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Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions
This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.
Spectral analysis and expansion of eigenfunctions in the fields of differential operators are important parts in the theory of ordinary differential equation boundary value problems. Generally speaking, the spectral parameter appears only in the equation. However, lots of problems in the mathematical physics require that eigenparameter appears not only in the equation but also in the boundary and transfer conditions. As is well known, many complicated physical phenomena with discontinuities can be transferred to operator problems with transfer conditions (also called point interactions, transmission conditions, interface conditions). Various physical applications of such problems arise in the theory of mechanics, heat, and mass transfer problems, etc. (see, for example, [1–23]).
Dirac system plays an important role in the theory of relativistic. Meanwhile, Dirac system with spectral parameter in the boundary conditions describes the behavior of a relativistic particle in an electromagnetic field. When an atomic system is subjected to an external electromagnetic field or a mechanical system to an external force, these would result in the discontinuity of origin system. Such as in geophysical problems, the reflection of transverse waves at the bottom of the earth’s crust jumps phenomena due to high-speed ions colliding with atomic systems. These reasons may cause the eigenfunctions in the equations describing the system to have discontinuities, that is to say, operators with transfer conditions [9, 24, 25].
Dissipative operator is an important class of non-self-adjoint operators, which can be traced back to the study of hyperbolic partial differential equations. For example, the telegraph operator equation can be converted into the study of dissipative differential operator . The dissipation of differential operators can be induced by many factors, such as the boundary conditions, coefficients in the equation, etc. In 1970s, Pavlov  proposed a new method for spectral analysis of dissipative operators. This method is based on building their self-adjoint dilation and the corresponding functional model of Sz.-Nagy-Foias type. Based on this, one could study spectral properties of the operator using the equivalence between the Lax-Phillips scattering matrix and the characteristic function. This equivalence has been used by many authors [2, 14, 15, 28–31]. Moreover, Behrndt et al.  investigated an alternative approach to the construction of the self-adjoint dilation of an m-dissipative operator as well as a connection between the characteristic function and scattering matrix. In this paper, we investigate a class of dissipative discontinuous Dirac operators with two singular endpoints (in Weyl’s limit circle case), and one of boundary conditions is linearly dependent on the eigenparameter, and general transfer conditions are imposed on the discontinuous points. Using the equivalence between the Lax-Phillips scattering matrix and the characteristic function, we investigate the discreteness of the spectrum and the completeness of the system consisting of eigenfunctions and associated functions. What calls for special attention is that the transfer conditions in this paper are in the sense of coupled, which are different from the usual, namely, the values of the solutions and their derivatives at the interior discontinuous points are not independent of each other. The transfer conditions in the sense of separated for dissipative Dirac operators have been investigated by Uğurlu [14, 15]. For self-adjoint Sturm–Liouville operators, this kind of problems have been investigated by Mukhtarov [16, 17, 19]. It should be noted that completeness properties and the Riesz basis property for strongly regular boundary value problems for Dirac operators on a finite interval have been established in [33–35].
The arrangement of this paper is as follows: in Section 2, we transfer the considered problem to a maximal dissipative operator In Section 3, the self-adjoint dilation, incoming and outgoing spectral representations, functional model, and characteristic function of this dissipative operator are derived. Our main results on the completeness theorem are investigated in Section 4.
2. Dissipative Operator
We study the following Dirac system consisting of the equation
where , and
We have the following basic hypotheses on the coefficients of Equation (2.1) and the interval
holds almost everywhere on ; and are real vector-valued locally integrable and Lebesgue measurable functions on .
Let be a Hilbert space which consists of all functions satisfying and equipped with inner product
Let Setting be a set such that for any are locally absolutely continuous functions on and
For arbitrary vectors we have
where . Hence, , exist and are finite by Equation (4).
Consider the boundary value transfer problem (BVTP)
where is a complex number with is a complex parameter, , are given real numbers, and
are the solutions of the system
and satisfy the conditions
are parts of defined on the interval respectively. By the property of the Wronskian and (12) we get that
Hence, any solutions of (11) can be represented as linear combination of and . Since Weyl’s limit circle cases hold for the Dirac system, , moreover,
Remark 1. In this article, we assume that with this hypothesis, the problem (5)–(9) can be transferred to the study of dissipative operators. Particularly, when the considered problem can be transferred to the self-adjoint case which is well known. Our interests focus on the non-self-adjoint case, hence, here we assume that
For convenience, the following notations will be used:
Then, for any ,
Let and be the solutions of (5) satisfying
Consider the operator with domain
and acts as
Theorem 1. is maximal dissipative in .
Proof. Let ThenBy (16), we haveUsing transfer conditions (8) and (9), we haveBy (7) and (17), one gets thatSubstituting (27)–(29) into (26) yields thatSince and , is dissipative in . Moreover, it can be easily proved thatTherefore, the result follows.
Using the same method in , it is easy to check that the following lemma holds.
Lemma 1. For the same eigenvalue , each chain of eigenvectors and associated vectors of BVTP (5)–(9) corresponds to the chain of eigenvectors and associated vectors of . In this case, the equalitytakes place.
3. Scattering Function
In this section, we derive self-adjoint dilation, to this end, the incoming channel and outgoing channel are added, and orthogonal sum is called main Hilbert space, where and
We consider the operator in with the domain , which is generated by the expression
where , are Sobolev spaces, and .
Theorem 2. is self-adjoint in .
Proof. Let . Integration by parts yieldswhich implies that is symmetric in , and .
In order to prove is self-adjoint, we only need to show . To do this, let , , and . Simple calculation giveswhere , . Analogously, let , then we havethrough integrating by parts with respect to Then the equality , holds by (37).
Through the definition of , we haveandComparing the coefficients of and yields thatTherefore, by (40) and (41). Above discussion implies .
Let , then it is an unitary group. Define the mappings and as follows:
Utilizing , then one can construct , which is a strongly continuous semigroup of nonunitary contractions on . is called the self-adjoint dilation of the generator of and
Theorem 3. is the self-adjoint dilation of
Proof. We only need to show To this end, we consider the following equalitywhere , and . Then we have and . Since , then , . Therefore, satisfies , and . Since is a dissipative operator, we get that if , then is not the eigenvalue of . Hence,Through we haveOn the other hand,Hence by (46) and (47), and result follows.
In what follows, let us consider the subspaces and
Lemma 2. The spaces and possess the following properties:(i) (ii) (iii) (iv)
Proof. Let , then for , we haveHence, if and , then
This gives that Hence, for , we have The similar discussion can be done for , thus the proof of property (i) is finished.
In the following, consider the semigroup of isometries , where and are in the form of
The generator of is
where and . It is known that the generator of the one-sided shift in is differential operator satisfying . Due to a semigroup is uniquely determined by its the generator, we have . Therefore,
The same proof can be done with respect to The proof of property (ii) is done.
If the restriction of on a subspace is the self-adjoint part, then for we have
This gives and by the boundary condition (7). It follows from and that . Utilizing the expansion theorem on in eigenfunctions of the self-adjoint operator , we have Thus is completely non-self-adjoint in which results in
Otherwise, there exists a nontrivial subspace which would invariant with respect to group and the restriction of to were unitary. Thus the restriction of on is a self-adjoint operator.
Let and be the solutions of Equation (5) satisfying
Setting the vectors
When is real, the vectors and do not belong to the space . Simple calculation gives that and satisfy and the boundary-transfer conditions of For , define the Fourier transformations
where are smooth, compactly supported functions. Let Then the equality
holds. Here are the Hardy classes in . Now consider the dense set in consisting of all vectors such that is compactly supported in and if , where is nonnegative number. Then if we obtain for and that , and their first components belong to . Therefore
Taking closure in (65), then the Parseval equality holds for the entire space . Furthermore, the inversion formula
results from the Parseval equality if all integrals are taken as limits in the mean of the intervals. Consequently,
The analogous argument can be used for Hence, and are isometrically identical with , which imply that The proof of property (iii) follows.
Utilizing the inner product in the property (iv) holds.
Evidently, is a meromorphic function in , and has a countable number of poles on by the definition of Moreover, for all and for all , except the real poles of
In the proof of Lemma 2, we have obtained that is the incoming (outgoing) spectral representation for the group , respectively. is transformed into .
Thus from (68) we have
According to the Lax-Phillips scattering theory , the following theorem holds.
Theorem 4. is the scattering function of (also of ).
Using one get thatTherefore,
Because is unitary equivalent under to Let be the orthogonal projection from onto , then we have is a semigroup of operators. Hence the generator of
is a maximal dissipative operator on The operator is called a model dissipative operator (see [2, 27, 37, 38]). Therefore, is the characteristic function of the operator Since the characteristic function of unitary equivalent dissipative operators coincides (see , Chapter VI), therefore, the following result is proved.
Theorem 5. is the characteristic function of the operator .
4. Completeness Theorem
Theorem 6. The characteristic function of is a Blaschke product except for a single value in the upper half-plane.
Proof. It can be verified that is an inner function in the upper half-plane. Therefore, can be written aswhere is a Blaschke product. By (73), we haveFrom (61) one gets thatUtilizing (74), we haveThus is zero except for a single point and this completes the proof.
Theorem 7. If Weyl’s limit-circle case holds for the Dirac system (2.1) at the endpoints and Then the BVTP (5)–(9) has purely discrete spectrum in the open upper half-plane and the possible limit points can only occur at infinity. For all with , all eigenvectors and associated vectors of BVTP (5)–(9) are complete in the space except possibly for a single value .
Boundary value problem with eigenparameter dependent boundary conditions and with discontinuities inside an interval has been extensively studied since its wide applications in engineering and mathematical physics. In this paper, we investigate a class of dissipative Dirac operators with discontinuities and eigenparameter dependent boundary conditions. For such a problem, we obtain the completeness theorem of this dissipative operator.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.
The authors thank the referees for his/her comments and detailed suggestions. These have significantly improved the presentation of this paper. The work of the authors is supported by the Nature Science Foundation of Shandong Province (nos. ZR2019MA034, ZR2017MA042), the Inner Mongolia Natural Science Foundation (nos. 2018MS01021, 2017MS0119), the National Nature Science Foundation of China (Nos. 11561050, 11671227, 11801286) and China Postdoctoral Science Foundation (Grant 2019M662313).
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