ISSN 1064�5624, Doklady Mathematics, 2014, Vol. 89, No. 1, pp. 8–10. © Pleiades Publishing, Ltd., 2014.Original Russian Text © M.Sh. Burlutskaya, A.P. Khromov, 2014, published in Doklady Akademii Nauk, 2014, Vol. 454, No. 1, pp. 15–17.

8

We consider the operators with involution

,and the related Dirac operator

,

where is a scalar function,

(T denotes transposition), , Q(x) =

, and is a complex�valued

function.Functional differential operators with involution

have a long history and have been extensively investi�gated (see, e.g., [1–3] and the bibliography therein).

The Dirac operator with an arbitrary differentiablepotential Q(x) is relatively easy to study, while the caseof a nonsmooth potential encounters great difficulties.Nevertheless, considerable advances have been madein the nondifferentiable case as well. For example, itwas shown in [4, 5] that, in the case of an arbitrarymatrix Q(x) with components in , the systemeigen� and associated functions forms a Riesz basis

with brackets in . In [6] the method of similaroperators was used to examine the spectral propertiesof the Dirac operator. Finally, a new technique basedon transformation operator formulas was proposed in[7–9] for studying the case of a nonsmooth potential.

In this paper, based on the specific properties of theoperators L± and L, more accurate results concerningthe behavior of eigenvalues and eigenfunctions in thedifficult case of periodic boundary conditions arederived assuming only that q(x) is continuous.

±

= ± − = ∈'( ) ( ) (1 ), (0) (1), [0, 1]L y y x q x y x y y x

= + = ∈( )( ) '( ) ( ) ( ), (0) (1), [0,1]Lz x Bz x Q x z x z z x

( )y x т

= 1 2( ) ( ( ), ( ))z x z x z x⎛ ⎞

= ⎜ ⎟−⎝ ⎠

1 0

0 1B

⎛ ⎞⎜ ⎟−⎝ ⎠

0 ( )

(1 ) 0

q x

q x( ) [0, 1]q x C∈

2[0, 1]L

22[0, 1]L

1. First, we consider the operators L+ and L.Lemma 1. The equation

is equivalent to the system, (1)

, (2)

where , , and F(x) = (f(x),f(1 – x))T.

Lemma 2. The equation isequivalent to system (1) with conditions (2) and

, where z(x) has the same sense as in Lemma 1.Remark 1. If system (1), (2) is considered with the

condition , then we obtain an arbitrary

Dirac system on the interval with Dirichlet

conditions, since q(x) and on are not

related.The result below can be obtained using the meth�

ods of [7, 8].

Theorem 1. The eigenvalues of the operator L+

that are sufficiently large in absolute value are simple,and the following asymptotic formulas hold for them:

.Remark 2. In view of Remark 1, more accurate

asymptotic formulas for (in fact, full asymptoticexpansions) can be derived using the methods of [7].

Lemma 3. If λ is such that the resolvent

(where E is the identity operator and

λ is a spectral parameter) exists and , then

, where and z2(x) =y(1 – x), satisfies the system

, ,

+ − = λ +

∈ 2

'( ) ( ) (1 ) ( ) ( ),

) [0, 1]( ,

y x q x y x y x

x Lf

f x

+ = λ +'( ) ( ) ( ) ( ) ( ),Bz x Q x z x z x F x ∈[0, 1]x

( ) ( )=1 21 12 2

z z

=1( ) ( )z x y x = −2( ) (1 )z x y x

+

= λ +( ) ( ) ( )L y x y x f x

=1 2(0) (0)z z

=1 2(0) (0)z z

0 12��,

−(1 )q x 0 12��,

+

λn

0 02 (1), , ( 1),n n i o n n n+

λ = π + = ± ± + …

+

λn

λ

+ + −

= − λ

1( )R L E

λ

+

=y R f

=

T1 2( ) ( ( ), ( ))z x z x z x =1( ) ( )z x y x

− λ =( )( ) ( ) ( )Lz x z x F x ∈[0, 1]x

Functional Differential Operators with Involutionand Dirac Operators with Periodic Boundary Conditions

M. Sh. Burlutskayaa and A. P. Khromovb

Presented by Academician V.A. Il’in May 6, 2013

Received June 17, 2013

DOI: 10.1134/S1064562414010037

a Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006 Russiae�mail: bmsh2001@mail.rub Saratov State University, ul. Astrakhanskaya 83, Saratov, 410026 Russiae�mail: KhromovAP@info.sgu.ru

MATHEMATICS

DOKLADY MATHEMATICS Vol. 89 No. 1 2014

FUNCTIONAL DIFFERENTIAL OPERATORS 9

with . Conversely, if Rλ = (L –

λE)–1 exists, then exists al well; moreover, if ,

then , where y(x) = z1(x) = [RλF]1 ([ ]1 is the firstcomponent of the vector ) and z2(x) = z1(1 – x).

Theorem 2 [9]. The eigenvalues of the operator Lform two infinite sequences with the asymptotics

where as . They are simple if and double if .

Theorem 3. For eigenvalues large in absolute value,the series of the operator L coincides with the eigen�values of the operator L+.

2. Now consider L– and L.Lemma 4. The equation

is equivalent to the system,

,

where , and

Φ(x) = .Lemma 5. If λ is such that the resolvent

exists and , then z(x) =

satisfies the system, ,

with . Conversely, if Rλ = (L –

λE)–1 exists, then exists as well; moreover, if

, then and.

Theorem 4. The eigenvalues of the operator L– coin�cide with the eigenvalues of the Dirac operator

considered on with Dirichlet

conditions and z1 = z2 . Conse�

quently, the refined asymptotic formulas [7] hold forthem.

Theorem 5. The eigenvalues of the operator L– arethe eigenvalues of the operator L.

3. In this section, we examine the Riesz projectorsPn of the operator L:

where with a sufficiently small.

= −

T( ) ( ( ), (1 ))F x f x f x

λ

+R λ=z R F

λ

+

=y R f

λR F

0 0

' ' '' ''2 , 2 ,

, ( 1), ,n n n nn i n i

n n n

λ = π + ε λ = π + ε

= ± ± + …

ε ε →' '', 0n n → ±∞n ε ≠ ε' ''n n

ε ε=' ''n n

λ 'n

− − = λ +'( ) ( ) (1 ) ( ) ( )y x q x y x y x f x

+ = λ + Φ'( ) ( ) ( ) ( ) ( ),Bz x Q x z x z x x ∈[0, 1]x

( ) ( )= −1 21 12 2

z z

= = − −

T T1 2( ) ( ( ), ( )) ( ( ), (1 ))z x z x z x y x y x

− −

T( ( ), (1 ))f x f x

λ

− − −

= − λ

1( )R L Eλ

−

=y R f

= − −

T T1 2( ( ), ( )) ( ( ), (1 ))z x z x y x y x

= λ + Φ( )( ) ( ) ( )Lz x z x x ∈[0, 1]x

Φ = − −T( ) ( ( ), (1 ))x f x f x

λ

−R

λ= Φz Rλ

−

λ= = = Φ1 1( ) ( ) ( ) [ ] ( )y x R f x z x R x= − −2 1( ) (1 )z x z x

−'( ) ( ) ( )Bz x Q x z x 0 12��,

=1 2(0) (0)z z 12��⎝ ⎠⎛ ⎞ 1

2��⎝ ⎠⎛ ⎞

λ

γ

= − λ

π∫1 ,

2n

nP R di

{ }γ = λ λ − π = δ| |2n n iδ > 0

Define

Then is a subspace of . Its orthogonal

complement is given by

Lemma 6. For any , , we have

,

where , ,

, and .

Lemma 7. It is true that

,

where denotes the inner product in and

; here, ( ) are theeigenfunctions of the operator L+ ((L+)*) corresponding to

the eigenvalues ( ), , and .Lemma 8. It is true that

,

where and ( ) arethe eigenfunctions of the operator L– ((L–)*) corresponding

to the eigenvalues ( ), , and .Theorem 6. It is true that

,

where is the inner product in ,

, and = ( (x),

⎯ (1 – x))T. The functions and are theeigenfunctions of the operator L corresponding to the

eigenvalues and .

Theorem 7. The systems and

are biorthogonal in and

form Riesz bases in . The systems and

( and ) are biorthogonal and form

Riesz bases in ( ).

Assume that the group contains all with indices and the remainingfinite number of eigen� and associated functions of Lwithout indices formed of the corresponding eigen�and associated functions of L+. The same is true of theother groups in G and G*.

Remark 3. The general potential Q(x) =

can be represented as Q(x) = Q1(x) + Q2(x),

T|

2

1 2 2 1 2

[0, 1] { ( )

( ( ), ( )) ( ) (1 ), ( ) [0, 1]}.k

H f x

f x f x f x f x f x L

=

= = − ∈

�

2[0, 1]H 22[0, 1]L

⊥

2 [0, 1]H

T| g

2

1 2 1 2 2

[0, 1] { ( )

( ( ), ( )) ( ) (1 ), ( ) [0, 1]}.k

H g x

g x g x g x g x x L

⊥=

= = − − ∈

�

1( )f x ∈2 2( ) [0, 1]f x L

= Φ= +�

T1 2( ) ( ( ), ( ( ))) ( )F xf x f f x xx

= −

T( ) ( ( ), (1 ))F x f x f x Φ = − −T( ) ( ( ), (1 ))x g x g x

+ −=

1 2( ) (1 )( )

2

f x f xf x

− −

=

1 2( ) (1 )( )

2

f x f xg x

+ +

= ψ( , ) ( )n n nFP f xv

(·,·) 2[0, 1]Lт+ + +

= ϕ ϕ −( ) ( ( ), (1 ))n n nx x xv+

ϕ ( )n x +

ψ ( )n x

+

λn+

λn+

λ = λ 'n n+ +

ϕ ψ =( , ) 1n n

− −

ψΦ = ( , ) ( )n n nP g xv

T( ) ( ( ), (1 ))n n nx x x− − −

= ϕ −ϕ −v−

ϕ ( )n x −

ψ ( )n x

−

λn−

λn−

λ = λ ''n n− −

ϕ ψ =( , ) 1n n

+ + − −

+=� � �( , ) ( ) ( , ) ( )n n n n nP w x wf f xf v v

(·,·) 22[0, 1]L

+ + +

= ψ ψ −

T1( ) ( ( ), (1 ))2

n n nw x x x −( )nw x 12�� ψn

–

ψn– +( )n xv

−( )n xv

λ 'n λ''n+ −

= ∪{ ( )} ({ )}n nG x xv v

+ −

= ∪* { ( )} { ( )}n nG w x w x 22[0, 1]L

22[0, 1]L +{ ( )}n xv

+{ ( )}nw x −{ ( )}nv x −{ ( )}nw x

2[0, 1]H ⊥

2 [0, 1]H+{ ( )}n xv

+( )n xv

= ± ± + …0 0, ( 1),n n n

⎛ ⎞⎜ ⎟⎝ ⎠

2

1

0 ( )

( ) 0

q x

q x

10

DOKLADY MATHEMATICS Vol. 89 No. 1 2014

BURLUTSKAYA, KHROMOV

where , ,

, and

.The Dirac system with the potential Q1(x) was stud�

ied above. Let us show that the Dirac system with thepotential Q2(x) is reduced to the considered case.Indeed, in the Dirac system

(3)we pass to a new unknown vector function

, where and. Then (3) becomes

where ; i.e., we have obtainedthe case studied above.

Remark 4. An arbitrary potential

has the form ,

if and only if

( ).The results in Section 1 were obtained by Bur�

lutskaya, the results in Section 2, by Khromov, whilethe results in Section 3 are due to both authors.

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research, project no. 13�01�00238.

REFERENCES

1. A. A. Andreev, Proceedings of the 2nd InternationalSeminar on Differential Equations and Applications(Samara, 1998), pp. 5–18.

2. M. Sh. Burlutskaya, V. P. Kurdyumov, A. S. Lukonina,and A. P. Khromov, Dokl. Math. 75, 399–402 (2007).

3. M. Sh. Burlutskaya and A. P. Khromov, Dokl. Math.84, 783–786 (2011).

4. P. V. Djakov and B. S. Mityagin, Russ. Math. Surv. 61,663–766 (2006).

5. P. Djakov and B. Mityagin, Math. Nachr. 283, 443–462(2010).

6. A. G. Baskakov, A. V. Derbushev, and A. O. Shcherba�kov, Izv. Math. 75, 445–469 (2011).

7. M. Sh. Burlutskaya, V. P. Kurdyumov, and A. P. Khro�mov, Dokl. Math. 85, 240–242 (2012).

8. M. Sh. Burlutskaya, V. P. Kurdyumov, and A. P. Khro�mov, Izv. Sarat. Univ. Nov. Ser. Mat. Mekh. Inf. 12 (3),22–30 (2012).

9. M. Sh. Burlutskaya, V. V. Kornev, and A. P. Khromov,Zh. Vychisl. Mat. Mat. Fiz. 52, 1621–1632 (2012).

Translated by I. Ruzanova

⎛ ⎞= ⎜ ⎟−⎝ ⎠

1

0 ( )( )

(1 ) 0

g xQ x

g x⎛ ⎞

= ⎜ ⎟− −⎝ ⎠2

0 ( )( )

(1 ) 0

q xQ x

q x

= + −2 11( ) ( ( ) (1 ))2

g x q x q x 21( ) ( ( )2

q x q x= −

1(1 ))q x−

+ = λ + =2'( ) ( ) ( ) ( ) ( ), (0) (1),Bz x Q x z x z x f x z z

=

T1 2( ) ( ( ), ( ))u x u x u x =1 1( ) ( )u x z x

= −2 2( ) ( )u x iz x

0 ( )'( ) ( )

(1 ) 0ˆ( ) ( ), (0) (1),

q xBu x i u x

q x

u x f x u u

⎛ ⎞+ ⎜ ⎟−⎝ ⎠

= λ + =

= −

T1 2

ˆ( ) ( ( ), ( ))f x f x if x

⎛ ⎞= ⎜ ⎟⎝ ⎠

2

1

0 ( )( )

( ) 0

q xQ x

q x⎛ ⎞

= ⎜ ⎟−⎝ ⎠

0 ( )( )

(1 ) 0

q xQ x

q x⎛ ⎛ ⎞⎞

=⎜ ⎜ ⎟⎟− −⎝ ⎝ ⎠⎠

0 ( )( )

(1 ) 0

q xQ x

q x= −

T( ) (1 )Q x Q x

= − −

T( ) (1 )Q x Q x