# Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

Authors: Xiaoyao Jia; Yan Zhao; Haoyu Zhai
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study the asymptotic behaviors of the discrete eigenvalue of Schrödinger operatorP
(
λ
)
=
P
0
+
λ
V with P
0
=
-
Δ
+
q
θ
/
r
2
. We obtain the leading terms of discrete eigenvalues of P
(
λ
) when the eigenvalues tend to 0. In particular, we obtain the asymptotic behaviors of eigenvalues when (
P
0
−
α
)
−
1 has singularity at α
=
0.


---

## Body

## 1. Introduction

This paper is devoted to the study of the asymptotic behaviors of the eigenvalues of a class of Schrödinger operators. This problem is related to low-energy spectral analysis for Schrödinger operators, which has been studied in many works (see [1–7]). References [1–3] are concerned with perturbation of a constant elliptic differential operator by a term decaying like O
(
|
x
|
-
2
-
ϵ
), ϵ
>
0, as |
x
|
→
+
∞. The spectral analysis of Schrödinger operators with potentials of critical decay (decaying like O
(
|
x
|
-
2
), as |
x
|
→
∞) is studied in [4–7]. The complex potentials are considered in [7].The asymptotic behaviors of the eigenvalues of Schrödinger operator have been studied by many authors. In [8], Klaus and Simon got the leading term of the eigenvalue of Schrödinger operator with fast decaying potential. The asymptotic behavior of the eigenvalue of Schrödinger operator with periodic potential has been studied by Fassari and Klaus [9]. We have studied the asymptotic behavior of the smallest eigenvalue of Schrödinger operator with potential of critical decay [6]. In this paper, we will consider the asymptotic behavior of all other eigenvalues. The main tool we used in this paper is Birman-Schwinger kernel which was originated in the seventies. But Birman-Schwinger kernel is still a very important tool for the spectrum problem of Schrödinger operators (see [10–14]).In this paper, we want to study the asymptotic behavior of discrete eigenvalue of Schrödinger operator(1)
P
(
λ
)
=
P
0
+
λ
V
,
for
λ
≥
0
.
L
2
(
ℝ
d
), d
≥
2. Here, P
0
=
-
Δ
+
q
(
θ
)
/
r
2. (
r
,
θ
) is the polar coordinate on ℝ
d, and q
(
θ
) is a real continuous function.The assumptions used in this paper are as follows. Suppose thatV
≤
0 is a nonzero real continuous function and satisfies
(2)
|
V
(
x
)
|
≤
C
〈
x
〉
-
ρ
0
,
for
some
ρ
0
>
2
.
Here, 〈
x
〉
=
(
1
+
|
x
|
2
)
1
/
2. The assumption on q
(
θ
) is that
(3)
-
Δ
s
+
q
(
θ
)
≥
-
1
4
(
d
-
2
)
2
,
on
⁡
L
2
(
𝕊
d
-
1
)
.
Here, Δ
s denotes Laplace operator on the sphere 𝕊
d
-
1. If (3) holds, then P
0
≥
0 in L
2
(
ℝ
d
) (see [5]).In Section 2 [6], it is shown that, under the assumption on V, P
(
λ
) has discrete eigenvalues when λ is large enough, and each discrete eigenvalue tends to zero at some λ
0. Set
(4)
σ
∞
=
{
ν
;
ν
=
λ
+
(
d
-
2
)
2
4
,
λ
∈
σ
(
-
Δ
s
+
q
(
θ
)
)
}
,
σ
k
=
σ
∞
∩
[
0
,
k
]
,
k
∈
ℕ
.
If 0
∉
σ
∞, then the Birman-Schwinger kernel |
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2 is a bounded operator for α
<
0 (see [6]). Since there is a one-to-one correspondence between the eigenvalues of P
(
λ
) and the eigenvalues of |
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2, the asymptotic expansion of the smallest eigenvalue of P
(
λ
) has been obtained through the asymptotic expansion of the eigenvalue of |
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2 in [6], if (2) holds with ρ
0
>
6. In this paper, we will get the asymptotic behaviors of all discrete eigenvalues of P
(
λ
) in the case of 0
∉
σ
∞. The eigenfunction corresponding to the smallest eigenvalue is a positive function; we can obtain the leading term of the smallest eigenvalue easily. The eigenfunctions corresponding to all other eigenvalues may not be positive. Therefore, it is much more difficult to obtain the leading term of the eigenvalue.If0
∈
σ
∞, (
P
0
-
α
)
-
1 has singularity at α
=
0 (see [5]). Thus, the Birman-Schwinger kernel |
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2 also has singularity at α
=
0. The other goal of this paper is to obtain the asymptotic behavior of discrete eigenvalue when 0
∈
σ
∞. The main difficulty of this situation is the singularity of |
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2. In Section 4, the discrete eigenvalue of |
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2 will be studied through the operator (
z
-
|
V
|
1
/
2
(
P
0
-
α
)
-
1
|
V
|
1
/
2
)
-
1 for some fixed z. Finally, we obtain the leading term of discrete eigenvalue of P
(
λ
), when the assumption (2) and (3)with ρ
0
>
6.The plan of this work is as follows. In Section2, we recall some known results for P
0, especially the asymptotic expansion of (
P
0
-
z
)
-
1 for z near 0. We obtain the asymptotic behavior of discrete eigenvalue of P
(
λ
) for the case of 0
∉
σ
∞ in Section 3. Section 4 concentrates on the asymptotic expansion of discrete eigenvalue in the case of 0
∈
σ
∞.Let us introduce some notations first.Notation 1.
The scalar product onL
2
(
ℝ
+
;
r
d
-
1
d
r
) and L
2
(
ℝ
d
) is denoted by 〈
·
,
·
〉 and that on L
2
(
𝕊
d
-
1
) by (
·
,
·
). H
r
,
s
(
ℝ
d
),r
∈
ℤ,s
∈
ℝ, denotes the weighted Sobolev space of order r with volume element 〈
x
〉
2
s
d
x. The duality between H
1
,
s and H
-
1
,
-
s is identified with L
2 product. Denote H
0
,
s
=
L
2
,
s. Notation ℒ
(
H
r
,
s
,
H
r
′
,
s
′
) stands for the space of continuous linear operators from H
r
,
s to H
r
′
,
s
′. The complex plans ℂ is slit along positive real axis so that z
ν
=
e
ν
ln
⁡
z and ln
⁡
z
=
ln
⁡
|
z
|
+
i
arg
z with 0
<
arg
z
<
2
π are holomorphic there.


## 2. Some Results forP
0

Consider the operator(5)
P
0
=
-
Δ
+
q
(
θ
)
r
2
on L
2
(
ℝ
d
),d
≥
2. Assume that (
r
,
θ
) is the polar coordinate on ℝ
d. Then, the condition
(6)
-
Δ
s
+
q
(
θ
)
≥
-
1
4
(
d
-
2
)
2
,
on
⁡
L
2
(
𝕊
d
-
1
)
implies that P
0
≥
0, in L
2
(
ℝ
d
) (see [5]). First, we recall some results on the resolvent of Schrödinger operator P
0. Define
(7)
σ
∞
=
{
ν
;
ν
=
λ
+
(
d
-
2
)
2
4
,
λ
∈
σ
(
-
Δ
s
+
q
(
θ
)
)
}
;
σ
k
=
σ
∞
∩
[
0
,
k
]
,
k
∈
ℕ
.
For ν
∈
σ
∞, let n
ν denote the multiplicity of λ
ν
=
ν
2
-
(
d
-
2
)
2
/
4 as the eigenvalue of -
Δ
s
+
q
(
θ
). Let φ
ν
(
j
),ν
∈
σ
∞,1
≤
j
≤
n
ν denote an orthogonal basis of L
2
(
𝕊
d
-
1
) consisting of eigenfunctions of -
Δ
s
+
q
(
θ
):
(8)
(
-
Δ
s
+
q
(
θ
)
)
φ
ν
(
j
)
=
λ
ν
φ
ν
(
j
)
,
(
φ
ν
(
i
)
,
φ
ν
(
j
)
)
=
δ
i
j
.
Let π
ν denote the orthogonal projection in L
2
(
𝕊
d
-
1
) onto the subspace spanned by the eigenfunctions of -
Δ
s
+
q
(
θ
) associated with the eigenvalue λ
ν:
(9)
π
ν
f
=
∑
j
=
1
n
ν
‍
(
f
,
φ
ν
(
j
)
)
⊗
φ
ν
(
j
)
,
f
∈
L
2
(
𝕊
n
-
1
)
.
Defined by ν
∈
σ
∞,
(10)
z
ν
=
{
z
ν
′
,
if
ν
∉
ℕ
,
z
ln
⁡
z
,
if
ν
∈
ℕ
.
Here, ν
′
=
ν
-
[
ν
] and [
ν
] is the largest integer which is not larger than ν. For ν
>
0, let [
ν
]
- be the largest integer strictly less than ν. When ν
=
0, set [
ν
]
-
=
0. Define δ
ν by δ
ν
=
1, if ν
∈
σ
∞
∩
ℕ, δ
ν
=
0, otherwise. One has [
ν
]
=
[
ν
]
-
+
δ
ν.Theorem 1 ([5] Theorem 2.2).
The following asymptotic expansion holds forz near 0: with ℑ
z
>
0,
(11)
R
0
(
z
)
=
δ
0
ln
⁡
z
G
0,0
π
0
+
∑
j
=
0
N
‍
z
j
F
j
+
∑
ν
∈
σ
N
‍
z
ν
∑
j
=
[
ν
]
-
N
-
1
‍
z
j
G
ν
,
j
+
δ
ν
π
ν
+
R
0
(
N
)
(
z
)
in ℒ
(
-
1
,
s
;
1
,
-
s
)
,
s
>
2
N
+
1. Here,
(12)
G
ν
,
j
(
r
,
τ
)
=
{
b
ν
′
,
j
(
r
τ
)
j
+
ν
′
f
j
-
[
ν
]
(
r
,
τ
;
ν
′
)
,
ν
∉
ℕ
;
-
(
i
r
τ
)
j
j
!
f
j
-
[
ν
]
(
r
,
τ
;
0
)
,
ν
∈
ℕ
;
F
j
∈
ℒ
(
-
1
,
s
;
1
,
-
s
)
,
s
>
2
j
+
1
;
R
0
(
N
)
(
z
)
=
O
(
|
z
|
N
+
ϵ
)
∈
ℒ
(
-
1
,
s
;
1
,
-
s
)
,
s
>
2
N
+
1
,
ϵ
>
0
.
Here,
(13)
b
ν
′
,
j
=
-
i
j
e
-
i
ν
′
π
/
2
Γ
(
1
-
ν
′
)
ν
′
(
ν
′
+
1
)
⋯
(
ν
′
+
j
)
,
for 0
≤
ν
′
<
1, and
(14)
f
j
(
r
,
τ
,
ν
)
=
(
r
τ
)
-
(
1
/
2
)
(
n
-
2
)
P
j
,
ν
(
ρ
)
,
with P
j
,
ν
(
ρ
) being a polynomial in ρ of degree j:
(15)
P
j
,
ν
(
ρ
)
=
i
j
a
ν
j
!
∫
-
1
1
‍
(
ρ
+
1
2
θ
)
j
(
1
-
θ
2
)
ν
-
1
/
2
d
θ
,
a
ν
=
-
e
-
i
π
ν
/
2
2
2
ν
+
1
π
1
/
2
Γ
(
ν
+
1
/
2
)
.Lete
1
(
λ
)
≤
e
2
(
λ
)
≤
⋯
≤
e
n
(
λ
)
≤
⋯ denote the negative eigenvalues of P
(
λ
) (counting multiplicity). Suppose that e
n
+
1
(
λ
)
=
0, for convenience, if there are at most n negative eigenvalues. Using the same discussion as in [6], we know that each negative eigenvalue tends to 0, as λ tends to some value. Suppose that e
i
(
λ
)
→
0, at λ
=
λ
i. Then, from the definition of e
n
(
λ
), one has λ
1
≤
λ
2
≤
⋯
≤
λ
n
≤
⋯. In [6], we have studied the asymptotic behavior of the smallest eigenvalue of P
(
λ
), namely, e
1
(
λ
), as e
1
(
λ
)
→
0. In this paper, we investigate the asymptotic behaviors of all eigenvalues of P
(
λ
). Suppose that there are exactly m eigenvalues of P
(
λ
) tending to 0 at λ
k. Without loss of generality, suppose that λ
k
-
1
<
λ
k
=
λ
k
+
1
=
⋯
=
λ
k
+
m
-
1
<
λ
k
+
m.As in [6], we define a family of Birman-Schwinger kernel operators, which are used to study the eigenvalues of P
(
λ
). For z
∉
σ
(
P
0
), set
(16)
K
(
z
)
=
|
V
|
1
/
2
(
P
0
-
z
)
-
1
|
V
|
1
/
2
;
K
(
0
)
=
|
V
|
1
/
2
F
0
|
V
|
1
/
2
.The following result shows the relation between the eigenvalues ofP
(
λ
) and the eigenvalues of K
(
α
).Proposition 2 ([6] Proposition 2.2).
Letα
<
0. Then,(a)
let
(17)
A
=
{
ψ
∈
L
2
(
ℝ
d
)
;
(
P
(
λ
)
-
α
)
ψ
=
0
}
,
B
=
{
ϕ
∈
L
2
(
ℝ
d
)
;
K
(
α
)
ϕ
=
λ
-
1
ϕ
}
.
Then,
|
V
|
1
/
2
is injective from
A
to
B
, and
(
P
0
-
α
)
-
1
|
V
|
1
/
2
is injective from
B
to
A.
(b)
The multiplicity of
α
as the eigenvalue of
P
(
λ
)
is exactly the multiplicity of
λ
-
1
as the eigenvalue of
K
(
α
).Proposition 3.
(a) Letλ
>
0. The negative eigenvalue of the P
(
λ
) is monotone decreasing and continuous about λ.
(b) Letα
<
0. The eigenvalue of the K
(
α
) is monotone increasing and continuous about α.Proof.
(a) Supposeλ
1,λ
2 are two arbitrary positive numbers. By Lemma 3.4 [6], one has
(18)
|
e
i
(
λ
1
)
-
e
i
(
λ
2
)
|
≤
∥
P
(
λ
1
)
-
P
(
λ
2
)
∥
≤
|
λ
1
-
λ
2
|
·
∥
V
∥
.
This means that the negative eigenvalue of P
(
λ
) is continuous about λ.
By min-max principle (Theorem XIII.1 [15]), the negative eigenvalue of P
(
λ
) has the following form:
(19)
e
n
(
λ
)
=
sup
⁡
ϕ
1
,
…
,
ϕ
n
-
1
inf
⁡
∥
ψ
∥
=
1
,
ψ
∈
[
ϕ
1
,
…
,
ϕ
n
-
1
]
⊥
〈
ψ
,
P
(
λ
)
ψ
〉
.
Since e
n
(
λ
)
<
0 and P
0
≥
0, it is easy to see that 〈
ψ
,
λ
V
ψ
〉
<
0. Hence, 〈
ψ
,
P
(
λ
)
ψ
〉 is monotone decreasing about λ. It follows that the eigenvalue of P
(
λ
) is monotone decreasing about λ.
(b) Forα
1
<
α
2
<
0,
(20)
K
(
α
1
)
-
K
(
α
2
)
=
(
α
1
-
α
2
)
|
V
|
1
/
2
(
P
0
-
α
1
)
-
1
(
P
0
-
α
2
)
-
1
|
V
|
1
/
2
<
0
.
This shows that K
(
α
1
)
<
K
(
α
2
). Hence, using min-max principle again, one has that the eigenvalue of K
(
α
) is monotone increasing about α.
Note that, for anyα
<
0, (
P
0
-
α
)
-
1 is a bounded operator in L
2
(
ℝ
d
). By (20) and Lemma 3.4 [6], we can get that, for any α
1
<
0,α
2
<
0, |
μ
i
(
K
(
α
1
)
)
-
μ
i
(
K
(
α
2
)
)
|
≤
C
|
α
1
-
α
2
| with some C large enough. Here, μ
i
(
K
(
α
)
) is the eigenvalue of K
(
α
). It implies that the eigenvalue of K
(
α
) is continuous about α.We give the definition of resonance which will be used to investigate the asymptotic behavior of the eigenvalue ofP
(
λ
) later.Definition 4.
Set𝒩
(
λ
)
=
{
u
;
P
(
λ
)
u
=
0
,
u
∈
H
1
,
-
s
,
∀
s
>
1
}. If 𝒩
(
λ
)
∖
L
2
≠
{
0
}, one says that 0 is the resonance of P
(
λ
). A nonzero function u
∈
𝒩
(
λ
)
∖
L
2 is called a resonant state of P
(
λ
) at 0.


## 3. The Case0
∉
σ
∞

Proposition 5.
Assume that0
∉
σ
∞. Then, λ
k
-
1 is the eigenvalue of K
(
0
), and the multiplicity of λ
k
-
1 is m.Proof.
e
k
(
λ
) is the eigenvalue of P
(
λ
); thus, λ
-
1 is the eigenvalue of K
(
e
k
(
λ
)
) by Proposition 2. Notice that e
k
(
λ
)
→
0, as λ
→
λ
k, and one gets that λ
k
-
1 is the eigenvalue of K
(
0
), by Lemma 3.5 [6].
Suppose that the multiplicity ofλ
k
-
1 is m
0. We will prove that m
0
=
m in the following. First, using Lemma 3.5 [6] again, we know that there are m
0 eigenvalues (counting multiplicity) of K
(
α
) tending to λ
k
-
1. Suppose that these m
0 eigenvalues are μ
1
(
α
)
,
μ
2
(
α
)
,
…
,
μ
m
0
(
α
). Since the eigenvalue of K
(
α
) is continuous and monotone increasing, for a fixed λ near λ
k with λ
>
λ
k, there exists a unique α
i
<
0
(
1
≤
i
≤
m
0
) such that μ
i
(
α
i
)
=
λ
-
1. It follows that α
i is the eigenvalue of P
(
λ
). This fact shows that m
0
≤
m, since there are m eigenvalues of P
(
λ
) that tend to 0 at λ
=
λ
k. On the other hand, note that, for any λ
>
0, e
i
(
λ
)
→
0 and e
i
(
λ
) is continuous and monotone decreasing. Therefore, for a fixed α
<
0, there exists a unique λ
i
~ such that e
i
(
λ
i
~
)
=
α
(
k
≤
i
≤
k
+
m
-
1
). It follows that λ
i
~
-
1 is the eigenvalue of K
(
α
). One has that K
(
α
)
→
K
(
0
) and λ
i
~
-
1
→
λ
k
-
1
(
k
≤
i
≤
k
+
m
-
1
), as α
→
0. It means that there are at least m eigenvalues (counting multiplicity) of K
(
α
) tending to λ
k
-
1. Thus, m
≤
m
0. Therefore, m
=
m
0. This ends the proof.SinceK
(
0
) is a compact operator, then we can suppose that all of the eigenvalues of K
(
0
) can be denoted by μ
1
,
μ
2
,
…
,
μ
n
,
… (counting multiplicity), and the corresponding eigenvectors are ϕ
1
,
ϕ
2
,
…
,
ϕ
n
,
…, respectively. Set
(21)
ψ
i
=
F
0
|
V
|
1
/
2
ϕ
i
.
Then,
(22)
|
V
|
1
/
2
ψ
i
=
|
V
|
1
/
2
F
0
|
V
|
1
/
2
ϕ
i
=
μ
i
ϕ
i
.
By Lemma 3.5 [6] and Proposition 5, we know that there are m eigenvalues (counting multiplicity) of K
(
α
) that tend to λ
k
-
1, as α
→
0. Without loss of generality, we can suppose that μ
1
=
μ
2
=
⋯
=
μ
m
=
λ
k
-
1 and the eigenvalues μ
1
(
α
)
,
μ
2
(
α
)
,
…
,
μ
m
(
α
) tend to λ
k
-
1, as α
→
0. Moreover, we can choose a set of eigenvectors, {
ϕ
j
(
α
)
(
1
≤
j
≤
m
)
}, of K
(
α
) such that 〈
ϕ
j
(
α
)
,
ϕ
l
(
α
)
〉
=
δ
j
l, K
(
α
)
ϕ
j
(
α
)
=
μ
j
(
α
)
ϕ
j
(
α
), and ϕ
j
(
α
)
→
ϕ
j, as α
→
0. Then, by Lemma 3.5 [6], ϕ
j is the eigenvector of K
(
0
) corresponding to λ
k
-
1. Thus, K
(
0
)
ϕ
j
=
λ
k
-
1
ϕ
j
(
1
≤
j
≤
m
). It follows that
(23)
P
0
ψ
j
=
P
0
F
0
|
V
|
1
/
2
ϕ
j
=
|
V
|
1
/
2
ϕ
j
=
λ
k
|
V
|
ψ
j
by (22). In the second step of the last equality, we use Proposition 3.2 [6]. Thus, P
(
λ
k
)
ψ
j
=
0. By Theorem 4.1 [4],
(24)
ψ
j
(
r
θ
)
=
∑
0
<
ν
≤
1
∑
s
=
1
n
ν
‍
-
1
2
ν
〈
λ
k
V
u
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
φ
ν
(
s
)
(
θ
)
r
(
d
-
2
)
/
2
+
ν
+
u
,
where u
∈
L
2. ψ
j
∈
𝒩
(
λ
k
)
∖
L
2 will be called a ζ
j-resonant state of P
(
λ
k
), if
(25)
ψ
j
(
r
θ
)
=
ϕ
(
θ
)
r
(
d
-
2
)
/
2
+
ν
+
o
(
1
r
(
d
-
2
)
/
2
+
ν
+
ϵ
)
,
r
⟶
∞
,
ϵ
>
0
,
for some ϕ
≠
0.Proposition 6.
(I)Assume
f
is a
ν
0
-resonant state of
P
(
λ
k
).Then
(a)
〈
V
f
,
G
ν
0
,
δ
ν
0
π
ν
0
V
f
〉
=
c
ν
0
∑
s
=
1
n
ν
0
|
〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
0
φ
ν
0
(
s
)
〉
|
2
≠
0
with
c
ν
0
be a nonzero constant depending only on
ν
0;
(b)
∀
g
∈
H
1
,
-
s,  ∀
ν
∈
σ
1
, with
ν
<
ν
0, 〈
V
g
,
G
ν
,
δ
ν
π
ν
V
f
〉
=
0.
(II) Assumef is an eigenvector of P
(
λ
k
). Then, ∀
ν
∈
σ
1, ∀
g
∈
H
-
1
,
s, 〈
V
g
,
G
ν
,
δ
ν
π
ν
V
f
〉
=
0.Proof.
(I) By the definition ofG
ν
,
δ
ν and π
ν, ∀
f
,
g
∈
H
1
,
-
s, ν
∈
σ
1,
(26)
〈
V
g
,
G
ν
,
δ
ν
π
ν
V
f
〉
=
c
ν
∑
s
=
1
n
ν
‍
〈
V
g
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
¯
.
Here, c
ν is a constant depending only on ν. If f is a resonant state of P
(
λ
k
), then, by (24),
(27)
f
(
r
θ
)
=
∑
0
<
ν
≤
1
∑
s
=
1
n
ν
‍
-
1
2
ν
〈
λ
k
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
×
φ
ν
(
s
)
(
θ
)
r
(
d
-
2
)
/
2
+
ν
+
u
,
where u
∈
L
2. If f is a ν
0-resonant state of P
(
λ
k
), then the leading part of f is ϕ
(
θ
)
/
r
(
d
-
2
)
/
2
+
ν
0. It follows that, for ν
∈
σ
1 with ν
<
ν
0,
(28)
〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
=
0
,
and there exists at least one s with 1
≤
s
≤
n
ν
0, such that
(29)
〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
0
φ
ν
0
(
s
)
〉
≠
0
.
Thus, (26) shows that
(30)
〈
V
g
,
G
ν
,
δ
ν
π
ν
V
f
〉
=
0
,
ν
∈
σ
1
,
ν
<
ν
0
;
〈
V
f
,
G
ν
0
,
δ
ν
0
π
ν
0
V
f
〉
=
c
ν
0
∑
s
=
1
n
ν
0
‍
|
〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
0
φ
ν
0
(
s
)
〉
|
2
≠
0
.
(II) By (24) and f
∈
L
2, it is easy to see that 〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
=
0 for ν
∈
σ
1. Thus, for ν
∈
σ
1,
(31)
〈
V
g
,
G
ν
,
δ
ν
π
ν
V
f
〉
=
c
ν
∑
s
=
1
n
ν
‍
〈
V
g
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
×
〈
V
f
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
¯
=
0
.Theorem 7.
Assume0
∉
σ
∞. ϕ
j, ψ
j, and ζ
j are defined as above. If ρ
0
>
6, one of three situations holds.(a)
e
l
(
λ
)
=
c
(
λ
-
λ
k
)
1
/
ζ
j
+
o
(
|
λ
-
λ
k
|
1
/
ζ
j
)
.
c is a nonzero constant independent of λ, ζ
j
∈
σ
1, and ζ
j
<
1.
(b)
e
l
(
λ
)
=
c
(
(
λ
-
λ
k
)
/
ln
⁡
⁡
(
λ
-
λ
k
)
)
+
o
(
|
(
λ
-
λ
k
)
/
ln
⁡
⁡
(
λ
-
λ
k
)
|
).  c is a nonzero constant independent of λ.
(c)
e
l
(
λ
)
=
c
(
λ
-
λ
k
)
+
o
(
|
λ
-
λ
k
|
)
c is a nonzero constant independent of λ.Proof.
Applying Theorem1 to R
0
(
α
)
=
(
P
0
-
α
)
-
1, we obtain the asymptotic expansion of R
0
(
α
) for α near 0, with α
<
0,
(32)
R
0
(
α
)
=
F
0
+
∑
0
<
ν
≤
1
‍
α
ν
G
ν
,
δ
ν
π
ν
+
α
F
1
+
O
(
|
α
|
1
+
ϵ
)
in ℒ
(
-
1
,
s
;
1
,
-
s
), s
>
3, ϵ
>
0. It follows that, if ρ
0
>
6,
(33)
K
(
α
)
=
K
(
0
)
+
∑
0
<
ν
≤
1
‍
α
ν
|
V
|
1
/
2
G
ν
,
δ
ν
π
ν
|
V
|
1
/
2
+
α
|
V
|
1
/
2
F
1
|
V
|
1
/
2
+
O
(
|
α
|
1
+
ϵ
)
in ℒ
(
0,0
;
0,0
). λ
k
-
1 is the eigenvalue of K
(
0
) with multiplicity m; then, there are m eigenvalues of K
(
α
) (counting multiplicity) tending to λ
k
-
1, as α
→
0. Suppose that the m eigenvalues of K
(
α
) are μ
1
(
α
)
,
μ
2
(
α
)
,
…
,
μ
m
(
α
). We study the asymptotic expansion of μ
i
(
α
) first.
SetK
1
(
α
)
=
∑
0
<
ν
≤
1
α
ν
|
V
|
1
/
2
G
ν
,
δ
ν
π
ν
|
V
|
1
/
2
+
α
|
V
|
1
/
2
F
1
|
V
|
1
/
2
+
O
(
|
α
|
1
+
ϵ
) and K
s
j
(
α
)
=
〈
ϕ
s
,
K
1
(
α
)
ϕ
j
〉. Then, by (22),
(34)
K
s
j
(
α
)
=
λ
k
〈
ϕ
s
,
K
1
(
α
)
|
V
|
1
/
2
ψ
j
〉
=
-
λ
k
〈
∑
0
<
ν
≤
1
‍
|
V
|
1
/
2
ϕ
s
,
(
∑
0
<
ν
≤
1
‍
α
ν
G
ν
,
δ
ν
π
ν
+
α
F
1
+
o
(
|
α
|
1
+
ϵ
)
)
V
ψ
j
〉
.
By Lemma 3.6 [6], the eigenvalue of K
(
α
), μ
j
(
α
)
(
1
≤
j
≤
m
), has the following form:
(35)
μ
j
(
α
)
=
λ
k
-
1
+
∑
n
=
0
∞
a
n
(
j
)
(
α
)
∑
n
=
0
∞
b
n
(
j
)
(
α
)
.
Here,
(36)
a
0
(
j
)
(
α
)
=
K
j
j
(
α
)
,
a
1
(
j
)
(
β
)
=
-
∑
{
s
;
μ
s
≠
λ
k
-
1
}
‍
(
μ
s
-
λ
k
-
1
)
-
1
K
j
s
(
α
)
K
s
j
(
α
)
,
a
2
(
j
)
(
α
)
=
∑
μ
s
≠
λ
k
-
1
≠
μ
l
‍
(
μ
s
-
λ
k
-
1
)
-
1
×
(
μ
l
-
λ
k
-
1
)
-
1
K
j
s
(
α
)
K
s
l
(
α
)
K
l
j
(
α
)
-
2
∑
{
s
;
μ
s
≠
λ
k
-
1
}
‍
(
μ
s
-
λ
k
-
1
)
-
1
×
K
j
s
(
α
)
K
s
j
(
α
)
K
j
j
(
α
)
,
a
n
(
j
)
(
α
)
=
-
(
-
1
)
n
2
π
i
×
∮
|
E
-
λ
k
-
1
|
=
δ
‍
(
λ
k
-
1
-
E
)
-
1
×
∑
i
1
,
i
2
,
…
,
i
n
‍
(
μ
i
1
-
E
)
-
1
⋯
(
μ
i
n
-
E
)
-
1
×
K
j
i
1
(
α
)
K
i
1
i
2
(
α
)
⋯
K
i
n
j
(
α
)
d
E
,
for
n
>
2
,
b
0
(
j
)
(
α
)
=
1
,
b
1
(
j
)
(
α
)
=
0
,
b
2
(
j
)
(
α
)
=
∑
{
s
;
μ
s
≠
λ
k
-
1
}
‍
(
μ
s
-
λ
k
-
1
)
-
2
K
j
s
(
α
)
K
s
j
(
α
)
,
b
n
(
j
)
(
α
)
=
-
(
-
1
)
n
2
π
i
×
∮
|
E
-
λ
k
-
1
|
=
δ
‍
(
μ
s
-
λ
k
-
1
)
-
2
×
∑
i
1
,
i
2
,
…
,
i
n
-
1
‍
(
μ
i
1
-
E
)
-
1
⋯
(
μ
i
n
-
1
-
E
)
-
1
×
K
j
i
1
(
α
)
K
i
1
i
2
(
α
)
⋯
K
i
n
-
1
j
(
α
)
d
E
,
for
n
>
2
.
Ifψ
j is ζ
j-resonant state of K
(
0
) with ζ
j
<
1, then by Proposition 6,
(37)
K
j
j
(
α
)
=
c
ζ
j
α
ζ
j
λ
k
2
∑
s
=
1
n
ζ
j
‍
|
〈
V
ψ
j
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
0
φ
ζ
j
(
s
)
〉
|
2
+
o
(
|
α
|
ζ
j
)
=
c
~
ζ
j
α
ζ
j
+
o
(
|
α
|
ζ
j
)
;
K
j
s
(
α
)
=
∑
0
<
ν
≤
1
‍
λ
k
〈
|
V
|
1
/
2
ϕ
s
,
G
ν
,
δ
ν
π
ν
V
ψ
j
〉
+
O
(
|
α
|
)
=
O
(
|
α
|
ζ
j
)
.
Hence,
(38)
a
2
(
j
)
=
O
(
|
α
|
2
ζ
j
)
.
Similarly,
(39)
a
n
(
j
)
=
O
(
|
α
|
2
ζ
j
)
;
b
n
(
j
)
=
O
(
|
α
|
2
ζ
j
)
,
for n
≥
2. It follows that
(40)
∑
n
=
0
∞
‍
a
n
(
j
)
=
c
~
ζ
j
α
ζ
j
+
o
(
|
α
|
ζ
j
)
;
∑
n
=
0
∞
‍
b
n
(
j
)
=
1
+
O
(
|
α
|
2
ζ
j
)
.
Thus, μ
j
(
α
)
=
λ
k
-
1
+
c
~
ζ
j
α
ζ
j
+
o
(
|
α
|
ζ
j
). By Proposition 2, for 1
≤
j
≤
m, ∃
l with k
≤
l
≤
k
+
m
-
1 such that μ
j
(
e
l
(
λ
)
)
=
λ
-
1. Thus, λ
-
1
=
λ
k
-
1
+
c
~
ζ
j
e
l
(
λ
)
ζ
j
+
o
(
|
e
l
(
λ
)
|
ζ
j
). Since
(41)
λ
-
1
=
λ
k
-
1
-
λ
k
-
2
(
λ
-
λ
k
)
+
O
(
|
λ
-
λ
k
|
2
)
,
we can get that the leading term of e
l
(
λ
) is c
(
λ
-
λ
k
)
1
/
ζ
j, with c
≠
0 being a constant independent of λ. This is in case (a).
Ifψ
j is 1-resonant state of K
(
0
), computing similarly, we obtain that μ
j
(
α
)
=
λ
k
-
1
+
c
~
α
ln
⁡
α
+
O
(
|
α
|
). Using again μ
j
(
e
l
(
λ
)
)
=
λ
-
1 and (41), we can get that the leading part of e
l
(
λ
) is c
(
λ
-
λ
k
)
/
ln
⁡
(
λ
-
λ
k
), with c
≠
0 being a constant independent of λ. This is in case (b).
Ifψ
j is the eigenfunction of K
(
0
) corresponding to eigenvalue λ
k
-
1, then, for k
≤
j
≤
k
+
m
-
1,
(42)
R
0
(
α
)
V
ψ
j
=
-
λ
k
-
1
(
ψ
j
+
α
R
0
(
α
)
ψ
j
)
;
F
0
V
ψ
j
=
-
F
0
|
V
|
1
/
2
|
V
|
1
/
2
ψ
j
=
-
λ
k
-
1
F
0
|
V
|
1
/
2
ϕ
j
=
-
λ
k
-
1
ψ
j
.
In the second equation, we use (22) and the definition of ψ
j. Proposition 6 shows that for ν
∈
σ
1, 〈
V
ψ
s
,
G
ν
,
δ
ν
π
ν
V
ψ
j
〉
=
0. Thus, by (42),
(43)
〈
V
ψ
j
,
F
1
V
ψ
j
〉
=
lim
⁡
α
→
0
α
-
1
〈
V
ψ
j
,
(
∑
ν
∈
σ
1
‍
R
0
(
α
)
-
F
0
-
∑
ν
∈
σ
1
‍
G
ν
,
δ
ν
π
ν
-
o
(
|
α
|
)
)
V
ψ
j
〉
=
lim
⁡
α
→
0
α
-
1
〈
V
ψ
j
,
(
R
0
(
α
)
-
F
0
)
V
ψ
j
〉
=
-
lim
⁡
α
→
0
λ
k
-
1
〈
V
ψ
j
,
R
0
(
α
)
ψ
j
〉
=
-
λ
k
-
2
∥
ψ
j
∥
2
.
Computing similarly as before, we obtain that μ
j
(
α
)
=
λ
k
-
1
+
c
~
α
+
o
(
|
α
|
). Thus, the leading part of e
l
(
λ
) is c
(
λ
-
λ
k
), with c
≠
0 being a constant independent of λ. This is in case (c). This completes the proof.


## 4. The Case0
∈
σ
∞

Assume that0
∈
σ
∞ in this section. By Theorem 1,
(44)
R
0
(
α
)
=
δ
0
ln
⁡
α
G
0,0
π
0
+
F
0
+
∑
ν
∈
σ
1
‍
α
ν
G
ν
,
δ
ν
π
ν
+
α
F
1
+
O
(
|
α
|
1
+
ϵ
)
in ℒ
(
-
1
,
s
;
1
,
-
s
)
,
s
>
3. Here,
(45)
G
0,0
π
0
=
1
2
〈
·
,
ϕ
0
〉
ϕ
0
with ϕ
0
=
φ
0
(
1
)
(
θ
)
r
-
(
d
-
2
)
/
2. (
r
,
θ
) is the polar coordinate on ℝ
d. Hence, K
(
α
) ban will be written as
(46)
K
(
α
)
=
l
(
α
)
P
+
K
(
0
)
+
K
1
(
α
)
.
Here, P is rank 1 projection with P
=
〈
·
,
ϕ
~
0
〉
ϕ
~
0, ϕ
~
0
=
(
∫
‍
|
V
|
ϕ
0
2
d
x
)
-
1
/
2
|
V
|
1
/
2
ϕ
0, l
(
α
)
=
(
1
/
2
)
ln
⁡
α
∫
‍
|
V
|
ϕ
0
2
d
x, and K
1
(
α
)
=
∑
0
<
ν
≤
1
α
ν
|
V
|
1
/
2
G
ν
,
δ
ν
π
ν
|
V
|
1
/
2
+
α
|
V
|
1
/
2
F
1
|
V
|
1
/
2
+
O
(
|
α
|
1
+
ϵ
).Note that asα
→
0, |
l
(
α
)
|
→
+
∞. It follows that K
(
α
) has singularity at α
=
0. Hence, it is more complicated than the case 0
∉
σ
∞. We first study the operator
(47)
K
~
(
α
)
=
l
(
α
)
P
+
K
(
0
)
,
as α
→
0. Set Q
=
1
-
P.We can obtain Propositions8, 10, and 11, in the same way that Lemmas 7.1 and 7.2 [8] were obtained.Proposition 8.
(a) Forα small enough with |
l
(
α
)
|
>
2
∥
K
(
0
)
∥, and for a fixed z with ∥
K
(
0
)
∥
<
z
<
|
l
(
α
)
|
-
∥
K
(
0
)
∥, (
z
-
K
~
(
α
)
)
-
1 converges to Q
(
z
-
Q
K
(
0
)
Q
)
-
1
Q in norm, as α
→
0.
(b) Forα small enough, and for fixed z with z
>
∥
K
(
0
)
∥,
(48)
(
z
-
K
~
(
α
)
)
-
1
=
(
z
-
K
(
0
)
)
-
1
-
ω
(
z
)
-
1
(
z
-
K
(
0
)
)
-
1
P
(
z
-
K
(
0
)
)
-
1
-
l
(
α
)
-
1
ω
(
z
)
-
2
(
z
-
K
(
0
)
)
-
1
P
(
z
-
K
(
0
)
)
-
1
+
O
(
|
l
(
α
)
|
-
2
)
.
Here, ω
(
z
)
=
〈
(
z
-
K
(
0
)
)
-
1
ϕ
~
0
,
ϕ
~
0
〉.Remark 9.
Proposition8 shows that, for fixed z, with ∥
K
(
0
)
∥
<
z
<
|
l
(
α
)
|
-
∥
K
(
0
)
∥,
(49)
Q
(
z
-
Q
K
(
0
)
Q
)
-
1
Q
=
(
z
-
K
(
0
)
)
-
1
-
ω
(
z
)
-
1
(
z
-
K
(
0
)
)
-
1
P
(
z
-
K
(
0
)
)
-
1
.Proposition 10.
For fixedz, with ∥
K
(
0
)
∥
<
z
<
|
l
(
α
)
|
-
∥
K
(
0
)
∥, (
z
-
λ
k
-
1
)
-
1 is the eigenvalue of Q
(
z
-
Q
K
(
0
)
Q
)
-
1
Q.This proposition follows thatλ
k
-
1 is the eigenvalue of Q
K
(
0
)
Q.Proposition 11.
Suppose that the multiplicity ofλ
k
-
1 as the eigenvalue of Q
K
(
0
)
Q is m
0. Set U
=
{
f
;
Q
K
(
0
)
Q
f
=
μ
0
f
}. Then, ∃
W
⊏
U with dim
⁡
W
=
m
0 or m
0
-
1, so that
(50)
(
z
-
K
~
(
α
)
)
-
1
f
≡
(
z
-
K
(
0
)
)
-
1
f
for f
∈
W. And for f
∈
W, K
(
0
)
f
=
λ
k
-
1
f.Suppose that all of the eigenvalues ofQ
K
(
0
)
Q can be denoted by μ
1
,
μ
2
,
…
,
μ
n
,
… (counting multiplicity), and the corresponding eigenvectors are ϕ
1
,
ϕ
2
,
…
,
ϕ
n
,
…, respectively. By Lemma 3.5 [6] and Proposition 5, we know that there are m
0 eigenvalues (counting multiplicity) of K
(
α
) tending to λ
k
-
1, as α
→
0. Without loss of generality, we can suppose that μ
1
=
μ
2
=
⋯
=
μ
m
0
=
λ
k
-
1 and the eigenvalues μ
1
(
α
)
,
μ
2
(
α
)
,
…
,
μ
m
0
(
α
) tend to λ
k
-
1, as α
→
0. Moreover, we can choose a set of eigenvectors, {
ϕ
j
(
α
)
(
1
≤
j
≤
m
0
)
}, of K
(
α
) such that 〈
ϕ
j
(
α
)
,
ϕ
l
(
α
)
〉
=
δ
j
l, K
(
α
)
ϕ
j
(
α
)
=
μ
j
(
α
)
ϕ
j
(
α
), and ϕ
j
(
α
)
→
ϕ
j, as α
→
0. Then, ϕ
j is the eigenvector of Q
K
(
0
)
Q corresponding to λ
k
-
1, by Lemma 3.5 [6]. Thus, Q
K
(
0
)
Q
ϕ
j
=
λ
k
-
1
ϕ
j.Theorem 12.
Assume0
∈
σ
∞. If ρ
0
>
6, one of the four situations holds.(a)
e
l
(
λ
)
=
c
1
e
-
1
/
c
2
(
λ
-
λ
k
)
+
o
(
e
-
1
/
c
2
(
λ
-
λ
k
)
)
.
c
1 and c
2 are two nonzero constants independent of λ.
(b)
e
l
(
λ
)
=
c
(
λ
-
λ
k
)
1
/
ζ
j
+
o
(
|
λ
-
λ
k
|
1
/
ζ
j
)
.
c is a nonzero constant independent of λ, ζ
j
∈
σ
1 and σ
1
<
1.
(c)
e
l
(
λ
)
=
c
(
(
λ
-
λ
k
)
/
ln
⁡
⁡
(
λ
-
λ
k
)
)
+
o
(
|
(
λ
-
λ
k
)
/
ln
⁡
(
λ
-
λ
k
)
|
)
.
c is a nonzero constant independent of λ.
(d)
e
l
(
λ
)
=
c
(
λ
-
λ
k
)
+
o
(
|
λ
-
λ
k
|
)
c is a nonzero constant independent of λ.Proof.
By Proposition10,
(51)
(
z
-
K
(
α
)
)
-
1
=
Q
(
z
-
Q
K
(
0
)
Q
)
-
1
Q
+
K
~
1
(
α
)
with
(52)
K
~
1
(
α
)
=
l
(
α
)
-
1
ω
(
z
)
-
2
(
z
-
K
(
0
)
)
-
1
×
P
(
z
-
K
(
0
)
)
-
1
+
O
(
|
l
(
α
)
|
-
2
)
+
(
z
-
K
~
(
α
)
)
-
1
∑
n
=
1
∞
‍
[
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
]
n
.
Lemma 3.6 [6] shows that the eigenvalue of (
z
-
K
(
α
)
)
-
1, (
z
-
μ
j
(
α
)
)
-
1
(
1
≤
j
≤
m
0
) has the form
(53)
(
z
-
μ
j
(
α
)
)
-
1
=
(
z
-
λ
k
-
1
)
-
1
+
∑
n
=
0
∞
a
n
(
j
)
(
α
)
∑
n
=
0
∞
b
n
(
j
)
(
α
)
.
As in Theorem 7, we should compute K
s
j
(
α
)
=
〈
ϕ
s
,
K
~
1
(
α
)
ϕ
j
〉,(
1
≤
j
≤
m
0
).
IfP
(
z
-
K
(
0
)
)
-
1
ϕ
j
≠
0, then
(54)
〈
ϕ
j
,
K
~
1
(
α
)
ϕ
j
〉
=
∥
P
(
z
-
K
(
0
)
)
-
1
ϕ
j
∥
2
×
ω
(
z
)
-
2
l
(
α
)
-
1
+
o
(
|
l
(
α
)
|
-
1
)
.
It follows that K
j
j
(
α
)
=
c
~
l
(
α
)
-
1
+
o
(
|
l
(
α
)
|
-
1
). Note that K
s
j
(
α
)
=
O
(
|
l
(
α
)
|
-
1
). It is easy to get that
(55)
∑
n
=
0
∞
‍
a
n
(
j
)
=
c
~
l
(
α
)
-
1
+
o
(
|
l
(
α
)
|
-
1
)
;
∑
n
=
0
∞
‍
b
n
(
j
)
=
1
+
O
(
|
l
(
α
)
|
-
2
)
.
It follows that (
z
-
μ
j
(
α
)
)
-
1
=
(
z
-
λ
k
-
1
)
-
1
+
c
~
l
(
α
)
-
1
+
o
(
|
l
(
α
)
|
-
1
) with some c
1
≠
0. Analyzing similarly as in Theorem 7, we can get the leading term of e
l
(
λ
)
=
c
1
e
-
1
/
c
2
(
λ
-
λ
k
). We are in case (a).
IfP
(
z
-
K
(
0
)
)
-
1
ϕ
j
=
0, then by Proposition 11 and the definition of W, ϕ
j
∈
W. Thus,
(56)
K
~
1
(
α
)
ϕ
j
=
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
ϕ
j
+
(
z
-
K
~
(
α
)
)
-
1
∑
n
=
2
∞
‍
[
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
]
n
ϕ
j
=
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
(
z
-
K
(
0
)
)
-
1
ϕ
j
+
(
z
-
K
~
(
α
)
)
-
1
∑
n
=
2
∞
‍
[
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
]
n
ϕ
j
=
(
z
-
λ
k
-
1
)
-
1
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
ϕ
j
+
(
z
-
K
~
(
α
)
)
-
1
∑
n
=
2
∞
‍
[
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
]
n
ϕ
j
.
In the second and third steps, we use Proposition 11 and the fact that ϕ
j
∈
W. Then,
(57)
〈
ϕ
j
,
(
z
-
λ
k
-
1
)
-
1
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
ϕ
j
〉
=
(
z
-
λ
k
-
1
)
-
2
〈
ϕ
j
,
K
1
(
α
)
ϕ
j
〉
=
(
z
-
λ
k
-
1
)
-
2
×
[
∑
ν
∈
σ
1
‍
α
ν
c
ν
∑
s
=
1
n
ν
‍
|
〈
|
V
|
1
/
2
ϕ
j
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
|
2
+
α
〈
|
V
|
1
/
2
ϕ
j
,
F
1
|
V
|
1
/
2
ϕ
j
〉
∑
ν
∈
σ
1
‍
]
+
o
(
|
α
|
)
.
If {
ν
∈
σ
1; ∃
1
≤
s
≤
n
ν, such that 〈
|
V
|
1
/
2
ϕ
j
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
≠
0
}
≠
∅, set ν
j
=
min
⁡
{
ν
∈
σ
1
;
〈
|
V
|
1
/
2
ϕ
j
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
≠
0
}. If ν
j
<
1, then, by (57),
(58)
〈
ϕ
j
,
(
z
-
λ
k
-
1
)
-
1
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
ϕ
j
〉
=
c
1
α
ν
j
+
o
(
|
α
|
ν
j
)
;
〈
ϕ
s
,
(
z
-
λ
k
-
1
)
-
1
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
ϕ
j
〉
=
O
(
|
α
|
ν
j
)
.
Computing similarly as before, we also can obtain that
(59)
〈
ϕ
s
,
(
z
-
K
~
(
α
)
)
-
1
[
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
]
n
ϕ
j
〉
=
o
(
|
α
|
ν
j
)
for
n
≥
2
.
By (53), one has
(60)
(
z
-
μ
j
(
α
)
)
-
1
=
(
z
-
λ
k
-
1
)
-
1
+
c
1
α
ν
j
+
o
(
|
α
|
ν
j
)
.
Computing similarly as in Theorem 7, we can get that the leading term of e
l
(
λ
) is c
(
λ
-
λ
k
)
ν
j. We are in case (b).
Ifν
j
=
1, computing similarly as in Theorem 7, we can get the leading term of e
l
(
λ
)
=
c
(
λ
-
λ
k
)
/
ln
⁡
(
λ
-
λ
k
), and we are in case (c).
If{
ν
∈
σ
1
;
〈
|
V
|
1
/
2
ϕ
j
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
≠
0
}
=
∅, then for ν
∈
σ
1,
(61)
G
ν
,
δ
ν
π
ν
|
V
|
1
/
2
ϕ
j
=
c
ν
∑
s
=
1
n
ν
‍
〈
|
V
|
1
/
2
ϕ
j
,
|
y
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
〉
|
x
|
-
(
d
-
2
)
/
2
+
ν
φ
ν
(
s
)
=
0
.
It follows that
(62)
〈
ϕ
j
,
|
V
|
1
/
2
α
F
1
|
V
|
1
/
2
ϕ
j
〉
=
〈
ϕ
j
,
|
V
|
1
/
2
(
∑
ν
∈
σ
1
‍
R
0
(
α
)
-
F
0
-
∑
ν
∈
σ
1
‍
α
ν
G
ν
,
δ
ν
π
ν
-
o
(
|
α
|
)
)
|
V
|
1
/
2
ϕ
j
〉
=
〈
ϕ
j
,
|
V
|
1
/
2
(
R
0
(
α
)
-
F
0
-
o
(
|
α
|
)
)
|
V
|
1
/
2
ϕ
j
〉
=
〈
ϕ
j
,
lim
⁡
β
→
0
|
V
|
1
/
2
[
R
0
(
α
)
-
(
R
0
(
β
)
-
ln
⁡
β
G
0,0
π
0
)
-
o
(
|
α
|
)
(
R
0
(
β
)
-
ln
⁡
β
G
0,0
π
0
)
]
|
V
|
1
/
2
ϕ
j
lim
⁡
β
→
0
〉
=
〈
ϕ
j
,
lim
⁡
β
→
0
[
|
V
|
1
/
2
(
α
-
β
)
R
0
(
α
)
R
0
(
β
)
|
V
|
1
/
2
ϕ
j
-
|
V
|
1
/
2
ln
⁡
β
G
0,0
π
0
|
V
|
1
/
2
ϕ
j
]
lim
⁡
β
→
0
〉
-
o
(
|
α
|
)
=
〈
ϕ
j
,
lim
⁡
β
→
0
[
(
α
-
β
)
|
V
|
1
/
2
R
0
(
α
)
×
(
F
0
+
ln
⁡
β
G
0,0
π
0
)
|
V
|
1
/
2
ϕ
j
]
lim
⁡
β
→
0
〉
-
o
(
|
α
|
)
=
〈
ϕ
j
,
lim
⁡
β
→
0
(
α
-
β
)
|
V
|
1
/
2
R
0
(
α
)
F
0
|
V
|
1
/
2
ϕ
j
〉
-
o
(
|
α
|
)
=
〈
ϕ
j
,
α
|
V
|
1
/
2
R
0
(
α
)
F
0
|
V
|
1
/
2
ϕ
j
〉
-
o
(
|
α
|
)
=
〈
ϕ
j
,
α
|
V
|
1
/
2
R
0
(
α
)
F
0
|
V
|
1
/
2
ϕ
j
-
o
(
|
α
|
)
〉
.
Thus,
(63)
〈
ϕ
j
,
|
V
|
1
/
2
F
1
|
V
|
1
/
2
ϕ
j
〉
=
lim
⁡
α
→
0
〈
F
0
|
V
|
1
/
2
ϕ
j
,
R
0
(
α
)
F
0
|
V
|
1
/
2
ϕ
j
〉
=
∥
F
0
|
V
|
1
/
2
ϕ
j
∥
2
≠
0
.
Computing as before, we obtain
(64)
〈
ϕ
j
,
(
z
-
λ
k
-
1
)
-
1
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
ϕ
j
〉
=
c
1
α
+
o
(
|
α
|
)
;
〈
ϕ
s
,
(
z
-
λ
k
-
1
)
-
1
(
z
-
K
~
(
α
)
)
-
1
K
1
(
α
)
ϕ
j
〉
=
O
(
|
α
|
)
;
〈
ϕ
s
,
(
z
-
K
~
(
α
)
)
-
1
[
K
1
(
α
)
(
z
-
K
~
(
α
)
)
-
1
]
n
ϕ
j
〉
=
o
(
|
α
|
)
for
n
≥
2
.
By (53), one has
(65)
(
z
-
μ
j
(
α
)
)
-
1
=
(
z
-
λ
k
-
1
)
-
1
+
c
1
α
+
o
(
|
α
|
)
.
Computing similarly as in Theorem 7, we can get the leading term of e
l
(
λ
)
=
c
(
λ
-
λ
k
). We are in case (d). This completes the proof.



---

*Source: 170397-2014-03-13.xml*