Abstract

A singular dissipative fourth-order differential operator in lim-4 case is considered. To investigate the spectral analysis of this operator, it is passed to the inverse operator with the help of Everitt's method. Finally, using Lidskiĭ's theorem, it is proved that the system of all eigen- and associated functions of this operator (also the boundary value problem) is complete.

1. Introduction

In 1910, Weyl showed that [1] the singular second-order differential operators can be fallen into two classes: operators in limit-circle case and operators in limit-point case. The operators in limit-circle case have the solutions that are all in square integrable space. However, in limit-point case, only one linearly independent solution can be in square integrable space. The development of this theory belongs to Titchmarsh [2]. After these fundamental works, second-order singular differential operators have been developed by many authors (e.g., see [3–6]).

Following the methods of Weyl and Titchmarsh, the theory for higher order equations and Hamiltonian systems was constructed in [7–19].

In [7], a regular self-adjoint fourth-order boundary value problem was investigated. Further Green’s function and the resolvent operator were constructed. In [8, 13], such a construction was done for the singular self-adjoint fourth order boundary value problem. In [9, 10], higher-order differential equations whose coefficients are complex-valued were studied. In 1974, Walker [20] showed that an arbitrary order self-adjoint eigenvalue problem can be represented as an equivalent self-adjoint Hamiltonian systems. Further, the developments in the theory of singular self-adjoint Hamiltonian systems were given in [14–19].

On the other hand, an important class of the nonself-adjoint operators is the class of the dissipative operators [21]. It is known that all eigenvalues of the dissipative operators lie in the closed upper half-plane, but this analysis is so weak. There are some methods to complete the analysis of the dissipative operators. Some of them are Livšic’s, Krein’s, and Lidskiĭ’s theorems and functional model [21, 22]. These methods were used in the literature for the second-order differential operators (see [22–26]). In this paper a singular dissipative fourth order differential operator in lim-4 case is investigated. In particular, using Lidskiĭ’s theorem, it is shown that the system of all eigen- and associated functions is complete in .

2. Preliminaries

Let denote the linear nonself-adjoint operator in the Hilbert space with the domain . The element ,  , is called a root function of the operator corresponding to the eigenvalue , if all powers of are defined on this element and for some .

The functions are called the associated functions of the eigenfunction if they belong to and the equalities ,  , hold.

The completeness of the system of all eigen- and associated functions of is equivalent to the completeness of the system of all root functions of this operator.

If, for the operator with dense domain in , the inequality () holds, then is called dissipative.

Theorem 1 (see [26]). Let be an invertible operator. Then, is dissipative if and only if the inverse operator of is dissipative.

It is known that a kernel of an integral operator defned by where , is called a Hilbert-Schmidt kernel if is integrable on

A kernel satisfying the property is called a Hermitian kernel. Properties of Hermitian kernels and related integral operators can be found in [27] (further see [28]). Now we shall remind some results. Let us consider the equation which is related with (1) as

The function which differs from zero is called a characteristic function of which corresponds to the characteristic value . If a Hermitian Hilbert-Schmidt kernel is not null then it posseses at least one characteristic value and this characteristic value (every characteristic value) is real. Further there is a finite orthonormal base of characteristic functions for each characteristic value of . Using the union of these bases one obtains an orthonormal system of characteristic functions of the kernel . If are distinct members of such a system belonging respectively to the characteristic value (not necessarily all different) then for , is an orthonormal system of two variables. For such a system the equivalence holds. Moreover if , , then the equivalence holds. Further the series in (5) are relatively uniformly absolutely convergent. However, equivalence can be replaced by equality. Before showing this, let us consider the equality Following the same idea of [27, pages 22] if is an -kernel such that is continuous and is an -function then is continuous.

Hence if is a continuous Hermitian -kernel (see [27, pages 127]) and , , then the equality holds and the series are uniformly absolutely convergent. In this case taking the series given in (4) converge to the continuous kernel uniformly in the variable for every . Hence from (4) one can get Integrating both side from to we obtain Consequently if is integrable on then the series converge. This implies that is of trace-class (nuclear). For the definition and properties of trace-class operators see, for example, [21].

Above arguments given in [27] hold for Hermitian kernels. However from arbitrary -kernels one can pass to the Hermitian kernels. For example and are Hermitian kernels. These Hermitian kernels are continuous and -kernels if is so. Hence above arguments hold for and if is continuous -kernel such that is integrable on . Further

Hence if and are trace-class kernels then so is . However this result has been given in [29, pages 526] as a definition.

Lidskiĭ’s Theorem (see [21, page 231]). If the dissipative operator is the trace class operator, then its system of root functions is complete in the Hilbert space .

3. Statement of the Problem

The fourth order differential expression is considered as where ,   is the regular point and is the singular point for , and is real-valued, Lebesgue measurable, and locally integrable function on .

Let be the Hilbert space consisting of all functions such that with the inner product .

Let where denotes the set of all locally absolutely continuous functions on . For arbitrary , Green’s formula is obtained as where and . Green’s formula implies the fact that if and both satisfy the equation for the same value of , then is independent of and depends only on . Further for arbitrary , at singular point , the limits and exist and are finite. The latter also follows from Green’s formula. In fact, it is sufficient to get the second factors with their complex conjugates on the left-hand side in Green’s formula.

In this paper it is assumed that satisfies the lim-4 case conditions at (see [30] and references therein). Lim-4 case is also known as Weyl’s limit-circle case [8].

Let us consider the solutions ,  ,  , and of the equation where is some complex parameter, satisfying the conditions where all ,  ,  , and are real numbers such that and . For the existence of these solutions, see [7, 8, 13]. Since lim- case holds for , all solutions and belong to . It is clear that , where is the Kronocker delta, and .

Let us set and ,  with . Then and become the real solutions of . Further, they belong to . Hence for arbitrary , the values ,  ,  , and exist and are finite.

Let be the set of all functions satisfying the boundary conditions where and are real numbers given as previously stated and and are some complex numbers such that and with ,  .

It should be noted that for any solutions of (14), conditions (16) and (17), respectively, can be written as

The operator is defined on as , ,  . The main aim of this paper is to investigate the spectral analysis of the operator (the boundary value problem (14)–(19)).

4. Completeness Theorems

Let be any solutions of . The notation denotes the Wronskian of order of this set of functions [7, 8, 31].

It is known that the following equality holds [7, 8, 13]: This equation also shows that of any four solutions of is independent of and depends only on [7, 8].

Now consider the solutions and of the equation ,  , satisfying the conditions For the existence of these solutions given with the previous conditions, see [13]. Clearly these solutions satisfy conditions (18) and (19), respectively. Now let us set Then becomes an entire function, and the zeros of coincide with the eigenvalues of the operator [13]. This implies that all zeros of (all eigenvalues of ) are discrete and that possible limit points of these zeros (eigenvalues of ) can only occur at infinity.

Using (22), we immediately have Hence, the Plücker identity for the fourth-order case is obtained (see [13, p. 435]): where .

Theorem 2. The operator is dissipative in .

Proof. For , we have Since , a direct calculation shows that Further, using (26) and conditions (18) and (19) one obtains Substituting (28) and (29) into (27) it is obtained that and this completes the proof.

Theorem 2 shows that all eigenvalues of lie in the closed upper half-plane.

Theorem 3. The operator has no real eigenvalue.

Proof. For , a direct calculation shows that Now let be a real eigenvalue of , and let be the corresponding eigenfunction. Then (31) and (30) gives . Using these equalities in (18) and (19) one gets that .
Let us consider the solution . Hence, using (26) it is obtained that This contradiction completes the proof.

From Theorem 3 it is obtained that all eigenvalues of lie in the open upper half-plane. In particular zero is not an eigenvalue of . Hence, the operator exists.

Consider the solutions ,  ,   and , where and . and satisfy conditions (16) and (17), respectively and and satisfy conditions (18) and (19), respectively.

The equation ,  ,  , is equivalent to the nonhomogeneous differential equation subject to the boundary conditions (compare the boundary conditions at with (20) and (21)). Using Everitt’s method [7] (further see [8, 13]) the general solution is obtained as where and

The operator defined by where , is the inverse operator of . Hence the completeness of the system of all eigen- and associated functions of is equivalent to the completeness of those of in .

Since is a continuous Hilbert-Schmidt kernel and is integrable on , the operator is of trace class.

Let us consider the operator . Since is dissipative in is also dissipative in . Thus all conditions are satisfied for Lidskiĭ’s theorem. So we have the following.

Theorem 4. The system of all root functions of (also ) is complete in .

Since the completeness of the system of root functions (eigen- and associated functions) of is equivalent to the completeness of those of , is obtained that the following.

Theorem 5. All eigenvalues of the problem (14)–(19) lie in the open upper half-plane and they are purely discrete. The system of all eigen- and associated functions of the problem (14)–(19) is complete in .