A class of dissipative di ﬀ erential operators of order three

: In this paper we ﬁnd a class of boundary conditions which determine dissipative di ﬀ erential operators of order three and prove that these operators have no real eigenvalues. The completeness of the system of eigenfunctions and associated functions is also established.


Introduction
In this paper we study a class of third order dissipative differential operators. Dissipative operators are of general interest in mathematics, for example in the study of the Cauchy problems in partial differential equations and in infinite dimensional dynamical systems. Even order dissipative operators and the boundary conditions generating them have been investigated by many authors, see [1][2][3][4][5][6][7][8][9] and their references. Odd order problems arise in physics and other areas of applied mathematics and have also been studied, e.g., in [10][11][12][13][14][15][16].
Non-self-adjointness of spectral problems can be caused by one or more of the following factors: the non-linear dependence of the problems on the spectral parameter, the non-symmetry of the differential expressions used, and the non-self-adjointness of the boundary conditions(BCs) involved. Many scholars focus on the non-self-adjoint differential operators caused by non-self-adjoint BCs. Bairamov, Uǧurlu, Tuna and Zhang et al. considered the even order dissipative operators and their spectral properties in [5][6][7][8][9], respectively. However, these results all restricted in some special boundary conditions. In 2012, Wang and Wu [2] found all boundary conditions which generate dissipative operators of order two and proved the completeness of eigenfunctions and associated functions for these operators. In [3] the authors studied a class of non-self-adjoint fourth order differential operators in Weyl's limit circle case with general separated BCs, and they proved the completeness of eigenfunctions and associated functions. Here we find a class of such general conditions for the third order case, which may help to classify the dissipative boundary conditions of third order differential operators.
As is mentioned above, there are many results for dissipative Sturm-Liouville operators and fourth order differential operators, however, there are few studies on the odd order dissipative operators. Thus, in this paper, we study a class of non-self-adjoint third order differential operators generated by the symmetric differential expression in Weyl's limit circle case together with non-self-adjoint BCs.
This paper is organized as follows. In Section 2 we introduce third order dissipative operators and develop their properties. Section 3 discusses some general properties of dissipative operators in Hilbert space and some particular properties of the third order operators studied here. The completeness of eigenfunctions and associated functions is given in Section 4. Brief concluding remarks on the obtained results in this present paper and the comparison with other works are reported in Section 5.

Third order boundary value problems
Consider the third order differential expression is a real-valued function on I and q(x) ∈ L 1 loc (I). Suppose that the endpoints a and b are singular, i.e., a = −∞ or for any c ∈ (a, b) q(x) is not absolutely integrable in (a, c] (the same statement holds for endpoint b), and Weyl's limit-circle case holds for the differential expression l(u), i.e., the deficiency indices at both endpoints are (3,3).
For all u, v ∈ Ω, we set where the bar over a function denotes its complex conjugate, and and R * v (x) is the complex conjugate transpose of R v (x). Let ψ j (x, λ), j = 1, 2, 3 represent a set of linearly independent solutions of the equation l(u) = λu, where λ is a complex parameter. Then ψ j (x, 0), j = 1, 2, 3 represent the linearly independent solutions of the equation l(u) = 0. From Naimark's Patching Lemma, we can choose the solutions above mentioned satisfying any initial conditions, for future conveniences, here we set z j (x) = ψ j (x, 0), j = 1, 2, 3 satisfying the condition From [17], the solutions z j (x), j = 1, 2, 3 as described above exist and are linearly independent. Since Weyl's limit-circle case holds for the differential expression l(u) on I, the solutions z j (x), j = 1, 2, 3 must belong to L 2 (I). Furthermore, because z j (x), j = 1, 2, 3 are solutions of equation l(u) = 0, thus according to the Green's formula, it is easy to get [z j , z k ] x = const for any x ∈ I, hence for any Let l(u) = λu and we consider the boundary value problem consisting of the differential equation and the boundary conditions: where λ is a complex parameter, r is a real number with |r| ≥ 1, θ ∈ (−π, π], γ j , j = 1, 2, 3, 4 are complex numbers with 2 γ 1 ≥ |γ 2 | 2 and 2 γ 3 ≤ |γ 4 | 2 , here denotes the real part of a value. In L 2 (I), let us define the operator L as Lu = l(u) on D(L), where the domain D(L) of L is given by Let Ψ(x) be the Wronskian matrix of the solutions z j (x), j = 1, 2, 3 in I, then ones have Now let us introduce several lemmas.
Then the conclusion can be obtained by left multiplying (Ψ * (x)) −1 and right multiplying Ψ −1 (x) on the two ends of the above equality.
This complete the proof.
Corollary 1. For arbitrary y 1 , y 2 , y 3 ∈ D(L), let Y(x) = (C y 1 (x), C y 2 (x), C y 3 (x)) be the Wronskian matrix of y 1 , y 2 , y 3 , then Proof. From Lemma 1 and Lemma 2, it is easy to calculate that This completes the proof.
Hence L is a dissipative operator in L 2 (I).

Lemma 4.
where C j , j = 1, 2, 3 are arbitrary constants and u * (x) is a special solution of l(u) = f (x). It can be obtained by the method of constant variation, where C j , j = 1, 2, 3 satisfies By proper calculation, we have then the solution can be written as substituting u(x) into the boundary conditions one obtains (4.5) , (4.6) thus , (4.8) then one obtains Now define the operator T as G(x, ξ)u(ξ)dξ, ∀u ∈ L 2 (I), (4.9) then T is an integral operator and T = L −1 , this implies that the root vectors of the operators T and L coincide, since z j (x) ∈ L 2 (I), j = 1, 2, 3, then the following inequality holds b a b a |G(x, ξ)| 2 dxdξ < +∞, (4.10) this implies that the integral operator T is a Hilbert-Schmidt operator [21]. The next theorem is known as Krein's Theorem.
Theorem 3. Let S be a compact dissipative operator in L 2 (I) with nuclear imaginary part S . The system of all root vectors of S is complete in L 2 (I) so long as at least one of the following two conditions is fulfilled: Proof. See [22]. Proof. See [23].
The operator T can be written as T = T 1 + iT 2 , where T 1 = T and T 2 = T , T and T 1 are Hilbert-Schmidt operators, T 1 is a self-adjoint operator in L 2 (I), and T 2 is a nuclear operator (since it is a finite dimensional operator) [22]. It is easy to verify that T 1 is the inverse of the real part L 1 of the operator L.
Since the operator L is dissipative, it follows that the operator −T is dissipative. Consider the operator −T = −T 1 − iT 2 , the eigenvalues of the operator −T 1 and L 1 coincide. Since the characteristic function of L 1 is an entire function, therefore using Theorem 4 and Krein's Theorem we arrive at the following results.
Theorem 5. The system of all root vectors of the operator −T (also of T ) is complete in L 2 (I).
Theorem 6. The system of all eigenvectors and associated vectors of the dissipative operator L is complete in L 2 (I).

Concluding remarks
This paper considered a class of third order dissipative operator generated by symmetric third order differential expression and a class of non-self-adjoint boundary conditions. By using the well known Krein's Theorem and theoretical analysis the completeness of eigenfunctions system and associated functions is proved.
The similar results already exist for second order S-L operators and fourth order differential operators, see e.g., [2] and [3]. For third order case, the corresponding discussions about dissipative operators can be found in most recent works in [15,19], where the maximal dissipative extension and the complete theorems of eigenvectors system are given. The boundary conditions at the present work are much general and the methods are different from those in [15,19]. These boundary conditions may help us to classify all the analytical representations of dissipative boundary conditions of third order differential operators.