Open access peer-reviewed chapter - ONLINE FIRST

Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

By Aleksandr Kholkin

Submitted: September 27th 2020Reviewed: January 5th 2021Published: January 27th 2021

DOI: 10.5772/intechopen.95820

Downloaded: 23

Abstract

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

Keywords

  • differential operators
  • spectrum
  • non-self-adjoint
  • block-triangular operator coefficients
  • Green’s function
  • resolvent

1. Introduction

The question of the generalization of the oscillatory Sturm theorem for scalar equations of higher orders and for equations with matrix coefficients for a long time remained open. Only in recent joint papers by F. Rofe-Beketov and A. Kholkin (see [1]) a connection was established between spectral and oscillatory properties for self-adjoint operators generated by equations of arbitrary even order with operator coefficients and boundary conditions of general form. Later, a Sturm-type oscillation theorem was proved [2] for a problem on finite and infinite intervals for a second-order equation with block-triangular matrix coefficients. In the case of non-self-adjoint differential operators, oscillation theorems have not been considered earlier.

Results turning out in self-adjoint and non-self-adjoint cases differentiate substantially. 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 [3]. In the study of the connection between spectral and oscillation properties of non-self-adjoint differential operators with block-triangular operator coefficients [2, 4] the question arises of the structure of the spectrum of such operators. For scalar non- self-adjoint differential operators these questions were studied in the papers [5, 6, 7, 8]. The theory of singular non-self-adjoint differential operators with matrix and operator coefficients is relatively new. In the context of the inverse scattering problem, for an operator with a triangular matrix potential decreasing at infinity, the first moment of which is bounded, the structure of the spectrum was established in [9, 10]. The theory of equations with block - triangular operator coefficients the first results were published in 2012 in the works of the author [11, 12, 13].

In this works we construct the fundamental system of solutions of differential equation with block-triangular operator potential that increases at infinity, one of that is decreasing at infinity, and the second growing. The asymptotics of the fundamental system of solutions of this equation is established. The Green’s function is constructed for a non-self-adjoint system with a block-triangular potential, the diagonal blocks of which are self-adjoint operators. We obtained a resolvent for a non-self-adjoint differential operator, using which the structure of the operator spectrum is set. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained. Here the rate of growth elements, not on the main diagonal, is subordinated to the rate of growth of the diagonal elements. In case of infringement of this condition, the operator can have spectral singularities [14].

2. The fundamental solutions for an non-self-adjoint differential operator with block – triangular operator coefficients.

Let us designate Hk,k=1,r¯as a finite-dimensional or infinite-dimensional separable Hilbert space with inner product and norm . Denote by H=H1H2Hr.Element h¯Hwill be written in the form of h¯=colh¯1h¯2h¯r, where h¯kHk, k=1,2,,r, Ik,I- are identity operators in Hkand Haccordingly.

We denote by L2H0the Hilbert space of vector-valued functions yxwith values in Hwith inner product yz=0yxzxdxand the norm .

Now let us consider the equation with block-triangular operator potential in BH

ly¯=y¯+Vxy¯=λy¯,0x<,E1

where

Vx=vxI+Ux,Ux=U11xU12xU1rx0U22xU2rx00Urrx,E2

vxis a real scalar function such that 0<vxmonotonically, as x, and it has monotone absolutely continuous derivative. Also, Uxis a relatively small perturbation, e. g. as xUxv1x0or Uv1LR+. The diagonal blocks Ukkx, k=1,r¯are assumed to be bounded self-adjoint operators in Hk.

In case where

vxCx2α,C>0,α>1,E3

we suppose that coefficients of the Eq. (1) satisfy relations:

0Utv12tdt<,E4
0v2tv52tdt<,0vtv32tdt<.E5

In case of vx=x2α,0<α1, we suppose that the coefficients of the Eq. (1) satisfy the relation

aUttαdt<,a>0.E6

2.1 Construction of the fundamental system of solutions for an operator differential equation with a rapidly increasing at infinity potential

Consider first the case where vxCx2α,C>0,α>1.

Condition (3) is performed, for example, quickly increasing functions ex,expexetc.

Rewrite the Eq. (1) in the form

y¯+vx+qxy¯=λ+qxIUxy¯,E7

where qxdetermined by a formula (cf. with the monograph [15])

qx=516vxvx214vxvx.E8

Now let us denote.

γ0xλ=14vx4exp0xvudu,γxλ=14vx4exp0xvudu.E9

It is easy to see that γ0xλ0,γxλas x. These solutions constitute a fundamental system of solutions of the scalar differential equation

z+vx+qxz=0,E10

in such a way that for all x0one has.

Wγ0γγ0xλγxλγ0xλγxλ=1.E11

Theorem 2.1 Under conditions (3), (4), (5)Eq. (1) has a unique decreasing at infinity operator solutionΦxλBH, satisfying the conditions

limxΦxλγ0xλ=IandlimxΦxλγ0xλ=I.E12

Also, there exists increasing at infinity operator solutionΨxλBHsatisfying the conditions

limxΨxλγxλ=IandlimxΨxλγxλ=I.E13

Proof a. Eq. (7) equivalently to integral equation

Φxλ=γ0xλI+xKxtλΦtλdt,E14

where

Kxtλ=Cxtλλ+qtIUt,E15
Cxtλ=γxλγ0tλγtλγ0xλ,E16

with Cxtλbeing the Cauchy function that in each variable satisfies Eq. (10) and the initial conditions Cxtλx=t=0,Cxxtλx=t=1Ctxtλx=t=1. Set χxλ=Φxλγ0xλto rewrite Eq. (14) in form

χxλ=I+xRxtλχtλdt,E17

where Rxtλ=Kxtλγ0tλγ0xλ. Thus

Cxtγ0tλγ0xλ=γ02tγγxλγ0xλγ0tλγtλ==12vtexp20tvuduexp20xvudu12vt==12vtexp2xtvudu1E18

and since with xtone has exp2xtvudu1, we deduce that

Cxtγ0tλγ0xλ1vt.E19

Hence.

Rxtλ=Cxtγ0tλγ0xγλ+qtIUt1vtλ+qt+Ut.E20

By virtue of (3)(5), (8),

1vtλ+qt+UtL0,E21

and therefore integral equation has a unique solution χxλand χxλconst. By (17), one has that limxχxλ=I, where the first part of formula (12) follows from.

Differentiable (14) to get Φxλγ0x=I+xSxtλχtλdt, where Sxtλ=Kxxtλγ0tλγ0xλ=Cxxtγ0tλγ0xλλ+qtIUt. We have similarly (18), that Cxxtγ0tλγ0xλ1vt, and therefore Sxtλ1vtλ+qt+UtL0, where the second part of formula (12) follows from.

b. Denote by Ψ̂xλBHblock-triangular operator solution of Eq. (1) that increases at infinity, ΨkkxλBHkHk,k=1,r¯-its diagonal blocks. Now Eq. (7) is equivalent to the integral equation

Ψ̂xλ=γxλI0xKxtλΨ̂tλdt,E22

where, just as in (14), the kernel Kxtλis given by (15). Now set χxλ=Ψ̂xλγxλto rewrite Eq. (22) in form

χxλ=I0xRxtλχtλdt,E23

where Rxtλ=Cxtλγtλγxλqt+λIUt. Similarly we can prove that the integral Eq. (23) has a unique solution χxλand χxλconst. Pass in (23) to a limit as xto get limxχxλ=I+C˜λwhere C˜λis block-triangular operator in H, that is

limxΨ̂xλγxλ=I+C˜λ.E24

Now consider another block-triangular operator solution Ψ˜xλthat increases at infinity diagonal blocks which are defined by.

Ψ˜kkxλ=ΦkkxλaxΦkk1tλΦkktλ1dt,k=1,r¯,a0,E25

Φkkxλare the diagonal blocks of operator solution Φxλas in Section a). In view (16) and the definition of the functions γ0x,γxcan be proved that

limxΨ˜kkxλγxλ=Ik,k=1,r¯.E26

Since Ψ̂xλand Ψ˜xλare the operator solutions of Eq. (1) that increase at infinity,

Ψ̂xλ=Ψ˜xλ+ΦxλC0λ,E27

where C0λis some block-triangular operator. Thus limxΨ̂xλγx=limxΨ˜xλγx, hence, by virtue (26), limxΨkkxλγx=Ik,k=1,r¯and in (24) has

C˜λ=0C12λC1rλ00C2rλ000.E28

The solution Ψxλgiven by Ψxλ=Ψ̂xλI+C˜λ1is subject to first from condition (13). Use (12) to differentiate (27), then find the asymptotes of Ψ˜xλas xsimilarly to (21) to obtain the second part of formula (13). Theorem is proved.

In this section, the fundamental system of solution is constructed for an operator differential equation with a rapidly increasing at infinity potential.

2.2 Asymptotic of the fundamental system solutions of equation with block-triangular potential

Now consider the case when vx=x2α,0<α1and coefficients of Eq. (1) satisfy the condition (6). Rewrite Eq. (1) in the form

y¯+x2αλ+qxλy¯=qxλIUxy¯,E29

where qxλdetermined by a formula

qxλ=5α24x2α1x2αλ2α2α1x2α22x2αλ.E30

Denote

γ0xλ=14x2αλ4expaxu2αλdu,E31
γxλ=14x2αλ4expaxu2αλdu.E32

There solutions constitute a fundamental system of solutions of the scalar differential equation z+x2αλ+qxλz=0, in such a way that for all x0one has Wγ0γγ0xλγxλγ0xλγxλ=1.

We are about to establish the asymptotics1 of γ0xλas x:

γ0xλ=2xα121λx2α14expaxuα1λu2α12du.E33

After expanding here the integral, we obtain the exponential as follows

expaxuα112λu2αk=2132k3k!2kλu2αkdu.E34

In case α+12α=nN, i.e. α=12n1, this expression after integration acquires the form:

cexpx1+α1+α+λ2x1α1α+k=2n1132k3k!λ2kx12k1α12k1α
 exp132n3n!λ2nlnx+o1=
=cexpx1+α1+α+λ2x1α1α+k=2n1132k3k!λ2kx12k1α12k1α
 x132n3n!λ2n1+o1.E35

The asymptotics of γ0xλas xis as follows:

γ0xλ=cexpx1+α1+α+λ2x1α1α+k=2n1132k3k!λ2kx12k1α12k1αx132n3n!λ2nα21+o1.E36

In particular, for α=1n=1, γ0xλhas the following asymptotics at infinity:

γ0xλ=cxλ12expx221+o1.E37

In case α+12αNwe set n=α+12α+1, with βbeing the integral part of β,to obtain the following asymptotics for γ0xλat infinity:

γ0xλ=cxα2expx1+α1+α+λ2x1α1α+k=2n1132k3k!λ2kx12k1α12k1αexp132n3n!λ2nxαα1+oxαE38

In particular, with α=12n=2one has

γ0xλ=cx14exp23x32+λx12λ22x121+ox12.E39

A similar procedure allows to establish the asymptotics of γxas x.If α+12α=nN, i.e. α=12n1, then

γxλ=cexpx1+α1+αλ2x1α1αk=2n1132k3k!λ2kx12k1α12k1α x132n3n!λ2n+α21+o1.E40

With α=1n=1, this becomes

γxλ=cxλ+12expx221+o1.E41

In case α+12αN, we set n=α+12α+1to get the asymptotics.

γxλ=cxα2expx1+α1+α+λ2x1α1α+k=2n1132k3k!λ2kx12k1α12k1αexp132n3n!λ2nxαα1+oxα.E42

In case α=12n=2, one has

γxλ=cx14exp23x32λx12+λ22x121+ox12.E43

Theorem 2.2 Under0<α1and condition (6), the statement of Theorem 2.1 is also valid for Eq. (1).

Proof is similar to Theorem 2.1. Moreover, note that

Cxtλγ0tλγ0xλ=γ02tλγxλγ0xλγ0tλγtλ==12t2αλexp2atu2αλduexp2axu2αλdu12t2αλ=12t2αλexp2xtu2αλdu1.E44

As xt, one has exp2xtu2αλdu1, and that is why

Cxtλγ0tλγ0xλ1t2αλ.E45

Hence

Rxtλ=Cxtλγ0tλγ0xλqtλIUt1t2αλqtλ+Ut.E46

By virtue of (6) and (30), 1t2αλqtλ+UtLaand therefore integral equation has a unique solution χxλand χxλconst. By (17), one has that limxχxλ=I, where the first part of formula (12) follows from.

The remaining statements of Theorem 2.1 are proved similarly.

From Theorem 2.2 and the asymptotic formulas (37), (39), (41), (43) follows.

Corollary 2.1 Ifα=1, i.e.vx=x2, then, under condition (6), the solutionsΦxλandΨxλhave common (known) asymptotics, as in the qualityγ0xλandγxλyou can take the following functions.

γ0xλ=xλ12expx22,..γxλ=xλ+12expx22.E47

Ifα=12, i.e. the coefficientvx=x, and the condition (6) holds, then.

γ0xλ=x14exp23x32+λx12,γxλ=x14exp23x32λx12.E48

Remark 2.1 It is known that scalar equation

φ+x2φ=λφE49

for λ=2n+1has the solution φnx=Hnxexpx22, where Hnxis the Chebyshev – Hermitre polynomial, that at xhas next asymptotics Hnx=2xn1+o1. Hence the solution φnxof the Eq. (49) at xwill have the following asymptotics at infinity: φnx=2xnexpx221+o1.

In the case of Ux=0,vx=x2in (2), the Eq. (1) is splitting into infinity system scalar equations of the form (49). The operator solution Φxλwill be diagonal in this case. Denote by φxλthe diagonal elements of the operator Φxλ. Then, by Corollary 2.1, the solution φxλwill have the following asymptotics at infinity: φxλ=xλ12expx221+o1. In particular, for λ=2n+1, this yields the solution proportional to φnx.

In this section, the asymptotics of the fundamental system of solutions for the Sturm-Liouville equation with block-triangular operator potential, increasing at infinity is established. One of the solutions is found decreasing at infinity, the other one increasing.

3. Green’s function for an operator differential equation with block – triangular coefficients

Let us suppose that at the x=0given boundary conditions

cosAy¯0sinAy¯0=0,E50

where A- the block-triangular operator of the same structure as the coefficients of the differential equation, Akk, k=1,r¯- bounded self-adjoint operators in Hk, which satisfy the conditions

π2Ik<<Akkπ2Ik.E51

Together with the problem (1), (50) we consider the separated system

lkyk=yk+vxIk+Ukkxyk=λyk,k=1,r¯E52

with the boundary conditions

cosAkkyk0sinAkkyk0=0,k=1,r¯.E53

Let Ldenote the minimal differential operator generated by differential expression ly¯(1) and the boundary condition (50), and let Lk, k=1,r¯denote the minimal differential operator on L2Hk0generated by differential expression lkykand the boundary conditions (53). Taking into account the conditions on coefficients, as well as sufficient smallness of perturbations Ukkxand conditions (51), we conclude that, for every symmetric operator Lk, there is a case of limit point at infinity. Hence their self-adjoint extensions Lkare the closures of operators Lkrespectively. The operators Lkare semi-bounded below, and their spectra are discrete.

Let Ldenote the operator extensions L, by requiring that L2H0be the domain of operator L.

The following theorem is proved in [4].

Theorem 3.1 Suppose that, for Eq. (1)conditions (3)-(5) are satisfied forα>1or condition (6) for0<α1. Then the discrete spectrum of the operatorLis real and coincides with the union of spectra of the self-adjoint operatorsLk,k=1,r¯, i.e.,σdL=k=1rσLk.

Comment 3.1 Note that this theorem contains a statement of the discrete spectrum of the non-self-adjoint operator Lonly and no allegations of its continuous and residual spectrum.

Along with the Eq. (1) we consider the equation

l1y¯=y¯+Vxy¯=λy¯E54

(Vxis adjoint to the operator Vx). If the space His finite-dimensional, then the Eq. (54) can be rewritten as

l˜y˜=y˜+y˜Vx=λy˜,E55

where y˜=y˜1y˜2y˜rand the equation is called the left.

For operator -functions Yxλ,ZxλBHlet

WZY=Zxλ¯YxλZxλ¯Yxλ.E56

If Yxλ- operator solution of the Eq. (1), and Zxλ- operator solution of Eq. (54), the Wronskian does not depend on x.

Now we denote Yxλand Y1xλthe solutions of the Eqs. (1) and (54), respectively, satisfying the initial conditions

Y0λ=cosA,Y0λ=sinA,Y10λ=cosA,Y10λ=sinA,λC.E57

Because the operator function Y1xλ¯satisfies equation

Y1xλ¯+Y1xλ¯Vx=λY1xλ¯,E58

the operator function Y˜xλY1xλ¯is a solution to the left of the equation

Y˜xλ+Y˜xλVx=λY˜xλE59

and satisfies the initial conditions Y˜0λ=cosA, Y˜0λ=sinA, λC.

Operator solutions of Eq. (54) decreasing and increasing at infinity will be denoted by Φ1xλ, Ψ1xλ, and the corresponding solutions of the Eq. (59) denote by Φ˜xλand Ψ˜xλ. For the system operator solutions Yxλ,Φ˜xλBHof the Eqs. (1) and (59), respectively, will take the form of Wronskian WΦ˜Y=Φ˜xλYxλΦ˜xλYxλand do not depend on x.

Let us designate

Gxtλ=YxλWΦ˜Y1Φ˜tλ0xtΦxλWY˜Φ1Y˜tλxt.E60

In the following theorem it is proved that the operator function Gxtλpossesses all the classical properties of the Green’s function.

Theorem 3.2 The operator functionGxtλis the Green’s function of the differential operatorL, i.e.:

  1. The functionGxtλis continuous for all valuesx,t0;

  2. For any fixedt, the functionGxtλhas a continuous derivative with respect toxon each of the intervals0tandt, and atx=tit has the jump

    Gxx+0xλGxx0xλ=I.E61

  • For a fixedt, the functionGxtλof the variablexis an operator solution of Eq. (1) on each of the intervals0t, t, and it satisfies the boundary condition (50) , and at a fixedxfunctionGxtλof the variabletis an operator solution of the Eq. (59) on each of the intervals0x,x, and it satisfies the boundary conditiony˜0cosAy˜0sinA=0.

  • Proof The function Gxtλis continuous with respect to xat each of the intervals 0tand t. Similarly to the variable t. To prove the continuity of the function Gxtλfor all x,t0, it is sufficient that the identity shown as

    YxλWΦ˜Y1Φ˜xλ+ΦxλWY˜Φ1Y˜xλ0.E62

    is satisfied for all x0.This identity shown as

    YxλΦ˜xλYxλΦ˜xλYxλ1Φ˜xλ
    ΦxλY˜xλΦxλY˜xλΦxλ1Y˜xλ0E63

    or

    YxλY1xλΦ˜1xλΦ˜xλ1Y˜1xλY˜xλΦxλΦ1xλ1,
    YxλY1xλΦ˜1xλΦ˜xλY˜1xλY˜xλΦxλΦ1xλ,E64

    which is equivalent to

    YxλY1xλY˜1xλY˜xλΦ˜1xλΦ˜xλΦxλΦ1xλE65

    or to.

    Y˜1xλY˜xλYxλY˜xλYxλY1xλΦ˜1xλΦ˜xλΦxλΦ˜xλΦ1xλΦ1xλ.E66

    This follows from the fact that WY˜Y=WΦ˜Φ=0.

    To make sure that the jump in the first derivative at t=xis equal to I, i.e., that the equality (61) holds, it is sufficient to prove the identity

    YxλWΦ˜Y1Φ˜xλ+ΦxλWY˜Φ1Y˜xλI.E67

    Now we consider the function

    Cxtλ=YxλWΦ˜Y1Φ˜tλ+ΦxλWY˜Φ1Y˜tλ,E68

    which is an analogue of the Cauchy function. This function is the solution of Eq. (1) of the variable x,and it is the solution of Eq. (59) of the variable t. By (62), we have Cxxλ0. But in this case Cxxt=x=VxλICt=x0, and, therefore, Cxxtλt=xΩ1λ, i.e.,

    YxλWΦ˜Y1Φ˜xλ+ΦxλWY˜Φ1Y˜xλΩ1λ.E69

    It shows that Ω1λ=I,we obtain (61).

    Since operator solutions Φxλand Ψxλform a fundamental system of solutions of Eq. (1), the operator solution Yxλof Eq. (1) satisfying the initial conditions (57), can be written as Yxλ=ΦxλAλ+ΨxλBλ, where Aλ=WΨ˜Y,Bλ=WΦ˜Y,

    Yxλ=ΨxλWΦ˜YΦxλWΨ˜Y.E70

    Similarly, operator solution Y˜xλof Eq. (59) can be represented in the form

    Y˜xλ=W˜Φ˜YΨ˜xλW˜Ψ˜YΦ˜xλ,E71

    where

    W˜Φ˜Y=sinAΦ0λcosAΦ0λ=Ω0λ=WY˜Φ.E72

    Similarly we get W˜Ψ˜Y=WY˜Ψ. Thus,

    Y˜xλ=WY˜ΨΦ˜xλWY˜ΦΨ˜xλ.E73

    Substituting (70) and (73) into the formula (69), using the fact that the equality (69) is performed on xidentically, we obtain

    Ω1λ=limxΨxλΦ˜xλΦxλΨ˜xλ.E74

    By Theorem 2.1, on the asymptotic behavior of functions Φxλand Ψxλat infinity, we have

    Ω1λ=limxγ0xλγxλγ0xλγxλI=Wγ0γI=I.E75

    This completes the proof of the formula (61), and with it the theorem 3.1.

    Corollary. By the definition (60) , functionGxtλis meromorphic of the parameterλwith the poles coincide with the eigenvalues of the operatorL.

    We constructed Green’s function for the non-self-adjoint differential operator.

    4. Resolvent for an non-self-adjoint operator differential equation with block – triangular coefficients

    We consider the operator Rλdefined in L2H0by the relation

    Rλf¯x=0Gxtλf¯tdt=E76
    =0xΦxλWY˜Φ1Y˜tλf¯tdt+xYxλWΦ˜Y1Φ˜tλf¯tdt.

    Theorem 4.1 The operatorRλis the resolvent of the operatorL.

    Proof One can directly verify that, for any function f¯xL2H0, the vector-function y¯xλ=Rλf¯xis a solution of the equation ly¯λy¯=f¯whenever λσL. We will prove that y¯xλL2H0.

    Since operator solutions Φxλand Ψxλform a fundamental system of solutions of Eq. (1), the operator solution Yxλof Eq. (1) satisfying the initial conditions (57), can be written as Yxλ=ΦxλAλ+ΨxλBλ, where Aλ=WΨ˜Y,Bλ=WΦ˜Y,

    Yxλ=ΦxλWΨ˜YΨxλWΦ˜Y.E77

    Similarly, the operator solution Y˜xλof Eq. (59) can be represented in the following form

    Y˜xλ=WY˜ΦΨ˜xλWY˜ΨΦ˜xλ.E78

    By using formulas (77) and (78), we can rewrite the relation (76) as follows:

    Rλfx=0aΦxλWY˜Φ1Y˜tλftdt+χ¯1xλχ¯2xλ+χ¯3xλχ¯4xλ,E79

    where a>0and

    χ¯1xλ=ΦxλWY˜Φ1WY˜ΨaxΦ˜tλf¯tdt,E80
    χ¯2xλ=ΦxλaxΨ˜tλf¯tdt,E81
    χ¯3xλ=ΦxλWΨ˜YWΦ˜Y1xΦ˜tλf¯tdt,E82
    χ¯4xλ=ΨxλxΦ˜tλf¯tdt.E83

    Let us show that each of these vector-functions χ¯1xλ,χ¯2xλ,χ¯3xλ,χ¯4xλbelongs to L2H0. Since the operator solution Φxλdecays fairly quickly as x, then ΦxλL20. It follows that

    χ¯1xλcλΦxλaxΦ˜tλf¯tdt
    cλΦxλaxΦ˜tλdt12axf¯tdt12<
    <cλΦxλaΦ˜tλdt12af¯tdt12c1λΦxλ,E84

    and therefore χ¯1xλL2H0. Similarly we get that χ¯3xλL2H0. First we prove the assertion for the function χ¯2xλ, when α>1and the coefficients of the Eq. (1) satisfy the conditions (3)-(5). In this case, we have χ¯2xλΦxλaxΨ˜tλf¯tdt.

    By virtue of the asymptotic formulas for the operator solutions Φxλand Ψxλwe obtain that

    χ¯2xλc1λγ0xλaxγtλf¯tdt.E85

    Let us rewrite this relation in the following form

    χ¯2xλc1λγ0xλγxλaxγtλγxλf¯tdt.E86

    By using the definition of the functions γ0xλand γxλ(see (9)) and by applying the Cauchy-Bunyakovskii inequality we obtain

    χ¯2xλ12c1λ1vxaxvxvtexp2txvududt120f¯t2dt12.E87

    Since tx, we get exp2txvudu1, and then the latter estimate for χ2xλcan be rewritten as follows

    χ¯2xλc2λ1vx4ax1vtdt12c2λ1vx4a1vtdt12.E88

    By formula (3), we get χ¯2xλc3λvx4, and hence, if α>1and the coefficients of the Eq. (1) satisfy the conditions (3)-(5), we have χ¯2xλL2H0. In the case of vx=x2α,0<α1, the assertion can be proved similarly.

    For the function χ¯4xλwe will conduct the proof for the case when vx=x2α,0<α1and the coefficients of the Eq. (1) satisfy the condition (6). As in (85) we have χ¯4xλc1λγxλxγ0tλf¯tdt, which can be rewritten as follows χ¯4xλc1λγ0xλγxλxγ0tλγ0xλf¯tdt.

    Let us use the asymptotics of the functions γ0xλand γxλ, for example, in the case α+12α=nN, i.e. α=12n1(see (36) and (40)). Setting aαλ=132n3n!λ2n, we obtain

    χ¯4xλc2λxαxγ0tλγ0xλf¯tdtc2λxαaxγ0tλγ0xλ2dt120f¯t2dt12,E89
    χ¯4xλc3λxαxtx2aαλαexp2xα+1txα+111+αdt12.E90

    Replacing variables t=xu, we get

    χ¯4xλc3λxα+121u2aαλαexp2xα+1uα+111+αdu12.E91

    Since the inequality expxα+1uα+111+αx2holds for all α01and u1with sufficiently large u1, we have

    χ¯4xλc3λxα121u2aαλαexpxα+1uα+111+αdu12.E92

    Hence it follows that χ¯4xλc4αλxα12, therefore χ¯4xλL2H0.In case, where 0<α1and α+12αNand where α>1, the proof is similar.

    Thus, Rλf¯L2H0for any function f¯L2H0.

    Since the resolvent Rλis a meromorphic function of λ, the poles of which coincide with the eigenvalues of the operator L, the statement of Theorem 3.1 can be refined.

    Theorem 4.2 If the conditions (3)-(5)) whereα>1or condition (6) where0<α1are satisfied for the Eq. (1), then the spectrum of the operatorLis real, discrete and coincides with the union of spectra of self-adjoint operatorsLk,k=1,m¯, i.e.σL=k=1rσLk.

    In this section, 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.

    5. Spectral singularities of differential operator with triangular matrix coefficients

    Remark 5.1 If the perturbationUxin Eq. (1) does not satisfy conditions (3)-(5) or condition (6), then the statement of Theorem 4.2 ceases to be true, which is shown by the following example.

    Example 5.1 Consider the equation:

    ly¯=y¯+x2qx0π2x2y¯=λy¯,0x<,y¯=y1y2E93

    with the boundary condition

    y¯0=0.E94

    Together with the problem (93), (94), consider the separated system

    l1y1=y1+x2y1=λy1,E95
    l2y2=y2+π2x2y2=λy2E96

    with the boundary conditions.

    y10=0,y20=0.E97

    As above, denote by L0the differential operator generated by the differential expression ly¯(93) and the boundary condition (94), and by L1,L2denote the minimal symmetric operators on L20, generated by the differential expressions l1y1,l2y2and the boundary conditions (97). Their self-adjoint extensions L1˜,L2˜are the closures of the operators L1,L2, respectively. The operators L1˜,L2˜are semi-bounded; let us denote their spectra by σ1=σL1˜,σ2=σL2˜.

    The Eq. (95) (cf. (49)) has the solution y1,nx=Hnxexpx22for λ=2n+1. Since H2n+10=0, the eigenvalues of the operator L1˜are λn=4n+3. The sets σ1and σ2do not intersect.

    Denote by Lthe extension of the operator L0generated by the requirement on the functions from the domain of the operator Lto belong to L2H20, and by σLits spectrum.

    Denote by Yxλ=y11xλy12xλ0y22xλthe matrix solution of the Eq. (93), satisfying the initial conditions Y0λ=0,Y0λ=I.

    If some λ0σL˜1, and yxλ0- is the corresponding eigenfunction of the operator L˜1, then the vector function y¯xλ0=yxλ00is the eigenfunction of the operator L, corresponding to the eigenvalue λ0, i.e. λ0σL. Moreover, λ0σL˜2is the eigenvalue of the operator Lif and only if the solution y12xλ0of the equation

    y12+x2y12+qxy22=λ0y12,E98

    satisfying the initial conditions y120λ=y120λ=0, belongs to L20. Let uxλ,vxλbe the solutions of the Eq. (95), satisfying the initial conditions u0λ=0,u0λ=1,v0λ=1,v0λ=0, and let Cxtλ=uxλvtλvxλutλ- be the Cauchy function of the Eq. (95). Then the solution y12xλ0is given by

    y12xλ0=0xqtCxtλ0y22tλ0dt.E99

    Choose the coefficient qx=y22xλ0exμ, where μ>2(for instance, μ=4), and show that the integral 0y122xλ0dxdiverges and, consequently, λ0σL. Indeed, since the solution y22xλ0has finitely many zeros, we conclude that, for any xN1>0,

    y22xλ0c1eαx2,α>0,E100

    and the Cauchy function decays no faster than ext2. Hence, if xt>N2, we have

    Cxtλ0c2ext2.E101

    In the case of x4tx2and xmax4N12N2, the inequalities (100) and (101) are fulfilled simultaneously, therefore, y12xλ0>c3x4x2et4e2αt2ext2dt. Since ext2ex24for tx2, we get y12xλ0>c3ex24x4x2et4e2αt2dt. If xis sufficiently large and tx4x2, we have et42αt2>e12t4ex432, hence for xy12xλ0>c3x4ex24+x432. It follows that y12xλ0L20and λ0σL.

    There arises the question on the nature of such values λ.

    Consider the equation with a triangular matrix potential:

    ly¯=y¯+pxqx0rxy¯=λy¯,0x<,y¯=y1y2,E102

    where px,qx,rxare scalar functions, px,rxare real functions and px,rxmonotonically as x.

    Let the boundary condition is given at x=0:

    cosAy¯0sinAy¯0=0,E103

    where Ais a triangular matrix, cosA=cosα11cosα120cosα22.

    Consider the separated system

    l1y1=y1+pxy1=λy1,E104
    l2y2=y2+rxy2=λy2.E105

    with the boundary conditions

    cosα11y10sinα11y10=0,E106
    cosα22y20sinα22y20=0.E107

    Let L0be the differential operator generated by the differential expression ly¯(103) and the boundary condition (104), and let L1,L2be minimal symmetric operators on L20generated by the differential expressions l1y1,l2y2and the boundary conditions (106), (108) respectively. Denote by L1˜,L2˜the self-adjoint extensions of the operators L1,L2respectively. The operators L1˜,L2˜are semi-bounded; let us denote their spectra by σ1and σ2respectively. Denote by Lthe extension of the operator L0and by σLits spectrum.

    Let uxλ,vxλbe the solutions of the Eq. (104) with the boundary conditions u0λ=0,u0λ=1,v0λ=1,v0λ=0.The general solution of the Eq. (104) has the form φxλ=uxλ+lvxλup to a constant. Choose an lsuch that the condition φbλ=0holds true. This equality is valid for l=lbλ=ubλvbλ(the solution vxλhas finitely many zeros for a fixed λ, hence vbλ0whenever bis sufficiently large). Put φ11bxλ=uxλ+lbλvxλ. Since for the operator L1there is the case of a limit point, then, as is known, lbλhas a unique limit mλas b, and the solution of the Eq. (104) satisfies φ11xλ=uxλ+mλvxλL20. Similarly we obtain that the solution of the Eq. (105) satisfies φ22xλL20.

    Denote by Φbxλ=φ11bxλφ12bxλ0φ22bxλthe matrix solution of the Eq. (103) satisfying the initial conditions Φbbλ=0,Φbbλ=I. We have φ11bxλφ11xλL20; φ22bxλφ22xλL20as b.

    The solution φ12bxλis given by φ12bxλ=0xqtCxtλφ22btλdt, where Cxtλ=uxλvtλvxλutλis the Cauchy function of the Eq. (104).

    Further, we have φ12bxλ0xqtCxtλφ22tλdtφ12xλas b. Put Φxλ=φ11xλφ12xλ0φ22xλ.

    Together with the Eq. (102), we consider the left equation.

    l˜y˜=y˜+y˜Vx=λy˜,y˜=y1y2.E108

    The matrix solutions of the Eq. (108) will be denoted by Φ˜bxλand Φ˜xλ.

    Denote by Yxλand Y˜xλthe solutions of the Eqs. (102) and (108) respectively satisfying the initial conditions

    Y0λ=cosA,Y0λ=sinA,Y˜0λ=cosA,Y˜0λ=sinA,λC.E109

    Put

    Gbxtλ=YxλWΦ˜bY1Φ˜btλ0xtΦbxλWY˜Φb1Y˜tλtxb.E110

    The function Gbxtλis the Green function of the operator Lb0generated by the problem (102), (103), yb=0, which spectrum coincides with the union of spectra of the operators Lb,10,Lb,20generated by the problems (104), (106), y1b=0and (105), (107), y2b=0respectively. Eigenvalues of the operators Lb,10and Lb,20tend to ones of the operators L˜1and L˜2respectively as b, ΦbxλΦxλ,Φ˜bxλΦ˜xλ, and

    WY˜Φb=cosAΦb0λsinAΦb0λcosAΦ0λsinAΦ0λ==WY˜Φ,WΦ˜bYWΦ˜Y,E111
    GbxtλGxtλ=YxλWΦ˜Y1Φ˜tλ0xtΦxλWY˜Φ1Y˜tλtx.E112

    Poles of the Green function Gxtλof the operator Lcoincide with the zero set of the determinant ΔλdetΩλ, where.

    Ωλ=WY˜Φx=0=cosAΦ0λsinAΦ0λ.E113

    Since the matrices cosA,sinA,Φ0λ,Φ0λare triangle, we have /Δλ=Δ1λΔ2λ, where Δkλ=cosαkkφkk0λsinαkkφkk0λ,k=1,2. On the other hand, zeros of the function Δkλare eigenvalues of the self-adjoint operator L˜k. Hence the poles of the Green function Gxtλof the operator Lare situated on the real axis, and their set coincides with the union of spectra of the operators L˜1and L˜2.

    Consider the operator Rλ,bdefined on L2H20bby.

    Rλ,bf¯x=0bGbxtλf¯tdt=0xΦbxλWY˜Φb1Y˜tλf¯tdt+
    +xbYxλWΦ˜bY1Φ˜tλf¯tdt.E114

    One can directly verify that the operator Rλ,bis the resolvent of the operator Lb0.

    Let f¯xbe an arbitrary vector function square integrable on 0. Choose a sequence of finite continuous vector functions f¯nxn=12converging in mean square to f¯x. Substituting f¯nfor f¯in (114) and letting first band then n, we obtain the following formula for the resolvent Rλof the operator L: Rλf¯x=0Gxtλf¯tdt, where the Green function of the operator Lis defined by the formula (112).

    Theorem 5.1 The operatorRλis the resolvent of the operatorL. The resolvent’s poles coincide with the union of the spectra of the self-adjoint operatorsL˜1andL˜2.

    Remark 5.2 As in Example 5.1, ifλ0σL˜2andφ12xλ0L20, thenλ0is the pole of the resolventRλof the operatorLbut it is not the eigenvalue of this operator, i.e.,λ0is the point of the spectral singularity of the operatorL.

    Theorem 5.1 implies that, if the rate of the coefficient’s growth qxof the Eq. (102) is subordinated to one of pxand rx, then the operator Lhas no spectral singularities, and its spectrum is real and coincides with the union of the spectra of the operators L˜1and L˜2.

    For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed. A special role of these points was found first by M.A. Naimark in [16]. The notion “spectral singularity” was introduced later due to J. Schwartz [17] (see also Supplement I in the monograph [3]).

    6. Conclusion

    We consider the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential. For him, built a fundamental system of solutions, one of which is decreasing at infinity, and the second is growing. The asymptotics of these solutions at infinity is defined. For non-self-adjoint operator generated by such differential expression obtained the Green’s function. A resolvent of such an operator is constructed. Sufficient conditions at which a spectrum of such non-self-adjoint differential operator is real and discrete are obtained.

    Notes

    • For α=1 and α=12, i.e., for vx=x2 and vx=x, the asymptotics of the functions γ0xλ and γ∞xλ is known.

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    Aleksandr Kholkin (January 27th 2021). Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients [Online First], IntechOpen, DOI: 10.5772/intechopen.95820. Available from:

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