# Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator Potential

Authors: Aleksandr Mikhailovich Kholkin
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.


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## Body

## 1. Introduction

The theory of non-self-adjoint singular differential operators, generated by scalar differential expressions, has been well studied. An overview on the theory of non-self-adjoint singular ordinary differential operators is provided in V. E. Lyantse’s Appendix  I to the monograph of Naimark [1]. In this regard the papers of Naimark [2], Lyantse [3], Marchenko [4], Rofe-Beketov [5], Schwartz [6], and Kato [7] should be noted. The questions regarding equations with non-Hermitian matrix or operator coefficients have been studied insufficiently. For a differential operator with a triangular matrix potential decreasing at infinity, which has a bounded first moment due to the inverse scattering problem, it is stated in [8, 9] that the discrete spectrum of the operator consists of a finite number of negative eigenvalues, and the essential spectrum covers the positive semiaxis. The questions regarding an operator with a block-triangular matrix potential that increases at infinity are considered in [10, 11]. In the future, by the author of this paper similar questions are considered for equations with block-triangular operator coefficients. In [11, 12] Green’s function of a non-self-adjoint operator is constructed.In this article we construct a resolvent for a non-self-adjoint differential operator, using which the structure of the operator spectrum is set.


## 2. Preliminary Notes

LetHk,k=1,2,…,r, be finite-dimensional or infinite-dimensional separable Hilbert space with inner product ·,· and norm ·, dim⁡Hk≤∞. Denote H=H1⊕H2⊕⋯⊕Hr. Element h∈H will be written in the form h=colh1,h2,…,hr, where hk∈Hk,k=1,r¯,  Ik,  I are identity operators in Hk and H accordingly.We denote byL2H,0,∞ the Hilbert space of vector-valued functions yx with values in H with inner product (1)y,z=∫0∞yx,zxdxand the corresponding norm ·.Consider the equation with block-triangular operator potential(2)ly=-y′′+Vxy=λy,0≤x<∞,where(3)Vx=vx·I+Ux,Ux=U11xU12x⋯U1rx0U22x⋯U2rx⋯⋯⋯⋯00⋯Urrx,vx is a real scalar function, and 0<vx→∞ monotonically, as x→∞, and it has monotone absolutely continuous derivative. Also, Ux is a relatively small perturbation; for example, Ux·v-1x→0 as x→∞ or Uv-1∈L∞R+. The diagonal blocks Ukkx,  k=1,r¯, are assumed as bounded self-adjoint operators in Hk,  Ukl:Hl→Hk.In case where(4)vx≥Cx2α,C>0,α>1,we suppose that coefficients of (2) satisfy relations(5)∫0∞Ut·v-1/2tdt<∞,∫0∞v′2t·v-5/2tdt<∞,∫0∞v′′t·v-3/2tdt<∞.Let us consider the functions(6)γ0x=14vx4·exp⁡-∫0xvudu,γ∞x=14vx4·exp⁡∫0xvudu.It is easy to see that γ0x→0,  γ∞x→∞ as x→∞. These solutions constitute a fundamental system of solutions of the scalar differential equation(7)-z′′+vx+qxz=0,where qx is determined by a formula (cf. with the monograph [13])(8)qx=516v′xvx2-14v′′xvx.In such a way for all x∈0,∞ one has(9)Wγ0,γ∞≔γ0x·γ∞′x-γ0′x·γ∞x=1.In case of vx=x2α,  0<α≤1, we suppose that the coefficients of (2) satisfy the relation(10)∫a∞Ut·t-αdt<∞,a>0.Now functions γ0x,λ and γ∞x,λ are defined as follows:(11)γ0x,λ=14x2α-λ4·exp⁡-∫axu2α-λdu,γ∞x,λ=14x2α-λ4·exp⁡∫axu2α-λdu.These functions also form a fundamental system of solutions of the scalar differential equation, which is obtained by replacing vx with vx-λ in formulas (7) and (8).In [10] the asymptotic behavior of the functions γ0x,λ and γ∞x,λ was established as x→∞. If α+1/2α=n∈N, that is, α=1/2n-1, then functions γ0x,λ and γ∞x,λ as x→∞ will have the following asymptotic behavior: (12)γ0x,λ=c·exp⁡-x1+α1+α+λ2·x1-α1-α+∑k=2n-11·3·…·2k-3k!·λ2k·x1-2k-1α1-2k-1α·x1·3·…·2n-3/n!·λ/2n-α/2·1+o1,γ∞x,λ=c·exp⁡x1+α1+α-λ2·x1-α1-α-∑k=2n-11·3·…·2k-3k!·λ2k·x1-2k-1α1-2k-1α·x-1·3·…·2n-3/n!·λ/2n+α/2·1+o1.In particular, with α=1n=1 one has (13)γ0x,λ=c·xλ-1/2·exp⁡-x221+o1,γ∞x,λ=c·x-λ+1/2·exp⁡x221+o1.In the case α+1/2α∉N, set n=α+1/2α+1, with [β] being the integral part of β, to obtain the following asymptotic behavior for γ0x,λ and γ∞x at infinity:(14)γ0x,λ=c·x-α/2exp⁡-x1+α1+α+λ2·x1-α1-α+∑k=2n-11·3·…·2k-3k!·λ2k·x1-2k-1α1-2k-1α·exp⁡-1·3·…·2n-3n!·λ2n·x-αα·1+ox-α,γ∞x,λ=c·x-α/2exp⁡x1+α1+α-λ2·x1-α1-α-∑k=2n-11·3·…·2k-3k!·λ2k·x1-2k-1α1-2k-1α·exp⁡1·3·…·2n-3n!·λ2n·x-αα·1+ox-α.In [10] for an equation with matrix coefficients, and in the furtherance for equations with operator coefficients, the following theorem is proved.Theorem 1.
If for (2) conditions (4)-(5) are satisfied for α>1 or condition (10) for 0<α≤1, then the equation has a unique decreasing at infinity operator solution Φx,λ, satisfying the conditions(15)limx→∞⁡Φx,λγ0x,λ=I,limx→∞⁡Φ′x,λγ0′x,λ=I.Also, there exists increasing at infinity operator solution Ψx,λ, satisfying the conditions(16)limx→∞Ψx,λγ∞x,λ=I,limx→∞Ψ′x,λγ∞′x,λ=I.Corollary 2.
Ifα=1, that is, vx=x2, then, under condition (10), the solutions Φx,λ and Ψx,λ have common (known) asymptotic behavior, as in the quality γ0x,λ and γ∞x,λ you can take the following functions:(17)γ0x,λ=xλ-1/2·exp⁡-x22,γ∞x,λ=x-λ+1/2·exp⁡x22.


## 3. Resolvent of the Non-Self-Adjoint Operator

Let the following boundary condition be given atx=0:(18)cos⁡A·y′0-sin⁡A·y0=0,where A is block-triangular operator of the same structure as the potential Vx (3) of the differential equation (2), and Akk,k=1,r¯, are the bounded self-adjoint operators in Hk, which satisfy the conditions (19)-π2Ik≪Akk≤π2Ik.Together with problem (2) and (18) we consider the separated system(20)lkyk=-yk′′+vxIk+Ukkxyk=λyk,k=1,r¯with the boundary conditions(21)cos⁡Akk·yk′0-sin⁡Akk·yk0=0,k=1,r¯.LetL′ denote the minimal differential operator generated by differential expression ly and the boundary condition (18), and let Lk′,  k=1,r¯, denote the minimal differential operator on L2H,0,∞ generated by differential expression lkyk and the boundary conditions (21). Taking into account the conditions on coefficients, as well as sufficient smallness of perturbations Ukkx, and conditions (19), we conclude that, for every symmetric operator Lk′,  k=1,r¯, there is a case of limit point at infinity. Hence their self-adjoint extensions Lk are the closures of operators Lk′, respectively. The operators Lk are semibounded below, and their spectra are discrete.LetL denote the operator extensions L′, by requiring that L2H,0,∞ be the domain of operator L.The following theorem is proved in [10].Theorem 3.
Suppose that for (2) conditions (4)-(5) are satisfied for α>1 or condition (10) for 0<α≤1. Then the discrete spectrum of the operator L is real and coincides with the union of spectra of the self-adjoint operators Lk,  k=1,r¯; that is, (22)σdL=⋃k=1rσLk.Comment 4.
Note that this theorem contains a statement of the discrete spectrum of the non-self-adjoint operatorL only and no allegations of its continuous and residual spectrum.Along with (2) we consider the equation(23)l1y=-y′′+V∗xy=λy(V∗x is adjoint to the operator Vx). If the space H is finite-dimensional, then (23) can be rewritten as(24)l~y~=-y~′′+y~Vx=λy~,where y~=y~1y~2…y~r and the equation is called the left.For operator functionsYx,λ,Zx,λ∈BH let(25)WZ∗,Y=Z∗′x,λ-Yx,λ-Z∗x,λ-Y′x,λ.If Yx,λ is operator solution of (2) and Zx,λ is operator solution of (23), the Wronskian does not depend on x.Now we denoteYx,λ and Y1x,λ as the solutions of (2) and (23), respectively, satisfying the initial conditions(26)Y0,λ=cos⁡A,Y′0,λ=sin⁡A,Y10,λ=cos⁡A∗,Y1′0,λ=sin⁡A∗,λ∈C.Because the operator function Y1∗x,λ- satisfies equation(27)-Y1∗′′x,λ-+Y1∗x,λ-·Vx=λY1∗x,λ-,the operator function Y~x,λ≕Y1∗x,λ- is a solution to the left of the equation(28)-Y~′′x,λ+Y~x,λ·Vx=λY~x,λand satisfies the initial conditions Y~0,λ=cos⁡A,  Y~′0,λ=sin⁡A,  λ∈C.Operator solutions of (23) decreasing and increasing at infinity will be denoted by Φ1x,λ, Ψ1x,λ, and the corresponding solutions of (28) are denoted by Φ~x,λ and Ψ~x,λ. The system operator solutions Yx,λ,Φ~x,λ∈BH of (2) and (28), respectively, will take the form of Wronskian WΦ~,Y=Φ~′x,λYx,λ-Φ~x,λY′x,λ.Let us designate(29)Gx,t,λ=Yx,λWΦ~,Y-1Φ~t,λ0≤x≤t-Φx,λWY~,Φ-1Y~t,λx≥t.It is proved in [12] that the operator function Gx,t,λ is Green’s function of the differential operator L; that is, it possesses all the classical properties of Green’s function. In particular, for a fixed t the function Gx,t,λ of the variable x is an operator solution of (2) on each of the intervals 0,t, t,∞, and it satisfies the boundary condition (18), and at a fixed x, the function Gx,t,λ satisfies (28) in the variable t on each of the intervals 0,x, x,∞, and it satisfies the boundary condition cos⁡A∗·y′0-sin⁡A∗·y0=0.By definition (28), function Gx,t,λ is meromorphic by parameter λ with the poles coinciding with the eigenvalues of the operator L.We consider the operatorRλ defined in L2H,0,∞ by the relation(30)Rλfx=∫0∞Gx,t,λftdt=-∫0xΦx,λWY~,Φ-1Y~t,λftdt+∫x∞Yx,λWΦ~,Y-1Φ~t,λftdt.Theorem 5.
The operatorRλ is the resolvent of the operator L.


## 4. Proof of Theorem5

One can directly verify that, for any functionfx∈L2H,0,∞, the vector-function yx,λ=Rλfx is a solution of the equation ly-λy=f whenever λ∉σL. We will prove that yx,λ∈L2H,0,∞.Since operator solutionsΦx,λ and Ψx,λ form a fundamental system of solutions of (2), the operator solution Yx,λ of (2) satisfying the initial conditions (26) can be written as Yx,λ=Φx,λAλ+Ψx,λBλ, where Aλ=WΨ~,Y,Bλ=-WΦ~,Y; that is,(31)Yx,λ=Φx,λWΨ~,Y-Ψx,λWΦ~,Y.Similarly, the operator solutionY~x,λ of (28) can be represented in the following form:(32)Y~x,λ=WY~,ΦΨ~x,λ-WY~,ΨΦ~x,λ.By using formulas (31) and (32), we can rewrite relation (30) as follows:(33)Rλfx=-∫0aΦx,λWY~,Φ-1Y~t,λftdt+χ1x,λ-χ2x,λ+χ3x,λ-χ4x,λ,where a>0 and(34)χ1x,λ=Φx,λWY~,Φ-1WY~,Ψ∫axΦ~t,λftdt,χ2x,λ=Φx,λ∫axΨ~t,λftdt,χ3x,λ=Φx,λWΨ~,YWΦ~,Y-1∫x∞Φ~t,λftdt,χ4x,λ=Ψx,λ∫x∞Φ~t,λftdt.Let us show that each of these vector-functions χ1x,λ, χ2x,λ, χ3x,λ, and χ4x,λ belongs to L2H,0,∞. Since the operator solution Φx,λ decays fairly quickly as x→∞, then Φx,λ∈L20,∞. It follows that (35)χ1x,λ≤cλ·Φx,λ·∫axΦ~t,λ·ftdt≤cλ·Φx,λ·∫axΦ~t,λdt1/2·∫axftdt1/2<cλ·Φx,λ·∫a∞Φ~t,λdt1/2·∫a∞ftdt1/2≤c1λ·Φx,λ,and therefore χ1x,λ∈L2H,0,∞. Similarly we get that χ3x,λ∈L2H,0,∞. First we prove the assertion for the function χ2x,λ, when α>1 and the coefficients of (2) satisfy the conditions (4)-(5). In this case, we have(36)χ2x,λ≤Φx,λ∫axΨ~t,λftdt.By virtue of the asymptotic formulas for the operator solutions Φx,λ and Ψx,λ we obtain that(37)χ2x,λ≤c1λγ0x,λ∫axγ∞t,λftdt.Let us rewrite this relation in the following form:(38)χ2x,λ≤c1λγ0x,λγ∞x,λ∫axγ∞t,λγ∞x,λftdt.By using the definition of the functions γ0x,λ and γ∞x,λ (see (6)) and by applying the Cauchy- Bunyakovsky inequality we obtain (39)χ2x,λ≤12c1λ1vx∫axvxvtexp⁡-2∫txvududt1/2∫0∞ft2dt1/2.Since t≤x, we get exp⁡-2∫txvudu≤1, and then the latter estimate for χ2x,λ can be rewritten as follows:(40)χ2x,λ≤c2λ1vx4∫ax1vtdt1/2≤c2λ1vx4∫a∞1vtdt1/2.By formula (4), we get(41)χ2x,λ≤c3λvx4,and hence if α>1 and the coefficients of (2) satisfy the conditions (4) and (5), we have χ2x,λ∈L2H,0,∞. In the case of vx=x2α,  0<α≤1, the assertion can be proved similarly.For the functionχ4x,λ we will conduct the proof for the case when vx=x2α,  0<α≤1, and the coefficients of (2) satisfy condition (10). As in (37) we have (42)χ4x,λ≤c1λγ∞x,λ∫x∞γ0t,λftdt,which can be rewritten as follows:(43)χ4x,λ≤c1λγ0x,λγ∞x,λ∫x∞γ0t,λγ0x,λftdt.Let us use the asymptotic behavior of the functions γ0x,λ and γ∞x,λ, for example, in the case α+1/2α=n∈N, that is, α=1/2n-1 (see (12)). Setting aα,λ=1·3·…·2n-3/n!·λ/2n, we obtain(44)χ4x,λ≤c2λx-α∫x∞γ0t,λγ0x,λftdt≤c2λx-α∫axγ0t,λγ0x,λ2dt1/2∫0∞ft2dt1/2,χ4x,λ≤c3λx-α∫x∞tx2aα,λ-αexp⁡-2xα+1t/xα+1-11+αdt1/2.Replacing variables t=xu, we get(45)χ4x,λ≤c3λx-α+1/2∫1∞u2aα,λ-αexp⁡-2xα+1uα+1-11+αdu1/2.Since the inequality exp⁡-xα+1uα+1-1/1+α≤x-2 holds for all α∈0,1 and u∈1,∞ with sufficiently large x, we have(46)χ4x,λ≤c3λx-α-1/2∫1∞u2aα,λ-αexp⁡-xα+1uα+1-11+αdu1/2.Hence it follows that χ4x,λ≤c4α,λx-α-1/2, and therefore χ4x,λ∈L2H,0,∞. In case, where 0<α≤1 and α+1/2α∉N, and where α>1, the proof is similar.Thus,Rλf∈L2H,0,∞ for any function f∈L2H,0,∞. This completes the proof.Since the resolventRλ is a meromorphic function of λ, the poles of which coincide with the eigenvalues of the operator L, the statement of Theorem 3 can be refined.Theorem 6.
If conditions (4)-(5) where α>1 or condition (10) where 0<α≤1 is satisfied for (2), then the spectrum of the operator L is real and discrete and coincides with the union of spectra of self-adjoint operators Lk, k=1,m¯; that is,(47)σL=⋃k=1rσLk.


## 5. Application

Here we consider (2) with matrix coefficients and use the same notation as in Section 3 (note that could be considered second-order equation with block-triangular coefficients of a more general form [14]). Suppose that every symmetric operator Lk′ is lower semibounded. Let L be an arbitrary extension of the operator L′, defined boundary condition at infinity, and Lk an arbitrary self-adjoint extension of the operator Lk′. If the conditions at infinity determine the Friedrichs extension Lk0 of the semibounded symmetric operator Lk′, the corresponding extension of L′ will be denoted L0. Besides, let us assume that coefficients of (2) for the problem of semiaxis are such that discrete spectrum of L operator coincides with the union of discrete spectra of Lk operators,  k=1,r¯, (sufficient conditions are specified above in Theorem 6).Denote bynula⁡T the algebraic multiplicity of 0 as an eigenvalue of T.Denote byNa0λ the number of eigenvalues λn0<λ<λeL0 of the operator L0 counted according to their algebraic multiplicities. Here λeL0 stands for the lower bound of the essential spectrum of the operator L0.In [14] is set oscillation theorem of Sturm for equations with block-triangular matrix potential.Theorem 7.
Suppose the operatorL0 is generated by the differential expression ly with matrix block-triangular potential, the boundary condition at 0 (18), and such boundary conditions at the infinity that one gets Friedrichs extensions for semibounded symmetric operators Lk′. Then for λ<λeL0 one has(48)∑x∈0,∞nulaYx,λ=Na0λ(the sum is in those x∈0,∞ for which nula⁡Yx,λ≠0).In the same article a theorem about the connection between spectral and oscillation properties for any extension of the minimal operator is also proved. These theorems are generalizations for non-self-adjoint operators of the classical Sturm type oscillation theorems and this problem was considered for the first time.


## 6. Conclusion

In this work a resolvent is constructed for the Sturm-Liouville operator with a block-triangular operator potential increasing at infinite. The structure of the spectrum of such an operator is obtained.



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