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MATHEMATICAL COMMUNICATIONS 65Math. Commun. 19(2014), 6573

Completeness of the system of root functions of

q-Sturm-Liouville operators

Huseyin Tuna1 and Aytekin Erylmaz2,

1 Department of Mathematics, Mehmet Akif Ersoy University, 15 100, Burdur, Turkey2 Department of Mathematics, Nevsehir University, 50 300, Nevsehir, Turkey

Received May 30, 2013; accepted November 30, 2013

Abstract. In this paper, we study q-Sturm-Liouville operators. We construct a spaceof boundary values of the minimal operator and describe all maximal dissipative, maxi-mal accretive, self-adjoint and other extensions of q-Sturm-Liouville operators in terms ofboundary conditions. Then we prove a theorem on completeness of the system of eigen-

functions and associated functions of dissipative operators by using the Lidskiis theorem.AMS subject classifications: 34L10, 39A13

Key words: q-Sturm-Liouville operator, dissipative operator, completeness of the systemof eigenvectors and associated vectors, Lidskiis theorem

1. Introduction

Quantum calculus was initiated at the beginning of the 19th century and recentlythere has been great interest therein. The subject of q-difference equations hasevolved into a multidisciplinary subject [6]. There are several physical models in-volving q-difference and their related problems (see [2, 5, 8, 9, 12, 23]). Manyworks have been devoted to some problems of a q-difference equation. In partic-ular, Advar and Bohner [1] investigated eigenvalues and spectral singularities ofnon-self-adjoint q-difference equations of second order with spectral singularities.Huseynov and Bairamov [16] examined the properties of eigenvalues and eigenvec-tors of a quadratic pencil of q-difference equations. In [2], Annaby and Mansourstudied a q-analogue of Sturm-Liouville eigenvalue problems and formulated a self-adjoint q-difference operator in a Hilbert space. They also discussed properties ofeigenvalues and eigenfunctions. One can see the reference [17] for some definitionsand theorems on q-derivative, q-integration,q-exponential function, q-trigonometricfunction,q-Taylor formula,q-Beta and Gamma functions, Euler-Maclaurin formula,etc.

Many problems in mechanics, engineering, and mathematical physics lead tothe study of completeness and basic properties of all or part of eigenvectors and

associated vectors corresponding to some operators. For instance, when we applythe method of separation of variables to solve an equation like the reduced waveequation or Helmholtz equation, we assume the solution expanded in a series of

Corresponding author. Email addresses: [email protected](H. Tuna),[email protected] (A. Erylmaz)

http://www.mathos.hr/mc c2014 Department of Mathematics, University of Osijek

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66 H. Tuna and A. Erylmaz

eigenfunctions of one of the variables. The coefficients depend upon the other vari-able. We substitute the expansion into the equation, thereby obtaining ordinarydifferential equations for the coefficients. The method relies upon completeness ofeigenfunctions corresponding to one of the variables [19]. Dissipative operators arean important part of non-self-adjoint operators. In the spectral analysis of a dissipa-tive operator, we should answer the question whether all eigenvectors and associatedvectors of a dissipative operator span the whole space or not. Many authors inves-tigated the problem of completeness of the system of eigenvectors and associatedvectors of boundary value problems for differential and difference operators (see[3, 4, 7, 13, 14, 15, 24]).

The organization of this paper is as follows: In Section 2, some preliminary con-cepts related to q-difference equation and Lidskiis theorem essentials are presentedfor the convenience of the readers. In Section 3, we construct a space of boundaryvalues of the minimal operator and describe all maximal dissipative, maximal ac-cretive, self-adjoint and other extensions ofq-Sturm-Liouville operators in terms ofboundary conditions. Finally, in Section 4, we proved a theorem on completeness

of the system of eigenvectors and associated vectors of dissipative operators underconsideration.

2. Preliminaries

In this section, we introduce some of the required q-notations and Lidskiis theoremessentials.

Following the standard notations in [7] and [17], let qbe a positive number with0 < q < 1, A R and a C. A q-difference equation is an equation that containsq-derivatives of a function defined on A.Let y (x) be a complex-valued function onx A. The q-difference operator Dq is defined by

Dqy (x) = y (qx) y (x)

(x) , for allx A, (1)

where (x) = (q 1) x. The q-derivative at zero is defined by

Dqy (0) = limn

y (qnx) y (0)

qnx , x A, (2)

if the limit exists and does not depend on x. A right inverse to Dq, the Jacksonq-integration is given by

x

0

f(t) dqt= x (1 q)

n=0

qnf(qnx) , x A, (3)

provided that the series converges, and

ba

f(t) dqt=

b0

f(t) dqt

a0

f(t) dqt, a, b A. (4)

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Completeness of the system of root functions of q-Sturm-Liouville operators 67

LetL2q (0, a) be the space of all complex-valued functions defined on [0, a] such that

f:= a

0

|f(x)| dqx1/2

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R2

|G(x, t)|2dxdt

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Lemma 1. For arbitraryy, z D, one has the equality

[y, z]x[u, v]x = [y, u]x[z, v]x [y, v]x[z, u]x, 0 x + (17)

Proof. Direct calculations verify equality (17).

Let us consider the functions y D satisfying the conditions

y (0) cos + Dq1y (0) sin = 0, (18)

[y, u]a h[y, v]a = 0, (19)

whereImh >0, R.

We recall that a triple (H, 1, 2) is called a space of boundary values of a closedsymmetric operator A on a Hilbert space H if 1, 2 are linear maps from D (A

)to Hwith equal deficiency numbers and

i) for everyf , g D (A)

(Af, g)H (f, Ag)H= (1f, 2g)H (2f, 1g)H ,

ii) for any F1, F2 H there is a vector f D (A) such that 1f = F1, 2f =

F2 ([11]).

Let us define by 1, 2 linear maps from D to C2 by the formula

1y=

y (0)[y, u]a

, 2y=

Dq1y (0)

[y, v]a

, y D. (20)

For any y, z D, using Lemma 1, we have

(1y, 2z)C2 (2y, 1z)C2 = y (0) Dq1z (0) + z (0) Dq1y (0)

+[y, u]a[z, v]a [z, u]a[y, v]a

= [y, z] (a) [y, z](0)

= (Ly,z) (y,Lz) . (21)

Theorem 3. The triple(C2, 1, 2

defined by (20) is a boundary space of the

operatorL0.

Proof. The proof is obtained from (21) and the definition of the boundary valuespace.

The following result is obtained from Theorem 1.6 in Chapter 3 of [11].

Theorem 4. For any contractionK inC2 the restriction of the operatorL to the

set of functionsy D satisfying either

(K I) 1y+ i (K+ I) 2y= 0 (22)

or(K I) 1y i (K+ I) 2y= 0 (23)

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is a maximal dissipative and accretive extension of the operator L0, respectively.Conversely, every maximally dissipative (accretive) extension ofL0 is the restrictionof L to the set of functions y D satisfying (22)((23)), and the contraction K isuniquely determined by the extension. Conditions (22)((23))) define self-adjointextensions ifKis unitary.

In particular, boundary conditions

cos y (0) + sin Dq1y (0) = 0 (24)

[y, u]a h[y, v]a = 0 (25)

with Imh >0,describe the maximal dissipative (self-adjoint) extensions ofL0 withseparated boundary conditions.

We know that all eigenvalues of a dissipative operator lie in the closed upperhalf-plane. By virtue of Theorem 4, all the eigenvalues ofL lie in the closed upperhalf-plane Im 0.

4. Completeness theorem for the q-difference operators

Theorem 5. The operatorL has no any real eigenvalue.

Proof. Suppose that the operator L has a real eigenvalue 0.Let 0(x) = (x, 0)be the corresponding eigenfunction. Since

Im (L0, 0) =I m

0 02

= I mh ([0, v]a)2

,

we get [0, v]a= 0. By boundary condition (25), we have [0, u]a= 0. Thus

[0(t, 0) , u]a= [0(t, 0) , v]a= 0. (26)

Let 0(t) =v (t, 0) . Then

1 = [0, 0]a = [0, u]a[0, v]a [0, v]a[0, u]a.

By equality (26), the right-hand side is equal to 0. This contradiction proves Theo-rem 5.

Definition 1. Let f be an entire function. If for each > 0 there exists a finiteconstantC >0, such that

|f() | Ce||, C, (27)

thenfis called an entire function of order1 of growth and minimal type [10].

Let (x, ) be a single linearly independent solution of the equation l (y) =y,and

1() : = [ (x, ) , u (x)]a,

2() : = [ (x, ) , v (x)]a,

() : =1() h2() .

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It is clear that

p(L) ={: C, () = 0} ,

where p(L) denotes the set of all eigenvalues ofL. Since (a, ) and Dq1 (a, )

are entire functions of of order 12

(see [2]), consequently, () have the sameproperty. Then() is entire functions of order 1 of growth, and of minimal type.It is clear that all zeros of() (all eigenvalues ofL) are discrete and possible limitpoints of these zeros (eigenvalues ofL) can only occur at infinity.

For y D (L) , Ly (x) = f(x)(

x (0, a), f(x) L2q(0, a)

is equivalent tothe nonhomogeneous differential equation l (y) = f(x) (x (0, a)) , subject to theconditions

cos y (0) + sin Dq1y (0) = 0,

[y, u]a h[y, v]a = 0, I mh >0.

By Theorem 5, there exists an inverse operator L1. In order to describe the

operator L1, we use the Greens function method. We consider the functions v (x)and (x) = u (x) hv (x) . These functions belong to the space L2q (0, a). TheirWronskian Wq(v, ) =1.

Let

G (x, t) =

v (x) (t) , 0 x t a

v (t) (x) , 0 t x a. (28)

Let us consider the integral operator Kdefined by the formula

Kf=

a0

G (x, t) f(t)dqt,(

fL2q(0, a)

. (29)

It is evident that K = L1

. Consequently, the root lineals of the operators Land K coincide and, therefore, the completeness in L2q(0, a) of the system of alleigenfunctions and associated functions of L is equivalent to the completeness ofthose for K.

We obtain that G (x, t) is a Hilbert-Schmidth kernel since v, L2q (0, a). Fur-thermore,G (x, t) is measurable and integrable on (0, a). Hence K is of trace class.Since L is a dissipative operator, Kis a dissipative operator by Theorem 1. Thusall conditions are satisfied for the Lidskiis theorem. Hence we have;

Theorem 6. The system of all root functions ofK (also K) is complete in L2q(0, a).

From all above conclusions, we have;

Theorem 7. All eigenvalues of the operator L lie in the open upper half-planeand they are purely discrete. The limit points of these eigenvalues can only occurat infinity. The system of all eigenfunctions and associated functions of the L iscomplete in the spaceL2q (0, a).

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