# A Classification of Fourth-Order Dissipative Differential Operators

Authors: Tao Wang; Ji-jun Ao; Mei-chun Yang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is devoted to the classification of the fourth-order dissipative differential operators by the boundary conditions. Subject to certain conditions, we determine some nonself-adjoint boundary conditions that generate the fourth-order differential operators to be dissipative. And under certain conditions, we prove that these dissipative operators have no real eigenvalues.


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## Body

## 1. Introduction

It is well known, the nonself-adjoint spectral problems have wide applications. For example, interesting nonclassical wavelets can be obtained from eigenfunctions and associated functions for nonself-adjoint spectral problems. Thus, such problems have received more and more attention by the scholars [1–6].The nonself-adjointness of spectral problems can be caused by one or more of the following factors: the nonlinear dependence of the problems on the spectral parameter, the nonsymmetry of the differential expressions used, and the nonself-adjointness of the boundary conditions (BCs) involved.Nonself-adjoint differential operators generated by symmetric differential expressions together with nonself-adjoint BCs have been investigated in [6–14]. The determinant of perturbation connected with the dissipative operator L generated in L2a,b by the Sturm–Liouville differential expression in Weyl’s limit-circle case has been studied by Bairamov and Ugurlu in [11]. By using the Livšic theorem, they showed the completeness of the system of eigenfunctions and associated functions of the operator.In recent years, there are several results for fourth-order dissipative operators [6, 12–14]. However, these studies only restricted into some special boundary conditions. For second order Sturm–Liouville differential expression, Wang and Wu in 2012 gave all the nonself-adjoint boundary conditions which generate the operators dissipative [8]. As far as we know, there are no such results on higher order cases.The dissipative operators have very important application backgrounds such as the scattering theory and telegraphist’s equation. As is mentioned above, the boundary conditions can make the operators to be dissipative. A general question is how to justify the operator as a dissipative operator from the boundary conditions? Motivated by this reason and [8], in this paper, we will consider the fourth-order case and show a classification of the fourth-order dissipative differential operators. The results here are more general than previous known results, and the process is much complicated.


## 2. Notation and Main Results

In this paper, we consider the following fourth-order differential expression:(1)ly=y4+qxy,x∈I=a,b,where −∞≤a<b≤+∞ and qx is a real-valued function on I and qx∈Lloc1I. Suppose that the endpoints a and b are quasi regular (regular or limit circle); hence, without loss of generality, we assume that Weyl’s limit-circle case holds for the differential expression ly throughout this paper.For anym,n∈ℕ, we use Mm,nℂ to denote the vector space of m by n complex matrices.Let(2)Ω=y∈L2I:y,y′,y″,y‴∈AClocI,ly∈L2I.Fory,z∈Ω, the so-called Lagrange bracket is given by(3)y,zx=−yz‴¯+y′z″¯−y″z′¯+y‴z¯=−Rz¯xQCyx,x∈I,where the bar over a function denotes complex conjugate, and(4)Q=000−100100−1001000,Rzx=zx,z′x,z″x,z‴x,Cz¯x=Rz∗x,Rz∗x is the complex conjugate transpose of Rzx.Letϕix,λ,i=1,2,3,4, represent a set of linearly independent solutions of the equation ly=λy, where λ is a complex parameter. From the well-known Naimark’s Patching Lemma, the solutions can be chosen to satisfied with any initial conditions; hence, we set vix=ϕix,0,λ=0,i=1,2,3,4, and satisfy the condition(5)vi,vja=J,i,j=1,2,3,4,where(6)J=00−10000−110000100.From [15], the solutions vix,i=1,2,3,4, as described above exist and are linearly independent. Since Weyl’s limit-circle case holds for the differential expression ly on I, the solutions vix,i=1,2,3,4, belong to L2I. Since vix,i=1,2,3,4, are solutions of equation ly=0, according to Green’s formula, we can get vi,vjx=const, so for any x∈I,vi,vjx=J.Now letly=λy, and we consider the boundary value problem consisting of the differential equation(7)y4+qxy=λy,x∈I,and the boundary condition (BC):(8)AYa+BYb=0,where(9)Ya=y,v1ay,v2ay,v3ay,v4a,Yb=y,v1by,v2by,v3by,v4b,A=a11a12a13a14a21a22a23a24a31a32a33a34a41a42a43a44,B=b11b12b13b14b21b22b23b24b31b32b33b34b41b42b43b44,and A,B∈M4,4ℂ.For the convenience to the next discussion we combine the matricesA and B as a whole and write it as the following block matrix form:(10)A:B=A1A2B1B2A3A4B3B4,where Ai,Bi,i=1,2,3,4∈M2,2ℂ. A:B is called as the boundary condition matrix throughout this paper.Remark 1.
When the endpointsa and b are both regular, the BCs are the same as given in (8), with Ya and Yb replaced by(11)Ya=yay′ay″ay‴a,Yb=yby′by″by‴b.
At this time the matricesA and B (or the boundary condition matrix A:B) are the same. There is a similar statement about one endpoint is regular and the other one is in limit-circle case. Hence, without loss of generality we will discuss the results in this paper under the BC (8), which is in limit-circle case.
Now inL2I, we define the operator L as Ly=ly on DL, where(12)DL=y∈ΩAYa+BYb=0.Definition 1 (see [1]).
A linear operatorL, acting in the Hilbert space L2I and having domain DL, is said to be dissipative if ℑLf,f≥0,∀f∈DL, where ℑ denotes the imaginary part of the value.
Then, we can state our main results.Theorem 1.
The fourth-order differential operatorL generated by the differential expression (1) and a BC (8) is dissipative if up to a GL4,ℂ factor on the left, where GL4,ℂ is the set of invertible complex matrices in dimension 4, the boundary condition matrix A:B has one of the following forms:(13)a11a1210−a31¯a41¯00a21a220−1a32¯−a42¯00a31a3200b31b3210a41a4200b41b420−1,where ai1,ai2,i=1,2,3,4,b3j,b4j,j=1,2∈ℂ and satisfy ℑa22−a11≥0,−4ℑa22ℑa11≥a21+a12¯2 and ℑb31−b42≥0,−4ℑb31ℑb42≥b32+b41¯2.(14)a11a121000a31¯−a41¯a21a220−100−a32¯a42¯a31a320010b33b34a41a42000−1b43b44,where ai1,ai2,i=1,2,3,4,b3j,b4j,j=3,4∈ℂ and satisfy ℑa22−a11≥0,−4ℑa22ℑa11≥a21+a12¯2 and ℑb44−b33≥0,−4ℑb33ℑb44≥b43+b34¯2.(15)a11a1210a41¯00a31¯a21a220−1−a42¯00−a32¯a31a3200b3110b34a41a4200b410−1b44,where ai1,ai2,i=1,2,3,4,b3j,b4j,j=1,4∈ℂ and satisfy ℑa22−a11≥0,−4ℑa22ℑa11≥a21+a12¯2 and ℑb41+b34≤0,4ℑb41ℑb34≥b44¯−b312.(16)a11a12100a41¯a31¯0a21a220−10−a42¯−a32¯0a31a32001b32b330a41a42000b42b43−1,where ai1,ai2,i=1,2,3,4,b3j,b4j,j=2,3∈ℂ and satisfy ℑa22−a11≥0,−4ℑa22ℑa11≥a21+a12¯2 and ℑb42+b33≤0,4ℑb42ℑb33≥b32¯−b432.(17)10a13a14a33¯−a43¯000−1a23a24−a34¯a44¯0000a33a34b31b321000a43a44b41b420−1,where ai3,ai4,i=1,2,3,4,b3j,b4j,j=1,2∈ℂ and satisfy ℑa13−a24≥0,−4ℑa13ℑa24≥a23+a14¯2 and ℑb31−b42≥0,−4ℑb31ℑb42≥b32+b41¯2.(18)10a13a1400−a33¯a43¯0−1a23a2400a34¯−a44¯00a33a3410b33b3400a43a440−1b43b44,where ai3,ai4,i=1,2,3,4,b3j,b4j,j=3,4∈ℂ and satisfy ℑa13−a24≥0,−4ℑa13ℑa24≥a14+a23¯2 and ℑb44−b33≥0,−4ℑb33ℑb44≥b43+b34¯2.(19)10a13a14−a43¯00−a33¯0−1a23a24a44¯00a34¯00a33a34b3110b3400a43a44b410−1b44,where ai3,ai4,i=1,2,3,4,b3j,b4j,j=1,4∈ℂ and satisfy ℑa13−a24≥0,−4ℑa13ℑa24≥a14+a23¯2 and ℑb41+b34≤0,4ℑb41ℑb34≥b44¯−b312.(20)10a13a140−a43¯−a33¯00−1a23a240a44¯a34¯000a33a341b32b33000a43a440b42b43−1,where ai3,ai4,i=1,2,3,4,b3j,b4j,j=2,3∈ℂ and satisfy ℑa13−a24≥0,−4ℑa13ℑa24≥a14+a23¯2 and ℑb42+b33≤0,4ℑb42ℑb33≥b32¯−b432.(21)1a12a13000−a33¯a43¯0a22a23−100−a32¯a42¯0a32a33010b33b340a42a4300−1b43b44,where ai2,ai3,i=1,2,3,4,b3j,b4j,j=3,4∈ℂ and satisfy ℑa22+a13≥0,4ℑa22ℑa13≥a12−a23¯2 and ℑb44−b33≥0,−4ℑb33ℑb44≥b43+b34¯2.(22)1a12a130a33¯−a43¯000a22a23−1a32¯−a42¯000a32a330b31b32100a42a430b41b420−1,where ai2,ai3,i=1,2,3,4,b3j,b4j,j=1,2∈ℂ and satisfy ℑa22+a13≥0,4ℑa22ℑa13≥a12−a23¯2 and ℑb31−b42≥0,−4ℑb33ℑb31≥b32+b41¯2. (23)1a12a1300−a43¯−a33¯00a22a23−10−a42¯−a32¯00a32a3301b32b3300a42a4300b42b43−1,where ai2,ai3,i=1,2,3,4,b3j,b4j,j=2,3∈ℂ and satisfy ℑa22+a13≥0,4ℑa22ℑa13≥a12−a23¯2 and ℑb42+b33≤0,4ℑb42ℑb33≥b43−b32¯2.(24)1a12a130−a43¯00−a33¯0a22a23−1−a42¯00−a32¯0a32a330b3110b340a42a430b410−1b44,where ai2,ai3,i=1,2,3,4,b3j,b4j,j=1,4∈ℂ and satisfy ℑa22+a13≥0,4ℑa22ℑa13≥a12−a23¯2 and ℑb41+b34≤0,4ℑb41ℑb34≥b31−b44¯2.(25)a1110a14a31¯−a41¯00a210−1a24a34¯−a44¯00a3100a34b31b3210a4100a44b41b420−1,where ai1,ai4,i=1,2,3,4,b3j,b4j,j=1,2∈ℂ and satisfy ℑa21+a14≥0,4ℑa21ℑa14≥a24−a11¯2 and ℑb31−b42≥0,−4ℑb31ℑb42≥b32+b41¯2.(26)a1110a1400−a34¯a44¯a210−1a2400−a31¯a41¯a3100a3410b33b34a4100a440−1b43b44,where ai1,ai4,i=1,2,3,4,b3j,b4j,j=3,4∈ℂ and satisfy ℑa21+a14≥0,4ℑa21ℑa14≥a24−a11¯2, and ℑb44−b33≥0,−4ℑb33ℑb44≥b43+b34¯2;(27)a1110a14−a44¯00−a34¯a210−1a24−a41¯00−a31¯a3100a34b3110b34a4100a44b410−1b44,where ai1,ai4,i=1,2,3,4,b3j,b4j,j=1,4∈ℂ and satisfy ℑa21+a14≥0,4ℑa21ℑa14≥a24−a11¯2 and ℑb41+b34≤0,4ℑb41ℑb34≥b31−b44¯2.(28)a1110a140−a44¯−a34¯0a210−1a240−a41¯−a31¯0a3100a341b32b330a4100a440b42b43−1,where ai1,ai4,i=1,2,3,4,b3j,b4j,j=2,3∈ℂ and satisfy ℑa21+a14≥0,4ℑa21ℑa14≥a24−a11¯2 and ℑb42+b33≤0,4ℑb42ℑb33≥b43−b32¯2.Proof.
It is given in Section3.Remark 2.
The BCs of the fourth-order dissipative differential operators given in Theorem 1 can be classified as separated, mixed, and coupled. LetrankA3:A4=rankB1:B2=r, then it can be concluded that when r=0, the BCs are separated, when r=1, the BCs are mixed, and when r=2, the BCs are coupled.Remark 3.
From Theorem 1, it can also be concluded that the fourth-order differential operators consist of the symmetric differential expression and certain types of BCs are dissipative when the BC matrix takes one of the forms (13)–(28). The BCs can be separated, mixed, or coupled. The classification includes all the separated cases; however, for the mixed BCs and the coupled BCs we still cannot say that the results include all the BCs which generate the fourth-order operators (7) and (8) to be dissipative. Furthermore, for the convenience to justify the dissipation of the fourth-order dissipative differential operators, the separated BCs can also be classified as follows, which is a supplementary explanation to the separated BCs given in Theorem 1.Theorem 2.
Suppose that the fourth-order differential operatorL is generated by the differential expression (1) and a BC (8) and suppose that the BC (8) is separated, then L is dissipative if up to a GL4,ℂ factor on the left the boundary condition matrix A:B takes one of the following forms:(29)1a120a1400000a22−1a24000000001b320b3400000b42−1b44,where a1i,a2i,i=2,4,b3j,b4j,j=2,4∈ℂ and satisfy ℑa12a22¯+a14a24¯≥0,4ℑa12a22¯ℑa14a24¯≥1−a12¯a24+a14a22¯2 and ℑb32¯b42+b34¯b44≥0,4ℑb32¯b42ℑb34¯b44≥1−b32b44¯+b34¯b422.(30)1a120a1400000a22−1a2400000000b311b3300000b410b43−1,where a1i,a2i,i=2,4,b3j,b4j,j=1,3∈ℂ and satisfy ℑa12a22¯+a14a24¯≥0,4ℑa12a22¯ℑa14a24¯≥1−a12¯a24+a14a22¯2 and ℑb31¯b41+b33¯b43≥0,4ℑb31¯b41ℑb33¯b43≥1−b31b43¯+b33¯b412.(31)a111a1300000a210a23−1000000001b320b3400000b42−1b44,where a1i,a2i,i=1,3,b3j,b4j,j=2,4∈ℂ and satisfy ℑa11a21¯+a13a23¯≥0,4ℑa11a21¯ℑa13a23¯≥1−a11¯a23+a13a21¯2 and ℑb32¯b42+b34¯b44≥0,4ℑb32¯b42ℑb34¯b44≥1−b32b44¯+b34¯b422.(32)a111a1300000a210a23−100000000b311b3300000b410b43−1,where a1i,a2i,i=1,3,b3j,b4j,j=1,3∈ℂ and satisfy ℑa11a21¯+a13a23¯≥0,4ℑa11a21¯ℑa13a23¯≥1−a11¯a23+a13a21¯2 and ℑb31¯b41+b33¯b43≥0,4ℑb31¯b41ℑb33¯b43≥1−b31b43¯+b33¯b412.Proof.
It is given in Section3.Remark 4.
The separated BCs given in Theorem 2 can be transformed into the separated BCs given in Theorem 1 by elementary row transformation of boundary condition matrixA:B, but the conditions satisfied by the coefficients are more complex than those in Theorem 1. Therefore, it is necessary to list them in brief additionally.Remark 5.
In the regular case, there are similar statements as Theorems1 and 2.


## 3. Proofs of the Main Results

In this section, we prove Theorems1 and 2 and present additional results about dissipative operators. Before proving these theorems, we need to introduce the following results. Since these results and their proofs are regular, we will state them without proofs. The readers may find them in [16].LetΦx be the Wronskian matrix of the solutions vix,i=1,2,3,4, in I, then we have(33)Φx=Cv1x,Cv2x,Cv3x,Cv4x.Lemma 1.
(34)
Q
=
−
Φ
∗
x
−
1
J
Φ
−
1
x
,
x
∈
I
.Proof.
(See [16]).Lemma 2.
For arbitraryy∈DL,(35)y,v1x,y,v2x,y,v3x,y,v4xT=JΦ−1xCyx,x∈I.Proof.
(See [16]).Corollary 1.
For arbitraryy1,y2,y3,y4∈DL, let(36)Ψx=Cy1x,Cy2x,Cy3x,Cy4x,be the Wronskian matrix of y1,y2,y3,y4, then(37)JΦ−1xΨx=y1,v1xy2,v1xy3,v1xy4,v1xy1,v2xy2,v2xy3,v2xy4,v2xy1,v3xy2,v3xy3,v3xy4,v3xy1,v4xy2,v4xy3,v4xy4,v4x,x∈I.Lemma 3.
For arbitraryy,z∈DL, we have(38)y,zx=y,v1xz,v3¯x+y,v2xz,v4¯x−y,v3xz,v1¯x−y,v4xz,v2¯x,x∈I.Proof.
By Lemma1 and Lemma 2, it follows(39)y,zx=−Rz¯xQCyx=Rz¯xΦ∗x−1JΦ−1xCyx=JΦ−1xCzx∗JJΦ−1xCyx=z,v1¯x,z,v2¯x,z,v3¯x,z,v4¯xJy,v1x,y,v2x,y,v3x,y,v4xT=y,v1xz,v3¯x+y,v2xz,v4¯x−y,v3xz,v1¯x−y,v4xz,v2¯x.
This completes the proof.
Now let us prove Theorems1 and 2.Proof of Theorem 1.
For arbitraryy∈DL, we have(40)2iℑLy,y=Ly,y−y,Ly=y,yb−y,ya.
Then, applying (38), one obtains(41)2iℑLy,y=y,v1by,v3¯b+y,v2by,v4¯b−y,v3by,v1¯b−y,v4by,v2¯b−y,v1ay,v3¯a+y,v2ay,v4¯a−y,v3ay,v1¯a−y,v4ay,v2¯a.
If the boundary condition matrixA:B is equivalent to the matrix in (13), i.e., the BC is(42)y,v3a=−a11y,v1a+a12y,v2a−a31¯y,v1b+a41¯y,v2b,y,v4a=a21y,v1a+a22y,v2a+a32¯y,v1b−a42¯y,v2b,y,v3b=−a31y,v1a+a32y,v2a+b31y,v1b+b32y,v2b,y,v4b=a41y,v1a+a42y,v2a+b41y,v1b+b42y,v2b.
Substituting (42) into (41), one obtains(43)2iℑLy,y=Ly,y−y,Ly=y,v1¯a,y,v2¯a,y,v1¯b,y,v2¯ba11¯−a11−a12−a21¯00a21+a12¯a22−a22¯0000b31−b31¯b32+b41¯00−b41−b32¯b42¯−b42y,v1ay,v2ay,v1by,v2b,and hence(44)2ℑLy,y=y,v1¯a,y,v2¯a,y,v1¯b,y,v2¯bdc00c¯s0000pg00g¯fy,v1ay,v2ay,v1by,v2b,where(45)d=−2ℑa11,s=2ℑa22,c=ia12+a21¯,p=2ℑb31,f=−2ℑb42,g=−ib32+b41¯.
Note that the 4 by 4 matrix in (44) is Hermitian. The eigenvalues of the Hermitian matrix are(46)d+s±d−s2+4c22,p+f±p−f2+4g22,and they are all nonnegative if and only if(47)d+s≥0,ds≥c2,p+f≥0,pf≥g2.
From condition (13), we have(48)ℑLy,y≥0,i.e., L is dissipative in L2I.
The other cases can be proved in the same way.Proof of Theorem 2.
If the boundary condition matrixA:B is equivalent to the matrix in (30), then(49)y,v1a=−a12y,v2a−a14y,v4a,y,v3a=a22y,v2a+a24y,v4a,y,v2b=−b31y,v1b−b33y,v3b,y,v4b=b41y,v1b+b43y,v3b.
Substituting (49) into (41), one obtains(50)2iℑLy,y=Ly,y−y,Ly=y,v2¯a,y,v4¯a,y,v1¯b,y,v3¯ba12a22¯−a22a12¯a14a22¯−a12¯a24+100a12a24¯−a14¯a22−1a14a24¯−a24a14¯0000−b31b41¯+b31¯b41−b33b41¯+b31¯b43−100−b31b43¯+b33¯b41+1−b33b43¯+b33¯b43y,v2ay,v4ay,v1by,v3b,and hence(51)2ℑLy,y=y,v2¯a,y,v4¯a,y,v1¯b,y,v3¯bd′c′00c′¯s′0000p′g′00g′¯f′y,v2ay,v4ay,v1by,v3b,where(52)d′=2ℑa12a22¯,s′=2ℑa14a24¯,c′=−ia14a22¯−a12¯a24+1,p′=2ℑb31¯b41,f′=2ℑb33¯b43,g′=−i−b33b41¯+b31¯b43−1.
Note that the 4 by 4 matrix in (51) is Hermitian. The eigenvalues of the Hermitian matrix are(53)d′+s′±d′−s′2+4c′22,p′+f′±p′−f′2+4g′22,and they are all nonnegative if and only if(54)d′+s′≥0,d′s′≥c′2,p′+f′≥0,p′f′≥g′2.
Since condition (30) holds, we have(55)ℑLy,y≥0,i.e., L is dissipative in L2I.
The other cases can be proved similarly.Finally, we will state a spectral property of the dissipative operatorL.Theorem 3.
Let the notations of (45) hold. And if d>0,s>0, and ds>c2, p>0,f>0, and pf>g2, then the operator L has no real eigenvalue.Proof.
Letλ0 be a real eigenvalue of L and let ϕ0x=ϕx,λ0≠0 be the corresponding eigenfunction, since(56)ℑLϕ0,ϕ0=ℑλ0ϕ02=0,from (44), it follows(57)ℑLϕ0,ϕ0=12ϕ0,v1¯a,ϕ0,v2¯a,ϕ0,v1¯b,ϕ0,v2¯bdc00c¯s0000pg00g¯fϕ0,v1aϕ0,v2aϕ0,v1bϕ0,v2b=0,since d>0,s>0, and ds>c2 and p>0,f>0, and pf>g2, the matrix dc00c¯s0000pg00g¯f is positive definite. So, it can be funded that ϕ0,v1a=0, ϕ0,v2a=0, ϕ0,v1b=0, and ϕ0,v2b=0. By the boundary condition matrix (42), one has that ϕ0,v3a=0, ϕ0,v4a=0, ϕ0,v3b=0, and ϕ0,v4b=0. Let τ0x=τx,λ0,η0x=ηx,λ0,δ0x=δx,λ0, and ϕ0x=ϕx,λ0 be the independent solutions of ly=λ0y. Then,(58)ϕ0,v1bτ0,v1bη0,v1bδ0,v1bϕ0,v2bτ0,v2bη0,v2bδ0,v2bϕ0,v3bτ0,v3bη0,v3bδ0,v3bϕ0,v4bτ0,v4bη0,v4bδ0,v4b=JΦ−1bCϕ0b,Cτ0b,Cη0b,Cδ0b.
It is evident that the determinant of the left-hand side is equal to zero, and the value of the Wronskian of the solutionsτ0x,η0x,δ0x, and ϕ0x is not equal to zero, so the determinant of the right hand side is not equal to zero. This is a contradiction, thus the theorem is proven.


## 4. Conclusion Remarks

In this paper, a classification of nonself-adjoint fourth-order dissipative differential operators generated by a fixed symmetric differential expression together with general two points BCs is investigated. This classification is based on the boundary conditions which generate the fourth-order differential operators to be dissipative. It seems that such a classification has not been introduced previously.We want to show all the classifications of the fourth-order dissipative differential operators; however, due to the complexity of the data of the boundary condition matrix, in this paper, we showed all the separated BCs and some of the mixed and coupled BCs which generate the fourth-order operators to be dissipative, and we still cannot show all the mixed and coupled BCs which generate the fourth-order operators to be dissipative. It seems an interesting problem; hence, we plan to continue the studies on this problem in our future works.



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