Dissipative Sturm-Liouville Operators with Transmission Conditions

and Applied Analysis 3 Proof. We first prove that L G is symmetric in H. Namely, (L G f, H − (f,LGg)H = 0. Let f, g ∈ D(LG), f = ⟨φ − , y, φ + ⟩ and g = ⟨ψ − , ?̂?, ψ + ⟩. Then we have (L G f, g) H − (f,L G g) H = (L G ⟨φ − , y, φ + ⟩ , ⟨ψ − , ?̂?, ψ + ⟩) − (⟨φ − , y, φ + ⟩ ,L G ⟨ψ − , ?̂?, ψ + ⟩)


Introduction
Spectral theory is one of the main branches of modern functional analysis and it has many applications in mathematics and applied sciences.There has recently been great interest in spectral analysis of Sturm-Liouville boundary value problems with eigenparameter-dependent boundary conditions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]).Furthermore, many researchers have studied some boundary value problems that may have discontinuities in the solution or its derivative at an interior point  [15][16][17][18][19].Such conditions which include left and right limits of solutions and their derivatives at  are often called "transmission conditions" or "interface conditions." These problems often arise in varied assortment of physical transfer problems [20].
The spectral analysis of non-self-adjoint (dissipative) operators is based on ideas of the functional model and dilation theory rather than the method of contour integration of resolvent which is studied by Naimark [21], but this method is not effective in studying the spectral analysis of boundary value problem.The functional model technique acts a part on the fundamental theorem of Nagy-Foias ¸.In 1960s independently from Nagy-Foias ¸ [22], Lax and Phillips [23] developed abstract scattering programme that is very important in scattering theory.Pavlov's functional model [24][25][26][27][28] has been extended to dissipative operators which are finite dimensional extensions of a symmetric operator, and the corresponding dissipative and Lax-Phillips scattering matrix was investigated in some detail [5-14, 22-27, 29, 30].This theory is based on the notion of incoming and outgoing subspaces to obtain information about analytical properties of scattering matrix by utilizing properties of original unitary group.By combining the results of Nagy-Foias ¸and Lax-Phillips, characteristic function is expressed with scattering matrix and the dilation of dissipative operator is set up.By means of different spectral representation of dilation, given operator can be written very simply and functional models are obtained.The eigenvalues, eigenvectors and spectral projection of model operator are expressed obviously by characteristic function.The problem of completeness of the system of eigenvectors is solved by writing characteristic function as factorization.
The purpose of this paper is to study non-self-adjoint Sturm-Liouville operators with transmission conditions.To do this, we constructed a functional model of dissipative operator by means of the incoming and outgoing spectral representations and defined its characteristic function, because this makes it possible to determine the scattering matrix of dilation according to the Lax and Phillips scheme [23].Finally, we proved a theorem on completeness of the system of eigenvectors and associated vectors of dissipative operators which is based on the method of Pavlov.While proving our results, we use the machinery of [5,[7][8][9][10].
In the space H, we consider the operator L  on the set (L  ), its elements consisting of vectors  = ⟨ − , ŷ,  + ⟩, generated by the expression satisfying the conditions , where  1  2 are Sobolev spaces and  2 := 2 Im ,  > 0.
Theorem 1.The operator L  is self-adjoint in H and it is a self-adjoint dilation of the operator L (=   ).
Definition 2. The linear operator  with domain () acting in the Hilbert space  is called completely non-self-adjoint (or simple) if there is no invariant subspace  ⊆ ()( ̸ = {0}) of the operator  on which the restriction  to  is selfadjoint.
To prove property (3) of the incoming and outgoing subspaces, let us prove following lemma.

Lemma 3. The operator L𝐺 is completely non-self-adjoint (simple).
Proof.Let   ⊂  be a nontrivial subspace in which L induces a self-adjoint operator L  with domain ( L Consequently, we have [ ŷ, ]  = 0. Using this result with boundary condition [ ŷ, V]  + [ ŷ, ]  = 0, we have [ ŷ, ]  = 0; that is, ŷ() = 0. Since all solutions of () =  belong to  2 (0, ∞), from this it can be concluded that the resolvent   ( L ) is a compact operator, and the spectrum of L is purely discrete.Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator L  we obtain   = {0}.Hence the operator L is simple.The proof is completed.Proof.Considering property (1) of the subspace  + , it is easy to show that the subspace H  = H ⊝ ( − +  + ) is invariant relative to the group {  } and has the form H  = ⟨0,   , 0⟩, where   is a subspace in .Therefore, if the subspace H  (and hence also   ) was nontrivial, then the unitary group {   } restricted to this subspace would be a unitary part of the group {  }, and hence, the restriction L  of L to   would be a self-adjoint operator in   .Since the operator L is simple, it follows that   = {0}.The lemma is proved.

Assume that
are solutions of () =  satisfying the conditions Let us adopt the following notations: () = ( ()  () + ) ( ()  () + ) where () is a meromorphic function on the complex plane C with a countable number of poles on the real axis.Further, it is possible to show that the function () possesses the following properties: Im () ≤ 0 for all Im  ̸ = 0, and () = () for all  ∈ C, except the real poles ().
We set We note that the vectors  −  (, , ) for real  do not belong to the space H.However,  −  (, , ) satisfies the equation L =  and the corresponding boundary conditions for the operator L ℎ .
We now extend the Parseval equality to the whole of  − .We consider in  − the dense set of   − of the vectors obtained as follows from the smooth, compactly supported functions in , where  =   is a nonnegative number depending on .If ,  ∈   − , then for  >   and  >   we have  − ,  −  ∈  − ; moreover, the first components of these vectors belong to  ∞ 0 (−∞, 0).Therefore, since the operators   ( ∈ R) are unitary, by the equality we have By taking the closure (35), we obtain the Parseval equality for the space  − .The inversion formula is obtained from the Parseval equality if all integrals in it are considered as limits in the integrals over finite intervals.Finally We note that the vectors  +  (, , ) for real  do not belong to the space H.However,  +  (, , ) satisfies the equation L =  and the corresponding boundary conditions for the operator L ℎ .With the help of vector  +  (, , ), we define the transformation  + :  → f+ () by ( + )() := f+ () := (1/ √ 2)(,  +  ) H on the vectors  = ⟨ − , ŷ,  + ⟩ in which  − (),  + (), and () are smooth, compactly supported functions.Lemma 6.The transformation  + isometrically maps  + onto  2 (R).For all vectors ,  ∈  + the Parseval equality and the inversion formula hold: where f+ () = ( + )() and g+ () = ( + )().
Proof.The proof is analogous to Lemma 6.
It is obvious that the matrix-valued function   () is meromorphic in C and all poles are in the lower half-plane.From (27), |  ()| ≤ 1 for Im  > 0, and   () is the unitary matrix for all  ∈ R. Therefore, it explicitly follows from the formulae for the vectors  −  and  +  that It follows from Lemmas 6 and 5 that  − =  + .Together with Lemma 5, this shows that  − =  + = H; therefore property (3) has been proved for the incoming and outgoing subspaces.Thus, the transformation  − isometrically maps  − onto  2 (R) with the subspace  − mapped onto  2 − and the operators   are transformed into the operators of multiplication by   .This means that  − is the incoming spectral representation for the group {  }.Similarly,  + is the outgoing spectral representation for the group {  }.It follows from (38) that the passage from the  − representation of an element  ∈ H to its  + representation is accomplished as f+ () =  −1  () f− ().Consequently, according to [22], we have proved the following.Let () be an arbitrary nonconstant inner function (see [19]) on the upper half-plane (the analytic function () and the upper half-plane C + is called inner function on C + if | ℎ ()| ≤ 1 for all  ∈ C + and | ℎ ()| = 1 for almost all  ∈ R).Define  =  2 + ⊝  2 + .Then  ̸ = {0} is a subspace of the Hilbert space  2 + .We consider the semigroup of operators   ( ≥ 0) acting in  according to the formula    = [  ],  = () ∈ , where  is the orthogonal projection from  2 + onto .The generator of the semigroup {  } is denoted by where  is a maximal dissipative operator acting in  and with the domain () consisting of all functions  ∈ , such that the limit exists.The operator  is called a model dissipative operator.Recall that this model dissipative operator, which is associated with the names of Lax-Phillips [23], is a special case of a more general model dissipative operator constructed by Nagy and Foias ¸ [22].The basic assertion is that () is the characteristic function of the operator .
Let  = ⟨0, , 0⟩, so that H =  − ⊕  ⊕  + .It follows from the explicit form of the unitary transformation  − under the mapping  − that The formulas (40) show that operator L is unitarily equivalent to the model dissipative operator with the characteristic function   ().We have thus proved the following theorem.
Theorem 8.The characteristic function of the maximal dissipative operator L coincides with the function   () defined by (27).

The Spectral Properties of Dissipative Sturm-Liouville Operators
By using characteristic function, the spectral properties of the maximal dissipative operator L (  ) can be investigated.The characteristic function of the maximal dissipative operator L is known to lead to information of completeness about the spectral properties of this operator.For instance, the absence of a singular factor () of the characteristic function   () in the factorization det   () = ()() (() is a Blaschke product) ensures completeness of the system of eigenvectors and associated vectors of the operator L (  ) in the space  2 (0, ∞) (see [10,21,30]).
Theorem 9.For all the values of  with Im  > 0, except possibly for a single value  =  0 , the characteristic function   () of the maximal dissipative operator L is a Blaschke product.The spectrum of L is purely discrete and belongs to the open upper half-plane.The operator L ( ̸ =  0 ) has a countable number of isolated eigenvalues with finite multiplicity and limit points at infinity.The system of all eigenvectors and associated vectors of the operator L is complete in the space .

Theorem 7 .
The function   () is the scattering matrix of the group {  } (of the self-adjoint operator L  ).
If the characteristic function   () has nontrivial singular factor, the system of eigenvectors and associated vectors of the operator L (  ) can fail to be complete.Because   () is smooth, the support of the corresponding singular measure  must be contained in the set of poles   ().But in this case the singular measure  is a simple step function.If we require   () to have no zeros of infinite multiplicity, then  = 0.So the singular factor vanishes.The characteristic function   () of the maximal dissipative operator L has the form