Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 924628, 15 pages
doi:10.1155/2012/924628
Research Article
Principal Functions of Nonselfadjoint
Discrete Dirac Equations with Spectral Parameter
in Boundary Conditions
Yelda Aygar,1 Murat Olgun,1 and Turhan Koprubasi2
1
2
Department of Mathematics, Ankara University, Tandogan, 06100 Ankara, Turkey
Department of Mathematics, Kastamonu University, 37100 Kastamonu, Turkey
Correspondence should be addressed to Yelda Aygar, yaygar@science.ankara.edu.tr
Received 18 July 2012; Accepted 12 September 2012
Academic Editor: Patricia J. Y. Wong
Copyright q 2012 Yelda Aygar 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.
2
2
1
1
1
Let L denote the operator generated in 2 N, C2 by an1 yn1 bn yn pn yn λyn , an−1 yn−1 1
bn yn
2
qn yn
2
λyn ,
2
boundary condition γ0 γ1 λy1 are complex sequences, γi , βi ∈ C,
1
β1 λy0
n ∈ N, and the
β0 0,
i 0, 1, and λ is
where an , bn ,pn , and qn , n ∈ N
an eigenparameter. In this paper we investigated the principal functions corresponding to the
eigenvalues and the spectral singularities of L.
1. Introduction
Consider the boundary value problem BVP
−y qxy λ2 y,
y0 0,
0 ≤ x < ∞,
1.1
in L2 R , where q is a complex-valued function and λ ∈ C is a spectral parameter. The
spectral theory of the above BVP with continuous and point spectrum was investigated
by Naı̆mark 1. He showed the existence of the spectral singularities in the continuous
spectrum of 1.1. Note that the eigen and associated functions corresponding to the spectral
singularities are not the elements of L2 R .
In 2, 3 the effect of the spectral singularities in the spectral expansion in terms
of the principal vectors was considered. Some problems related to the spectral analysis of
2
Abstract and Applied Analysis
difference equations with spectral singularities were discussed in 4–7. The spectral analysis
of eigenparameter dependent nonselfadjoint difference equation was studied in 8, 9.
Let us consider the nonselfadjoint BVP for the discrete Dirac equations
2
2
1
1
n ∈ N,
1
1
2
2
n ∈ N,
an1 yn1 bn yn pn yn λyn ,
an−1 yn−1 bn yn qn yn λyn ,
2 1
γ0 γ1 λ y1 β0 β1 λ y0 0,
where
γ0 β1 − γ1 β0 /
0,
1.2
γ1 /
a−1
0 β0 ,
1.3
, n ∈ N are vector sequences, an / 0, bn / 0 for all n ∈ N, γi , βi ∈ C, and i 0, 1
2
1
yn
yn
and, λ is a spectral parameter.
In 10 the authors proved that eigenvalues and spectral singularities of 1.2-1.3
have a finite number with finite multiplicities, if the condition,
∞
exp εnδ |1 − an | |1 bn | pn qn < ∞
1.4
n1
holds, for some ε > 0 and 1/2 ≤ δ < 1.
In this paper, we aim to investigate the principal functions corresponding to the
eigenvalues and the spectral singularities of the BVP 1.2-1.3.
2. Discrete Spectrum of 1.2-1.3
Let for some ε > 0 and 1/2 ≤ δ < 1,
∞
exp εnδ |1 − an | |1 bn | pn qn < ∞,
2.1
n1
be satisfied. It has been shown that 10 under the condition 2.1, 1.2 has the solution
fn z 1
fn z
2
fn z
αn I Anm e
m1
1
f0 z
∞
α11
0
e
iz/2
1
∞
imz
eiz/2 inz
e ,
−i
imz
A11
0m e
−i
m1
∞
n 1, 2, . . . ,
2.2
imz
A12
0m e
2.3
,
m1
for λ 2 sinz/2 and z ∈ C : {z ∈ C : Im z ≥ 0}, where
αn 12
α11
n αn
22
α21
n αn
,
1 0
I
,
0 1
Anm 12
A11
nm Anm
22
A21
nm Anm
.
2.4
Abstract and Applied Analysis
ij
3
ij
Note that αn and Anm i, j 1, 2 are uniquely expressed in terms of an , bn , pn , and
qn , n ∈ N as follows
α11
n
∞
−1
−1
n−k
bk ak−1
,
kn1
α12
n 0,
α22
n
bn
−1
∞
−1
n−k1
bk ak−1
,
kn1
α21
n
α22
n
∞
pn pk qk
,
kn1
∞
pk qk ,
A12
n1 −
kn1
A11
n1 ∞ ak1 ak bk2 − pk qk pk qk A12
k1 − 2 ,
kn1
2
12
−
1
a
a
A
A11
A22
n1
n
n1
n1
n1 ,
A21
n1 −
∞ qk1 A12
k1
ak1 ak qk1 pk1 qk1 qk1 A12
k1
kn
∞ 2
12
11
2
qk A22
bk1
A11
k1,1 − 1 − Ak1 1 Ak1
k1 − bk pk ,
kn1
12
12 11
12
21
A12
n2 − an1 an qn1 An1 An1 An1 An1 − An1 ,
A11
n2 ∞ 12
12
22
bk2 − 1 A11
q
A
−
a
a
A
−
A
k1
k
k1
k1
k1
k1,1
k1,1
kn1
11
12
12
21
12 12
22
− pk − A12
q
−
q
A
A
−
A
A
A
A
−
A
k
k
k1
k1
k1
k2
k1
k1 k2
k1 ,
12
12
22
12 12
11
11
A22
n2 − an1 an qn1 An1 An1,1 an1 an An1,1 An1 An2 − An1 An2 ,
A21
n2 ∞ 21
11
11
21
A12
qk1 A12
k1 Ak2 Ak2 − ak1 ak
k1 Ak1,1 − Ak1,1
kn
−
∞ kn1
qk A12
k−1,1
11
11
2 21
22
21
qk A12
k2 − Ak1 Ak2 bk Ak2 − pk Ak2 Ak1 ,
2.5
4
Abstract and Applied Analysis
and for m ≥ 3
A12
nm − an1 an
11
21
qn1 A12
n1 An1,m−2 An1,m−2
11
12
21
A12
n1 An,m−1 An,m−1 − An,m−1 ,
A11
nm −
∞
ak1 ak
12
22
qk1 A12
k1 Ak1,m−1 − Ak1,m−1
kn1
−
∞ pk − A12
k1
∞ 12
12
2
qk A11
b
A
−
A
−
1
A11
k,m−1
k,m−1
km
k
k,m−1
kn1
−
∞
kn1
qk A21
k,m−1 kn1
∞
12
A12
k1 Akm −
kn1
A22
nm − an1 an
A21
nm −
∞
∞
A22
k,m−1 ,
2.6
kn1
11
22
12 12
11
11
qn1 A12
n1 An1,m−1 − An1,m−1 An1 Anm Anm − An,m−1 ,
ak1 ak
11
21
qk1 A12
k1 Ak1,m−1 − Ak1,m−1
kn
−
∞ qk − A12
k−1,1
∞ 11
22
qk A21
bk2 − 1 A12
km Ak,m−1 − Akm −
km
kn1
∞
kn1
22
A12
k1 Akm kn
∞
qk A22
km kn1
∞
kn
A12
km −
∞
A21
k,m−1 .
kn1
Moreover
ij Anm ≤ C
∞
|1 − ak | |1 bk | pk qk 2.7
kn|m/2|
holds, where |m/2| is the integer part of m/2 and C > 0 is a constant. Therefore fn is vectorvalued analytic function with respect
in C : {z ∈ C : Im z > 0} and continuous in C
to z 10. The solution fz fn z 1
fn z
2
fn z
is called Jost solution of 1.2.
Let us define
z 2
z 1
Fz γ0 2γ1 sin
f1 z β0 2β1 sin
f z.
2
2 0
2.8
It follows 2.2 and 2.3 that the function F is analytic in C , continuous up to the real axis,
and
Fz 4π Fz.
2.9
Abstract and Applied Analysis
5
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 10
z
σd λ : λ 2 sin , z ∈ P0 , Fz 0 ,
2
z
σss λ : λ 2 sin , z ∈ 0, 4π, Fz 0 ,
2
2.10
where P0 : {z : z ∈ C, z x iy, 0 ≤ x ≤ 4π, y > 0}. The finiteness of the multiplicities
of eigenvalues and spectral singularities has been proven in 10. Using 2.2, 2.3, and 2.8
we obtain
2
f1 z
Fz γ0 γ1 −i eiz/2 − e−iz/2
1
f0 z
β0 β1 −i eiz/2 − e−iz/2
22
11
iz/2
22
11
iz
i −γ0 α22
iα11
0 β1 γ1 α1 α0 β0 e
1 γ1 α1 − α0 β1 e
22
i3z/2
2iz
γ0 α21
− iγ1 α21
1 − γ1 α1 e
1 e
∞
12 im−1/2z
α11
i
0 β1 A0m e
m1
∞ ∞ 12
11
11
imz
−α11
0 β0 A0m α0 β1 A0m e
m1
12
22 22
11
11
11
12
im1/2z
γ1 α21
1 A1m γ1 α1 A1m α0 β0 A0m − α0 β1 A0m e
m1
i
∞ m1
∞ 2.11
12
22 22
21 11
22 21
−γ0 α21
1 A1m − γ0 α1 A1m γ1 α1 A1m γ1 α1 A1m
11
im1z
−α11
β
A
1
0
0m e
11
22 21
21 12
22 22
im3/2z
γ0 α21
1 A1m γ0 α1 A1m − γ1 α1 A1m − γ1 α1 A1m e
m1
i
∞ 11
22 21
im2z
−γ1 α21
.
1 A1m − γ1 α1 A1m e
m1
i
∞ 11
22 21
im2z
−γ1 α21
.
1 A1m − γ1 α1 A1m e
m1
Definition 2.1. The multiplicity of a zero of F in P : P0 ∪ 0, 4π is called the multiplicity of
the corresponding eigenvalue or spectral singularity of the BVP 1.2, 1.3.
3. Principal Functions
In this section we also assume that 2.1 holds.
Let λ1 , λ2 , . . . , λk and λk1 , λk2 , . . . , λν denote the zeros of F in P0 and 0, 4π with
multiplicities m1 , m2 , . . . , mk and mk1 , mk2 , . . . , mν , respectively.
6
Abstract and Applied Analysis
Let us define :
y
y
where
2
n
n
2
1
an1 yn1 bn yn pn yn ,
1
an−1 yn−1
1
bn yn
2
qn yn ,
n ∈ N,
3.1
n ∈ N.
Definition 3.1. Let λ λ0 be an eigenvalue of L. If the vectors yn , d/dλyn , d2 /dλ2 yn , . . . ,
dν /dλν yn ,
dj
y :
dλj
dj
yn
dλj
,
j 0, 1, . . . , ν; n ∈ N ,
3.2
n∈N
satisfy the equations
dj
y
dλj
− λ0
n
y
n
− λ0 yn 0,
dj
dj−1
y
−
yn 0,
n
dλj
dλj−1
j 1, 2, . . . , ν, n ∈ N,
3.3
then the vector yn is called the eigenvector corresponding to the eigenvalue λ λ0 of L. The
vectors d/dλyn , d2 /dλ2 yn , . . . , dν /dλν yn 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
⎞
⎛ j
1 d 1
En λ
⎟
⎜
⎟
⎜ j! dλj
dj
λλi ⎟
⎜
V
λ
⎟,
⎜
n i
⎟
⎜ 1 dj
dλj
2
⎠
⎝
E
λ
n
j! dλj
λλ
3.4
i
n ∈ N,
j 0, 1, . . . , mi − 1,
i 1, 2, . . . , k, k 1, . . . , ν,
where λ 2 sinz/2 and
λ ⎞
1
2 arcsin
En λ
⎜
2 ⎟
⎠ : fn 2 arcsin λ ⎜
En λ ⎝
⎟.
⎝
2
2
λ ⎠
2
En λ
fn 2 arcsin
2
⎛
⎞
⎛
1
fn
3.5
Abstract and Applied Analysis
7
If
yλ yn λ :
1
yn λ
2
yn λ
3.6
n∈N
is a solution of 1.2, then
dj
yλ dλj
dj
dλj
yn λ
n∈N
⎫
⎧
⎪
⎪
dj
⎪
⎪
1
⎪
yn λ⎪
⎪
⎪
⎪
⎪
j
⎬
⎨ dλ
: ⎪
⎪
⎪
⎪
dj
⎪
⎪
2
⎪
⎪
y
λ
⎪
⎪
n
⎭
⎩
j
dλ
3.7
satisfies
⎞
dj 2
dj 2
dj 1
⎜an−1 j yn1 λ bn j yn λ pn j yn λ⎟
dλ
dλ
dλ
⎟
⎜
⎟
⎜
j
j
j
⎝
d 1
d 1
d 2 ⎠
an−1 j yn−1 λ bn j yn λ qn j yn λ
dλ
dλ
dλ
⎛
⎞
dj 1
dj−1 1
⎜λ j yn λ j j−1 yn λ⎟
dλ
⎜ dλ
⎟
⎜
⎟.
j−1
⎝ dj 2
⎠
d
2
λ j yn λ j j−1 yn λ
dλ
dλ
⎛
3.8
From 3.4 and 3.8 we get that
V λi n − λ0 Vn λi 0,
dj
dj
dj−1
V λi − λ0 j Vn λi −
Vn λi 0,
j
dλ
dλ
dλj−1
n
n ∈ N,
j 1, 2, . . . , mi − 1,
3.9
i 1, 2, . . . , ν.
The vectors dj /dλj Vn λi , j 0, 1, 2, . . . , mi − 1, i 1, 2, . . . , k and dj /dλj Vn λi , j 0, 1, 2, . . . ,
mi − 1, i k 1, k 2, . . . , ν are the principal vectors of eigenvalues and spectral singularities
of L, respectively.
Theorem 3.2.
dj
Vn λi ∈ 2 N, C2 , j 0, 1, 2, . . . , mi − 1, i 1, 2, . . . , k,
j
dλ
j
d
2
N,
C
, j 0, 1, 2, . . . , mi − 1, i k 1, k 2, . . . , ν.
V
∈
/
λ
n
i
2
dλj
3.10
8
Abstract and Applied Analysis
Proof. Using 3.5 we get that
dj 1
En λ
dλj
dj 2
En λ
dλj
j
λλi
Ct
t0
λλi
&
dt 1
fn z
dλt
'
,
n ∈ N,
zzi
& t
'
j
d 2
Dt
fn z
,
dλt
zzi
t0
3.11
n ∈ N,
where λi 2 sin zi /2, zi ∈ P P0 ∪ 0, 4, i 1, 2, . . . , k and Ct , Dt are constant depending on
λ. From 2.2 we obtain that
&
dt 1
fn z
dλt
'
zzi
1 t izi n1/2
t
α11
i
e
n
n
2
∞
1 t imn1/2zi
11
t
αn A11
i
e
m
n
nm
2
m1
3.12
t1
−A12
m nt eimnzi ,
nm i
&
dt 2
fn z
dλt
'
zzi
1 t izi n1/2
inzi
e
− iint α22
n
n e
2
∞
1 t imn1/2zi
21
t
αn A11
e
nm i m n 2
m1
t
α21
n i
t1
−A12
m nt eimnzi
nm i
∞
α22
n
m1
t
A21
nm i
3.13
1 t imn1/2zi
e
mn
2
t1
−A22
m nt eimnzi .
nm i
For the principal vectors dj /dλj Vn λi {dj /dλj Vn λi }n∈N , j 0, 1, . . . , mi − 1, i 1, 2,
. . . , k corresponding to the eigenvalues of L we get
1
j!
1
j!
dj 1
En λ
dλj
λλi
& t
'
j
1j d 1
Ct
f
,
z
i
n
j! t0
dλt
j 0, 1, . . . , mi − 1, i 1, 2, . . . , k,
& t
'
j
dj 2
d 2
1j E
D
f
,
λ
z
t
i
n
n
j! t0
dλt
dλj
λλi
j 0, 1, . . . , mi − 1,
i 1, 2, . . . , k.
3.14
Abstract and Applied Analysis
9
Since Im λi > 0 for i 1, 2, . . . , k from 3.14 we obtain that
⎛ (
(2
2 2 ⎞
∞
1 dj
( dj
(
1 dj
(
(
⎠
1
2
⎝
E
E
λ
λ
( j Vn ( n
n
j
j
( dλ
(
j!
j!
dλ
dλ
n1
λλi
λλi
⎛
⎞
2 & t
& t
'2 '2
j
j
∞
1
d
d
1
2
⎝ Ct
fn zi Dt
fn zi ⎠
t0
j! n1 t0
dλt
dλt
2 & t
'2
'2 j & t
j ∞
1
d
d
1
2
Ct
Dt
≤
fn zi fn zi t
t
j! n1
dλ
dλ
t0
t0
2 ' & t
'2
& t
j
∞ d 2
d 1
1
≤
max{|Ct |, |Dt |} fn zi fn zi ,
t
t
j!
dλ
dλ
n1 t0
3.15
or
(2
(
(
( dj
(
(
( j Vn (
(
( dλ
j
2 ∞ 1
≤
max{|Ct |, |Dt |}
j!
n1 t0
1 t −n1/2 Im zi
22 t −n Im zi
11 21 αn |n| e
×
αn αn n e
2
j
∞ 1 t −mn1/2 Im zi
11 21 11 max{|Ct |, |Dt |}
αn αn Anm m n e
2
t0
m1
t −mn Im zi
A12
nm |m n| e
j
∞ 22 21 |Dt |
αn Anm m n t0
1 t −mn1/2 Im zi 22 e
Anm |m n|t
2
2
m1
×e−mn Im zi
.
3.16
From 3.16,
⎫
⎧
t
−n1/2 Im z ⎪
11 21 ⎪
1
⎪
⎪
2 i
j
∞ ⎨ αn αn n e
⎬
1
2
max{|Ct |, |Dt |}
⎪
⎪
j! n1 t0
⎪
⎪
t −n Im z
⎩
⎭
i
e
α22
|n|
n
∞
1
1 2
1 j
A 1 n
e−n1/2 Im zi
n
··· n ≤ 2
2
2
2
j! n1
10
Abstract and Applied Analysis
1 n n2 · · · nj e−n Im zi
2 ∞ A j 1 1 j −n1/2 Im zi
j −n Im zi
< ∞.
≤ 2
e
n e
n
2
j!
n1
3.17
Holds, where
A max{|Ct |, |Dt |} max
11 21 22 αn αn , αn .
3.18
Now we define the function
gn z j
t0
∞ 11 21 11 max{|Ct |, |Dt |}
αn αn Anm m n m1
1 t −mn1/2 Im zi
e
2
t −mn Im zi
A12
nm |m n| e
j
t0
∞ 22 21 |Dt |
αn Anm m n m1
1 t −mn1/2 Im zi
e
2
t −mn Im zi
A22
nm |m n| e
.
3.19
So we get
j
2 ∞ ∞ 1
11 21 11 max{|Ct |, |Dt |}
αn αn Anm m n j! n1 t0
m1
1 t −mn1/2 Im zi
e
2
t −mn Im zi
A12
nm |m n| e
j
t0
∞ 22 21 |Dt |
αn Anm m n m1
1 t −mn1/2 Im zi
e
2
t −mn Im zi
A22
nm |m n| e
2 ∞
1
gn z.
j! n1
3.20
Abstract and Applied Analysis
11
ij
ij
Using the boundness of Anm and αn , i, j 1, 2 we obtain that
j ∞
m n gn z ≤ max{|Ct |, |Dt |}M
t0 m1
1 t −mn1/2 Im zi
e
2
t −mn Im zi
|m n| e
3.21
,
where
M max
11 21 11 22 21 αn αn Anm , αn Anm ,
11 21 12 22 22 αn αn Anm , αn Anm .
3.22
If we take max{|Ct |, |Dt |}M N, we can write
j
∞
1 t −m Im zi
t −m Im zi
−n Im zi
gn z ≤ N e
e
m n e
mn
2
t0
m1
∞
∞
1
−n Im zi
−m Im zi
−m Im zi
m n · · ·
Ne
2e
e
mn
2
m1
m1
∞
1 j
j
−m Im zi
e
m n
mn
2
m1
≤ Ne
−n Im zi
j
∞ e−m Im zi
m1 t0
1
mn
2
t
3.23
m n
t
≤ Be−n Im zi ,
where
BA
j
∞ m1 t0
e
−m Im zi
1
mn
2
t
m n
t
.
3.24
Therefore we have
2 2 ∞
∞
1
1
gn z ≤
Be−n Im zi < ∞.
j! n1
j! n1
From 3.17 and 3.25 dj /dλj Vn λi ∈ 2 N, C2 , j 0, 1, . . . , mi − 1, i 1, 2, . . . , k.
3.25
12
Abstract and Applied Analysis
On the other hand, since Im zi 0 for j 0, 1, . . . , mi − 1; i k 1, k 2, . . . , ν using
3.12 we find that
2
∞ 1 t izi n1/2 11 t
e
αn i n ∞,
2
n1
3.26
1
but the other terms in 3.12 belongs 2 N, C2 , so dj /dλj En λ ∈
/ 2 N, C2 . Similarly from
2
j
j
2
j
3.13 we get d /dλ En λ ∈
/ 2 N, C , then we obtain that d /dλj Vn λi ∈
/ 2 N, C2 ,
j 0, 1, . . . , mi − 1; i k 1, k 2, . . . , ν.
Let us introduce Hilbert space j 0, 1, 2, . . .
1 2 yn
2 2
−2j 1 :
H−j N y 1 |n|
yn yn < ∞ ,
2
yn
n∈N
with
2 ( (2
2 2
−2j 1 (y( .
1
|n|
y
y
n
n −j
3.27
3.28
n∈N
Theorem 3.3. dj /dλj Vn λi ∈ H−j1 N, j 0, 1, 2, . . . , mi − 1, i k 1, k 2, . . . , ν.
Proof. Using 3.4, 3.14 we have
⎛ 2 2 ⎞
1 dj
1 dj
⎠
1
2
E
E
1 |n|−2j1 ⎝
λ
λ
n
n
j
j
j!
j!
dλ
dλ
n∈N
λλi
λλi
⎧
⎫
& t
& t
'2 '2 ⎬
j
j
1 |n|−2j1 ⎨
d 1
d 2
f zi Dt
f zi Ct
2
t n
t n
⎭
⎩
dλ
dλ
j!
t0
t0
n∈N
⎧
⎫
& t
'2 '2 ⎬
j & t
j ∞
⎨
1
Ct d fn1 zi Dt d fn2 zi ≤ 2 1 |n|−2j1
⎭,
t
⎩
dλ
dλt
j! n1
t0
t0
3.29
for j 0, 1, 2, . . . , mi − 1, i k 1, k 2, ν. Since Im zi 0, using 3.29 we obtain
j &
'2
1
dt 1
1 |n|
2
Ct dλt fn zi j!
t0
n∈N
j
∞
1 1 t 11 −j1
2
n
1 |n|
αn |Ct |
2
j! n1 t0
−2j1
2
∞ 1 t 12 11 11 −j1
t
|Ct |αn 1 |n|
Anm m n
Anm m n 2
t0
m1
j
Abstract and Applied Analysis
⎧
2
j
∞ ⎨ 1 t 11 1 −j1
n
2
1 |n|
αn |Ct |
2
j! n1⎩ t0
21 j 2 |n|−2j1 α11
n t0
1
n
2
t
13
|Ct |
j
∞ 1 t 12 11 t
×
Anm m n
|Ct | Anm m n 2
t0
m1
2 ⎫
t ∞ ⎬
1
11 t
A12
.
|Ct |1 |n|−j1 α11
Anm m n n nm m n
⎭
2
t0
m1
j
3.30
Using 3.30, 2.1, and 2.7 we first obtain that
j
2
∞
1 t 12 11 11 −j1
t
Anm m n
|Ct |αn 1 |n|
Anm m n 2
t0
m1
j
∞
1 t
−j1
≤4
C
m
n
1
|n|
|Ct |α11
n m1
2
t0
∞
×
⎞2
−εj δ εj δ
1 − aj 1 bj pj qj e e ⎠
jnm/2
≤4
j ∞
1 t
n m δ
11 −j1
m
n
C
exp
−ε
1
|n|
αn 2
4
t0
m1
∞
×
eεj
jnm/2
≤ C1
j
1 |n|
δ
⎫2
⎬
1 − aj 1 bj pj qj ⎭
−j1
t0
≤ C1
j
t0
∞ m1
1 |n|
−j1
∞ m1
1
mn
2
1
mn
2
t
t
exp −ε
exp −ε
n m δ 2
4
n m 1/2 2
4
⎞⎞⎞2
⎛
⎛
√
1
1
t
j
∞ 2⎜
1
⎟⎟⎟
⎜
⎜
≤ C1 ⎝ 1 |n|−j1
exp⎝−ε
mn
⎝n 2 m 2 ⎠⎠⎠
2
4
t0
m1
⎛
14
Abstract and Applied Analysis
√
j
2
∞ 2 1/2
2 1/2
1 t
n
m
exp −ε
exp −ε
mn
C1 1 |n|
4
2
4
t0 m1
⎞
⎛
√ 1
2
⎟
⎜
n 2 ⎠1 |n|−2j1 ,
G exp⎝−ε
4
√
−2j1
3.31
where
⎛
C1 ⎝2Cα11
n ⎞2
δ
eεj 1 − aj 1 bj pj qj ⎠ ,
∞
jnm/2
j
2
√
∞ 2 1/2
1 t
m
exp −ε
.
G C1
mn
2
4
t0 m1
3.32
So we get from 3.31
j
2
∞
∞ 1 t 12 11 −j1 11 t
Anm m n
|Ct |1 |n|
αn Anm m n 2
t0
n1
m1
√
∞
2 1/2
n
≤ G exp −ε
1 |n|−2j1 < ∞.
4
n1
3.33
Secondly, using 3.30 and 3.31 we obtain that
j
∞
1 t
−j1
11
2
n
αn |Ct |1 |n|
2
t0
n1
j
∞
1 t 11 11 −j1
t 12 ×
mn
|Ct |αn 1 |n|
Anm m n Anm 2
t0
m1
≤
j
∞ n1
1 |n|
−2j1
t0
1
n
2
t
3.34
2 1/2
1/2
G
< ∞,
n
exp −ε
4
√
and also the first part of the 3.31 obviously convergent so, we get from 3.33 and 3.34
n∈N
1 |n|
−2j1
1
2
j!
j &
'2
t
Ct d fn1 zi < ∞,
dλt
t0
3.35
Abstract and Applied Analysis
15
and similarly
n∈N
−2j1
1 |n|
1
2
j!
j &
'2
t
Dt d fn2 zi < ∞.
dλt
3.36
t0
Finally dj /dλj Vn λi ∈ H−j1 N, j 0, 1, 2, . . . , mi − 1, i k 1, k 2, . . . , ν.
References
1 M. A. Naı̆mark, “Investigation of the spectrum and the expansion in eigenfunctions of a nonselfadjoint differential operator of the second order on a semi-axis,” American Mathematical Society
Translations, vol. 16, pp. 103–193, 1960.
2 V. E. Lyance, “A differential operator with spectral singularities, I, II,” American Mathematical Society
Translations, vol. 2, no. 60, pp. 185–225, 1967.
3 M. Adıvar, “Quadratic pencil of difference equations: jost solutions, spectrum, and principal vectors,”
Quaestiones Mathematicae, vol. 33, no. 3, pp. 305–323, 2010.
4 M. Adıvar and M. Bohner, “Spectral analysis of q-difference equations with spectral singularities,”
Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 695–703, 2006.
5 M. Adivar and M. Bohner, “Spectrum and principal vectors of second order q-difference equations,”
Indian Journal of Mathematics, vol. 48, no. 1, pp. 17–33, 2006.
6 E. Bairamov and A. O. Çelebi, “Spectrum and spectral expansion for the non-selfadjoint discrete Dirac
operators,” The Quarterly Journal of Mathematics, vol. 50, no. 200, pp. 371–384, 1999.
7 E. Bairamov and C. Coskun, “Jost solutions and the spectrum of the system of difference equations,”
Applied Mathematics Letters, vol. 17, no. 9, pp. 1039–1045, 2004.
8 E. Bairamov, Y. Aygar, and T. Koprubasi, “The spectrum of eigenparameter-dependent discrete SturmLiouville equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4519–4523,
2011.
9 M. Adıvar and E. Bairamov, “Spectral properties of non-selfadjoint difference operators,” Journal of
Mathematical Analysis and Applications, vol. 261, no. 2, pp. 461–478, 2001.
10 E. Bairamov and T. Koprubasi, “Eigenparameter dependent discrete Dirac equations with spectral
singularities,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4216–4220, 2010.
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