# Dissipative Sturm-Liouville Operators with Transmission Conditions

Authors: Hüseyin Tuna; Aytekin Eryılmaz
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

In this paper we study dissipative Sturm-Liouville operators with transmission conditions. By using Pavlov’s method (Pavlov 1947, Pavlov 1981, Pavlov 1975, and Pavlov 1977), we proved a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators with transmission conditions.


---

## Body

## 1. 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 [1–14]). Furthermore, many researchers have studied some boundary value problems that may have discontinuities in the solution or its derivative at an interior point c [15–19]. Such conditions which include left and right limits of solutions and their derivatives at c 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-Foiaş. In 1960s independently from Nagy-Foiaş [22], Lax and Phillips [23] developed abstract scattering programme that is very important in scattering theory. Pavlov’s functional model [24–28] 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-Foiaş 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, 7–10].


## 2. Self-Adjoint Dilation of Dissipative Sturm-Liouville Operator

Consider the differential expression(1)l(y)=-y′′+q(x)y,x∈[a,c)∪(c,b),
where I1:=[a,c),I2:=(c,b) and I=I1∪I2, q(x)is real-valued function on I, and q∈Lloc1(I). The points a and c are regular and b is singular for the differential expressionl(y). Moreover q(c±):=limx→c±q(x) one-sided limits exist and are finite.To pass from the differential expressionl(y) to operators, we introduce the Hilbert space H=L2(I1)⊕L2(I2) with the inner product
(2)〈f,g〉H:=γ1γ2∫ac‍f1g1¯dx+δ1δ2∫cb‍f2g2¯dx,
where
(3)f(x)={f1(x),f2(x),x∈I1x∈I2∈H,g(x)={g1(x),g2(x),x∈I1x∈I2∈H,
and γ1, γ2, δ1, and δ2 are some real numbers with γ1γ2>0and δ1δ2>0.LetL0denote the closure of the minimal operator generated by (1) and by D0 its domain. Besides, we denote the set of all functions f(x) from Hsuch that f,f′∈ACloc(I),f(c±),f′(c±) one-sided limits exist and are finite and l(y)∈H;D is the domain of the maximal operator L. Furthermore, L=L0*[21].For two arbitrary functionsy(x),  z(x)∈D, we have Green’s formula
(4)∫ab‍l(y)z¯dx-∫ab‍yl(z)¯dx=γ1γ2[y,z]c--γ1γ2[y,z]a+δ1δ2[y,z]b-δ1δ2[y,z]c+,
where [y,z]x=y(x)z′(x)¯-y′(x)z(x)¯(x∈I),[y,z]b=limx→b[y,z]x exists and is finite.Suppose that Weyl’s limit circle case holds for the differential expressionl(y) on I. There are several sufficient conditions in which Weyl’s limit circle case holds for a differential expression [21]. Denote by
(5)θ(x)={θ1(x),x∈I1θ2(x),x∈I2,φ(x)={φ1(x),x∈I1φ2(x),x∈I2,
the solutions of the equation l(y)=λy,x∈I, satisfying the initial conditions
(6)θ1(a,λ)=cosα,θ1′(a,λ)=sinα,φ1(a,λ)=-sinα,φ1′(a,λ)=cosα
and transmission conditions
(7)θ2(c+,λ)=γ1δ1θ1(c-,λ),θ2′(c+,λ)=γ2δ2θ1′(c-,λ),φ2(c+,λ)=γ1δ1φ1(c-,λ),φ2′(c+,λ)=γ2δ2φ1′(c-,λ),
where α,γ1,γ2,δ1, andδ2∈ℝ with γ1γ2>0and δ1δ2>0.The solutions θ(x,λ) and φ(x,λ) belong to H. Let
(8)u(x)={u1(x)=θ1(x,0),x∈I1u2(x)=θ2(x,0),x∈I2,v(x)={v1(x)=φ1(x,0),x∈I1,v2(x)=φ2(x,0),x∈I2,u(x), and v(x) be solutions of the equation l(y)=0,x∈I, satisfying the initial conditions
(9)u1(a)=cosα,u1′(a)=sinα,v1(a)=-sinα,v1′(a)=cosα
and transmission conditions
(10)u2(c+)=γ1δ1u1(c-),u2′(c+)=γ2δ2u1′(c-),v2(c+)=γ1δ1v1(c-),v2′(c+)=γ2δ2v1′(c-).All the maximal dissipative extensionsℒG of the operator L0 are described by the following conditions (see [4, 15, 18]):
(11)R1(y):γ1y(c-)-δ1y(c+)=0,R2(y):γ2y′(c-)-δ2y′(c+)=0,R3(y):y(a)cosα+y′(a)sinα=0,R4(y):[y,v]b-G[y,u]b=0,ImG>0.Let us add the “incoming” and “outgoing” subspacesD-=L2(-∞,0) and D+=L2(0,∞) to H. The orthogonal sum H=D-⊕H⊕D+ is called main Hilbert space of the dilation.In the spaceℋ, we consider the operator ℒG on the set D(ℒG), its elements consisting of vectors w=〈φ-,y^,φ+〉, generated by the expression
(12)ℒG〈φ-,y^,φ+〉=〈idφ-dξ,l(y^),idφ+dξ〉
satisfying the conditions φ-∈W21(-∞,0), φ+∈W21(0,∞), y^∈H, R1(y^)=0,R2(y^)=0,R3(y^)=0, [y^,v]b-G[y^,u]b=(C/δ1δ2)φ-(0), [y^,v]b-G¯[y^,u]b=(C/δ1δ2)φ+(0), where W21 are Sobolev spaces and C2:=2ImG, C>0.Theorem 1.
The operatorℒG is self-adjoint in ℋand it is a self-adjoint dilation of the operator L~G(=LK).Proof.
We first prove thatℒGis symmetric in ℋ. Namely, (ℒGf,g)ℋ-(f,ℒGg)ℋ=0. Let f,g∈D(ℒG), f=〈φ-,y^,φ+〉 and g=〈ψ-,z^,ψ+〉. Then we have
(13)(ℒGf,g)ℋ-(f,ℒGg)ℋ=(ℒG〈φ-,y^,φ+〉,〈ψ-,z^,ψ+〉)-(〈φ-,y^,φ+〉,ℒG〈ψ-,z^,ψ+〉)=∫-∞0iφ-′ψ¯-dξ+(l(y^),z^)H+∫0∞iφ+′ψ¯+dξ-∫-∞0iψ-′φ¯-dξ-(y^,l(z^))H-∫0∞iψ+′φ¯+dξ=∫-∞0iφ-′ψ¯-dξ+δ1δ2[y^,z^]b+∫0∞iφ+′ψ¯+dξ-∫-∞0iψ-′φ¯-dξ-∫0∞iψ+′φ¯+dξ=iψ-(0)φ¯-(0)-iφ+(0)ψ¯+(0)+δ1δ2[y^,z^]b.
We obtain by direct computation
(14)iψ-(0)φ¯-(0)-iφ+(0)ψ¯+(0)+δ1δ2[y^,z^]b=0.
Thus, ℒG is a symmetric operator. To prove that ℒG is self-adjoint, we need to show that ℒG⊆ℒG*. Take g=〈ψ-,z^,ψ+〉∈D(ℒG*). Let ℒG*g=g*=〈ψ-*,z^*,ψ+*〉∈ℋ, so that
(15)(ℒGf,g)ℋ=(f,ℒG*g)ℋ=(f,g*)ℋ.
By choosing elements with suitable components as the f∈D(ℒG) in (15), it is not difficult to show that ψ-∈W21(-∞,0), ψ+∈W21(0,∞),g∈D(ℒ), and g*=ℒGg; the operator ℒG is defined by (12). Therefore (15) is obtained from(ℒGf,g)ℋ=(f,ℒGg)ℋ for all f∈D(ℒG*).Furthermore, g∈D(ℒG*) satisfies the conditions
(16)[y^,v]b+G[y^,u]b=Cδ1δ2φ-(0),[y^,v]b+G¯[y^,u]b=Cδ1δ2φ+(0).
Hence,D(ℒG*)⊆D(ℒG); that is, ℒG=ℒG*.
The self-adjoint operatorℒG generates on ℋ a unitary group Ut=exp(iℒGt) (t∈ℝ+=(0,∞)). Let us denote by P:ℋ→H and P1:H→ℋ the mapping acting according to the formulae P:〈φ-,y^,φ+〉→y^ and P1:y^→〈0,y^,0〉. Let Zt:=PUtP1, t≥0, by using Ut. The family {Zt}(t≥0) of operators is a strongly continuous semigroup of completely nonunitary contraction on H. Let us denote by BG the generator of this semigroup:BGy^=limt→+0(it)-1(Zty^-y^). The domain of BG consists of all the vectors for which the limit exists. The operator BG is dissipative. The operator ℒG is called the self-adjoint dilation of BG (see [10, 21, 30]). We show that BG=L~G; hence ℒG is self-adjoint dilation of BG. To show this, it is sufficient to verify the equality
(17)P(ℒG-λI)-1P1y^=(L~G-λI)-1y^,y^∈H,Imh<0.
For this purpose, we set (ℒG-λI)-1P1y^=g=〈ψ-,z^,ψ+〉 which implies that (ℒG-λI)g=P1y^, and hence l~(z^)-λz^=y^,ψ-(ξ)=ψ-(0)e-iλξ and ψ+(ξ)=ψ+(0)e-iλξ. Since g∈D(ℒG), then ψ-∈W21(-∞,0); it follows that ψ-(0)=0, and consequently z satisfies the boundary condition [y^,v]b-G[y^,u]b=0. Therefore z^∈D(L~G), and since point λ with Imλ<0 cannot be an eigenvalue of dissipative operator, then z^=(L~G-λI)-1y^. Thus we have
(18)(ℒG-λI)-1P1y^=〈0,(L~G-λI)-1y^,δ1δ2C([y^,v]b+G¯[y^,u]b)e-iλξ〉
for y^ and Imλ<0. By applying onto the mapping P, we obtain (17), and
(19)(L~G-λI)-1=P(ℒG-λI)-1P1=-iP∫0∞Ute-iλtdtP1=-i∫0∞Zte-iλtdt=(BG-λI)-1,Imλ<0,
so this clearly shows that L~G=BG.


## 3. Functional Model of Dissipative Sturm-Liouville Operator

The unitary group{Ut} has an important property which makes it possible to apply it to the Lax-Phillips [23]. It has orthogonal incoming and outgoing subspaces D-=〈L2(-∞,0),0,0〉 and D+=〈0,0,L2(0,∞)〉 having the following properties:(1)
UtD-⊂D-, t≤0 and UtD+⊂D+, t≥0,(2)
∩t≤0UtD-=∩t≥0UtD+={0},(3)
∪t≥0UtD-¯=∪t≤0UtD+¯=ℋ,(4)
D-⊥D+.Property (4) is clear. To be able to prove property (1) for D+ (the proof for D- is similar), we set ℛλ=(ℒG-λI)-1. For all λ, with  Imλ<0 and for any f=〈0,0,φ+〉∈D+, we have
(20)ℛλf=〈0,0,-ie-iλξ∫0ξeiλsφ+(s)ds〉,
as ℛλf∈D+. Therefore, if g⊥D+, then
(21)0=(ℛλf,g)ℋ=-i∫0∞e-iλt(Utf,g)ℋdt,Imλ<0,
which implies that (Utf,g)ℋ=0 for all t≥0. Hence, for t≥0, UtD+⊂D+, and property (1) has been proved.In order to prove property (2), we define the mappings P+:ℋ→L2(0,∞) and P1+:L2(0,∞)→D+ as follows: P+:〈φ-,y^,φ+〉→φ+ and P1+:φ→〈0,0,φ〉, respectively. We take into consider that the semigroup of isometries Ut+:=P+UtP1+(t≥0) is a one-sided shift in L2(0,∞). Indeed, the generator of the semigroup of the one-sided shift Vt in L2(0,∞) is the differential operator i(d/dξ) with the boundary condition φ(0)=0. On the other hand, the generator S of the semigroup of isometries Ut+(t≥0) is the operator Sφ=P+ℒGP1+φ=P+ℒG〈0,0,φ〉=P+〈0,0,i(d/dξ)φ〉=i(d/dξ)φ, where φ∈W21(0,∞) and φ(0)=0. Since a semigroup is uniquely determined by its generator, it follows that Ut+=Vt, and, hence,
(22)⋂t≥0UtD+=〈0,0,⋂t≤0VtL2(0,∞)〉={0},
so the proof is completed.Definition 2.
The linear operatorA with domain D(A) acting in the Hilbert space H is called completely non-self-adjoint (or simple) if there is no invariant subspace M⊆D(A)(M≠{0}) of the operator A on which the restriction A to M is self-adjoint.To prove property (3) of the incoming and outgoing subspaces, let us prove following lemma.Lemma 3.
The operatorL~G is completely non-self-adjoint (simple).Proof.
LetH′⊂H be a nontrivial subspace in which L~G induces a self-adjoint operator L~G′ with domain D(L~G′)=H′∩D(L~G). If f^∈D(L~G′), then f^∈D(L~G*) and
(23)0=ddt∥eiL~G′tf^∥H2=ddt(eiL~G′tf^,eiL~G′tf^)H=-C2([eiL~G′tf^,u]b)H2.
Consequently, we have [y^,u]b=0.Using this result with boundary condition [y^,v]b+G[y^,u]b=0,we have [y^,u]b=0; that is, y^(λ)=0. Since all solutions of l(y)=λybelong to L2(0,∞), from this it can be concluded that the resolvent Rλ(L~G) is a compact operator, and the spectrum of L~G is purely discrete. Consequently, by the theorem on expansion in the eigenvectors of the self-adjoint operator L~G′ we obtain H′={0}. Hence the operator L~G is simple. The proof is completed.Let us defineH-=∪t≥0UtD-¯, H+=∪t≤0UtD+¯.Lemma 4.
The equalityH-+H+=ℋ holds.Proof.
Considering property (1) of the subspace D+, it is easy to show that the subspace ℋ′=ℋ⊝(H-+H+) is invariant relative to the group {Ut} and has the form ℋ′=〈0,H′,0〉, where H′ is a subspace in H. Therefore, if the subspace ℋ′(andhencealsoH′) was nontrivial, then the unitary group {Ut′} restricted to this subspace would be a unitary part of the group {Ut}, and hence, the restriction L~G′ of L~G to H′ would be a self-adjoint operator in H′. Since the operator L~G is simple, it follows that H′={0}. The lemma is proved.Assume that(24)φ(x,λ):={φ1(x,λ),x∈I1φ2(x,λ),x∈I2,ψ(x,λ):={ψ1(x,λ),x∈I1ψ2(x,λ),x∈I2
are solutions of l(y)=λy satisfying the conditions
(25)φ1(a,λ)=cosα,φ1′(a,λ)=sinα,ψ1(a,λ)=-sinα,ψ1′(a,λ)=cosα,φ2(c+,λ)=γ1δ1φ1(c-,λ),φ2′(c+,λ)=γ2δ2φ1′(c-,λ).
Let us adopt the following notations:
(26)K(λ)=[φ,v]b[ψ,u]b,M(λ)=-[ψ,u]b[φ,u]b,(27)SG(λ)=(M(λ)K(λ)+G)(M(λ)K(λ)+G¯)-1,
where M(λ) 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 M(λ) possesses the following properties: ImM(λ)≤0 for all Imλ≠0, and M¯(λ)=M(λ¯) for all λ∈ℂ, except the real poles M(λ).We set(28)Uλ-(x,ξ,ζ)=〈{Cδ1δ2}e-iλξ,Cδ1δ2M(λ)[(M(λ)K(λ)+G)[ψ,u]b]-1φ(x,λ),SG¯(λ)e-iλζ{Cδ1δ2}〉.
We note that the vectors Uλ-(x,ξ,ζ) for real λ do not belong to the space ℋ. However, Uλ-(x,ξ,ζ) satisfies the equation ℒU=λU and the corresponding boundary conditions for the operator ℒh.By means of vectorUλ-(x,ξ,ζ), we define the transformation F-:f→f-~(λ) by
(29)(F-f)(λ):=f-~(λ):=12π(f,Uλ¯)ℋ
on the vectors f=〈φ-,y^,φ+〉 in which φ-(ξ),φ+(ζ),y(x) are smooth, compactly supported functions.Lemma 5.
The transformationF- isometrically maps H- onto L2(ℝ). For all vectors f,g∈H- the Parseval equality and the inversion formulae hold:
(30)(f,g)ℋ=(f-~,g-~)L2=∫-∞∞f-~(λ)g-~(λ)¯dλ,f=12π∫-∞∞f-~(λ)Uλ¯dλ,wheref-~(λ)=(F-f)(λ)andg-~(λ)=(F-g)(λ).Proof.
Forf,g∈D-, f=〈φ-,0,0〉, g=〈ψ-,0,0〉, with Paley-Wiener theorem, we have
(31)f-~(λ)=12π(f,Uλ¯)ℋ=12π∫-∞0φ-(ξ)e-iλξdξ∈H-2,
and by using usual Parseval equality for Fourier integrals,
(32)(f,g)ℋ=∫-∞∞φ-(ξ)ψ-(ξ)¯dξ=∫-∞∞f-~(λ)g-~(λ)¯dλ=(F-f,F-g)L2.
Here, H±2 denote the Hardy classes in L2(ℝ) consisting of the functions analytically extendible to the upper and lower half-planes, respectively.
We now extend the Parseval equality to the whole ofH-. We consider in H- the dense set of H-′ of the vectors obtained as follows from the smooth, compactly supported functions in D-:f∈H-′ if f=UTf0, f0=〈φ-,0,0〉, φ-∈C0∞(-∞,0), where T=Tf  is a nonnegative number depending on f. If f,g∈H-′, then for T>Tf and T>Tg we have U-Tf,U-Tg∈D-; moreover, the first components of these vectors belong to C0∞(-∞,0).Therefore, since the operators Ut(t∈ℝ) are unitary, by the equality
(33)F-Utf=(Utf,Uλ-)ℋ=eiλt(f,Uλ-)ℋ=eiλtF-f,
we have
(34)(f,g)ℋ=(U-Tf,U-Tg)ℋ=(F-U-Tf,F-U-Tg)L2,(35)(eiλTF-f,eiλTF-g)L2=(f~,g~)L2.
By taking the closure (35), we obtain the Parseval equality for the space H-. 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 F-H-=∪t≥0F-UtD-¯=∪t≥0eiλtH-2¯=L2(ℝ); that is, F- maps H- onto the whole of L2(ℝ). The lemma is proved.We set(36)Uλ+(x,ξ,ζ)=〈{Cδ1δ2}SG(λ)e-iλξ,Cδ1δ2M(λ)[(M(λ)K(λ)+G¯)[ψ,u]b]-1φ(x,λ),e-iλζ{Cδ1δ2}〉.
We note that the vectors Uλ+(x,ξ,ζ) for real λ do not belong to the space ℋ. However, Uλ+(x,ξ,ζ) satisfies the equation ℒU=λU and the corresponding boundary conditions for the operator ℒh. With the help of vector Uλ+(x,ξ,ζ), we define the transformation F+:f→f+~(λ) by (F+f)(λ):=f+~(λ):=(1/2π)(f,Uλ+)ℋ on the vectors f=〈φ-,y^,φ+〉 in which φ-(ξ),φ+(ζ), and y(x) are smooth, compactly supported functions.Lemma 6.
The transformationF+ isometrically maps H+ onto L2(ℝ). For all vectors  f,g∈H+  the Parseval equality and the inversion formula hold:
(37)(f,g)ℋ=(f+~,g+~)L2=∫-∞∞f+~(λ)g+~(λ)¯dλ,f=12π∫-∞∞f+~(λ)Uλ+dλ,
wheref+~(λ)=(F+f)(λ)and g+~(λ)=(F+g)(λ).Proof.
The proof is analogous to Lemma6.It is obvious that the matrix-valued functionSG(λ) is meromorphic in ℂ and all poles are in the lower half-plane. From (27), |SG(λ)|≤1 for Imλ>0, and SG(λ) is the unitary matrix for all λ∈ℝ. Therefore, it explicitly follows from the formulae for the vectors Uλ- and Uλ+ that
(38)Uλ+=SG(λ)Uλ-.
It follows from Lemmas 6 and 5 that H-=H+. Together with Lemma 5, this shows that H-=H+=ℋ; therefore property (3) has been proved for the incoming and outgoing subspaces.Thus, the transformationF- isometrically maps H- onto L2(ℝ) with the subspace D- mapped onto H-2 and the operators Ut are transformed into the operators of multiplication by eiλt. This means that F- is the incoming spectral representation for the group {Ut}. Similarly, F+ is the outgoing spectral representation for the group {Ut}. It follows from (38) that the passage from the F- representation of an element f∈ℋ to its F+representation is accomplished as f+~(λ)=SG-1(λ)f-~(λ). Consequently, according to [22], we have proved the following.Theorem 7.
The functionSG¯(λ) is the scattering matrix of the group {Ut} (of the self-adjoint operator ℒG).LetS(λ) be an arbitrary nonconstant inner function (see [19]) on the upper half-plane (the analytic function S(λ) and the upper half-plane ℂ+ is called inner function on ℂ+ if |Sh(λ)|≤1 for all λ∈ℂ+ and |Sh(λ)|=1 for almost all λ∈ℝ). Define K=H+2⊝SH+2. Then K≠{0} is a subspace of the Hilbert space H+2. We consider the semigroup of operators Zt(t≥0) acting in K according to the formula Ztφ=P[eiλtφ], φ=φ(λ)∈K, where P is the orthogonal projection from H+2 onto K. The generator of the semigroup {Zt} is denoted by
(39)Tφ=limt→+0(it)-1(Ztφ-φ),
where T is a maximal dissipative operator acting in K and with the domain D(T) consisting of all functions φ∈K, such that the limit exists. The operator T 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 Foiaş [22]. The basic assertion is that S(λ) is the characteristic functionof the operator T.LetK=〈0,H,0〉, so that ℋ=D-⊕K⊕D+. It follows from the explicit form of the unitary transformation F- under the mapping F- that
(40)ℋ→L2(ℝ),f→f-~(λ)=(F-f)(λ),D-→H-2,D+→SGH+2,K→H+2⊝SGH+2,Ut→(F-UtF--1f-~)(λ)=eiλtf-~(λ).
The formulas (40) show that operator L~G is unitarily equivalent to the model dissipative operator with the characteristic function SG(λ). We have thus proved the following theorem.Theorem 8.
The characteristic function of the maximal dissipative operatorL~G coincides with the function SG(λ) defined by (27).


## 4. The Spectral Properties of Dissipative Sturm-Liouville Operators

By using characteristic function, the spectral properties of the maximal dissipative operatorL~G(LK) can be investigated. The characteristic function of the maximal dissipative operator L~G is known to lead to information of completeness about the spectral properties of this operator. For instance, the absence of a singular factor s(λ) of the characteristic function SG(λ) in the factorization detSG(λ)=s(λ)B(λ)  (B(λ) is a Blaschke product) ensures completeness of the system of eigenvectors and associated vectors of the operator L~G(LK)in the space L2(0,∞) (see [10, 21, 30]). If thecharacteristic function SG(λ) has nontrivial singular factor, the system of eigenvectors and associated vectors of the operator L~G(LK)can fail to be complete. Because SG(λ)is smooth, the support of the corresponding singular measure μ must be contained in the set of poles SG(λ). But in this case the singular measure μ is a simple step function. If we require SG(λ)to have no zeros of infinite multiplicity, then μ=0.So the singular factor vanishes. The characteristic function SG(λ) of the maximal dissipative operator L~G has the form
(41)SG(λ):=M(λ)K(λ)+GM(λ)K(λ)+G¯,
where ImG>0.Theorem 9.
For all the values ofG with ImG>0, except possibly for a single value G=G0, the characteristic function SG(λ) of the maximal dissipative operator L~G is a Blaschke product. The spectrum of L~G is purely discrete and belongs to the open upper half-plane. The operator L~G(G≠G0) 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~G is complete in the space H.Proof.
From (35), it is clear that SG(λ) is an inner function in the upper half-plane, and it is meromorphic in the whole complex λ-plane. Therefore, it can be factored in the form
(42)SG(λ)=eiλcBG(λ),c=c(G)≥0,
where BG(λ) is a Blaschke product. It follows from (42) that
(43)|SG(λ)|=|eiλc||BG(λ)|≤e-b(G)Imλ,Imλ≥0.
Further, expressing nG(λ):=M(λ)K(λ) in terms of SG(λ), we find from (35) that
(44)nG(λ)=G¯SG(λ)-G1-SG(λ).
For a given value G (ImG>0), if  c(G)>0, then (43) implies that limt→+∞SG(it)=0, and then (44) gives us that limt→+∞nG(it)=G0. Since nG(λ) does not depend on G, this implies that c(G) can be nonzero at not more than a single point G=G0 (and further G0=-limt→+∞nG(it)). This completes the proof.



---

*Source: 248740-2013-05-07.xml*