Electronic Journal of Differential Equations, Vol. 2010(2010), No. 133, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SPECTRAL CONCENTRATION IN STURM-LIOUVILLE
EQUATIONS WITH LARGE NEGATIVE POTENTIAL
BERNARD J. HARRIS, JEFFREY C. KALLENBACH
Abstract. We consider the spectral function, ρα (λ), associated with the linear second-order question
y 00 + (λ − q(x))y = 0
in [0, ∞)
and the initial condition
y(0) cos(α) + y 0 (0) sin(α) = 0,
α ∈ [0, π).
in the case where q(x) → −∞ as x → ∞. We obtain a representation of ρ0 (λ)
as a convergent series for λ > Λ0 where Λ0 is computable, and a bound for
the points of spectral concentration.
1. Introduction
We consider the linear differential equation
y 00 + (λ − q(x))y = 0
(1.1)
on the interval [0, ∞) where the potential, q, is a real-valued function of C 3 [0, ∞)
and q(x) → −∞ as x → ∞. When augmented with the boundary condition
y(0) cos(α) + y 0 (0) sin(α) = 0
α ∈ [0, π)
(1.2)
2
Equation (1.1) leads to a self-adjoint operator on the Hilbert space L [0, ∞)
and an associated spectral function ρα (λ). The function ρα (λ), in particular ρ0 (λ),
is our primary concern here. For a detailed account of its definition we refer to
[1, 3, 8].
It is known that if q satisfies
Z ∞
Z ∞
Z ∞
0 2
−5/2
00
−3/2
(q ) |q|
dt < ∞,
|q | |q|
dt < ∞,
|q|−1/2 dt = ∞, (1.3)
then ρα (λ) is absolutely continuous on (−∞, ∞). This condition is fulfilled, for
example, when q(x) = −xc where 0 < c ≤ 2. In this article, we derive an expression,
in the form of a uniformly absolutely convergent series, for ρ00 (λ) in the case where
λ is positive and sufficiently large. Our results hold under conditions that are
somewhat more restrictive than those of (1.3). In particular, if q(x) = −xc , they
hold in the case 0 < c ≤ 1.
2000 Mathematics Subject Classification. 34L05, 34L20.
Key words and phrases. Spectral theory; Schrodinger equation.
c
2010
Texas State University - San Marcos.
Submitted August 27, 2009. Published September 14, 2010.
1
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B. J. HARRIS, J. C. KALLENBACH
EJDE-2010/133
Representations of ρ00 (λ) have been obtained before, notably in [1] in the case
when q is m-times differentiable, but these have been asymptotic results whereas
ours hold for all λ greater than some Λ0 which is, in principle, computable.
A secondary goal of this article is to establish bounds for the points of spectral
concentration of (1.1), (1.3). For a discussion of spectral concentration in general
we refer to [5], but a point of spectral concentration may broadly be defined as a
value of λ which is a local maximum of ρ0α (λ) and is thus a point at which ρα (λ)
is increasing relatively rapidly. Specifically we show the existence of a Λ1 ≥ Λ0 for
which ρ000 (λ) is of one sign.
In our analysis we suppose that the parameter λ is positive and that λ−q(x) > 0
for all x ∈ [0, ∞) and Λ0 ≤ λ. This can clearly be done if q(x) is bounded above.
Our choice of Λ0 will be increased as necessary throughout the paper.
Our main result concerning spectral concentration is the following.
Theorem 1.1. If q ∈ C 3 [0, ∞) satisfies
q(x) → −∞ as x → ∞
q(x) < 0 for all x ∈ [0, ∞)
q 0 (x) < 0, q 00 (x) ≥ 0, q 000 (x) ≤ 0 for all x ∈ [0, ∞)
5
3
q 00 /|q| 2 −ε and (q 0 )2 /|q| 2 −ε ∈ L1 [0, ∞) for some ε > 0
R∞
R
00
∞ |q |
(q 0 )2
|q(s)|−1/2 s |q|
3/2 + |q|5/2 dt ds < ∞
0
supx∈[0,∞) |q 0 (x)|λ − q(x)|2 → 0 as λ → ∞.
R ∞ q00 (t)
R ∞ (q0 (t))2
(vii) 0 (λ−q(t))
dt are o(1) as λ → ∞
2 dt and
0 (λ−q(t))3
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Then there exists Λ1 such that ρ0 (λ) has no points of spectral concentration in
[Λ1 , ∞).
We note that by writing λ = (λ − λ0 ) + λ0 in (1.1) condition (ii) effectively
requires that q be bounded above.
Our principal tool, as in [2, 4, 6] is the connection between (1.1) and the Riccati
equation
v 0 + v 2 + (λ − q) = 0.
(1.4)
Let v(x, λ) be the unique complex-valued solution of (1.4) which exists for all
x ∈ [0, ∞). For α = 0 and ξ ∈ C+ , v(x, ξ) is the logarithmic derivative with
respect to x of the Weyl solution u(x, ξ) of y 00 + (ξ − q(x))y = 0. That is:
v(x, ξ) = u0 (x, ξ)/u(x, ξ).
It follows that v(0, ξ) = m(ξ, 0) where m(ξ, 0) is the Dirichlet Titchmarsh-Weyl
m-function. For the class of potentials considered, the solution v(x, ξ) of (1.4) is
continuously extendable onto the real λ-axis as ξ = λ + iε ↓ λ. It then follows that
ρ00 (λ) =
1
Im{v(0, λ)}.
π
(1.5)
Our strategy then is to identify a suitable solution of (1.4) which is complexvalued for λ real and suitably large. Consequently we have from (1.5) that
ρ000 (λ) =
1 ∂
{Im v(0, λ)}.
π ∂λ
(1.6)
EJDE-2010/133
SPECTRAL CONCENTRATION
3
2. Preliminaries
To derive our main result it is convenient to show that the conditions imposed
on the potential, q, imply the existence of a function I(x, λ) which satisfies the
conclusion of the following lemma.
Lemma 2.1. If q(x) satisfies (i)–(iv), (vi) of Theorem 1.1, then there exists a
real-valued function I(x, λ) such that I(x, λ) > 0 for x ∈ [0, ∞), λ > 0 and
(i) I(·, λ) ∈ L1 [0, ∞)
I(x,λ)
(ii) (λ−q(x))
1/2 is a decreasing function of x for each λ > 0
R∞
(iii) 0 I(x, λ)dx → 0 as λ → ∞.
R ∞ 2i R t (λ−q(s)) ds q00 (t)
5(q 0 (t)2
(iv) (λ − q(x))1/2 e x
dt ≤ I(x, λ).
3/2 +
5/2
x
4(λ−q(t)
16(λ−q(t)
Proof. We set
I(x, λ) :=
q 00 (x)
5(q 0 (x))2
+
3/2
2(λ − q(x))
8(λ − q(x))5/2
and show that this choice of I(x, λ) satisfies (i)–(iv) if q satisfies the conditions of
Theorem 1.1. Part (i) follows from Theorem 1.1 (iv).
Differentiation with respect to x of (λ − q(x))−1/2 I(x, y) and Theorem 1.1(iii)
shows each of the terms is decreasing (in
R ∞x) for each λ which establishes (ii).
To see (iii) we rewrite the terms of 0 I(x, λ) dx as
Z
Z
1 ∞
q 00 (x)
1 −ε ∞ q 00 (x)
λ
dx
≤
dx.
2 0 (λ − q(x))ε (λ − q(x))3/2−ε
2
|q(x)|3/2−ε
0
The other terms in the sum is treated similarly.
5(q 0 )2
q 00
To prove (iv) we note that 4(λ−q)
2 + 16(λ−q)3 is decreasing so, by the Second
Mean Value Theorem,
Z
∞ 2i R t (λ−q(s))1/2 ds q 00
5(q 0 )2
1/2 (λ − q(x))
e x
+
dx
3/2
5/2
4(λ
−
q)
16(λ
−
q)
x
Z ∞
Z t
1/2 1 1/2
= (λ − q(x))
2(λ − q(t)) cis 2
(λ − q(s))1/2 ds
2 x
x
q 00
5(q 0 )2 ×
+
dt
4(λ − q)2
16(λ − q)3
1
q 00 (x)
5
(q 0 (x))2 =
+
2 4(λ − q(x))3/2
16 (λ − q(x))5/2
Z t
Z ∞
×
2(λ − q)1/2 cos(2
(λ − q(s))1/2 ds dt
ξ1
∞
Z
+i
x
2(λ − q)1/2 sin 2
ξ2
≤2
Z
t
x
1/2
(λ − q(s)
ds dt
5
q 0 (x))2
q 00 (x)
+
= I(x, λ).
3/2
5/2
16 (λ − q(x))
4(λ − q(x))
The proof is complete.
To obtain the required complex-valued solution of the Riccati equation (1.4), we
proceed as in [2, 4]. Based on the asymptotic representation established in [6], we
4
B. J. HARRIS, J. C. KALLENBACH
EJDE-2010/133
seek a solution in the form of
∞
X
1
v(x, λ) = i(λ − q(x))1/2 + q 0 (x)(λ − q(x))−1 +
vn (x, λ).
4
n=1
(2.1)
Substitution of (2.1) into (1.4) gives
∞ X
1
vn0 + 2 i(λ − q)1/2 + q 0 (λ − 2)−1 vn
4
n=1
= −Q − v12 −
∞ X
2
vn−1
+ 2vn−1
n=3
where
Q :=
n−2
X
vm ,
m=1
5(q 0 )2
q 00
+
.
4(λ − q) 16(λ − q)2
We choose v1 , v2 . . . so that
v10 + 2i(λ − q)1/2 +
q0
v1 = −Q
2(λ − q)
q0
v2 = −v12
v20 + 2i(λ − q)1/2 +
2(λ − q)
n−2
X
q0
2
vn0 + 2i(λ − q)1/2 +
vn = −vn−1
− 2vn−1
vm
2(λ − q)
m=1
for n = 3, 4, . . . . The required solution to (2.1) is
Z ∞
Rt
1/2
1/2
v1 (x, λ) = (λ − q(x))
(λ − q(t))−1/2 e2i x (λ−q) ds Q(t, λ)dt
Z x∞
Rt
1/2
1/2
v2 (x, λ) = (λ − q(x))
(λ − q(t))−1/2 e2i x (λ−q) ds v1 (t, λ)2 dt
x
Z ∞
Rt
1/2
vn (x, λ) = (λ − q(x))1/2
(λ − q(t))−1/2 e2i x (λ−q) ds
2
× vn−1
+2
x
n−2
X
(2.2)
(2.3)
vm vn−1 dt
m=1
for n = 3, 4, . . . .
Lemma 2.2. If Λ0 is so large that for all λ ≥ Λ0 ,
Z ∞
9
I(t, λ) dt ≤ 1
0
then for n = 1, 2, 3, . . . ,
|vn (x, λ)| ≤ I(x, λ)/2n−1
for all x ∈ [0, ∞).
Proof. We use induction on n. When n = 1 this is Lemma 2.1 (iv). For n = 2,
Z ∞
|v2 (x, λ)| ≤ (λ − q(x))1/2
(λ − q(t))−1/2 I(t, λ)2 dt
x
Z ∞
≤ I(x, λ)
I(t, λ) dt,
0
EJDE-2010/133
SPECTRAL CONCENTRATION
5
by Lemma 2.1 (ii). If n ≥ 3 then, by the induction hypothesis:
Z ∞
∞
h I(t, λ)2 X
1 i
1/2
(λ − q(t))−1/2
|vn (x, λ)| ≤ (λ − q(x))
dt
2n−2 m−1 2m−1
x
Z ∞
I(t, λ) 1
1/2
(λ − q(t))−1/2 n−1
≤ (λ − q(x))
+ 8 I(t, λ) dt
n−2
2
2
Z ∞x
I(x, λ)
≤ n−1 · 9
I(t, λ) dt
2
0
and the result follows.
P∞
The uniform, absolute convergence P
of n=1 vn (x, λ) follows from Lemma 2.2.
∞
The uniform absolute convergence of n=1 vn0 (x, λ) which justifies the term differentiation used to derive the series solution, follows from the bound for the vn
obtained in Lemma 2.2 and the representation of the vn0 in (2.2). Since, for example,
q 0 (x)
|vn (x, λ)|
|vn0 (x, λ)| ≤ |2(λ − q(x)1/2 | + 2(λ − q(x))
+ |vn−1 (x, λ)|2 + 2|vn−1 |
n−2
X
|vm |.
m=1
q0
I(x, λ) · 1
2(λ − q)
2n−1
∞
I(x, λ) 2
I(x)2 X 1
for n = 3, 4, . . .
+
+ 2 n−2
n−2
2
2
2m−1
m=1
P
It follows readily that |vn0 (x, λ)| is uniformly absolutely convergent for x ∈ [0, ∞)
and λ > Λ0 . We have proved the following result.
≤ 2|λ − q|1/2 + Theorem 2.3. Let qR satisfy the conditions of Theorem 1.1. If Λ0 is so large that
∞
for all λ ≥ Λ0 > 0, 9 0 I(t, λ) dt ≤ 1 and (λ − q(x) > 0 for all x ∈ [0, ∞) then
ρ00 (λ) =
∞
1X
1
(λ − q(0))1/2 +
Im(vn (0, λ)).
π
π n=1
for all λ > Λ0 .
3. Spectral Concentration
We seek the second derivative of ρ0 (λ). Our strategy is to differentiate the
equations of (2.2) with respect to λ, justify the equality of the mixed second order
n
partial derivatives and derive expressions for ∂v
∂λ akin to (2.3) which we then bound
as in Lemma 2.2.
Differentiating the first equation of (2.2) with respect to λ gives
∂v
∂ 2 v1 q0
∂Q
1
1
+ 2i(λ−q)1/2 +
=−
−i(λ−q)−1/2 v1 + q 0 (λ−q)−2 v1 (3.1)
∂λ∂x
2(λ − q) ∂λ
∂λ
2
1
We note from (2.3) that v1 (x, λ) is continuous and so, by (2.2), is ∂v
∂x . It remains
∂v1
to show that ∂λ is continuous. We do this by differentiating the first equation of
6
B. J. HARRIS, J. C. KALLENBACH
EJDE-2010/133
(2.3) under the integral sign to obtain
Z ∞
Rt
1/2
∂v1
5(q 0 )2
1
q 00
+
dt
= (λ − q(x))−1/2
e2i x (λ−q(s)) ds
3/2
5/2
∂λ
2
4(λ − q)
16(λ − q)
x
Z ∞ Z t
Rt
1/2
2i
(λ − q(s))−1/2 ds e2i x (λ−q(s)) ds
+ (λ − q(x))1/2
x
x
q 00
5(q 0 )2
×
+
dt
4(λ − q)3/2
16(λ − q)5/2
Z ∞
Rt
1/2
e2i x (λ−q(s)) ds
+ (λ − q(x))1/2
x
×
3q 00
25
−
(λ − q)−5/2 − (q 0 )2 (λ − q)−7/2 dt
8
32
(3.2)
providing that the differentiation under the integral sign is justified. To ensure
that it is, we note that under the conditions of Theorem 1.1, the integrand in the
expression for v1 (x, λ) in (2.3) is continuously differentiable with respect to λ, and
that each term of the integrand in (3.2) is integrable with respect to t ∈ R+ ; to see
this in the case of the second term, note that by a change in the order of integration
Z
Z
Rt
∞ t
1/2
5(q 0 )2
q 00
+
dt
(λ − q(s))−1/2 ds e2i x (λ−q(s)) ds
3/2
5/2
4(λ − q)
16(λ − q)
x
x
Z ∞
Z ∞
5(q 0 )2
q 00
≤
+
dt ds.
(λ − q(s))−1/2
4(λ − q)3/2
16(λ − q)5/2
x
s
1
It now follows from (3.2) that ∂v
∂λ is continuous in x and λ, so the equality of the
∂ 2 v1
∂ 2 v1
mixed partial derivatives is established. We may therefore replace ∂λ∂x
by ∂x∂λ
in
(3.1), then integrate with respect to x to obtain a more suitable representation of
∂v1
∂λ . This yields
Z ∞
Rt
q 00
1/2
∂v1
(x, λ) = (λ − q(x))1/2
(λ − q)−1/2 e2i x (λ−q(s)) ds − (λ − q)−2
∂λ
4
x
0
q
− 5/8(q 0 )2 (λ − q)−3 + [−i(λ − q)−1/2 v1 ] + [ (λ − q)−2 v1 ] dt
2
=: I1 + I2 + I3
(3.3)
This provides
a
convenient
first
step
for
an
iterative
scheme
to
establish
upper
n
bounds on ∂v
∂λ for x ≥ 0 and λ sufficiently large. To this end we note that
q 00
(λ − q)−2 + 5/8(q 0 )2 (λ − q)−3 ≤ (λ − q)−1/2 I(x, λ) so
4
Z ∞
|I1 | ≤
sup (λ − q(x))−1/2 (λ − q(x))1/2
(λ − q(t))−1/2 I(t, λ)dt
x∈[0,∞)
x
Also,
Z
∞
(λ − q(t))−1 I(t, λ) dt
x
Z
≤
sup (λ − q(x))−1/2 (λ − q(x))1/2
|I2 | ≤ (λ − q(x))1/2
x∈[0,∞)
∞
x
(λ − q(t))−1/2 I(t, λ) dt
EJDE-2010/133
SPECTRAL CONCENTRATION
7
and
1
|I3 | ≤
sup |q 00 (x)|(λ − q(x))−2 (λ − q(x))1/2
2 x∈[0,∞)
Z
∞
(λ − q(t))−1/2 I(t, λ)dt.
x
It follows that
∂v1
1
(x, λ) ≤ 2 sup (λ − q(x))−1/2 +
sup |q 0 (x)(λ − q(x))−2 |
∂λ
2 x∈[0,∞)
x∈[0,∞)
Z ∞
(λ − q(t))−1/2 I(t, λ) dt.
× (λ − q(x))1/2
(3.4)
x
Lemma 3.1. If Λ1 ≥ Λ0 > 0 is so large that for all λ ≥ Λ1 ,
Z ∞
1
16
I(t, λ)dt + 2 sup (λ − q(x))−1/2 +
sup |q 0 (x)(λ − q(x))−2 | ≤ 1,
2 x∈[0,∞
x∈[0,∞)
0
then for x ∈ [0, ∞), λ > Λ, and n = 1, 2, 3, . . . ,
Z ∞
∂vn
1
(x, λ) ≤ n−1 (λ − q(x))1/2
(λ − q(t))−1/2 I(t, λ) dt .
∂λ
2
x
(3.5)
n
Proof. We use induction on n to prove the hypothesis: ∂v
∂λ (x, λ) is continuous in
x and λ, for x ∈ [0, ∞), λ > Λ and inequality (3.4) holds.
The case n = 1 follows from (3.4) since the hypothesis of the lemma implies the
asserted bound. The case n = 2 will follow from the general case, the difference
being that some of the series terms are vacuous.
∂vn−1
1
In the general case, suppose the induction hypothesis holds for ∂v
∂λ , . . . , ∂λ .
∂v1
As in the case for ∂λ , we differentiate (2.2) with respect to λ, show the equality
n
of the mixed second order derivatives and obtain an integral representation for ∂v
∂λ
which we bound.
n
The function ∂v
∂λ is continuous from (2.2) and, differentiating (2.2) with respect
∂ 2 vn
n
to λ shows that ∂λ∂x is continuous if ∂v
∂λ is. This we now show by differentiating
(2.3) with respect to λ under the integral.
Z ∞
Rt
1/2
1
1
∂vn
−1
1/2
= (λ − q(x)) vn − (λ − q(x))
(λ − q)−3/2 eei x (λ−q) ds
∂λ
2
2
x
n−2
X
2
× vn−1
+2
vm vn−1 dt
m=1
1/2
Z
∞
−1/2
+ (λ − q(x))
(λ − q)
i
x
2
× vn−1
+2
n−2
X
t
Z
Rt
1/2
(λ − q(s))−1/2 ds e2i x (λ−q) ds
x
vm vn−1 dt
m=1
+ (λ − q(x))1/2
Z
∞
(x − q)−1/2 e2i
Rt
x
(λ−q)1/2 ds
x
n−2
X
∂ 2
×
vn−1 + 2
vm vn−1 dt.
∂λ
m=1
(3.6)
8
B. J. HARRIS, J. C. KALLENBACH
EJDE-2010/133
The continuity of all but the third term is clear. This consists of a sum of terms
which are
Z
Z ∞
t
−1/2
1/2
(λ − q(s))−1/2 ds I(t, λ)2 dt
(λ − q(t))
O (λ − q(x))
x
x
Z ∞
Z ∞
−1/2
1/2
(λ − q(t))−1/2 I(t, λ)2 dt ds
(λ − q(s))
= O (λ − q(x))
s
Zx∞
Z ∞
1/2
−1
(λ − q(s)) I(s, λ)
= O (λ − q(x))
I(t, λ) dt ds .
x
0
By Lemma 2.1 (i) and (ii), the continuity of the third term follows.
By the induction hypothesis the fourth term consists of a sum of terms each of
which is
Z ∞
Z ∞
O (λ − q(x))1/2
I(t, λ)
(λ − q(s))−1/2 I(s, λ) ds dt
x
t
and so is bounded by Lemma 2.1 (i).
∂ 2 vn
n
The continuity of ∂v
∂λ , and hence of ∂λ∂x now follows and, by the equality of the
second order mixed partial derivatives, we have from (2.2):
Z ∞
Rt
1/2
∂vn
= (λ − q(x))1/2
e2i x (λ−q(s)) ds (λ − q(t))−1/2
∂λ
x
n−1
n−2
n
X ∂vn−1
X
q0
∂vm o
× i(λ − q)−1/2 vn −
v
+
2
v
+
2
v
dt
n
m
n−1
2(λ − q)2
∂λ
∂λ
m=1
m=1
=: I1 + · · · + I4
(3.7)
From Lemma 2.2,
1
Z
∞
(λ − q(x))
(λ − q(t))−1 I(t, λ) dt
2n−1
x
Z ∞
1 −1/2
1/2
≤ n−1
sup (λ − q(x))
(λ − q(x))
(λ − q(t))−1/2 I(t, λ) dt.
2
0≤x<∞
x
|I1 | ≤
1/2
∞
q 0 (t)
−1/2 I(t, λ) dt
(λ
−
q(x))
(λ
−
q(t))
2n−1
2(λ − q(t))2
x
Z ∞
1
q 0 (x)
(λ − q(x))1/2
(λ − q(t))−1/2 I(t, λ) dt.
≤ n−1 sup 2
2
2(λ
−
q(x))
0≤x<∞
x
|I2 | ≤
1
|I3 | ≤ 2(λ − q(x))1/2
1/2
Z
∞
Z
Z
x
∞
t
n−1
X 1
I(s, λ) ds
I(t,
λ)
dt
n−2
2
2m−1
m=1
Z ∞
1/2
(λ
−
q(x))
8
I(t,
λ)
(λ − q(s))−1/2 I(s, λ) ds dt
2n−1
x
t
Z ∞
Z s
8
(λ − q(s))−1/2 I(s, λ)
I(t, λ) dt ds
= n−1 (λ − q(x))1/2
2
x
Zx∞
Z ∞
1
1/2
−1/2
≤ n−1 (λ − q(x))
(λ − q(s))
I(s, λ) ds 8
I(t, λ) dt .
2
x
0
≤
1
Z
∞
(λ − q(s))−1/2
EJDE-2010/133
SPECTRAL CONCENTRATION
9
Z
n−2
X 1
I(t, λ) ∞
−1/2
(λ
−
q(s))
I(s,
λ)
ds dt
n−2
2
2m−1
x
t
m=1
Z ∞
Z ∞
8
1/2
≤ n−1 (λ − q(x))
I(t, λ)
(λ − q(s))−1/2 I(s, λ) ds dt
2
x
t
Z ∞
Z ∞
1
(λ − q(s))−1/2 I(s, λ) ds 8
I(t, λ) dt .
≤ n−1 (λ − q(x))1/2
2
x
0
The result now follows since for λ ≥ Λ1 ,
Z ∞
q 0 (x)
+ sup |λ − q(x)|−1/2 ≤ 1.
16
I(t, λ) dt + sup 2
2(λ
−
q(x))
0≤x<∞
0≤x<∞
0
|I4 | ≤ 2(λ − q(x))1/2
Z
∞
4. Proof of Theorem 1.1
If q satisfies the conditions of Theorem 1.1 then there exists a function I(x, λ)
satisfying the conclusions of Lemma 2.1 and hence Lemmas 2.2 and 3.1. Thus, for
λ > Λ1 the following representation of ρ00 (λ) holds
ρ00 (λ) =
∞
1
1X ∂
(λ − q(0))−1/2 +
Im(vn (0, λ))
2π
π n=1 ∂λ
and
∞
X
00
∂
ρ (λ) − 1 (λ − q(0))−1/2 ≤ 1
vn (0, λ)
2π
π n=1 ∂λ
Z ∞
2
1/2
≤ (λ − q(0))
(λ − q(t))−1/2 I(t, λ) dt.
π
0
Thus ρ00 (λ) > 0 if λ is so large that
Z ∞
(λ − q(t))−1/2 I(t, λ) dt < 1.
4(λ − q(0))
0
With the function I(t, λ) from Lemma 2.1 this is satisfied if, in addition to the
requirements of Lemmas 2.2 and 3.1, λ is so large that
Z ∞
5(q 0 (t))2
q 00 (t)
2(λ − q(0))
+
dt < 1.
(4.1)
(λ − q(t))2
4(λ − q(t))3
0
Acknowledgments. Some material in the paper is part of the dissertation submitted by the second author in partial fulfillment of the requirements for the Ph.
D. at Northern Illinois University citek1. Both authors want to express their gratitude to Dr. D. J. Gilbert at Dublin Institute of Technology for her careful reading
of the dissertation and helpful suggestions. The authors are also grateful to the
anonymous referee for his/her helpful comments.
References
[1] M. S. P. Eastham; The asymptotic form of the spectral function in Sturm-Liouville problems
with a large potential like −xc (c ≤ 2). Proc. Royal Soc. Edinburgh 128A (1998), 37-45.
[2] D. J. Gilbert, B. J. Harris; Bounds for the points of spectral concentration of Sturm-Liouville
problems. Mathematika, 47 (2000), 327-337.
[3] D. J. Gilbert, B. J. Harris; Connection formulae for spectral functions associated with singular
Sturm-Liouville expressions. Proc. Royal Soc. Edinburgh, 130A (2000), 25-34.
10
B. J. HARRIS, J. C. KALLENBACH
EJDE-2010/133
[4] D. J. Gilbert, B. J. Harris, S. M. Riehl; The spectral function for Sturm-Liouville equations
where the potential is of Wigner-von Neumann type or slowly decaying. Journal of Differential
Equations 201, (2004), 139-159.
[5] D. J. Gilbert, B. J. Harris, S. M. Riehl; Higher derivatives of spectral functions associated
with one-dimensional Schrödinger operators. Operator Theory: Advances and Applications.
186 (2008), 217-228.
[6] B. J. Harris; The form of the spectral functions associated with Sturm-Liouville equations
with large negative potential. J. Math Analysis and Appl. 222 (1998), 569-584.
[7] J. C. Kallenbach; Ph. D. Dissertation Northern Illinois University 2007.
[8] E. C. Titchmarsh; Eigenfunction Expansions, part 1. Clarendon Press: Oxford, 2nd ed. 1962.
Bernard J. Harris
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115,
USA
E-mail address: harris@math.niu.edu
Jeffrey C. Kallenbach
Department of Mathematical Sciences, Siena Heights University, Adrian, MI 49221, USA
E-mail address: jkallenb@siennaheights.edu