principal functions of non-selfadjoint difference...

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)


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


)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


)ei(m+1)z

+∞∑

m=1


)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.


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.


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.


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.


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)


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.


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[9] E. Bairamov, Y. Aygar, and T. Koprubasi, “The spectrumof eigenparameter-dependent discrete Sturm-Liouville equations,” Journal of Computational and Applied Mathematics, vol. 235, pp. 4519–4523, 2011.

[10] G. S. Guseınov, “Determination of an infinite Jacobi matrix from scattering data,” Doklady AkademiiNauk SSSR, vol. 227, no. 6, pp. 1289–1292, 1976.

[11] G. S. Guseinov, “The inverse problem of scattering theory for a second order difference equation onthe whole axis,” Doklady Akademii Nauk SSSR, vol. 17, pp. 1684–1688, 1976.

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