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Original Research ARTICLE

Front. Phys., 10 July 2019 | https://doi.org/10.3389/fphy.2019.00099

Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator

  • 1Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, Ankara, Turkey
  • 2The Theoretical Physics Laboratory, Institute of Space Sciences, Magurele-Bucharest, Romania

In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential expression and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal non-selfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.

2000 Mathematics Subject Classification: Primary 47A45, 47E05; Secondary 47A20

1. Introduction

A model operator may be regarded as an equivalent operator to another operator in a certain sense. Such an equivalent representation has been constructed by Szokefalvi-Nagy and Foiaş [1] for a contractive operator. The main idea for this construction is to obtain the unitary dilation of the contraction. In fact, if the following equality holds

Tny=PUny,

where T is a contraction on the Hilbert space H and U is the operator on H, yH, n ≥ 0 and P is the orthogonal projection of H onto H, then U is called a dilation of T. U is called unitary provided that U is a unitary operator and in this case U is called unitary dilation of T. There exists a geometric meaning of the dilation space. This meaning has been given by Sarason [2]. Sarason showed that U is a dilation of T if and only if H has the representation

H=G*HG,

where UGG and U*G*G*. This representation is closely related with incoming and outgoing spaces in the scattering theory [3]. In the case that

H=span{UnH,n}

then G and G are uniquely determined and U is called minimal. If U is unitary minimal dilation of T then one may consider the decomposition [4]

U=[PG*UG*00DT*V**T0-VT*V**VDTUG],

where DT* and DT are so called defect operators of T defined by

DT=(I-TT*)1/2,DT*=(I-T*T)1/2,

𝔇T and 𝔇T* are the defect spaces defined by

𝔇T=closDTH,𝔇T*=closDT*H,

V is a partial isometry with the initial space 𝔇T and final space E = GUG and V is a partial isometry with initial space 𝔇T* and the final space E*=G*U*G*.

Now consider the transformations

π:L2(E) nUn(GUG),nznennUnυen,

and

π*:L2(E*)nUn(G*U*G*),   nznen*nUn+1υ*en*,

where υ and υ are the unitary mappings defined by

υ:EGUG

and

υ*:E*G*U*G*.

In the literature the operators π and π are called functional embeddings. The function π**π acting from E into E is called the characteristic function of the contraction T. If υ and υ are chosen as the unitary identifications between 𝔇T and E, and, 𝔇T* and E, respectively, then the characteristic function can be introduced as

π**π=ΘT(ζ)a=V*(-T+ζDT*(I-ζT*)-1DT)V*a,aE.

Nagy and Foiaş introduced the characteristic function of a contraction as [1]

ΘT=-T+ζDT*(I-ζT*)-1DT

and this can be obtained from the previous equation by choosing E = 𝔇T and E*=𝔇T*.

Solomyak [5] using the connection between dissipative operators and their Cayley transforms introduced an effective way to obtain the characteristic function of both dissipative operators and related contractions generated by Cayley transforms. By a dissipative operator it is meant an operator A with a dense domain D(A) acting on a Hilbert space K satisfying

Im(Ah,h)0,hD(A).

An immediate result on dissipative operators is that all eigenvalues lie in the closed upper half-plane. If a dissipative operator does not have a proper dissipative extension then A is called maximal dissipative. The Cayley transform of a dissipative operator

C(A)=(A-iI)(A+iI)-1

is a contraction from (A + iI)D(A) onto (AiI)D(A), i.e.,

C(A)1.

It is known that a dissipative operator is maximal if and only if C(A) is a contraction such that domain of C(A) is the Hilbert space K and 1 can not belong to the point spectrum of C(A). Solomyak used these connections and boundary spaces associated with A to construct the characteristic function SA(λ) with the rule

SA(λ)=P*(A*-λI)-1(A-λI)P-1,    (1.1)

where P and P are the natural projections. To be more precise we should note that for a maximal dissipative operator A the Hermitian part AH of A is defined as the restriction of A to the following subspace

GA={fD(A)D(A*):Af=A*f}.

The natural projection P is defined by

P:D(A)D(A)/GA,

where D(A)/GA is the quotient space. Similarly P is defined by

P*:D(A*)D(A*)/GA.

On the quotient spaces the following inner products are defined

Pφ,Pψ=i2((φ,Pψ)-(Pφ,ψ)),φ,ψD(A)

and

P*φ*,P*ψ*=i2((φ*,P*ψ*)-(P*φ*,ψ*)),φ*,ψ*D(A*).

Let F(A) be the completion of the quotient space D(A)/GA with respect to the norm

PφF2=Im(Aφ,φ).

In a similar way one may define F*(A):=F(-A*) and F(A) is equipped with the norm

P*φ*F*2=Im(A*φ*,φ*).

These spaces F(A) and F(A) are called boundary spaces. Solomyak showed for a maximal dissipative operator A and its Cayley transform C(A) that there exist isometric isomorphisms

ρ:F(A)𝔇C,ρ*:F*(A)𝔇C*

with the rules

ρP(I-C)=𝔇C,ρ*P*(I-C*)=𝔇C*.

Then fixing arbitrary isometric isomorphisms Ω : E → 𝔇C, Ω*:E*𝔇C* the characteristic function ΘC of the Cayley transform C(A) can be introduced by

ΘC(ζ)=Ω**(-C+ζDC*(I-ζC*)DC)Ω.

Finally taking Ω = ρ, Ω = ρ, E = F(A), E = F(A) one obtains (1.1).

In this paper using the results of Solomyak we investigate some spectral properties of a regular third-order dissipative operator. We should note that such an investigation with the aid of Solomyak's approach has not been introduced for the third-order case. In fact, the literature has less works on odd-order operators than on even-order equations even if there exists some results in the literature [613]. The main reason is the confusion of imposing the boundary conditions because as Everitt says in [9] that it is impossible to impose separated boundary conditions for the solutions of a third-order equation. Consequently, this paper may give an idea to use Solomyak's method for the odd-order dissipative or accumulative operators.

2. Maximal Dissipative Operator

Throughout the paper we consider the following third-order differential expression

(y)=1w{-i(q0(q0y))-(p0y)+i[q1y+(q1y)]+p1y}              ,x[a,b],

where qj, pj, j = 0, 1, w are real-valued and continuous functions on [a, b] and q0 > 0 or q0 < 0 and w > 0 on [a, b].

The quasi-derivative y[r] of the function y is defined by

y[0]=y,y[1]=-1+i2q0y,y[2]=iq0(q0y)+p0y-iq1y.

Let H denote the Hilbert space with the usual inner product

(y,z)=abyz¯wdx

and with the norm

y2=(y,y).

Now consider the subspace D of H which consists of the functions yH such that y[r], 0 ≤ r ≤ 2, is locally absolutely continuous on [a, b] and ℓ(y) ∈ H. The maximal operator L is defined on D by

Ly=(y),yD,x[a,b].

For y, zD following Lagrange's formula can be introduced

(Ly,z)-(y,Lz)=[y,z](b)-[y,z](a),    (2.1)

where

[y,z]:=yz[2]¯-y[2]z¯+iy[1]z[1]¯.

(2.1) particulary implies the meaning of [y, z](a) and [y, z](b) for y, zD.

Let D0 be a set of D that consists of those functions yD such that y has a compact support on [a, b]. The operator L0 which is the restriction of L to D0 is a densely defined symmetric operator and therefore it admits the closure. Let L0 be the closure of L0. L0 then becomes a densely defined, symmetric operator with domain D0 that consists of the functions yD satisfying

y[r](a)=y[r](b)=0,0r2.

Moreover one has L0*=L [14, 15].

For the symmetric operators there exists a useful theory called deficiency indices theory to construct the extensions. In fact, let M be a symmetric operator on a Hilbert space B and Rλ denoted the range of M − λI, where λ is a parameter and I is the identity operator in B. The deficiency spaces Nλ and Nλ¯ are defined by Naimark [14]

Nλ=BRλ,Nλ¯=BRλ¯.

The deficiency indices (m, n) of the operator M are defined by

(m,n)=(dimNi,dimN-i).

Note that the deficiency indices of L0 are (3, 3).

To describe the extensions of a closed, symmetric operator with equal deficiency indices one may use the boundary value space. Boundary value space of the closed symmetric operator M is a triple (K, σ1, σ2) such that σ1, σ2 are linear mappings from D(M*) (domain of M*) into K and following holds:

(i) for any f, gD(M*)

(M*f,g)-(f,M*g)=(σ1f,σ2g)K-(σ2f,σ1g)K,

(ii) for and F1, F2K, there exists a vector fD(M*) such that σ1f = F1 and σ2f = F.

Now for yD consider the following mappings

σ1y=(y[2](a),12y[1](a)+i2y[1](b),y(b))

and

σ2y=(y(a),iy[1](a)+y[1](b),y[2](b)).

Then we have the following Lemma.

Lemma 2.1. Fory, zD

(σ1y,σ2z)3(σ2y,σ1z)3=[y,z](b)[y,z](a).

Proof: Let y, zD. Then

(σ1y,σ2z)3(σ2y,σ1z)3=σ1y(σ2z)σ2y(σ1z)=y[2](a)z(a)¯+(12y[1](a)+i2y[1](b))(iz[1](a)¯+z[1](b))+y(b)z[2](b)¯[y(a)z[2](a)¯+(iy[1](a)+y[1](b))(12z[1](a)¯i2z[1](b)¯)+y[2](b)z(b)¯]=[y,z](b)[y,z](a).

This completes the proof.

One of our aim is to impose some suitable boundary conditions for the solution y of the equation

-i(q0(q0y))-(p0y)+i[q1y+(q1y)]+p1y=λwy,x[a,b],    (2.2)

where λ is the spectral parameter and yD. We should note that the Equation (2.2) has a unique solution χ(x,λ) satisfying the initial conditions

χ[r](c,λ)=lr,0r2,

where lr is a complex number. This fact follows from the assumptions on the coefficients q0, q1, p0, p1, w, and following representation

Y=A(x,λ)Y,    (2.3)

where

Y=[y[0]y[1]y[2]],  A=[0-2(1+i)q00-1+i2q1q0ip0q02-2(1+i)q0p1-λw-1+i2q1q00].

Then the theory on ordinary differential equations may be applied to the first-order system (2.3), where the elements of A are integrable on each compact subintervals of [a, b].

Now the next Lemma can be introduced with the aid of Naimark's patching Lemma [14].

Lemma 2.2. There existsyD satisfying

y[r](a)=αr,y[r](b)=βr,0r2,

where αr, βr are arbitrary complex numbers.

Now we may introduce the following.

Theorem 2.3. (3,σ1,σ2) is a boundary value space for L0.

Proof: Since L0*=L we obtain for y, zD that

(L0*y,z)-(y,L0*z)=[y,z](b)-[y,z](a).

Therefore, Lemma 2.1 and Lemma 2.2 complete the proof.

Let S be a contraction and N be a selfadjoint operator on ℂ3. Then using the Theorem of Gorbachuks' [16], p. 156, the following abstract Theorem can be introduced.

Theorem 2.4. Let fD. Then the conditions

(sinN)σ1f-(cosN)σ2f=0,(S-I)σ1f+i(S+I)σ2f=0,(S-I)σ1f-i(S+I)σ2f=0,

describe, respectively, the maximal selfadjoint, maximal dissipative, and maximal accumulative extensions of L0.

Since we will investigate the spectral properties of the maximal dissipative extension of L0 we shall introduce the following.

Corollary 2.5. For yD the maximal dissipative extension of L0 is described by

y(a)+h1y[2](a)=0,Im h10,iy[1](a)+y[1](b)+h2,(12y[1](a)+i2y[1](b))=0,Im h2,0,y[2](b)+h3y(b)=0,Im h30.

Corollary 2.6. For yD the conditions

y(a)+h1y[2](a)=0,Im h1=0,(i+h2)y[1](a)+(1+ih2)y[1](b)=0,Im h2>0,h2i,y[2](b)+h3y(b)=0,Im h3>0,    (2.4)

where h2 = h2,∗/2, describe the maximal dissipative extension of L0.

Remark 2.7. As may be seen in the next sections, the caseh2 = i may give rise to some complications. Therefore, we exclude this case.

Now let D(L) be a set consisting of all functions yD satisfying the conditions (2.4). Let us define the operator L on D(L) with the rule

Ly=(y),yD(L),x[a,b].

Then L is a maximal dissipative operator on H.

The adjoint operator L* of L is given by

L*y=(y),yD(L*),x[a,b],

where D(L*) is the domain of L* consisting of all functions yD satisfying

y(a)+h1y[2](a)=0,Im h1=0,(i+h¯2)y[1](a)+(1ih¯2)y[1](b)=0,Im h2>0,h2i,y[2](b)+h¯3y(b)=0,Im h3>0.

Theorem 2.8. L is totally dissipative (simple) in H.

Proof: This follows from choosing h2 and h3 with positive imaginary parts. Indeed, for yD(L) one gets

Im(Ly,y)=2 Im h2|1+ih2|2|y[1](a)|2+Im h3|y(b)|2.    (2.5)

If L had a selfadjoint part Ls in HsH then from (2.5) one would get

y(b)=y[1](b)=y[2](b)=0

and therefore y ≡ 0. This completes the proof.

3. Contractive Operator

There exists a connection between dissipative operator L and the contractive operator C. This connection can be given by the following relation

C=(L-iI)(L+iI)-1.

Since L is maximal dissipative the domain of C is the whole Hilbert space H.

An important class of contractions on a Hilbert space consists of completely non-unitary (c.n.u.) contractions. A contraction C is said to be c.n.u. if there exists no non-zero reducing subspace H0 such that CH0 is a unitary operator.

From the simplicity of L we have the following.

Theorem 3.1. C is a c.n.u. contraction onH.

Proof: Let (L+iI)-1f=y, where yD(L) and fH. Then we get

(LiI)y2<(L+iI)y2    (3.1)

because

Im(Ly,y)>0,yD(L).

(3.1) implies that

C<1    (3.2)

and this completes the proof.

Now we define the defect operators of C as

DC=(I-C*C)1/2,DC*=(I-CC*)1/2

and the defect spaces of C as

𝔇C=DCH¯,𝔇C*=DC*H¯.

The numbers 𝔡C and 𝔡C* defined by

𝔡C=dim𝔇C,𝔡C*=dim𝔇C*

are called the defect indices of C.

Theorem 3.2. 𝔡C=𝔡C*=2.

Proof: Consider the equation

DC2f=(L+iI)y-(L*+iI)z,

where f=(L+iI)y, yD(L), fH and zD(L*). Then

z=(L*-iI)-1(L-iI)y

or

(L*-iI)z=(L-iI)y.    (3.3)

Equation (3.3) implies that 𝔇C is spanned by two independent solutions. In fact, let φ(x, λ) and φ~(x,λ) be two solutions of (2.2) satisfying

φ(a,λ)=-h1,φ[2](a,λ)=1,φ[1](a,λ)=c(0),    (3.4)

where c is a constant and

φ~(a,λ)=-h1,φ~[2](a,λ)=1,φ[1](a,λ)=0.

(2.5) needs the solutions of (2.2) satisfying the condition

y(a)+h1y[2](a)=0,h1    (3.5)

Clearly φ and φ~ satisfies (3.5) and φ can not be represented by a constant of φ~. If there exists any other solution ψ(x, λ) of (2.2) satisfying

ψ(a,λ)=-h1,ψ[2](a,λ)=1,ψ[1](a,λ)=c1,

where c1 is another constant different from c then ψ~(x,λ):=(c1/c)φ(x,λ) becomes a solution of (2.2) satisfying (3.5) and ψ(x, λ) may be introduced by φ(x, λ).

Therefore,

DC2f=(+iI)(y-z)=2i(d1φ+d2φ~),

where d1 and d2 are constants and 𝔇C is spanned by φ(x, i) and φ~(x,i).

With a similar argument one may see that

DC*2f=(-iI)(y-z)=-2i(d1φ+d2φ~),

and therefore 𝔇C* is spanned by φ(x, −i) and φ~(x,-i).

This completes the proof.

Definition 3.3. [17] The classesC0. andC.0 are defined as

C0.={T:T1,limnTnf=0 for allf},   C.0={T:T1,limnT*nf=0 for allf}.

C00 is defined by C00 = C0.C.0.

Theorem 3.4. CC00.

Proof: This follows from (3.2), C=C* and the equalities

CnfCnf,

and

C*nfC*nf.

The class C0 consists of those c.n.u. contractions T for which there exists a non-zero function uH (Hp denotes the Hardy class) such that u(T) = 0. Since C belongs to the class C00 with finite defect numbers this implies the following [1].

Theorem 3.5. CC0.

4. Characteristic Function

We shall consider the inner product on the quotient space D(L)/GL as follows

(Py,Pz)=i2((y,Ly)-(Ly,y)),y,zD(L),

where P is the natural projection with P:D(L)D(L)/GL. The completion of D(L)/GL is denoted by F(L) with respect to the corresponding norm. Similarly F*(L):=F(-L*) and P is defined by P*:D(L*)D(L*)/GL. One has

PyF2=Im(Ly,y),PyF2=Im(Ly,y).    (4.1)

F(L) and F(L*) are the boundary spaces of L. From (4.1) we get

PyF2=2 Im h2|1+ih2|2|y[1](a)|2+Im h3|y(b)|2    (4.2)

and

P*zF*2=2 Im h2|1+ih2|2|z[1](a)|2+Im h3|z(b)|2.    (4.3)

From (4.2) and (4.3) we may set

Py=[2 Im h2|1+ih2|y[1](a)Im h3y(b)],P*z=[2 Im h2|1+ih2|z[1](a)Im h3z(b)].

Setting E=E*=2 we define the following isometric isomorphisms

Ψ:E       F(L),c=[c1c2]Ψ(c)=Py=[2 Im h2|1+ih2|y[1](a)Im h3y(b)],    (4.4)

where yD(L), y[1](a)=c2|1+ih2|(2 Im h2)-1/2, y(b)=c1(Im h3)-1/2 and

Ψ*:E*    F*(L),c=[c1c2]Ψ*(c)=P*z=[2 Im h2|1+ih2|z[1](a)Im h3z(b)],    (4.5)

where zD(L*), z[1](a)=c2|1+ih2|(2 Im h2)-1/2, z(b)=c1(Im h3)-1/2. Then we may introduce the characteristic function of L.

Theorem 4.1. The characteristic matrix-function ΘL of L is given by

ΘL(λ)=[-i+h2¯i+h2(1+ih2)φ[1](b)-(i+h2)φ[1](a)(1-ih2¯)φ[1](b)-(-i+h2¯)φ[1](a)00φ[2](b)+h3φ(b)φ[2](b)+h3¯φ(b)],Im λ>0.

Proof: Consider the equation

ΘL(λ)c=Ψ**P*(L*-λI)-1(L-λI)P-1Ψc.    (4.6)

(4.4) implies that yP−1 Ψc with yD(L) and therefore

(L*-λI)-1(L-λI)y=z,    (4.7)

where zD(L*) and

(L-λI)y=(L*-λI)z.    (4.8)

Using (4.6) and (4.7) we obtain

Ψ**P*z=[2 Im h2|1+ih2|z[1](a)Im h3z(b)].    (4.9)

From (4.8) we should find a solution u = zy of the Equation (2.2) satisfying (3.5). Therefore, we may set u=B(λ)φ(x,λ),, where φ is the solution of (2.2) satisfying the conditions in (3.4). Consider the equation

(z-y)[2](b)=φ[2](b,λ)φ(b,λ)(z-y)(b).    (4.10)

Since yD(L) and zD(L*) we get from (4.10)

z(b)(φ[2](b,λ)+h3¯φ[2](b,λ))=y(b)(φ[2](b,λ)+h3φ[2](b,λ)).    (4.11)

Similarly the equation

(z-y)[1](a)=φ[1](a,λ)φ[1](b,λ)(z-y)[1](b)

gives

z[1](a)-i+h2¯1-ih2¯(φ[1](a,λ)-1-ih2¯-i+h2¯φ[1](b,λ))   =y[1](a)i+h21+ih2(1+ih2i+h2φ[1](b,λ)-φ[1](a,λ)).    (4.12)

(4.6) and (4.9) show that

ΘL(λ)[c1c2]=[z[1](a)y[1](a)00z(b)y(b)][c1c2].    (4.13)

Consequently (4.11) – (4.13) complete the proof.

Remind that a function Θ(ζ) whose values are bounded operators from a Hilbert space ℍ to a Hilbert space ℍ, both separable and which has a power series expansion

Θ(ζ)=k=0ζkΘk

whose coefficients are bounded operators from ℍ to ℍ. Moreover assume that

Θ(ζ)const.

Such a function with the spaces ℍ and ℍ is called bounded analytic function. If const = 1 then it is called contractive analytic function. The contractive analytic function Θ is said to be inner if Θ(eit) is isometry from ℍ into ℍ for almost all t.

Since there exists a connection between the characteristic function ΘL of L and the characteristic function ΘC of C with the rule

ΘL(λ)=ΘC(λ-iλ+i)

we have the following.

Corollary 4.2. The characteristic function ΘC of C is given by

ΘC(μ)=ΘL(λ),μ=λ-iλ+i,Imλ>0.

Since C is a c.n.u. contraction belonging to the class C.0 we have the following.

Theorem 4.3. ΘC(μ) is inner.

Corollary 4.4. det ΘC(μ) is inner.

An operator A ≥ 0 on a Hilbert space H is said to be of finite trace if A is compact and its eigenvalues is finite. This sum is called the trace of A [1, 18]. A contraction C on a Hilbert space H is called weak contraction if

(i) its spectrum does not fill the unit disc D,

(ii) IC*C is of finite trace.

Since every C0 contraction with finite-multiplicity is a weak contraction [18], p. 437, we may introduce the following.

Theorem 4.5. I-C*C is of finite trace.

The following Theorem is obtained from Nikolskiĭ [17], p. 134.

Theorem 4.6. The followings are satisfied:

(i) The root functions of C are complete in H,

(ii) The roots functions of C* are complete in H,

(iii)det(C*C)=det|ΘC(0)|2=μ|μ|2d(μ),

where μ belongs to the point spectrum of  C and d(μ) is the rank of the Riesz projection at a point μ in the set of point spectrum.

Proof: The proof follows from the fact detΘC is a Blashke product. So we shall prove this fact.

By Corollary 4.4 we may write

-i+h2¯i+h2(1+ih2)φ[1](b)-(i+h2)φ[1](a)(1-ih2¯)φ[1](b)-(-i+h2¯)φ[1](a)φ[2](b)+h3φ(b)φ[2](b)+h3¯φ(b)=𝔹(λ)eiλb,    (4.14)

where b > 0, Im λ > 0 and 𝔹(λ) is a Blashke product in the upper half-plane. Hence

|(1+ih2)φ[1](b)-(i+h2)φ[1](a)(1-ih2¯)φ[1](b)-(-i+h2¯)φ[1](a)φ[2](b)+h3φ(b)φ[2](b)+h3¯φ(b)|e-b Imλ.    (4.15)

For λs = is we have from (4.15) that the following possibilities may occur:

(i) φ[2](b)+h3φ(b)φ[2](b)+h3¯φ(b)0 as s → ∞,

(ii) (1+ih2)φ[1](b)-(i+h2)φ[1](a)(1-ih2¯)φ[1](b)-(-i+h2¯)φ[1](a)0 as s → ∞,

(iii) φ[2](b)+h3φ(b)φ[2](b)+h3¯φ(b)0 and (1+ih2)φ[1](b)-(i+h2)φ[1](a)(1-ih2¯)φ[1](b)-(-i+h2¯)φ[1](a)0 as s → ∞.

In fact (iii) is possible because in this case λ is an eigenvalue of L and this implies that λ is an eigenvalue of the operator L or equivalently 1 is an eigenvalue of the c.n.u. contraction C. However the latter one is not possible. Therefore, this completes the proof.

Definition 4.7. Let all root functions of the operatorL span the Hilbert spaceH. Such an operator is called complete operator. If everyLinvariant subspace is generated by root vectors of L belonging to the subspace then it is said L admits spectral synthesis.

Since every complete operator in C0 admits spectral synthesis [17], we obtain the following.

Theorem 4.8. C admits spectral synthesis.

Since the root functions of L span H then those of C must span H [19] (p. 42). Consequently we may introduce the following.

Theorem 4.9. Root functions of L associated with the point spectrum of L in the open upper half-planeIm λ > 0 span the Hilbert space H.

5. Dilation Operator and Its Eigenfunctions

In this section we investigate the properties of selfadjoint dilation of the operator L and eigenfunctions of selfadjoint dilation.

5.1. Selfadjoint Dilation of the Maximal Dissipative Operator

Following theorem gives the selfadjoint operator with free parameters [5].

Theorem 5.1.1. The minimal selfadjoint dilationL of the maximal dissipative operatorL in the space

HL=L2(-,E*)HL2(+,E)

has the form

L[φ-fφ+]=[iφ-i{2(I-C)-1[f-i2DC*Ω*φ-(0)]-ρ}iφ+]

and the domain of L is given by the conditions

φ-W21(-,E*),φ+W21(+,E),f-i2DC*Ω*φ-(0)(I-C)H=D(L),2iDC(I-C)-1[f-i2DC*Ω*φ-(0)]=C*Ω*φ-(0)+Ωφ+(0),

where W21 denotes the Sobolev space.

The isometries Ω:EDC, Ω*:E*DC* are the free parameters. In the case that dim𝔇C<, dim𝔇C*< then one may consider the boundary spaces F(L) and F*(L) instead of 𝔇C and 𝔇C*. Then following Lemma gives a direct approach for the dilation [5].

Lemma 5.1.2. The minimal selfadjoint dilationL in the spaceHL of the maximial dissipative operatorL inH with finite defects has the form

L[φ-fφ+]=[iφ-L(f-i2[Ψ*φ-(0)])+i2L*[Ψ*φ-(0)]iφ+]

where Ψ:EF(L) and Ψ*:E*F*(L) are the isometric isomorphisms and the domain of L is given by the conditions

φW21(,E),φ+W21(+,E),fi2[Ψφ(0)]D(L),fi2[Ψφ(0)]+i2[Ψφ+(0)]GL.

If GL is dense in H one may consider GL=D(L)D(L*) and

L˜:​​={L on D(L), L on D(L).

The following Corollary now may be introduced [5].

Corollary 5.1.3. The selfadjoint dilation L of the maximal dissipative operator L with finite defects such that GL is dense in H has the form

L[φ-fφ+]=[iφ-L~fiφ+] , L~=(LGL)*,

and the domain of L is given by the conditions

φW21(,E),φ+W21(+,E),fi2[Ψφ(0)]D(L),f+i2[Ψφ+(0)]D(L).

Now using Corollary 5.1.3 we may introduce the following.

Theorem 5.1.4. The selfadjoint dilation L of the maximal dissipative operator L in the space

HL=L2(-;2)HL2(+;2)

is given by the rule

L[φ-fφ+]=[iφ-(f)iφ+]

whose domain is given by the conditions

(i+h2)f[1](a)+(1+ih2)f[1](b)=(1-|h2|)(1+ih2)Im h2|1+ih2|φ-(1)(0),f[2](b)+h3f(b)=-2 Im h3φ-(2)(0),(-i+h2¯)f[1](a)+(1-ih2¯)f[1](b)=(|h2|-1)(1-ih2¯)Im h2|1+ih2|φ+(1)(0),f[2](b)+h3¯f(b)=-2 Im h3φ+(2)(0),

where

φ±:​​=[φ±(1)φ±(2)]W21(±;2).

Proof: Let yD(L) with y[1](a)=(2 Im h2)-1/2|1+ih2|φ+(1)(0), y(b)=(Im h3)-1/2φ+(2)(0) and zD(L*) with z[1](a)=(2 Im h2)-1/2|1+ih2|φ-(1)(0), z(b)=(Im h3)-1/2φ-(2)(0). Then f-i2-1/2[Ψ*φ-(0)]D(L) if and only if

f[2](b)-i2z[2](b)=-h3(f(b)-i2z(b))    (5.1)

and

(i+h2)(f[1](a)-i2z[1](a))=-(1+ih2)(f[1](b)-i2z[1](b)).    (5.2)

(5.1) gives

f[2](b)+h3f(b)=-2Im h3φ-(2)(0)

and (5.2) implies

(i+h2)f[1](a)+(1+ih2)f[1](b)=(1-|h2|)(1+ih2)Im h2|1+ih2|φ-(1)(0).

Similarly f+i2-1/2[Ψφ+(0)]D(L*) if and only if

f[2](b)+i2z[2](b)=-h3¯(f(b)+i2z(b))    (5.3)

and

(-i-h2¯)(f[1](a)+i2z[1](a))=-(1-ih2¯)(f[1](b)+i2z[1](b)).    (5.4)

(5.3) shows that

f[2](b)+h3¯f(b)=-2Im h3φ+(2)(0)

and (5.4) shows

(-i+h2¯)f[1](a)+(1-ih2¯)f[1](b)=-(1-|h2|)(1+ih2)Im h2|1+ih2|φ+(1)(0).

Therefore the proof is completed.

5.2. Eigenfunctions of the Dilation

As is pointed out in Solomyak [5] the generalized eigenfunctions of the dilation L may be introduced by incoming eigenfunctions

[ΘL(λ)exp(-iλr)ci2((L*-λ¯I)-1(L-λ¯I)-I)P-1Ψcexp(-iλs)c]

and outgoing eigenfunctions

[exp(-iλr)c~-i2((L-λI)-1(L*-λI)-I)P*-1Ψ*c~ΘL*(λ)exp(-iλs)c~],

where r ∈ ℝ, s ∈ ℝ+, cE, c~E* and λ ∈ ℝ.

Therefore we may introduce the following.

Theorem 5.2.1. The incoming and outgoing eigenfunction of L can be introduced by

[ΘL(λ)exp(-iλr)c[-2 Im h3φ[2](b,λ¯)+h3¯φ(b,λ¯)00(1+ih2i+h2φ[1](b,λ¯)φ[1](a,λ¯)-1-1φ[1](a,λ¯))|1+ih2|2 Im h2]exp(-iλs)c~],[exp(-iλr)c[-2 Im h3φ[2](b,λ¯)+h3¯φ(b,λ¯)00(1+ih2i+h2φ[1](b,λ¯)φ[1](a,λ¯)-1-1φ[1](a,λ¯))]ΘL*(λ)exp(-iλs)c~]

or

[ΘL(λ)exp(-iλr)c[2 Im h3φ[2](b,λ)+h3φ(b,λ)00-(1-ih2¯-i+h2¯φ[1](b,λ)φ[1](a,λ)-1-1φ[1](a,λ))]exp(-iλs)c~],[exp(-iλr)c[2 Im h3φ[2](b,λ)+h3φ(b,λ)|1+ih2|2 Im h200-(1-ih2¯-i+h2¯φ[1](b,λ)φ[1](a,λ)-1-1φ[1](a,λ))]ΘL*(λ)exp(-iλs)c~],

wherer ∈ ℝ, s ∈ ℝ+, λ ∈ ℝ.

Proof: Consider the equation

((L*-λ¯I)-1(L-λ¯I)-I)P-1Ψc=B(λ¯)φ(x,λ¯)c,    (5.5)

where z-y=B(λ¯)φ(x,λ¯), zD(L*) and yD(L).

One gets

(z-y)(b)=2i Im h3φ(b,λ¯)φ[2](b,λ¯)+h3¯φ(b,λ¯)y(b)

or

B(λ¯)φ(b,λ¯)=i2Im h3φ[2](b,λ¯)+h3¯φ(b,λ¯)φ(b,λ¯)c1

and

(z-y)[1](a)=(1+ih2i+h2φ[1](b,λ¯)-φ[1](a,λ¯)-1)y[1](a)

or

B(λ¯)φ[1](a,λ¯)=(1+ih2i+h2φ[1](b,λ¯)-φ[1](a,λ¯)-1)|1+ih2|2 Im h2c2.

Therefore the left-hand side of (5.5) can be introduced as

B(λ¯)[i2 Im h3φ[2](b,λ¯)+h3¯φ(b,λ¯)00(1+ih2i+h2φ[1](b,λ¯)-φ[1](a,λ¯)-1)|1+ih2|2 Im h2][c1c2].

Now consider the equation

((L-λI)-1(L*-λI)-I)P*-1Ψ*c=B(λ)φ(x,λ),

where yz = B(λ)φ(x,λ), yD(L) and zD(L*). A similar argument completes the proof.

6. Conclusion and Remarks

This paper provides a new method to analyze the spectral properties of some third-order dissipative boundary value problems and it seems that such a method has not been introduced previously for third-order case. This method is very effective and can be applied for other odd-order dissipative operators generated by suitable odd-order differential equation and boundary conditions.

Finally we should note that the differential expression ℓ can also be handled as the following

(y)=1w{-i[(ry)+(ry)]-(p0y)+i[q1y+(q1y)]+p1y},

where r is a suitable function. Then with some modifications a similar boundary value problem as (2.2), (2.4) can be analyzed.

Data Availability

All datasets generated for this study are included in the manuscript and the supplementary files.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: coordinate-free approach, model operator, characteristic function, spectral analysis, dissipative operator

Citation: Uğurlu E and Baleanu D (2019) Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator. Front. Phys. 7:99. doi: 10.3389/fphy.2019.00099

Received: 05 April 2019; Accepted: 21 June 2019;
Published: 10 July 2019.

Edited by:

Cosmas K. Zachos, Argonne National Laboratory (DOE), United States

Reviewed by:

Ebenezer Bonyah, University of Education, Winneba, Ghana
Daniel Luiz Nedel, Universidade Federal da Integração Latino-Americana, Brazil

Copyright © 2019 Uğurlu and Baleanu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Ekin Uğurlu, ekinugurlu@cankaya.edu.tr