Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2011, Article ID 608329, 10 pagesdoi:10.1155/2011/608329
Research ArticlePrincipal Functions of Non-SelfadjointDifference Operator with Spectral Parameter inBoundary Conditions
Murat Olgun, Turhan Koprubasi, and Yelda Aygar
Department of Mathematics, Ankara University, 06100 Ankara, Turkey
Correspondence should be addressed to Murat Olgun, [email protected]
Received 21 January 2011; Accepted 6 April 2011
Academic Editor: Svatoslav Stanek
Copyright q 2011 Murat Olgun et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
We investigate the principal functions corresponding to the eigenvalues and the spectralsingularities of the boundary value problem (BVP) an−1yn−1 + bnyn + anyn+1 = λyn, n ∈ � and(γ0 + γ1λ)y1 + (β0 + β1λ)y0 = 0, where (an) and (bn) are complex sequences, λ is an eigenparameter,and γi, βi ∈ � for i = 0, 1.
1. Introduction
Let us consider the (BVP)
−y′′ + q(x)y = λ2y, 0 ≤ x < ∞,
y′(0) − hy(0) = 0(1.1)
in L2(�+), where q is a complex-valued function and λ ∈ � is a spectral parameter andh ∈ � . The spectral theory of the above BVP with continuous and point spectrum wasinvestigated by Naımark [1]. He showed that the existence of the spectral singularities in thecontinuous spectrum of the BVP. He noted that the spectral singularities that belong to thecontinuous spectrum are the poles of the resolvents kernel but they are not the eigenvaluesof the BVP. Also he showed that eigenfunctions and the associated functions (principalfunctions) corresponding to the spectral singularities are not the element of L2(�+). Thespectral singularities in the spectral expansion of the BVP in terms of principal functions havebeen investigated in [2]. The spectral analysis of the quadratic pencil of Schrodinger, Dirac,
2 Abstract and Applied Analysis
and Klein-Gordon operators with spectral singularities was studied in [3–8]. The spectralanalysis of a non-selfadjoint difference equation with spectral parameter has been studied in[9]. In this paper, it is proved that the BVP
an−1yn−1 + bnyn + anyn+1 = λyn, n ∈ �, (1.2)
(γ0 + γ1λ
)y1 +
(β0 + β1λ
)y0 = 0 (1.3)
has a finite number of eigenvalues and spectral singularities with a finite multiplicities if
supn∈�
[exp
(εnδ
)(|1 − an| + |bn|)
]< ∞ (1.4)
for some ε > 0 and 1/2 ≤ δ ≤ 1.Let L denote difference operator of second order generated in �2(�) by
(�y
)n= an−1yn−1 + bnyn + anyn+1, n ∈ � (1.5)
and with boundary condition
(γ0 + γ1λ
)y1 +
(β0 + β1λ
)y0 = 0, γ0β1 − γ1β0 /= 0, γ1 /=a−1
0 β0, (1.6)
where {an}n∈�, {bn}n∈� are complex sequences and an /= 0 for all n ∈ � ∪ {0} and γi, βi ∈ � fori = 0, 1.
In this paper, which is extension of [9], we aim to investigate the properties of theprincipal functions corresponding to the eigenvalues and spectral singularities of the BVP(1.2)-(1.3).
2. Discrete Spectrum of (1.2)-(1.3)
Let
supn∈�
[exp
(εnδ
)(|1 − an| + |bn|)
]< ∞ (2.1)
for some ε > 0 and 1/2 ≤ δ ≤ 1. The following result is obtained in [10, 11]: under thecondition (2.1), equation (1.2) has the solution
en(z) = αneinz
(
1 +∞∑
m=1
Anmeimz
)
, n ∈ � ∪ {0} (2.2)
Abstract and Applied Analysis 3
for λ = 2 cos z, where z ∈ � + := {z : z ∈ � , Im z ≥ 0} and αn, Anm are expressed in terms of(an) and (bn) as
αn =
( ∞∏
k=n
ak
)−1,
An,1 = −∞∑
k=n+1
bk,
An,2 = −∞∑
k=n+1
(1 − a2
k
)+
∞∑
k=n+1
bk∞∑
p=k+1
bp,
An,m+2 =∞∑
k=n+1
(1 − a2
k
)Ak+1,m
∞∑
k=n+1
bkAk,m+1 +An+1,m.
(2.3)
Moreover, Anm satisfies
|Anm| ≤ C∞∑
k=n+�m/2(|1 − ak| + |bk|), (2.4)
where �m/2 is the integer part ofm/2 andC > 0 is a constant. So e(z) = {en(z)} is continuousin Im z = 0 and analytic in � + := {z : z ∈ � , Im z > 0}with respect to z.
Let us define f(z) using (2.2) and the boundary condition (1.3) as
f(z) =(γ0 + 2γ1 cos z
)e1(z) +
(β0 + 2β1 cos z
)e0(z). (2.5)
The function f is analytic in � + , continuous in � +, and f(z) = f(z + 2π).We denote the set of eigenvalues and spectral singularities of L by σd(L) and σss(L),
respectively. From the definition of the eigenvalues and spectral singularities, we have[12]
σd(L) = {λ : λ = 2 cosz, z ∈ P0, F(z) = 0},
σss(L) ={λ : λ = 2 cosz, z ∈
[−π2,3π2
], F(z) = 0
}\ {0}.
(2.6)
4 Abstract and Applied Analysis
From (2.2) and (2.5), we get
f(z) =[γ0 + γ1
(eiz + e−iz
)][
α1eiz
(
1 +∞∑
m=1
A1meimz
)]
+[β0 + β1
(eiz + e−iz
)][
α0
(
1 +∞∑
m=1
A0meimz
)]
= α0β1e−iz + γ1α1 + α0β0 +
(γ0α1 + α0β1
)eiz + γ1α1e
i2z
+∞∑
m=1
α0β1A0mei(m−1)z +
∞∑
m=1
(γ1α1A1m + α0β0A0m
)eimz
+∞∑
m=1
(γ0α1A1m + α0β1A0m
)ei(m+1)z +
∞∑
m=1
γ1α1A1mei(m+2)z.
(2.7)
Let
F(z) = f(z)eiz = α0β1 +(γ1α1 + α0β0
)eiz +
(γ0α1 + α0β1
)e2iz + γ1α1e
3iz
+∞∑
m=1
α0β1A0meimz +
∞∑
m=1
(γ1α1A1m + α0β0A0m
)ei(m+1)z
+∞∑
m=1
(γ0α1A1m + α0β1A0m
)ei(m+2)z +
∞∑
m=1
γ1α1A1mei(m+3)z;
(2.8)
then the function F is analytic in � + , continuous in � +, and F(z) = F(z + 2π). It follows from(2.6) and (2.8) that
σd(L) = {λ : λ = 2 cos z, z ∈ P0, F(z) = 0},
σss(L) ={λ : λ = 2 cosz, z ∈
[−π2,3π2
], F(z) = 0
}\ {0}.
(2.9)
Definition 2.1. The multiplicity of a zero of F in P is called the multiplicity of the correspond-ing eigenvalue or spectral singularity of the BVP (1.2) and (1.3).
3. Principal Functions
Let λ1, λ2, . . . , λp and λp+1, λp+2, . . . , λq denote the zeros of F in P0 := {z : z ∈ � , z = x +iy, −π/2 ≤ x ≤ 3π/2, y > 0} and [−π/2, 3π/2] with multiplicities m1, m2, . . . , mp andmp+1, mp+2, . . . , mq, respectively.
Abstract and Applied Analysis 5
Definition 3.1. Let λ = λ0 be an eigenvalue of L. If the vectors y(0), y(1), . . . , y(s); y(k) ={y(k)
n }n∈� k = 0, 1, . . . , s satisfy the equations
(ly(0)
)
n− λ0y
(0)n = 0,
(ly(k)
)
n− λ0y
(k)n − y
(k−1)n = 0, k = 1, 2, ..., s; n ∈ �,
(3.1)
then vector y(0) is called the eigenvector corresponding to the eigenvalue λ = λ0 of L. Thevectors y(1), . . . , y(s) are called the associated vectors corresponding to λ = λ0. The eigenvectorand the associated vectors corresponding to λ = λ0 are called the principal vectors of theeigenvalue λ = λ0.
The principal vectors of the spectral singularities of L are defined similarly.We define the vectors
V(k)n
(λj
)=
1k!
{dk
dλkEn(λ)
}∣∣∣∣∣λ=λj
, k = 0, 1, . . . , mj − 1; j = 1, 2, . . . , p,
V(k)n
(λj
)=
1k!
{dk
dλkEn(λ)
}∣∣∣∣∣λ=λj
, k = 0, 1, . . . , mj − 1; j = p + 1, p + 2, . . . , q,
(3.2)
where λ = 2 cosz, z ∈ P0, and
{En(λ)} :={en
(arccos
λ
2
)}, n ∈ � . (3.3)
Moreover, if y(λ) = {yn(λ)}n∈� is a solution of (1.2), then (dk/dλk)y(λ) = {(dk/dλk)yn(λ)}n∈�satisfies
an−1dk
dλkyn−1(λ) + bn
dk
dλkyn(λ) + an
dk
dλkyn+1(λ) = λ
dk
dλkyn(λ) + k
dk−1
dλk−1yn(λ). (3.4)
From (3.2) and (3.4), we get that
(�V (0)(λj
))
n− λjV
(0)n
(λj
)= 0,
(�V (k)(λj
))
n− λjV
(k)n
(λj
) − V(k−1)n
(λj
)= 0, k = 1, 2, . . . , mj − 1; j = 1, 2, . . . , q.
(3.5)
Consequently, the vectors V(k)n (λj); k = 0, 1, . . . , mj − 1, j = 1, 2, . . . , p and V
(k)n (λj); k =
0, 1, . . . , mj − 1, j = p + 1, p + 2, . . . , q are the principal vectors of eigenvalues and spectralsingularities of L, respectively.
6 Abstract and Applied Analysis
Theorem 3.2.
V(k)n
(λj
) ∈ �2(�) ; k = 0, 1, . . . , mj − 1, j = 1, 2, . . . , p,
V(k)n
(λj
)/∈ �2(�); k = 0, 1, . . . , mj − 1, j = p + 1, . . . , q.
(3.6)
Proof. Using En(λ) = en(arccos(λ/2)), we obtain that
{dk
dλkEn(λ)
}∣∣∣∣∣λ=λj
=k∑
ν=0
Cν
{dν
dλνen(z)
}
z=zj, n ∈ �, (3.7)
where λj = 2 cos zj ; zj ∈ P = P0 ∪ [−π/2, 3π/2], j = 1, 2, . . . , q; Cν is a constant depending onλj .
From (2.2), we find that
{dν
dzνen(z)
}
z=zj= αne
inzj
{
(in)ν +∞∑
m=1
[i(n +m)]νAnmeimzj
}
= αneinzj (in)ν + αne
inzj∞∑
m=1
[i(n +m)]νAnmeimzj .
(3.8)
For the principal vectors V(k)n (λj) = {V (k)(λj)}n∈�, k = 0, 1, . . . , mj − 1, j = 1, 2, . . . , p,
corresponding to the eigenvalues λj = 2 cos zj , j = 1, 2, . . . , p, of L, we get
{dk
dλkEn(λ)
}∣∣∣∣∣λ=λj
=k∑
ν=0
Cν
{
αneinzj (in)ν + αne
inzj∞∑
m=1
[i(n +m)]νAnmeimzj
}
; (3.9)
then
V(k)n
(λj
)=
1k!
{k∑
ν=0
Cν
[
αneinzj (in)ν + αne
inzj∞∑
m=1
[i(n +m)]νAnmeimzj
]}
(3.10)
for k = 0, 1, . . . , mj − 1, j = 1, 2, . . . , p.
Abstract and Applied Analysis 7
Since Imλj > 0, j = 1, 2, . . . , p from (3.10) we obtain that
∞∑
n=1
∣∣∣∣∣1k!
k∑
ν=0
Cναneinzj (in)ν
∣∣∣∣∣
2
≤ 1
(k!)2
[ ∞∑
n=1
k∑
ν=0|Cν ||αn|e−n Im zj |nν|
]2
≤ A
(k!)2
[ ∞∑
n=1
e−n Imzj(1 + n + n2 + · · · + nk
)]2
≤ A
(k!)2(k + 1)2
( ∞∑
n=1
e−n Im zjnk
)2
< ∞,
(3.11)
where A is a constant. Now we define the function
gn(z) =1k!
k∑
ν=0
αneinzj
∞∑
m=1
[i(n +m)]νAnmeimzj , j = 1, 2, . . . , p. (3.12)
From (2.4), we obtain that
∣∣gn(z)
∣∣ ≤
k∑
ν=0|αn|e−n Im zj
∞∑
m=1
|n +m|ν |Anm|e−m Im zj
≤ |αn|e−n Imzj
[ ∞∑
m=1
|Anm|e−m Imzj +∞∑
m=1
(n +m)|Anm|e−m Im zj
+ · · · +∞∑
m=1
(n +m)k|Anm|e−m Im zj
]
< Be−n Im zj ,
(3.13)
where B = |αn|∑∞
m=1∑k
ν=0 |Anm|e−m Imzj (n +m)ν . Therefore, we have
∞∑
n=1
∣∣gn(z)∣∣2 ≤
∞∑
n=1
B2e−2n Imzj , j = 1, 2, . . . , p
< ∞.
(3.14)
It follows from (3.11) and (3.14) that V (k)n (λj) ∈ �2(�), k = 0, 1, . . . , mj − 1, j = 1, 2, . . . , p.
8 Abstract and Applied Analysis
If we consider (3.10) for the principal vectors corresponding to the spectral singular-ities λj = 2 cos zj , j = p + 1, p + 2, . . . , q, of L and consider that Imzj = 0 for the spectralsingularities, then we have
V(k)n
(λj
)=
1k!
{k∑
ν=0
Cναneinzj (in)ν + αne
inzjk∑
ν=0
∞∑
m=1
[i(n +m)]νAnmeimzj
}
(3.15)
for k = 0, 1, . . . , mj − 1, j = p + 1, p + 2, . . . , q.Since Imλj = 0, j = p + 1, . . . , q from (3.15) we find that
1k!
∞∑
n=1
∣∣∣∣∣
k∑
ν=0
Cναneinzj (in)ν
∣∣∣∣∣
2
= ∞. (3.16)
Now we define tn(z) =∑k
ν=0∑∞
m=1 [i(n +m)]νAnmeimzj , and using (2.4) we get
|tn(z)| ≤k∑
ν=0
∞∑
m=1
∣∣(n +m)ν∣∣|Anm|
≤k∑
ν=0
∞∑
m=1
(n +m)νC∞∑
k=n+�m/2(|1 − ak | + |bk|)
≤ Ck∑
ν=0
∞∑
m=1
(n +m)ν∞∑
k=n+�m/2exp(−εk) exp(εk)(|1 − ak | + |bk|)
≤ Ck∑
ν=0
∞∑
m=1
(n +m)ν exp[−ε4(n +m)
] ∞∑
k=n+�m/2exp(εk)(|1 − ak| + |bk|)
≤ C1
k∑
ν=0
∞∑
m=1
(n +m)ν exp[−ε4(n +m)
]
= C1e(−ε/4)n
∞∑
m=1
k∑
ν=0(n +m)ν exp
(−ε4m
)
= Ae(−ε/4)n,
(3.17)
where
A = C1
∞∑
m=1
k∑
ν=0(n +m)ν exp
(−ε4m
). (3.18)
Abstract and Applied Analysis 9
If we use (3.17), we obtain that
1k!
∞∑
n=1
∣∣∣∣∣αne
inzjk∑
ν=0
∞∑
m=1
[i(n +m)]νAnmeimzj
∣∣∣∣∣
2
≤ 1k!
∞∑
n=1
α2nA
2e−εn/2
< ∞.
(3.19)
So V(k)n /∈ �2(�), k = 0, 1, . . . , mj − 1, j = p + 1, p + 2, . . . q.
Let us introduce Hilbert spaces
Hk(�) =
{
y ={yn
}n∈� :
∑
n∈�(1 + |n|)2k∣∣yn
∣∣2 < ∞}
,
H−k(�) =
{
u = {un}n∈� :∑
n∈�(1 + |n|)−2k|un|2 < ∞
}
, k = 0, 1, 2, . . . ,
(3.20)
with ‖y‖2k =∑
n∈�(1 + |n|)2k|yn|2, ‖u‖2−k =∑
n∈�(1 + |n|)−2k|un|2, respectively. It is obvious thatH0(�) = �2(�) and
Hk+1(�) �Hk(�) � l2(�) � H−k(�) �H−(k+1)(�), k = 1, 2, . . . (3.21)
Theorem 3.3. V (k)n (λj) ∈ H−(k+1)(�), k = 0, 1, . . . , mj − 1, j = p + 1, . . . , q.
Proof. From (3.15), we have
∞∑
n=1
(1 + |n|)−2(k+1)∣∣∣∣∣1k!
k∑
ν=0
Cναneinzj (in)ν
∣∣∣∣∣
2
< ∞,
∞∑
n=1
(1 + |n|)−2(k+1)∣∣∣∣∣1k!
k∑
ν=0
αneinzj
∞∑
m=1
[i(n +m)]νAnmeimzj
∣∣∣∣∣
2
< ∞
(3.22)
for k = 0, 1, . . . , mj − 1, j = p + 1, p + 2, . . . , q. Therefore, we obtain that V (k)n (λj) ∈ H−(k+1)(�),
k = 0, 1, . . . , mj − 1, j = p + 1, p + 2, . . . , q.
Let us choose m0 = max{mp+1, mp+2, . . . , mq}. By Theorem 3.2 and (3.21), we get thefollowing.
Theorem 3.4. V (k)n (λj) ∈ H−m0(�), k = 0, 1, . . . , mj − 1, j = p + 1, p + 2, . . . , q.
Proof. The proof of theorem is trivial.
10 Abstract and Applied Analysis
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