was treated by [19] and the case N ≥ 2 was given by
[29] who used a rearrangement and an inequality due to Strauss to prove the
compactness of the imbedding of radial symmetric functions u ∈ H 1 (RN ) into
Ls (RN ), 2 < s < ∞ if N = 2, and 2 < s < 2N/(N − 2) if N ≥ 3 (see also
(1.9), (1.10)). The Euler equation connected with the infimum of ΛN,θ becomes
2N
,
N − 2θ
which can be scaled into the form (2.1) with λN,θ given by (2.2). The following
relations between λN,θ and the following norms of v̄N,θ (x1 , ..., xN ) = vN,θ (|x|)
hold (cf. [24, p. 151], where the factor “(n − 2)” has to be skipped in the last
line on that page)
(2.8)
(2.9)
(2.10)
−2
−r
r−2
−θk∇uk−2
u = 0, r =
2 ∆u + (1 − θ)kuk2 u − kukr |u|
kv̄N,θ k22
= L(1 − θ),
L=θ
−N/2
(1 −
k∇v̄N,θ k22
= Lθ, kv̄N,θ krr
N/θ
θ)−N (1−θ)/(2θ) λN,θ .
= L,
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
Since (2.1) is nonlinear the value of v(0) has to be chosen properly to satisfy
limr→∞ v(r) = 0.
Remark 2.1. We note that the existence of solutions of (2.1) has been proved
by many authors: it is just the range 0 < θ < 1, see [17]. The uniqueness for
the full θ-range has been proved by Kwong, see [11], after preliminary work by
[17], and [18]. A proof based on geometrical arguments has been given by [5].
See for related work also [12].
Remark 2.2. Numerical information for λN,θ for N = 2, 3 can be obtained
from [15, Appendix], where curves for L1γ,N (see (1.27)) are given (0 ≤ γ ≤ 2.8,
N = 2, 3). By (1.27) we have
(2.11)
λN,θ = θθ/2 (1 − θ)(1−θ)/2 (L1γ,N )−θ/N , γ = N (1 − θ)/(2θ).
Comparison with (2.10) learns that L1γ,N = 1/L. Besides, the following two
values for λN,θ are known based on numerical calculations
1
−1
(2.12)
λ2,1/2 '
' 0.642988, ([29], after (I.5))
π(1.86225 · · · )
→ λ2,1/2 ' 1.55524,
(2.13)
λ32,2/3 ' 4.5981, ([13], p. 185)
→ λ2,2/3 ' 1.66287.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Proof of Theorem 1.1. We estimate h(u, u), see (1.3), as follows. All integrals
Page 19 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
are over RN .
Z
(2.14) h(u, u) =
k∇uk22
≥
k∇uk22
+
Z
−
q|u|2 dx
q− |u|2 dx
(2.15)
≥ k∇uk22 − kq− kp kuk2r
(2.16)
2θ
≥ k∇uk22 − kq− kp λ−2
N,θ k∇uk2 kuk2
[r = 2p/(p − 1) = 2N/(N − 2θ)]
2(1−θ)
.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
Apply now (1.12) with
P = 1/θ, a = θ−θ k∇uk2θ
2 ,
and
2(1−θ)
2θ
ab = kq− kp λ−2
N,θ k∇uk2 kuk2
E.J.M. Veling
.
Then
2(1−θ)
θ
b = λ−2
N,θ θ kq− kp kuk2
Title Page
,
Contents
and finally we find
(2.17)
−2/(1−θ)
h(u, u) = −bQ /Q = −(1 − θ)θθ/(1−θ) λN,θ
kq− k1/(1−θ)
kuk22 ,
p
which is the bound of Theorem 1.1. To prove the optimality part we observe
that in such a case we need
(2.18)
(2.19)
(2.20)
(2.21)
q = q−
q− = (const)|u|2/(p−1)
u(x1 , ..., xN ) = (const)vN,θ (|x|)
P
Q
a =b ,
JJ
J
II
I
Go Back
by (2.14),
Close
by (2.15),
by (2.16),
Quit
Page 20 of 49
by (2.17).
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
that is
−2/(1−θ) θ/(1−θ)
θ−1 k∇uk22 = λN,θ
θ
kq− k1/(1−θ)
kuk22 .
p
If one takes
(2.22)
u(x1 , ..., xN ) = vN,θ (|x|),
and
(2.23)
q(x1 , ..., xN ) = −q− (x1 , ..., xN ) = − [vN,θ (|x|)]2/(p−1) ,
then (2.1) becomes −∆u + qu = −u; this means that the Schrödinger equation
and the Euler equation for ΛN,θ are the same if e1 = −1. This is true because
for these scalings the lower bound becomes:
−2/(1−θ)
−(1 − θ)θθ/(1−θ) λN,θ
−2/(1−θ)
[kv̄N,θ krr ]2θ/(N (1−θ)) by (2.23),
= −1
by (2.9), (2.10).
Finally, (2.21) is implied also by (2.9) and (2.10). It means that the infimum in
(1.22) over q− ∈ Lp (RN ) is actually attained. In addition to (2.9) there holds
that for q as chosen as in (2.23)
(2.24)
E.J.M. Veling
Title Page
Contents
kq− kp1/(1−θ)
= −(1 − θ)θθ/(1−θ) λN,θ
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
kq− kpp = L.
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
d
Only the case θ = 1/2, N = 1 deserves special attention since dx
v1,1/2 (x) is
not continuous at x = 0. We take the following sequences (see [27])
(2.25)
qj (x) = −(j + 1)[cosh(jx)]−2 , kqj k1 = 1 + 1/j,
(2.26)
uj (x) = [cosh(jx)]−1/j ,
then uj , qj satisfy
−
d2
uj + qj uj = −uj ,
dx2
so
R∞
ku0j k22 + −∞ q|uj |22 dx
2
(2.27)
kqj k−2
1 = −(1 + 1/j) /4 > −1/4 = l(1, 1/2).
2
kuj k2
For these sequences, j → ∞, the bound can be approached arbitrarily close.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
Remark 2.3. As one can observe the proof does not work for θ = 1, i.e. p =
N/2, however, in that case we can estimate (N ≥ 3)
Z
2
h(u, u) = k∇uk2 + q|u|2 dx
Z
2
≥ k∇uk2 − q− |u|2 dx
≥ k∇uk22 − kq− kN/2 kuk22N/(N −2)
≥ k∇uk22 1 − kq− kN/2 λ−2
N,1 .
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
So, if
2
= πN (N − 2)[Γ(N/2)/Γ(N )]2/N , N ≥ 3,
(2.28) kq− kN/2 < λ2N,1 = CN,2
it follows that σd (H) = ∅, i.e. there are no isolated eigenvalues. This is a
well-known result, see [15, (4.24)].
Proof of Theorem 1.2. i) By the Hölder inequality we have
00
00
(2.29) kvkr < kvkαr0 kvk1−α
, 0 < α < 1, 1/r = α/r0 + (1 − α)/r , r0 6= r ,
r00
00
which inequality is strict, since r0 6= r . Therefore, by the conditions specified
under i)
ΛN,θ (v) =
k∇vkθ2 kvk1−θ
2
kvkr
0
>
(2.30)
k∇vkθ2 kvk1−θ
2
kvkr0
E.J.M. Veling
Title Page
!α
0
00
k∇vkθ2 kvk1−θ
2
kvkr00
00 !1−α
= ΛαN,θ0 (v)Λ1−α
(v),
N,θ00
and we find (1.30), which is also strict, since both infima are attained.
ii) This result is given by [13, (1.5)], by making the transformation w = v 1/θ
for v > 0 in (1.15) as follows
CN,s ≤
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
k∇wks
k∇v 1/θ ks
1/θkv (1−θ)/θ ∇vks
=
=
kwkt
kv 1/θ kt
kv 1/θ kt
[t = sN/(N − s)]
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1/s
(∇v)s v s(1−θ)/θ dx
1/t
R
v t/θ dx
R
1/(sP ) R
1/(sQ)
sP
(∇v)
dx
v Qs(1−θ)/θ dx
1
≤
1/t
R
θ
v t/θ dx
1/2 R Qs(1−θ)/θ (2−s)/(2s)
R
2
v
dx
1 (∇v) dx
=
1/t
R
θ
v t/θ dx
1
=
θ
=
R
(1−θ)/θ
1 k∇vk2 kvk2
1/θ
θ
kvkr
=
[apply Hölder inequality,
1/P + 1/Q = 1]
[take P = 2/s,
Q = 2/(2 − s)]
[take s = 2θ, and
r = t/θ = 2N/(N − 2θ)]
1
(ΛN,θ (v))1/θ ,
θ
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
for the choice s = 2θ. We have to restrict θ to the interval 1/2 ≤ θ ≤ 1 to
give the right-hand side of (31) a meaning. Again, the inequality is strict since
θ
w = vN,θ
does not equal a function wN,s (see (1.21)), with s = 2θ.
00
iii) Combining i) with θ = 0 and ii) one finds
(2.31)
ΛN,θ > (θ0 CN,2θ0 )θ , 0 < θ < 1, θ ≤ θ0 , 1/2 ≤ θ0 < 1.
This motivates the determination of the maximum of θCN,2θ = (N/(2p))CN,N/p
on 1/2 ≤ θ < 1. There holds by (1.17), (1.18)
Title Page
Contents
JJ
J
II
I
Go Back
Close
(2.32)
N2
N
CN,N/p =
2p
2p
p−1
N −p
(N −p)/N
Quit
Page 24 of 49
1/N
× [N ωN B(p, N + 1 − p)]
,
1 < p < N,
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1
1/N
CN,1 = (N/2)ωN ,
2
(2.33)
p = N,
θ = 1/2.
The maximum of (2.33) is found by putting the logaritmic derivative of (2.33)
with respect to p equal to zero, which is equation (1.34). It can be proven that
d
(1.34) has a unique solution pN , 1 < pN < N , because dp
M (N, p) ≤ 0. For
0
this last inequality we use the fact that ψ (z) < 1/z + 1/(2z 2 ) + 3/(4z 3 ). So,
with θN = N/(2pN ) and for 0 < θ ≤ θN , there holds ΛN,θ > (θN CN,2θN )θ , and
for the remaining interval θN ≤ θ < 1, λN,θ > (θCN,2θ )θ .
iv) Since limp→N M (N, p) = −∞, it follows that θCN,2θ > CN,2 for θ in a
neighbourhood of θ = 1. So (1.33) follows from (2.31).
00
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
0
Remark 2.4. Application of Theorem 1.2 i) with θ = 0, α = θ/θ , gives
2θ/θ0
λ2N,θ ≥ λN,θ0 ,
(2.34)
θ0 > θ.
Title Page
Contents
[15, (2.21)] give the inequality
L1γ,N
(2.35)
≤
L1γ−1,N (γ/(γ
+ N/2)), γ > 2 − N/2.
By (1.27) this is equivalent with
(2.36)
JJ
J
II
I
Go Back
2θ/θ0
λ2N,θ ≥ λN,θ0 F (θ, θ0 ), θ = N/(2p),
θ0 = N/(2(p − 1)),
Close
Quit
with
Page 25 of 49
0
0
θ(1−θ0 )/θ0
F (θ, θ ) = [(1 − θ)/(1 − θ )]
0 θ
(θ/θ ) .
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
For θ0 > θ it will be proved that F (θ, θ0 ) < 1, which means that i) of Theorem
1.2 (equation (2.34)) is better than (2.35). F (θ, θ0 ) < 1 is equivalent with
(2.37)
θ0
[θ(1 − θ0 )/(θ0 (1 − θ))] < (1 − θ0 )/(1 − θ),
and (2.37) is true by the inequality (1 − a)b < 1 − ab, 0 < a < 1, b < 1, where
a = (θ0 − θ)/(θ0 (1 − θ)), b = θ0 .
Remark 2.5. To show the merits Theorem 1.2 of ii) we compare two known
values for λN,θ , see (2.12), (2.13), by the estimate (1.31)
(2.38)
λ2,1/2 ' 1.55524 > 1.33134 · · · = π 1/4 = (1/2 C2,1 )1/2 ,
(2.39)
λ2,2/3 ' 1.66287 > 1.63696 · · · = (2π/3)2/3 = (2/3 C2,4/3 )2/3 .
Note that in the work of Levine [13, p. 183, third line] the lower bound (2.39)
is not calculated correctly. The lower bound C1 for his variable C (which is
λ32,2/3 ) should be C1 = 4π 2 /9 ' 4.38649, in stead of C1 = 2π 3/2 /9 ' 1.237
([13, p. 183, eighth line]). This corrected value for C1 is a much better lower
bound, since numerically we found C = λ32,2/3 ' 1.662873 ' 4.5981. See also
Section 3 and Table 1.
Remark 2.6. Approximate solutions pN of (1.34) for N = 2, 3 and N → ∞
are
(2.40)
(2.41)
p2 ' 1.647, θ2 ' 0.6070,
p3 ' 2.304, θ3 ' 0.6509,
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
(2.42)
pN = 2N/3 + 5/18 + O(1/N ),
θN = 3/4 − 5/(16N ) + O(1/N 2 ), N → ∞.
The knowledge of (2.40) allows us to improve (2.38) as follows
(2.43)
λ2,1/2 ' 1.55524 > 1.46436 · · · = (1/1.647 C2,1.2140 )1/2 .
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
3.
Numerical Experiments
In order to assess the quality of the estimates (1.31), (1.32), (1.36) and (1.37)
we have calculated the numbers λN,θ for N = 2, 3 and θ = 0.1 + (i − 1)0.005,
i = 1, 2, 3, · · · , 180, and for N = 4, 5, 10, and θ = 0.0125 + (i − 1)0.025,
i = 1, 2, 3, · · · , 40. For N = 2 we had to exclude θ ≥ 0.945 due to numerical
overflow. The method to find λN,θ consists of a shooting technique to find that
value v(0) = v0 such that v(r) is a positive solution of (2.1) with limr→∞ v(r) =
0. Therefore, we transformed the interval r ∈ (0, ∞) into s = r/(1 + r) ∈
(0, 1). The transformed differential equation becomes, with v(r) = u(s), 0 <
s < 1,
2
(N − 1)
d
4 d
3
(1 − s) 2 u +
− 2 (1 − s)
u
ds
s
ds
(3.1)
−u|u|(N +2θ)/(N −2θ)−1 − u = 0,
d
u(0) = 0.
u(0) = v0 ,
ds
We solved the transformed differential equation (3.1) by means of a numerical
integration method (Runge-Kutta of the fourth order) with a self-adapting stepsize routine such that a prescribed maximal relative error (εrel ) in each compod
nent (u(s), ds
u(s)) has been satisfied. We made the choice εrel = 10−15 . For every value of v0 the numerical integrator will find some point s = s(v0 ) ∈ (0, 1)
d
where either u(s) < 0, or ds
u(s) > 0. At that point s the integration will
be stopped. This integrator is coupled to a numerical zero-finding routine (see
[4]), which can also be applied for finding a discontinuity. The function f for
which such a discontinuity has to been found is specified by if u(s(v0 )) < 0,
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
d
f (v0 ) = −(1−s(v0 )) else (that means thus ds
u(s(v0 )) > 0) f (v0 ) = (1−s(v0 )).
The sought value v0 has been found if this numerical routine has come up with
two values v0 and v01 such that |v0 − v01 | < rp |v0 | + ap , (with rp = ap = 10−15
relative and absolute precisions, respectively) and |f (v0 )| ≤ |f (v01 )|, while
sign(f (v0 ) = −sign(f (v01 )). During the integration processes the norms in
(2.9) will be calculated. As a check upon this procedure the following expressions
(3.2)
kv̄N,θ k22 /(1 − θ),
k∇v̄N,θ k22 /θ,
kv̄N,θ krr ,
are compared. They should be all equal, see (2.9). In the Table 1 the value
for λN,θ are given with one digit less than the number of equal digits in this
comparison; between brackets the next digit is given.
The results of the calculations are shown in the Figures 1, 3, 5, 6, 7. For
N = 2, 3 part of the θ-range has been enlarged to show better the approximations and the infimum of the functional, see Figures 2, 4. (All figures appear in
Appendix A at the end of this paper.)
In Fig. 13 the value v(0) of the minimizer v(r) of the functional ΛN,θ as
function of θ for N = 2, 3, 4, 5, 10 has been shown. Note the logarithmic ordinate axis for v(0).
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
N
θ
p
s
ρ
2
1/3
3
1/2
1
λN,θ
numerical
1.379427(6)
λN,θ
lower bnd.
1.28953
N.A.
1.37026
1.35157
2
1/2
2
1
2
1.55524
1.555239(5)
1.46436
1.33134
1.51739
1.51739
2
2/3
3/2
2
4
1.66287
1.663066(0)
1.63696
1.63696
1.55436
1.61962
3
3/4
2
1
2
2.2258(9)
2.21005
2.21005
2.05668
2.05668
Comment
numerical, this work
see (1.32), this work
see (1.31), this work
see (1.42), Nasibov
see (1.45), Nasibov
numerical (2.12),
based on Weinstein [29]
numerical, this work
see (1.32), this work
see (1.31), this work
see (1.42), Nasibov
see (1.45), Nasibov
numerical (2.13),
based on Levine [13]
numerical, this work
see (1.32), this work
see (1.31), this work
see (1.42), Nasibov
see (1.45), Nasibov
numerical, this work
see (1.32), this work
see (1.31), this work
see (1.42), Nasibov
see (1.45), Nasibov
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Table 1: Comparison of some cases for λN,θ ; p = N/(2θ); s = 2θ/(N − 2θ)
(notation Weinstein); ρ = 4θ/(N − 2θ) (notation Nasibov).
Page 30 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
4.
Discussion
In this article the infimum of the spectrum of the Schrödinger operator τ =
−∆ + q in RN has been expressed in the infimum λN,θ of the functional ΛN,θ ,
and known estimates for λN,θ have been optimized and applied to supply estimates of the infimum of the spectrum. Moreover, numerical experiments have
been done to calculate λN,θ as function of θ for N = 2, 3, 4, 5, and 10. These
results have been used to compare the estimates found in this article with these
found by Nasibov [20].
Except for N = 2, in general, the estimate of Nasibov is better for the lower
half of the θ-interval, while the estimate in this article is better for the upper
half. For N = 2 there is an interval (θ− ,θ+ ) (with θ− ∈ (0.615, 0.620), and
θ+ ∈ (0.745, 0.750)) where the bound in this article is better, while the opposite
is true outside that interval, see Fig. 8. For 0 < θ ≤ θ0 (where θ0 ∈ (0.55, 0.65)
is depending on the value of N , N = 3, 4, 5, 10), the lower bound by Nasibov is
better, but the bounds are of the same order of magnitude and very close to the
actual value of λN,θ ; for θ0 < θ < 1, the bound of Nasibov is worse, see Figs.
9, 10, 11, and 12.
The ratio of the estimate in this article with λN,θ , for θ → 1, N ≥ 3, approaches the value 1, since λN,1 = CN,2 , N ≥ 3 (see just after (1.16) and the
Figs. 9, 10, 11, and 12).
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 31 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
5.
Acknowledgment
The author thanks Dr. N.G. Lloyd (Aberystwyth, Wales) for his invitation to
visit the Gregynog Conference on Differential Equations (1983) which stimulated this research (SERC GR/C/26958), and he is grateful for the hospitality of the University of Birmingham (U.K.) during the spring of 1984 (SERC
GR/C/77660) through the kind invitation of Prof. W.N. Everitt, where the first
stage of this article was written. He also acknowledges Dr. J. Gunson (Birmingham, U.K.) for several stimulating discussions and Dr. H. Kaper (Argonne
National Laboratories, U.S.A.) for a number of suggestions to improve the presentation of this article.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 32 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
References
[1] R.A. ADAMS, Sobolev Spaces, Academic Press, New York 1975.
[2] T. AUBIN, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11 (1976), 573–598.
[3] C. BENNEWITZ AND E.J.M. VELING, Optimal bounds for the spectrum of a one-dimensional Schrödinger operator, in General Inequalities 6
(Edited by W. WALTER), International Series of Numerical Mathematics,
vol. 103, Birkhäuser Verlag, Basel, 6th International Conference on General Inequalities, December 9-15, 1990, Oberwolfach 1992, pp. 257–268.
[4] J.C.P. BUS AND T.J. DEKKER, Two efficient algorithms with guaranteed
convergence for finding a zero of a function, ACM Trans. Math. Software,
1(4) (1975), 330–345.
[5] C.B. CLEMONS AND C.K.R.T. JONES, A geometric proof of the KwongMcLeod uniqueness result, SIAM J. Math. Anal., 24(2) (1993), 436–443.
[6] M.S.P. EASTHAM AND H. KALF, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, vol. 65, Pitman, London
1982.
[7] D.E. EDMUNDS AND W.D. EVANS, Spectral Theory and Differential
Operators, Oxford Mathematical Monographs, Oxford University Press,
Oxford 1990.
[8] W.N. EVERITT, On the spectrum of a second order linear differential
equation with a p-integrable coefficient, Appl. Anal., 2 (1972), 143–160.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 33 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
[9] J. GUNSON, Inequalities in Mathematical Physics, in Inequalities, Fifty
Years On from Hardy, Littlewood and Pólya (Edited by W. N ORRIE
E VERITT), Lecture Notes in Pure and Applied Mathematics Series, no.
129, London Mathematical Society, Marcel Dekker, Inc., New York,
Basel, Hong Kong, Proceedings of the International Conference, July 1317, 1987, University of Birmingham, U.K. 1991, pp. 53–79.
[10] B.J. HARRIS, Lower bounds for the spectrum of a second order linear
differential equation with a coefficient whose negative part is p-integrable,
Proc. Roy. Soc. Edinburgh Sect. A, 97 (1984), 105–107.
[11] MAN KAM KWONG, Uniqueness of positive solutions of
∆u − u + up = 0 in RN , Arch. Ration. Mech. Anal., 105(3) (1989),
243–266.
[12] MAN KAM KWONG AND LIQUN ZHANG, Uniqueness of the positive
solution of ∆u + f (u) = 0 in an annulus, Differential Integral Equations,
4(3) (1991), 583–599.
[13] H.A. LEVINE, An estimate for the best constant in a Sobolev inequality
involving three integral norms, Ann. Mat. Pura Appl. (4), 124 (1980), 181–
197.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
[14] E.H. LIEB, Sharp constants in the Hardy-Littlewood-Sobolev and related
inequalities, Ann. of Math., 118(2) (1983), 349–374.
Close
[15] E.H. LIEB AND W.E. THIRRING, Inequalities for the moments of the
eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev
Page 34 of 49
Quit
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
inequalities, in Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann (Edited by E. H. L IEB , B. S IMON AND A. S. W IGHTMAN ), Princeton University Press, Princeton 1976, pp. 269–303.
[16] E.H. LIEB, Bounds on Schrödinger Operators and Generalized SobolevType Inequalities with Applications in Mathematics and Physics, in Inequalities, Fifty Years On from Hardy, Littlewood and Pólya (Edited by
W. N ORRIE E VERITT), Lecture Notes in Pure and Applied Mathematics
Series, no. 129, London Mathematical Society, Marcel Dekker, Inc., New
York, Basel, Hong Kong, Proceedings of the International Conference,
July 13-17, 1987, University of Birmingham, U.K. 1991, pp. 123–133.
[17] K. McLEOD AND J. SERRIN, Uniqueness of solutions of semilinear Poisson equations, Proc. Nat. Acad. Sci. USA, 78(11) (1981), 6592–6595.
[18] K. McLEOD AND J. SERRIN, Uniqueness of positive radial solutions of
∆u + f (u) = 0 in RN , Arch. Ration. Mech. Anal., 99 (1987), 115–145.
[19] B. v. Sz. NAGY, Űber Integralungleichungen zwischer einer Funktion und
ihrer Ableitung, Acta Sci. Math. (Szeged), 10 (1941), 64–74.
[20] Sh. M. NASIBOV, On optimal constants in some Sobolev inequalities and
their application to a nonlinear Schrödinger equation, Soviet. Math. Dokl.,
40(1) (1990), 110–115, translation of Dokl. Akad. Nauk SSSR 307(3)
(1989), 538-542.
[21] M. REED AND B. SIMON, Methods of Modern Mathematical Physics II:
Fourier Analysis, Self-Adjointness, Academic Press, New York 1975.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 35 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
[22] M. REED AND B. SIMON, Methods of Modern Mathematical Physics I:
Functional Analysis, Academic Press, New York 1980, Revised and enlarged edn.
[23] G. ROSEN, Necessary conditions on potential functions for nonrelativistic
bound states, Phys. Rev. Lett., 49 (1982), 1885–1887.
[24] W.A. STRAUSS, Existence of solitary waves in higher dimensions,
Comm. Math. Phys., 55 (1977), 149–162.
[25] G. TALENTI, Best constant in Sobolev inequality, Ann. Mat. Pura Appl.,
110(4) (1976), 353–372.
[26] W. THIRRING, A Course in Mathematical Physics III. Quantum Mechanics of Atoms and Molecules, Springer, New York 1981.
[27] E.J.M. VELING, Optimal lower bounds for the spectrum of a second order
linear differential equation with a p-integrable coefficient, Proc. Roy. Soc.
Edinburgh Sect. A, 92 (1982), 95–101.
[28] E.J.M. VELING, Transport by Diffusion, Ph.D. thesis, University of Leiden, Leiden, The Netherlands (1983).
[29] M.I. WEINSTEIN, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567–576.
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 36 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
A.
Figures
1.8
FUNCTIONAL
1.6
1.4
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
b
1.2
a
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
e: Nasibov-2
a>d
e
c
1.0
e
E.J.M. Veling
Title Page
b=c
Contents
d
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 2, THETA -->
0.7
0.8
0.9
1
Figure 1: N = 2: λ2,θ with four approximations; Approximation-1 corresponds
with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii), Nasibov-1 with
(1.43), Nasibov-2 with (1.46).
JJ
J
II
I
Go Back
Close
Quit
Page 37 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.8
1.7
a
e
FUNCTIONAL
1.6
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
b=c
1.5
c
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
e: Nasibov-2
1.4
b
E.J.M. Veling
d
Title Page
1.3
1.2
0.50
Contents
0.55
0.60
0.65
0.70
N = 2, THETA -->
0.75
0.80
0.85
Figure 2: N = 2: λ2,θ with four approximations; Approximation-1 corresponds
with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii), Nasibov-1 with
(1.43), Nasibov-2 with (1.46).
JJ
J
II
I
Go Back
Close
Quit
Page 38 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
2.6
a
2.4
b=c
2.2
e
FUNCTIONAL
2.0
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
1.8
1.6
b
d
E.J.M. Veling
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
e: Nasibov-2
1.4
1.2
Title Page
Contents
1.0
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 3, THETA -->
0.7
0.8
0.9
1
Figure 3: N = 3: λ3,θ with four approximations; Approximation-1 corresponds
with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii), Nasibov-1 with
(1.43), Nasibov-2 with (1.46).
JJ
J
II
I
Go Back
Close
Quit
Page 39 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
2.6
2.5
a
2.4
b=c
2.3
2.2
FUNCTIONAL
2.1
2.0
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
e
1.9
1.8
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
e: Nasibov-2
1.7
1.6
b
1.5
E.J.M. Veling
d
Title Page
1.4
Contents
1.3
1.2
0.50
0.55
0.60
0.65
0.70
0.75
N = 3, THETA -->
0.80
0.85
0.90
0.95
Figure 4: N = 3: λ3,θ with four approximations; Approximation-1 corresponds
with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii), Nasibov-1 with
(1.43), Nasibov-2 with (1.46).
JJ
J
II
I
Go Back
Close
Quit
Page 40 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
3.4
a
3.2
3.0
b=c
2.8
FUNCTIONAL
2.6
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
2.4
d
2.2
2.0
E.J.M. Veling
b
1.8
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
1.6
1.4
Title Page
Contents
1.2
1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 4, THETA -->
0.7
0.8
0.9
1
Figure 5: N = 4: λ4,θ with three approximations; Approximation-1 corresponds with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii),
Nasibov-1 with (1.43).
JJ
J
II
I
Go Back
Close
Quit
Page 41 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
4.0
a
3.6
b=c
3.2
d
FUNCTIONAL
2.8
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
2.4
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
1.6
1.2
0.8
E.J.M. Veling
b
2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 5, THETA -->
0.7
0.8
Title Page
Contents
0.9
1
Figure 6: N = 5: λ5,θ with three approximations; Approximation-1 corresponds with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii),
Nasibov-1 with (1.43).
JJ
J
II
I
Go Back
Close
Quit
Page 42 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
6.4
6.0
a>b=c
5.6
5.2
4.8
d
FUNCTIONAL
4.4
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
4.0
3.6
3.2
E.J.M. Veling
2.8
b
2.4
2.0
1.6
a: Functional
b: Approximation-1
c: Approximation-2
d: Nasibov-1
Title Page
Contents
1.2
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 10, THETA -->
0.7
0.8
0.9
1
Figure 7: N = 10: λ10,θ with three approximations; Approximation-1
corresponds with Theorem 1.2-(ii), Approximation-2 with Theorem 1.2-(iii),
Nasibov-1 with (1.43).
JJ
J
II
I
Go Back
Close
Quit
Page 43 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.00
c
0.95
RATIO
0.90
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
0.85
E.J.M. Veling
0.80
a: Ratio Approximation-2 with Functional
b: Ratio Nasibov-1 with Functional
c: Ratio Nasibov-2 with Functional
a
Title Page
0.75
Contents
b
0.70
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 2, THETA -->
0.7
0.8
0.9
1
JJ
J
II
I
Go Back
Figure 8: N = 2: Ratio of three approximations with λ2,θ : Approximation-2
(Theorem 1.2-(iii)), Nasibov-1 (1.43), and Nasibov-2 (1.46).
Close
Quit
Page 44 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.00
a
0.95
0.90
0.85
0.80
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
c
RATIO
0.75
0.70
E.J.M. Veling
0.65
0.60
b
a: Ratio Approximation-2 with Functional
b: Ratio Nasibov-1 with Functional
c: Ratio Nasibov-2 with Functional
0.55
Title Page
0.50
Contents
0.45
0.40
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 3, THETA -->
0.7
0.8
0.9
1
JJ
J
II
I
Go Back
Figure 9: N = 3: Ratio of three approximations with λ3,θ : Approximation-2
(Theorem 1.2-(iii)), Nasibov-1 (1.43), and Nasibov-2 (1.46).
Close
Quit
Page 45 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.00
a
0.95
0.90
0.85
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
RATIO
0.80
0.75
0.70
E.J.M. Veling
a: Ratio Approximation-2 with Functional
b: Ratio Nasibov-1 with Functional
0.65
Title Page
0.60
0.50
Contents
b
0.55
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 4, THETA -->
0.7
0.8
0.9
1
JJ
J
II
I
Go Back
Figure 10: N = 4: Ratio of two approximations with λ4,θ : Approximation-2
(Theorem 1.2-(iii)) and Nasibov-1 (1.43).
Close
Quit
Page 46 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.00
a
0.95
0.90
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
RATIO
0.85
0.80
a: Ratio Approximation-2 with Functional
b: Ratio Nasibov-1 with Functional
E.J.M. Veling
0.75
Title Page
0.70
Contents
0.65
0.60
b
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 5, THETA -->
0.7
0.8
0.9
1
JJ
J
II
I
Go Back
Figure 11: N = 5: Ratio of two approximations with λ5,θ : Approximation-2
(Theorem 1.2-(iii)) and Nasibov-1 (1.43).
Close
Quit
Page 47 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.00
a
RATIO
0.95
0.90
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
a: Ratio Approximation-2 with Functional
b: Ratio Nasibov-1 with Functional
E.J.M. Veling
Title Page
0.85
Contents
b
0.80
0
0.1
0.2
0.3
0.4
0.5
0.6
N = 10, THETA -->
0.7
0.8
0.9
1
JJ
J
II
I
Go Back
Figure 12: N = 10: Ratio of two approximations with λ10,θ : Approximation-2
(Theorem 1.2-(iii)) and Nasibov-1 (1.43).
Close
Quit
Page 48 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au
1.E+6
1.E+5
N=2
N=3
N=4
N=5
v(0)
1.E+4
Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities
N = 10
1.E+3
E.J.M. Veling
N = 10
1.E+2
Title Page
N=5
1.E+1
Contents
N=2
1.E+0
0
0.1
0.2
0.3
0.4
0.5
THETA -->
0.6
0.7
0.8
0.9
1
JJ
J
II
I
Go Back
Figure 13: The value v(0) of the minimizer v(r) of the functional ΛN,θ as function of θ for N = 2, 3, 4, 5, 10.
Close
Quit
Page 49 of 49
J. Ineq. Pure and Appl. Math. 3(4) Art. 63, 2002
http://jipam.vu.edu.au