Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 608329, 10 pages
doi:10.1155/2011/608329
Research Article
Principal Functions of Non-Selfadjoint
Difference Operator with Spectral Parameter in
Boundary Conditions
Murat Olgun, Turhan Koprubasi, and Yelda Aygar
Department of Mathematics, Ankara University, 06100 Ankara, Turkey
Correspondence should be addressed to Murat Olgun, molgun@science.ankara.edu.tr
Received 21 January 2011; Accepted 6 April 2011
Academic Editor: Svatoslav Staněk
Copyright q 2011 Murat Olgun et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We investigate the principal functions corresponding to the eigenvalues and the spectral
singularities of the boundary value problem BVP an−1 yn−1 bn yn an yn1 λyn , n ∈
and
γ0 γ1 λy1 β0 β1 λy0 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 qxy λ2 y,
0 ≤ x < ∞,
y 0 − hy0 0
1.1
in L2 , where q is a complex-valued function and λ ∈ is a spectral parameter and
h ∈ . The spectral theory of the above BVP with continuous and point spectrum was
investigated by Naı̆mark 1. He showed that the existence of the spectral singularities in the
continuous spectrum of the BVP. He noted that the spectral singularities that belong to the
continuous spectrum are the poles of the resolvents kernel but they are not the eigenvalues
of the BVP. Also he showed that eigenfunctions and the associated functions principal
functions corresponding to the spectral singularities are not the element of L2 . The
spectral singularities in the spectral expansion of the BVP in terms of principal functions have
been investigated in 2. The spectral analysis of the quadratic pencil of Schrödinger, Dirac,
2
Abstract and Applied Analysis
and Klein-Gordon operators with spectral singularities was studied in 3–8. The spectral
analysis of a non-selfadjoint difference equation with spectral parameter has been studied in
9. In this paper, it is proved that the BVP
n ∈ ,
an−1 yn−1 bn yn an yn1 λyn ,
1.2
γ0 γ1 λ y1 β0 β1 λ y0 0
1.3
has a finite number of eigenvalues and spectral singularities with a finite multiplicities if
sup exp εnδ |1 − an | |bn | < ∞
n∈
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−1 yn−1 bn yn an yn1 ,
n∈
1.5
and with boundary condition
γ0 γ1 λ y1 β0 β1 λ y0 0,
γ0 β1 − γ1 β0 /
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 ∈ for
i 0, 1.
In this paper, which is extension of 9, we aim to investigate the properties of the
principal functions corresponding to the eigenvalues and spectral singularities of the BVP
1.2-1.3.
2. Discrete Spectrum of 1.2-1.3
Let
sup exp εnδ |1 − an | |bn | < ∞
n∈
2.1
for some ε > 0 and 1/2 ≤ δ ≤ 1. The following result is obtained in 10, 11: under the
condition 2.1, equation 1.2 has the solution
en z αn e
inz
1
∞
Anm e
m1
imz
,
n ∈ ∪ {0}
2.2
Abstract and Applied Analysis
for λ 2 cos z, where z ∈
an and bn as
3
: {z : z ∈ ,
αn ∞
Im z ≥ 0} and αn , Anm are expressed in terms of
−1
ak
,
kn
An,1 −
∞
bk ,
kn1
An,2 −
∞ 1 − a2k kn1
An,m2 ∞ ∞
kn1
∞
bk
2.3
bp ,
pk1
∞
bk Ak,m1 An1,m .
1 − a2k Ak1,m
kn1
kn1
Moreover, Anm satisfies
|Anm | ≤ C
∞
|1 − ak | |bk |,
2.4
knm/2
where m/2 is the integer part of m/2 and C > 0 is a constant. So ez {en z} is continuous
in Im z 0 and analytic in : {z : z ∈ , Im z > 0} with respect to z.
Let us define fz using 2.2 and the boundary condition 1.3 as
fz γ0 2γ1 cos z e1 z β0 2β1 cos z e0 z.
2.5
The function f is analytic in , continuous in , and fz fz 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 cos z, z ∈ P0 , Fz 0},
π 3π
σss L λ : λ 2 cos z, z ∈ − ,
, Fz 0 \ {0}.
2 2
2.6
4
Abstract and Applied Analysis
From 2.2 and 2.5, we get
∞
iz
−iz
iz
imz
1
α1 e
A1m e
fz γ0 γ1 e e
m1
∞
iz
−iz
imz
β0 β1 e e
α0 1 A0m e
m1
α0 β1 e−iz γ1 α1 α0 β0 γ0 α1 α0 β1 eiz γ1 α1 ei2z
∞
α0 β1 A0m eim−1z m1
2.7
∞ γ1 α1 A1m α0 β0 A0m eimz
m1
∞
∞
γ0 α1 A1m α0 β1 A0m eim1z γ1 α1 A1m eim2z .
m1
m1
Let
Fz fzeiz α0 β1 γ1 α1 α0 β0 eiz γ0 α1 α0 β1 e2iz γ1 α1 e3iz
∞
α0 β1 A0m eimz m1
∞ γ1 α1 A1m α0 β0 A0m eim1z
m1
2.8
∞
∞
γ0 α1 A1m α0 β1 A0m eim2z γ1 α1 A1m eim3z ;
m1
m1
then the function F is analytic in , continuous in , and Fz Fz 2π. It follows from
2.6 and 2.8 that
σd L {λ : λ 2 cos z, z ∈ P0 , Fz 0},
σss L π 3π
, Fz 0 \ {0}.
λ : λ 2 cos z, z ∈ − ,
2 2
2.9
Definition 2.1. The multiplicity of a zero of F in P is called the multiplicity of the corresponding eigenvalue or spectral singularity of the BVP 1.2 and 1.3.
3. Principal Functions
Let λ1 , λ2 , . . . , λp and λp1 , λp2 , . . . , 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 and
mp1 , mp2 , . . . , mq , respectively.
Abstract and Applied Analysis
5
Definition 3.1. Let λ λ0 be an eigenvalue of L. If the vectors y0 , y1 , . . . , ys ; yk k
{yn }n∈ k 0, 1, . . . , s satisfy the equations
ly 0
0
n
− λ0 yn 0,
k
k−1
ly k − λ0 yn − yn
0,
n
3.1
k 1, 2, ..., s; n ∈ ,
then vector y0 is called the eigenvector corresponding to the eigenvalue λ λ0 of L. The
vectors y1 , . . . , ys are called the associated vectors corresponding to λ λ0 . The eigenvector
and the associated vectors corresponding to λ λ0 are called the principal vectors of the
eigenvalue λ λ0 .
The principal vectors of the spectral singularities of L are defined similarly.
We define the vectors
k Vn λ j
1
k!
1
k Vn λ j k!
dk
E
,
λ
n
k
dλ
λλ
k 0, 1, . . . , mj − 1; j 1, 2, . . . , p,
j
dk
E
,
λ
n
dλk
λλ
3.2
k 0, 1, . . . , mj − 1; j p 1, p 2, . . . , q,
j
where λ 2 cos z, z ∈ P0 , and
{En λ} :
λ
,
en arccos
2
n ∈ .
3.3
Moreover, if yλ {yn λ}n∈ is a solution of 1.2, then dk /dλk yλ {dk /dλk yn λ}n∈
satisfies
an−1
dk
dk
dk
dk
dk−1
y
y
y
y
yn λ.
b
a
λ
k
λ
λ
λ
λ
n−1
n
n
n
n1
n
dλk
dλk
dλk
dλk
dλk−1
3.4
From 3.2 and 3.4, we get that
0 − λj Vn λj 0,
V 0 λj
n
V
k λj
n
−
k λ j Vn λ j
−
k
k−1 Vn
λj
3.5
0,
k 1, 2, . . . , mj − 1; j 1, 2, . . . , q.
k
Consequently, the vectors Vn λj ; k 0, 1, . . . , mj − 1, j 1, 2, . . . , p and Vn λj ; k 0, 1, . . . , mj − 1, j p 1, p 2, . . . , q are the principal vectors of eigenvalues and spectral
singularities of L, respectively.
6
Abstract and Applied Analysis
Theorem 3.2.
k Vn
λj ∈ 2 ;
k Vn λ j
k 0, 1, . . . , mj − 1, j 1, 2, . . . , p,
∈
/ 2 ;
k 0, 1, . . . , mj − 1, j p 1, . . . , q.
3.6
Proof. Using En λ en arccosλ/2, we obtain that
ν
k
d
dk
E
C
e
,
n λ ν
n z
dλν
dλk
zz
j
ν0
λλ
n ∈ ,
3.7
j
where λj 2 cos zj ; zj ∈ P P0 ∪ −π/2, 3π/2, j 1, 2, . . . , q; Cν is a constant depending on
λj .
From 2.2, we find that
dν
en z
dzν
αn e
zzj
inzj
ν
in ∞
ν
in m Anm e
imzj
m1
αn einzj inν αn einzj
∞
3.8
in mν Anm eimzj .
m1
k
For the principal vectors Vn 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
k
∞
dk
ν
ν
inzj
inzj
imzj
;
En λ Cν αn e in αn e
in m Anm e
dλk
ν0
m1
λλ
3.9
j
then
k Vn λ j
k
∞
1 ν
ν
inzj
inzj
imzj
Cν αn e in αn e
in m Anm e
k! ν0
m1
for k 0, 1, . . . , mj − 1, j 1, 2, . . . , p.
3.10
Abstract and Applied Analysis
7
Since Im λj > 0, j 1, 2, . . . , p from 3.10 we obtain that
2
2
∞ k
∞ k
1 1
ν
inzj
−n Im zj ν
C α e in ≤
|Cν ||αn |e
|n |
k! ν0 ν n
k!2 n1 ν0
n1
∞
A
≤
k!2
A
≤
e
−n Im zj
1 n n ··· n
2
k
n1
2
k!2
k 1
∞
2
3.11
2
e
−n Im zj
n
k
n1
< ∞,
where A is a constant. Now we define the function
gn z k
∞
1
αn einzj
in mν Anm eimzj ,
k! ν0
m1
j 1, 2, . . . , p.
3.12
From 2.4, we obtain that
k
∞
gn z ≤
|αn |e−n Im zj
|n m|ν |Anm |e−m Im zj
ν0
m1
≤ |αn |e
−n Im zj
∞
|Anm |e−m Im zj m1
···
∞
n m|Anm |e−m Im zj
3.13
m1
∞
n mk |Anm |e−m Im zj
m1
< Be−n Im zj ,
where B |αn |
∞
m1
k
ν0
|Anm |e−m Im zj n mν . Therefore, we have
∞ ∞
gn z2 ≤
B2 e−2n Im zj ,
n1
n1
j 1, 2, . . . , p
3.14
< ∞.
It follows from 3.11 and 3.14 that Vn λj ∈ 2 , k 0, 1, . . . , mj − 1, j 1, 2, . . . , p.
k
8
Abstract and Applied Analysis
If we consider 3.10 for the principal vectors corresponding to the spectral singularities λj 2 cos zj , j p 1, p 2, . . . , q, of L and consider that Im zj 0 for the spectral
singularities, then we have
k Vn λ j
k
k ∞
1 ν
ν
inzj
inzj
imzj
Cν αn e in αn e
in m Anm e
k! ν0
ν0 m1
3.15
for k 0, 1, . . . , mj − 1, j p 1, p 2, . . . , q.
Since Im λj 0, j p 1, . . . , q from 3.15 we find that
2
∞ k
1
ν
inzj
Cν αn e in ∞.
k! n1 ν0
Now we define tn z |tn z| ≤
k ∞
ν0
m1
3.16
in mν Anm eimzj , and using 2.4 we get
k ∞ n mν |Anm |
ν0 m1
≤
k ∞
ν0 m1
≤C
k ∞
k ∞
|1 − ak | |bk |
knm/2
∞
n mν
ν0 m1
≤C
∞
n mν C
exp−εk expεk|1 − ak | |bk |
knm/2
n mν exp
ν0 m1
∞
−ε
expεk|1 − ak | |bk |
n m
4
knm/2
3.17
−ε
≤ C1
n m exp
n m
4
ν0 m1
k ∞
C1 e−ε/4n
ν
k
∞ n mν exp
m1 ν0
−ε
m
4
Ae−ε/4n ,
where
A C1
∞ k
m1 ν0
n mν exp
−ε
m .
4
3.18
Abstract and Applied Analysis
9
If we use 3.17, we obtain that
2
∞ k ∞
∞
1
1
ν
α2 A2 e−εn/2
in m Anm eimzj ≤
αn einzj
k! n1 k! n1 n
ν0 m1
3.19
< ∞.
k
So Vn
∈
/ 2 , k 0, 1, . . . , mj − 1, j p 1, p 2, . . . q.
Let us introduce Hilbert spaces
Hk 2
y yn n∈ :
1 |n| yn < ∞ ,
n∈
H−k u {un }n∈ :
2k 3.20
1 |n|−2k |un |2 < ∞ ,
n∈
k 0, 1, 2, . . . ,
with y
2k n∈ 1 |n|2k |yn |2 , u
2−k n∈ 1 |n|−2k |un |2 , respectively. It is obvious that
H0 2 and
Hk1 Hk l2 H−k H−k1 ,
k 1, 2, . . .
3.21
Theorem 3.3. Vn λj ∈ H−k1 , k 0, 1, . . . , mj − 1, j p 1, . . . , q.
k
Proof. From 3.15, we have
∞
2
k
1
ν
inzj
Cν αn e in < ∞,
k! ν0
−2k1 1 |n|
n1
2
k
∞
−2k1 1
ν
inzj
imzj αn e
1 |n|
in m Anm e
<∞
k! ν0
n1
m1
∞
3.22
for k 0, 1, . . . , mj − 1, j p 1, p 2, . . . , q. Therefore, we obtain that Vn λj ∈ H−k1 ,
k 0, 1, . . . , mj − 1, j p 1, p 2, . . . , q.
k
Let us choose m0 max{mp1 , mp2 , . . . , mq }. By Theorem 3.2 and 3.21, we get the
following.
Theorem 3.4. Vn λj ∈ H−m0 , k 0, 1, . . . , mj − 1, j p 1, p 2, . . . , q.
k
Proof. The proof of theorem is trivial.
10
Abstract and Applied Analysis
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