A CERTAIN CLASS OF PERTURBED SYMMETRIC OPERATORS

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A CERTAIN CLASS OF PERTURBED SYMMETRIC OPERATORS

A. HEBBECHE

Received 3 February 2004 and in revised form 20 May 2004

The generalized resolvents for a certain class of perturbed symmetric operators with equal and finite deficiency indices are investigated. Using the Weinstein-Aronszajn formula, we give a classification of the spectrum.

1. Introduction

The present paper is concerned with the study of spectral properties for a certain class of linear symmetric operator T, defined in the Hilbert spaceH of the formT =A+ B, whereAis a closed linear symmetric operator, with nondensely defined domain in general,D(A)⊂H, andBis a finite-rank operator of the form

B f = n k=1

a_kf,y_ky_k, (1.1)

where y1,y2,. . .,y_n is a linearly independant system inH,a1,a2,. . .,a_n∈R. We remark that the operatorT can be considered as a perturbation of the operatorAby the finite- rank operatorB.

The case whenA is a first-order or second-order diﬀerential operator in the spaces L²(0, 2π),L²(0,∞) or in the Hilbert space of vector-valued functions, andB is a one- dimensional perturbation (n=1), has been studied by many authors (see, e.g., [9,20, 24]).

In particular, certain integrodiﬀerential equations of the above type occur in quantum mechanical scattering theory [8].

In this paper, the generalized resolvents of perturbed symmetric operatorTwith equal and finite deficiency indices are investigated. Using the Weinstein-Aronszajn formula (see, e.g., [18]), we give a classification of the spectrum. Finally, the obtained results are applied to the study of two classes of first-order and second-order diﬀerential operators.

We note that the spectral theory of perturbed symmetric and selfadjoint operators have been investigated using various methods by many authors [3,4,5,6,11,12,13,14, 15,16,17,21,22].

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2. Preliminaries

LetA be a closed symmetric operator with nondensely defined domain in a separable Hilbert spaceHwith equal deficiency indices (m,m), andm <∞. We denote byρ(A) the resolvent set of the operatorA, the resolvent operatorRλ(A) ofAis defined asRλ(A)= (A−λI)⁻¹. The complement ofρ(A) in the complex plane is called the spectrum ofA and denoted byσ(A). There is a decomposition of the spectrumσ(A) into three disjoint subsets, at least one of which is not empty [1,2,10]:

σ(A)=Pσ(A)∪Cσ(A)∪PCσ(A), (2.1) Pσ(A) is called the point spectrum,Cσ(A) the continuous spectrum, andPCσ(A) the point-continuous spectrum. We denote the essential spectrum of the operatorAbyσ_e(A)

=Cσ(A)∪PCσ(A).

For arbitraryλ∈C, we denotePλ=Nλ∩(D(A)⊕N_λ), whereNλ=HΘ(A−λI)D(A) is the deficiency subspace of the operatorA[1,2].

It is known [23] thatP_λ= {0} if and only ifD(A)=H, and if D(A)=H, then the subset

Gλ=

[ϕ,ψ]∈Nλ×N_λ:ϕ−ψ∈D(A) (2.2) is a graph of the isometric operatorXλwith domainPλand values inP_λ.

We denote bythe set of linear operatorsFdefined fromNitoN₋i, such that F ≤ 1. For each analytic operator-valued functionF(λ) inC⁺, withC⁺= {λ: Imλ >0}, and values in, we introduce the setΩF(∞) consisting of elementsh∈Nisuch that

λ→∞lim,λ∈C⁺ε

|λ|

h −F(λ)h<∞, (2.3) whereC_ε⁺= {λ∈C⁺:ε <argλ < π−ε}, 0< ε < π/2.

It is known [27] thatΩF(∞) is a vector space and for eachh∈ΩF(∞),

λ→∞lim,λ∈C⁺ε

F(λ)h=F0(∞)h (2.4)

exists in the sense of the strong topology, andF0(∞) is an isometric operator.

According to the theory of ˇStraus [28], the generalized resolvents ofAare given by the formula

R_λ(A)=R_λ=

A_F(λ)−λI⁻¹, R_λ=R^∗_λ, λ∈C⁺, (2.5) whereA_F(λ)is an extension ofAwhich is determined by the functionF(λ), whose values are operators from the deficiency subspaceN_i to the deficiency subspaceN₋_isuch that

F(λ) ≤1 andF(λ) satisfy the condition

F0(∞)ψ=Xiψ, forψ=0 only, (2.6)

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thenA_F(λ) is a restriction onH of a selfadjoint operator defined in a certain extended Hilbert space and is called quasiselfadjoint extension of the operatorA[28] defined on D(AF(λ))=D(A) + (F(λ)−I)Niby

A_F(λ)f+F(λ)ϕ−ϕ=A f+iF(λ)ϕ+iϕ, f ∈D(A),ϕ∈N_i. (2.7) For selfadjoint extensions with exit in the space in which acts the considered operators, see, for example, [12,21] and the references therein.

We denote byℵ the set of analytic operator functionsF(λ) in C⁺ with values in satisfying the condition (2.6).

Remark 2.1. To each selfadjoint extension of the operatorAcorresponds a certain constant operator functionF(λ)=V, whereVis an isometric operator defined fromN_iover N₋isatisfying the conditionV ψ=Xiψforψ=0 only, and reciprocally.

We denote byA˚a selfadjoint extension ofAand we introduce the operator

U˚λλ0=A˚−λ0IA˚−λI⁻¹, Imλ >0. (2.8) We note that (see [19])

U˚λλ0N_λ₀=N_λ, (Imλ)Imλ0

=0. (2.9)

We denote by

ϕ⁽¹⁾_i ,ϕ⁽²⁾_i ,. . .,ϕ^(m)_i (2.10) a basis ofN₋i. From (2.9),ϕ^(k)_λ =U˚λiϕ^(k)_i ,k=1, 2,. . .,mform a basis forN_λ. In particular, the vectors

ϕ^(k)₋_i =Uϕ˚ ^(k)_i , k=1, 2,. . .,m, (2.11) whereU˚=U˚₋iiis the Cayley transform [1,2] ofA, form an orthogonal basis of˚ Ni.

To get a convenient formula of the generalized resolvents ofA, we will need the following notation:

Φλµ=(λ−µ)ϕ^(k)_λ ,ϕ^(s)_µ ^m_k,s₌₁, C(λ)=Φ⁻_λi¹Φλ(−i), (2.12) whereE is the identity matrix of orderm, Ω(λ) is an analytic matrix function in C⁺ corresponding, in the bases (2.10) and (2.11), to the operator functionF(λ)∈ ℵandϕλ= (ϕ⁽¹⁾_λ ,. . .,ϕ^(m)_λ )^t, (f,ϕ_λ)^t=((f,ϕ⁽¹⁾_λ ),. . ., (f,ϕ^(m)_λ )),tdenotes the transpose, and (ϕ_λ,g) is defined analogously.

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In what follows, we denote byΦthe set of matricesΩ(λ),λ∈C⁺, associated in the bases (2.10) and (2.11) to the operator functionsF(λ)∈ ℵ.

According to the notation used in [7], the generalized resolvents ofAare given by Rλ(A)f =Rλf =R˚λf +f,ϕ_λ^tE−Ω(λ)C(λ)Ω(λ)−E⁻¹Φ⁻λi¹ϕλ,

R_λ=R^∗_λ, λ∈C⁺, (2.13)

whereR˚λis the resolvent ofA˚andΩ(λ)∈Φ.

Remark 2.2. The formula (2.13) defines a resolvent of a selfadjoint extension ofAif and only ifΩ(λ) is a unitary constant matrix.

3. Resolvent and spectrum of a symmetric perturbed operator

LetT=A+Bbe defined onD(T)=D(A), whereAis a linear closed symmetric operator inHandBis a finite-rank operator.

Lemma3.1. Forλ∈ρ(A)∩ρ(T), the resolventRλ(T)of the operatorTis given by

Rλ(T)=Rλ(A)−Rλ(A)I+BRλ(A)⁻¹BRλ(A). (3.1) Proof. Forλ∈ρ(A)∩ρ(T), the operator

Rλ(A)I+BRλ(A)⁻¹=Rλ(T) (3.2) exists and is bounded. Then, we get

(T−λI)Rλ(A)−Rλ(A)I+BRλ(A)⁻¹BRλ(A)

=(A−λI+B)Rλ(A)−Rλ(A)I+BRλ(A)⁻¹BRλ(A)

=I+BR_λ(A)−

I+BR_λ(A)I+BR_λ(A)⁻¹BR_λ(A)=I

(3.3)

as required.

Remark 3.2. If BR_λ(A) <1, then from (3.1), we obtain Rλ(T)=Rλ(A)I+BRλ(A)⁻¹=Rλ(A)

∞ k=0

(−1)^kBRλ(A)^k. (3.4) Now, the aim is to give a convenient expression of (I+BR_λ(A))⁻¹in a more specific case.

So, we study in detail the case whenBis a finite-rank operator. Then, B f =

n k=1

a_kf,y_ky_k, f ∈H, (3.5) wherea1,a2,. . .,an∈R;{y1,y2,. . .,yn}is a linearly independent system inH. If we put

I+BRλ(A)⁻¹BRλ(A)f =y, (3.6)

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we have

y=BRλ(A)f−BRλ(A)y, (3.7)

then,y∈ImB, so that

y= n k=1

ckyk. (3.8)

From (3.7) and (3.8), we get n k=1

c_ky_k=BR_λ(A)f− n k=1

c_kBR_λ(A)y_k, (3.9)

with

c_k+a_k n j=1

c_jR_λ(A)y_j,y_k=a_kR_λ(A)f,y_k. (3.10)

The determinant∆(λ) of the system (3.10) is given by

∆(λ)=detδ_{k j}+a_kR_λ(A)y_j,y_kⁿ_k,j₌₁, (3.11) whereδk j is the Kronecker symbol. If we suppose that∆(λ)=0, the solution of (3.10) is given by

ck=ck(λ;f)=

f,∆k(λ)

∆(λ) , k=1, 2,. . .,n, (3.12) where ∆k(λ) is the determinant obtained from ∆(λ) by replacing the kth column by [ajR_λ(A)yj]ⁿ_j₌₁. So, from (3.1), we have

Rλ(T)f =Rλ(A)f− n k=1

f,∆k(λ)

∆(λ) Rλ(A)yk. (3.13)

This completes the proof of the following theorem.

Theorem3.3. Let λ∈ρ(A)such that∆(λ)=0. Then,λ∈ρ(T)and the resolvent of the operatorTis given by (3.13).

Remark 3.4. From (3.13), we note that the resolventR_λ(T) is a perturbation ofR_λ(A) by a finite-rank operator.

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Remark 3.5. For the particular casen=1 anda1=1, the formula (3.13) was established in [9].

Remark 3.6. Ifλ∈ρ(A) such that∆(λ)=0, thenλis an eigenvalue of the operatorT.

Proof. We can show that there exists an element ψ=

n k=1

α_ky_k (3.14)

such thatR_λ(A)ψis an eigenvector of the operatorT, corresponding to the eigenvalueλ.

Consequently, we have a_k

n j=1

α_jR_λ(A)y_j,y_k+α_k=0, k=1,n. (3.15)

Since the determinant of this system∆(λ)=0, it admits a nontrivial solution, which gives

the desired result.

Theorem3.7. Letµbe a fixed complex number. Then, the following holds.

(a)Ifµ∈ρ(A)and∆(µ)=0, thenµ∈ρ(T).

(b)Ifµ∈ρ(A)and∆(µ)=0, thenµ∈Pσ(T)and the multiplicity ofµas an eigenvalue ofTis equal to the order of the zero of∆(λ)atµ.

(c)Ifµ∈Pσ(A)andµof multiplicityk >0and ifµis a pole of∆(λ)of multiplicity p (k≥p), then

(1)fork > p, it holds thatµ∈Pσ(T)of multiplicity(k−p), (2)fork=p, it holds thatµ∈ρ(T).

(d)Ifµ∈Pσ(A)is neither a zero, nor a pole of∆(λ), thenµ∈Pσ(T).

(e)Ifµ∈Pσ(A)of multiplicityk andµis a root of the function∆(λ)of order p, then µ∈Pσ(T)of order(k+p).

(f)The essential spectraσe(A)andσe(T), respectively of the operatorsAandT, coincide.

Proof. It is suﬃcient to evaluate the function

C(λ)=detI+BR_λ(A). (3.16)

To this end, lety∈ImB. Then, BRλ(A)y=

n k=1

ak

y,R^∗_λ(A)yk

yk, (3.17)

it is clear thatC(λ)=∆(λ), and the function∆(λ) is meromorphic inρ(A)∪Pσ(A). From the formula of Weinstein and Aronszajn [18], we have

ϑ(λ;T)=ϑ(λ;A) +ϑ(λ;∆), (3.18)

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where

ϑ(λ;A)=











0 ifλ∈ρ(A),

k ifλ∈Pσ(A) and of multiplicityk, +∞ otherwise,

ϑ(λ;∆)=











k ifλis a zero of∆(λ) of orderk,

−k ifλis a pole of∆(λ) of orderk, 0 for otherλ∈Ω,

(3.19)

which gives the desired result.

4. Generalized resolvents

Now, we suppose thatAis a symmetric operator with deficiency indices (m,m),m <∞. Lemma4.1. Letλ∈Csuch thatImλ >0andϕλ(A)∈N_λ(A). Then, the elementϕλ(T), defined by the formula

ϕ_λ(T)=D(λ)ϕ_λ(A)=ϕ_λ(A)− n k=1

ϕ_λ(A),˚g_k(λ)

˚∆(λ) R_λA˚y_k, (4.1) is an element of the deficiency subspaceN_λ(T), where

D(λ)=I−R_λA˚I+BR_λA˚⁻¹B=I−R_λT˚B, g˚_k(λ)=A˚−λI∆˚k(λ), (4.2)

˚∆(λ)and˚∆k(λ)are defined similarly as∆(λ)and∆k(λ)in the formula (3.13) by putting the operatorA˚instead of the operatorA.

Proof. Since the operatorsA˚andT˚=A˚+Bare selfadjoint and λis nonreal, then λ∈ ρ(A)˚ ∩ρ(T). In addition, from˚ Theorem 3.3 we have˚∆(λ)=0. Furthermore, for each

f ∈D(A)=D(T), we have

T˚−λIf,D(λ)ϕ_λ(A)=

D^∗(λ)T˚−λIf,ϕ_λ(A)

=

I−BR_λT˚T˚−λIf,ϕ_λ(A)

=A˚−λIf,ϕλ(A)

=0,

(4.3)

and the equality

ϕ_λ(T)=ϕ_λ(A)− n k=1

ϕλ(A),g˚k(λ)

˚∆(λ) R_λA˚y_k (4.4)

results from (3.13).

Remark 4.2. We note that ifϕ_λ(A)=0, thenϕ_λ(T)=0.

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Proof. If we suppose the contrary, we obtainR_λ(T)Bϕ˚ _λ(A)=ϕ_λ(A), which givesAϕ˚ _λ(A)= λϕλ(A). This leads to a contradiction, since a selfadjoint operator can not have nonreal

eigenvalues.

Remark 4.3. IfD(A) is dense inH, thenϕλ(A) andϕλ(T) are, respectively, eigenfunctions of the operatorsA^∗andT^∗, corresponding to the eigenvaluesλ.

Let ϕ^(k)_i (T)=D(i)ϕ^(k)_λ (A), k=1, 2,. . .,m, defined by the formula (4.1). If ϕ⁽¹⁾_i (A), ϕ⁽²⁾_i (A),. . .,ϕ^(m)_i (A) is a basis of the deficiency subspaceNi(A) of the operatorA, then ϕ⁽¹⁾_i (T),ϕ⁽²⁾_i (T),. . .,ϕ^(m)_i (T) is a basis of the deficiency subspaceN_i(T) of the operatorT.

Putting

U˚_λλ0(T)˚ =T˚−λ0IR_λT˚, ϕ^(k)_λ (T)=U˚_λiT˚ϕ^(k)_i (T), k=1, 2,. . .,m, ϕλ(T)=

ϕ⁽¹⁾_λ (T),. . .,ϕ^(m)_λ (T)^t, Φλµ(T)=(λ−µ)ϕ^(k)_λ (T),ϕ^(j)µ (T)^m_k,j₌₁, (4.5) C(λ)₌Φ⁻_λi¹(T)Φλ(−i)(T) denotes the characteristic matrix of the operatorT, andω(λ) the corresponding matrix of orderm×m, in the basesϕ⁽¹⁾_i (T),ϕ⁽²⁾_i (T),. . .,ϕ^(m)_i (T) and ϕ⁽¹⁾₋_i(T),ϕ⁽²⁾₋_i(T),. . .,ϕ^(m)₋_i (T).

Theorem4.4. The set of all generalized resolvents of the operatorTis given by

R_λ(T)f =R_λT˚f+f,ϕ_λ(T)^tE−ω(λ)C(λ)ω(λ)−E⁻¹Φ⁻_λi¹(T)ϕ_λ(T), ∀f ∈H, (4.6) where

RλT˚f =RλA˚f− n k=1

f,∆˚k(λ)

˚∆(λ) RλA˚yk. (4.7) Proof. The proof results fromLemma 4.1and formula (2.13).

We denote, respectively, byAωandTωthe quasiselfadjoint extensions of operatorsA andTcorresponding to the operator functionF(λ)∈ , defined by the matrixω(λ).

Remark 4.5. To selfadjoint extensions of these operators correspond the constant unitary matricesω=[ωi j].

Theorem4.6. Suppose thaty1,y2,. . .,yn∈ImA,µis an eigenvalue of the quasiselfadjoint extensionA_ωof the operatorA,µ∈Pσ(A_ω). Ifµ∈ρ(A)˚ and˚∆(µ)=0,thenµis an eigen- value of the operatorTω=Aω+Band the corresponding eigenfunctionϕµ(Tω)is given by

ϕ_µT_ω=D(µ)ϕ_µA_ω=ϕ_µA_ω− n k=1

ϕ_µA_ω,˚g_k(µ)

˚∆(µ) R_µA˚y_k, (4.8) whereϕ_µ(A_ω)is the eigenfunction of the operatorA_ω, corresponding to the eigenvalueµ.

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Proof. Sincey1,y2,. . .,y_n∈ImA, thenBϕ_µ(A)∈ImA. We also have ϕµ

Tω

=D(µ)ϕµ

Aω

=ϕµ

Aω

−Rµ

T˚Bϕµ(A)=ϕµ

Aω

−ψµ, (4.9) where

ψ_µ=R_µT˚Bϕ_µ(A)∈D(A). (4.10) Then,

Tωϕµ Tω

=Tω ϕµ

Aω

−ψµ

=

A_ω+Bϕ_µA_ω−T_ωR_µT˚Bϕ_µ(A)

=µϕ_µA_ω+Bϕ_µA_ω−Bϕ_µA_ω+µR_µT˚Bϕ_µ(A)

=µϕ_µT_ω.

(4.11)

5. Applications

5.1. Perturbed first-order diﬀerential operator. Consider inL²(0, 2π) the operatorT= A+B, whereAis defined byAy=iywith domainD(A)=H₀¹(0, 2π) andBis given by

(B y)(x)= n k=1

ak

y,yk

yk(x), (5.1)

wherey1,y2,. . .,yn∈L²(0, 2π) andak∈R, for allk=1,n. From [1,2], the operatorAis regular symmetric of deficiency indices (1, 1) and each selfadjoint extension ofAhas a discrete spectrum.

Theorem5.1. The generalized resolventRλ(Tθ)ofT, corresponding to the functionω(λ)= θ(λ), is an integral operator with kernel

K(x,t)=

1[x,2π](x) + 1 θ(λ)e^2πλi+ 1

e^iλ(t⁻^x)+ n k=1

θk(λ,x)φk(λ,t), (5.2) where1[x,2π](x)is the characteristic function of the interval[x, 2π],

φ_k(λ,t)=

∆^θ_k(λ)(t), θ_k(λ,x)=

R_λA_θy_k(x)

∆^θ(λ) , (5.3)

whereR_λ(A_θ), associated to the functionθ(λ), is given by Rλ

Aθ y(x)=

_x

0 y(t)e^iλ(t⁻^x)dt− 1 θ(λ)e^2πti+ 1

_2π

0 y(t)e^λi(t⁻^x)dt (5.4) with

∆^θ(λ)= δkj+ak

Rλ

Aθ

yj,yk

, (5.5)

and∆^θ_kis the determinant obtained from∆^θ(λ)replacing thekth column by[a_kR_λ(A_θ)y_k]ⁿ₁.

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Proof. The proof results from [26] andTheorem 3.3.

Corollary 5.2. Let T_θ be a selfadjoint extension of T corresponding to the functionθ,

|θ| =1.

(1)The spectrum of Tθ is simple if and only if the roots of∆^θ(λ) are simple and for k=0,±1,±2,. . .,∆^θ(1/2 +k−ϕ0/2π)=0, where{1/2 +k−ϕ0/2π}is the spectrum ofAθ, andϕ0=argθ.

(2)σ(Tθ)=Pσ(Tθ)=E1∪E2, whereE1is the set of points ofσ(Aθ)= {1/2 +k−ϕ0/2π, k=0,±1,±2,. . .}in which∆^θ(λ)is analytic,E2is the set of roots of∆^θ(λ).

Proof. The proof results from (5.4),Theorem 3.7, andLemma 4.1.

5.2. Perturbed second-order diﬀerential operator. Consider in L²(0,∞) the operator T=A+B, whereAis defined by

Ay= −y+x²y (5.6)

with domainD(A) consisting of all variablesywhich satisfy (i) y∈L²(0,∞),

(ii) yis absolutely continuous on all compact subintervals of [0,∞[, (iii)Ay∈L²(0,∞),

(IV)y(0)=y(∞)=limx→∞y(x)=0,y(0)=y(∞)=0, andBis given by

(B y)(x)= n k=1

ak

y,yk

yk(x), (5.7)

wherey1,y2,. . .,yn∈L²(0, 2π) andak∈IR, for allk=1,n.

From [1,2], the operatorAis symmetric of deficiency indices (1, 1). Letu1,u2be two solutions of (5.6), satisfying the initial conditions

u1(0,λ)=1, u₁(x,λ)_x₌₀=0,

u2(0,λ)=0, u₂(x,λ)_x₌₀= −1. (5.8) There exists a functionm(λ) [29] analytic inC\Rsuch that

ψ(x,λ)=u2(x,λ) +m(λ)u1(x,λ)∈L²(0,∞). (5.9) Theorem5.3. The generalized resolventsR_λ(T_θ)of the operatorTare defined by

R_λT_θy=R_λA_θy− n k=1

y,∆^θ_k(λ)

∆^θ(λ) R_λA_θy_k, Imλ >0, (5.10)

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where

Rλ

Aθ

y=ψ(x,λ) _x

0 y(s)u1(s,λ)ds+u1(x,λ) _∞

x y(s)ψ(s,λ)ds

− ψ(x,λ) θ(λ) +m(λ)

_∞

0 y(s)ψ(s,λ)ds,

(5.11)

∆^θ(λ)=detσjk+ak

Rλ

Aθ

yj,yk

, λ∈C⁺, (5.12)

withθ(λ) an arbitrary function analytic inC⁺ and such that Imθ(λ)≥0 or θ(λ)is an infinite constant.

Proof. First, we show that forλ∈C⁺,∆^θ(λ)=0 (then,∆^θ=0). We know (see [1,2]) that for each quasiselfadjoint extension of a symmetric operator,C⁺is contained in the set of regular points of this operator. Then, ifλ∈C⁺, we haveλ∈ρ(Aθ) andλ∈ρ(Tθ). If we suppose thatλ∈C⁺and∆^θ(λ)=0, fromTheorem 3.7, we obtainλ∈Pσ(T_θ), which is a contradiction. The formula (5.11) results from [25]. UsingTheorem 3.3, we end the

proof.

Corollary5.4. LetTθ be a selfadjoint extension associated toθ∈IR, letλ1,λ2,. . .be the roots of∆^θ(λ)inρ(A_θ)and letz1,z2,. . .be the poles of∆^θ(λ). Then,

PσT_θ=

PσA_θ\

z_i^∞₁∪

λ_j^∞₁. (5.13)

Proof. The proof results from (b) and (c) ofTheorem 3.7.

Acknowledgment

The author is grateful to the editor and the anonymous referees for their valuable com- ments and helpful suggestions which have much improved the presentation of the paper.

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A. Hebbeche: Laboratoire Equations Differentielles, Departement de Mathematiques, Faculté des Sciences, Université Mentouri, 25000 Constantine, Algeria

E-mail address:[email protected]