On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval
We study representations of solutions of the Dirac equation, properties of spectral data, and inverse problems for the Dirac operator on a finite interval with discontinuity conditions inside the interval.
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| Date: | 2005 |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509746536120320 |
|---|---|
| author | Amirov, R. Kh. Амиров, Р. Х. Амиров, Р. Х. |
| author_facet | Amirov, R. Kh. Амиров, Р. Х. Амиров, Р. Х. |
| author_sort | Amirov, R. Kh. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:32Z |
| description | We study representations of solutions of the Dirac equation, properties of spectral data, and inverse problems for the Dirac operator on a finite interval with discontinuity conditions inside the interval. |
| first_indexed | 2026-03-24T02:46:00Z |
| format | Article |
| fulltext |
UDK 517.9
R.�X.�Amyrov (Baku, AzerbajdΩan)
O SYSTEME DYFFERENCYAL|NÁX URAVNENYJ
DYRAKA S USLOVYQMY RAZRÁVA VNUTRY YNTERVALA
We study representations of solutions of the Dirac equation, properties of spectral data, and inverse
problems of the Dirac operator on a finite interval with conditions of discontinuity inside the interval.
Vyvçagt\sq zobraΩennq rozv’qzkiv rivnqnnq Diraka, vlastyvosti spektral\nyx danyx ta ober-
neni zadaçi operatora Diraka na skinçennomu intervali z umovamy rozryvu vseredyni intervalu.
1. Vvedenye. Rassmotrym kanonyçeskug systemu dyfferencyal\n¥x uravne-
nyj Dyraka
B y ′ + Ω ( x ) y = λ y , 0 < x < π , (1)
na koneçnom yntervale. Zdes\
B =
0 1
1 0−
, Ω ( x ) =
p x q x
q x p x
( ) ( )
( ) ( )−
, y ( x ) =
y x
y x
1
2
( )
( )
.
PredpoloΩym, çto p ( x ) y q ( x ) — dejstvytel\n¥e funkcyy yz prostran-
stva L2 ( 0 , π ) .
Oboznaçym çerez L kraevug zadaçu, poroΩdennug uravnenyem (1) s hranyç-
n¥my uslovyqmy
y1 ( 0 ) = 0, (2)
y1 ( π ) = 0, (3)
a takΩe uslovyqmy razr¥va vo vnutrennej toçke x = a yntervala ( 0 , π ) :
y ( a – 0) = A y ( a + 0) , A =
α
α
0
0 1−
, a ∈ ( 0 , π ) . (4)
Zdes\ α > 0 — dejstvytel\noe çyslo y α ≠ 1.
Yz (2) y (3) vydno, çto v toçkax x = 0 y x = π hranyçn¥e uslovyq ymegt
specyal\n¥j vyd. MoΩno rassmatryvat\ y obwye samosoprqΩenn¥e hranyçn¥e
uslovyq, a takΩe sluçaj, kohda v uslovyy (4) matryca A — lgbaq samosoprq-
Ωennaq matryca vtoroho porqdka y det A = 1.
Cel\g nastoqwej rabot¥ qvlqetsq yssledovanye prqm¥x y obratn¥x zadaç
spektral\noho analyza dlq kraevoj zadaçy L .
Dlq klassyçeskyx operatorov Íturma – Lyuvyllq, Dyraka, uravnenyq Íre-
dynhera y hyperbolyçeskyx uravnenyj prqm¥e y obratn¥e zadaçy dostatoçno
polno yzuçen¥ (sm. [1 – 6] y pryvedennug v nyx byblyohrafyg). Nalyçye uslo-
vyq razr¥va vnutry yntervala vnosyt kaçestvenn¥e yzmenenyq v yssledovanye.
Nekotor¥e aspekt¥ prqm¥x y obratn¥x zadaç dlq operatorov Íturma – Lyu-
vyllq y Dyraka s uslovyqmy razr¥va yzuçalys\ v [7 – 12].
2. Predstavlenye reßenyq. Oboznaçym çerez Y 0 ( x , λ ) matryçnoe reße-
nye uravnenyq (1) pry Ω ( x ) ≡ 0, udovletvorqgwee uslovyg Y0 ( 0 , λ ) = I ( I —
edynyçnaq matryca) . Tohda Y0 ( x , λ ) ymeet vyd
Y0 ( x , λ ) =
e x a
e A e x
B x
B x a Ba
−
− − − −
< <
< <
λ
λ λ π
, ,
, .( )
0
01
(5)
Pust\ Y ( x , λ ) — matryçnoe reßenye uravnenyq (1), udovletvorqgwee uslo-
© R.=X.=AMYROV, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5 601
602 R.=X.=AMYROV
vyg Y ( 0 , λ ) = I . DokaΩem, çto reßenye Y ( x , λ ) moΩno predstavyt\ v vyde
Y ( x , λ ) = Y0 ( x , λ ) + K x t e dtBt
x
x
( , ) −
−
∫ λ . (6)
Netrudno pokazat\, çto yntehral\noe uravnenye dlq reßenyq Y ( x , λ ) ymeet vyd
Y ( x , λ ) = Y0 ( x , λ ) + Y x Y t B t Y t dt
x
0 0
1
0
( , ) ( , ) ( ) ( , )λ λ λ−∫ Ω . (7)
Dlq toho çtob¥ funkcyq vyda (6) udovletvorqla uravnenyg (7), dolΩno v¥-
polnqt\sq ravenstvo
K x t e dtBt
x
x
( , ) −
−
∫ λ = Y x Y t B t Y t dt
x
0 0
1
0
0
( , ) ( , ) ( ) ( , )λ λ λ−∫ Ω +
+ Y x Y t B t K t s e ds dtBs
t
tx
0 0
1
0
( , ) ( , ) ( ) ( , )λ λ λ− −
−
∫∫ Ω , (8)
y naoborot, esly matryca-funkcyq K ( x , t ) udovletvorqet πtomu ravenstvu, to
matryca-funkcyq Y ( x , λ ) udovletvorqet uravnenyg (7). Preobrazuem pravug
çast\ ravenstva (8) tak, çtob¥ ona ymela vyd, analohyçn¥j levoj çasty. Vvedem
sledugwye oboznaçenyq:
K± ( x , t ) = 1
2
K x t B K x t B( , ) ( , )±[ ] ( K ( x , t ) = K+ ( x , t ) + K– ( x , t ) ) .
Yz vyda matryc-funkcyj K+ ( x , t ) y K– ( x , t ) qsno, çto ony udovletvorqgt sle-
dugwym svojstvam:
B K+ ( x , t ) = 1
2
B K x t K x t B( , ) ( , )−[ ] = – K+ ( x , t ) B ,
B K– ( x , t ) = 1
2
B K x t K x t B( , ) ( , )+[ ] = K– ( x , t ) B .
Uçyt¥vaq, çto
Y x Y t0 0
1( , ) ( , )λ λ− =
e t x a
e
e t a x
e a t x
B x t
B x t
B a x t
B x t
− −
+ − −
− − − −
− −
< <
+
+
−
< <
< <
λ
λ
λ
λ
α
α
( )
( )
( )
( )
, ,
, ,
, ,
1 0
0 1
2
hde α± = 1
2
1
α
α±
, y preobrazuq pravug çast\ ravenstva (8) dlq matryc-funk-
cyj K+ ( x , t ) y K– ( x , t ) , poluçaem sledugwug systemu yntehral\n¥x uravnenyj:
K+ ( x , t ) = 1
2 2 2
B
x t
B K t x d
x t
x
Ω Ω+
+ + −−
+
∫ ( ) ( , )
( )/
ξ ξ ξ ξ , esly t < x < a ,
K– ( x , t ) = B K t x d
x t
x
Ω( ) ( , )
( )/
ξ ξ ξ ξ+
+
+ −∫
2
, esly t < x < a ,
K+ ( x , t ) = α α ξ ξ ξ ξ
+
+
−
+
+
+ − +∫2 2 2
B
x t
B K t x d
x t
a
Ω Ω( ) ( , )
( )/
+
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
O SYSTEME DYFFERENCYAL|NÁX URAVNENYJ DYRAKA … 603
+ α ξ ξ ξ ξ−
+
− −−
+ + −∫
1 0
0 1
2
2 2
B K t x a d
a x t
a
Ω( ) ( , )
( )/
+
+ B K t x d
a
x
Ω( ) ( , )ξ ξ ξ ξ− + −∫ , esly x > a , – x < t < 2a – x ,
K+ ( x , t ) = α
+ +
2 2
B
x tΩ +
+ B K t x d
x t
x
Ω( ) ( , )
( )/
ξ ξ ξ ξ−
+
+ −∫
2
, esly x > a , 2a – x < t < x ,
K– ( x , t ) = α ξ ξ ξ ξ+
+
+
− +∫ B K t x d
x t
a
Ω( ) ( , )
( )/2
+
+ α ξ ξ ξ ξ−
−
− −−
+ + −∫
1 0
0 1
2
2 2
B K t x a d
a x t
a
Ω( ) ( , )
( )/
+
+ B K t x d
a
x
Ω( ) ( , )ξ ξ ξ ξ+ + −∫ , esly x > a , – x < t < x – 2a ,
K– ( x , t ) = α
−
−
− +
2
1 0
0 1
2
2
B
t x aΩ +
+ α ξ ξ ξ ξ+
+
+
− +∫ B K t x d
x t
a
Ω( ) ( , )
( )/2
+
+ α ξ ξ ξ ξ−
−
− −−
+ + −∫
1 0
0 1
2
2 2
B K t x a d
a x t
a
Ω( ) ( , )
( )/
+
+ B K t x d
a
x
Ω( ) ( , )ξ ξ ξ ξ+ + −∫ , esly x > a , x – 2a < t < 2a – x ,
K– ( x , t ) = α
−
−
− +
2
1 0
0 1
2
2
B
t x aΩ + α
− + −
−
2
2
2
1 0
0 1
B
x a tΩ +
+ B K t x d
x t
x
Ω( ) ( , )
( )/
ξ ξ ξ ξ+
+
+ −∫
2
, esly x > a , 2a – x < t < x .
Prymenqq metod posledovatel\n¥x pryblyΩenyj (sm. [2, 5]), poluçaem sle-
dugwug teoremu.
Teorema 1. Pust\
|| ||∫ Ω( )x dx
0
π
< + ∞ .
Tohda reßenye Y ( x , λ ) uravnenyq (1), udovletvorqgwee naçal\nomu uslovyg
Y ( 0 , λ ) = I , moΩno predstavyt\ v vyde (6), pryçem evklydova norma || ||K x t( , )
udovletvorqet neravenstvu
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
604 R.=X.=AMYROV
|| ||
−
∫ K x t dt
x
x
( , ) ≤ eC xσ( ) − 1,
hde σ ( x ) = || ||∫ Ω( )t dt
x
0
, C — poloΩytel\naq postoqnnaq.
Dalee, esly funkcyq Ω ( x ) dyfferencyruema, to qdro K ( x , t ) udovletvo-
rqet sootnoßenyqm
B Kx + Ω ( x ) K = – Kt B ,
K x B( , )0
0
1−
= 0,
Ω ( x ) + B K ( x , x ) = K ( x , x ) B , esly 0 < x < a ,
α+
Ω ( x ) + B K ( x , x ) = K ( x , x ) B , esly x > a ,
B [ K ( x , 2a – x + 0 ) – K ( x , 2a – x – 0 ) ] =
= – [ K ( x , 2a – x + 0 ) – K ( x , 2a – x – 0 ) ] B , esly x < a ,
α−
−
Ω( )x
1 0
0 1
+ B [ K ( x , 2a – x + 0 ) – K ( x , 2a – x – 0 ) ] =
= – [ K ( x , 2a – x + 0 ) – K ( x , 2a – x – 0 ) ] B , esly x > a .
Otmetym, çto esly p ( x ) ∈ L2 ( 0 , π ) , q ( x ) ∈ L2 ( 0 , π ) , to πlement¥ matryc¥-
funkcyy K ( x , t ) = ( )( , ) ,K x tij i j=1
2
pry kaΩdom fyksyrovannom x ∈ [ 0 , π ] pry-
nadleΩat prostranstvu L2 ( 0 , π ) po peremennoj t .
3. Svojstva spektra. V πtom punkte yzuçagtsq svojstva spektra zadaçy L .
V=sluçae Ω ( x ) ≡ 0 zadaçu L oboznaçym çerez L0 . Netrudno pokazat\, çto
reßenye ϕ0 ( x , λ ) =
ϕ λ
ϕ λ
01
02
( , )
( , )
x
x
uravnenyq B y ′ = λ y s naçal\n¥m uslovyem
ϕ0 ( 0 , λ ) =
0
1−
y uslovyqmy (4) ymeet vyd
ϕ0 ( x , λ ) =
sin
cos
, ,
sin
cos
sin ( )
cos ( )
, .
λ
λ
α
λ
λ
α
λ
λ
π
x
x
x a
x
x
a x
a x
a x
−
< <
−
+
−
×
×
−
− −
< <
+ −
0
1 0
0 1
2
2
Oboznaçym çerez ∆0 ( λ ) xarakterystyçeskug funkcyg zadaçy L0 :
∆0 ( λ ) ≡ α+
sin λπ + α–
sin λ ( 2a – π ) .
Korny πtoho uravnenyq λn
0
qvlqgtsq sobstvenn¥my znaçenyqmy zadaçy L0 .
PredpoloΩym, çto λn
0 > 0, esly n > 0, λn
0 = 0 y λ−n
0 = – λn
0 , n = 1, 2, … .
Lemma 1. Ymeet mesto sootnoßenye inf
n m n m≠
| |−λ λ0 0 = β > 0, t .$e. korny
xarakterystyçeskoho uravnenyq ∆0 ( λ ) = 0 otdelen¥.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
O SYSTEME DYFFERENCYAL|NÁX URAVNENYJ DYRAKA … 605
Dokazatel\stvo. Dopustym protyvnoe, t.=e. pust\ suwestvugt podposle-
dovatel\nosty λnk
0{ } y λ̂nk
0{ } posledovatel\nosty λn
0{ } takye, çto λnk
0 ≠ λ̂nk
0 ,
λnk
0 , λ̂nk
0 → ∞ pry k → + ∞ y
lim ˆ
k n nk k→∞
−λ λ0 0 = 0.
Yspol\zuq ortohonal\nost\ sobstvenn¥x funkcyj ϕ λ0
0x nk
,( ) y ϕ λ0
0x nk
, ˆ( ) za-
daçy L0 v prostranstve L2 ( 0 , π ; R
2
) , poluçaem
0 = ϕ λ ϕ λ
π
0
0
0
0
0
x x dxn nk k
, , , ˆ( ) ( )∫ = ϕ λ ϕ λ
π
0
0
0
0
0
x x dxn nk k
, , ,( ) ( )∫ +
+ ϕ λ ϕ λ ϕ λ
π
0
0
0
0
0
0
0
x x x dxn n nk k k
, , , ˆ ,( ) ( ) − ( )[ ]∫ ≥
≥ ϕ λ ϕ λ0
0
0
0
0
x x dxn n
a
k k
, , ,( ) ( )∫ +
+ ϕ λ ϕ λ ϕ λ
π
0
0
0
0
0
0
0
x x x dxn n nk k k
, , , ˆ ,( ) ( ) − ( )[ ]∫ ≥
≥ sin cos2 0 2 0
0
λ λn n
a
k k
x x dx+( )∫ +
+ ϕ λ ϕ λ ϕ λ
π
0
0
0
0
0
0
0
x x x dxn n nk k k
, , , ˆ ,( ) ( ) − ( )[ ]∫ =
= a + ϕ λ ϕ λ ϕ λ
π
0
0
0
0
0
0
0
x x x dxn n nk k k
, , , ˆ ,( ) ( ) − ( )[ ]∫ .
Zdes\ çerez 〈 ⋅ , ⋅ 〉 oboznaçeno skalqrnoe proyzvedenye v evklydovom prostran-
stve R
2.
Takym obrazom, poluçym neravenstvo
0 ≥ ϕ λ ϕ λ ϕ λ
π
0
0
0
0
0
0
0
x x x dx an n nk k k
, , , ˆ ,( ) ( ) − ( ) +∫ . (9)
Yz vyda reßenyq ϕ0 ( x , λ ) sleduet, çto
lim , ˆ ,
k n n R
x x
k k→∞ ( ) − ( )ϕ λ ϕ λ0
0
0
0
2 = 0 ( || ||⋅
R2 = 〈⋅ ⋅〉, )
ravnomerno po x ∈ [ 0 , π ] . Poπtomu, perexodq k predelu pry k → ∞ v nera-
venstve (9), ymeem 0 ≥ a . Poluçennoe protyvoreçye dokaz¥vaet spravedlyvost\
lemm¥.
Oboznaçym çerez ∆ ( λ ) , { λn } y { αn } sootvetstvenno xarakterystyçeskug
funkcyg, posledovatel\nost\ sobstvenn¥x znaçenyj y normyrovoçn¥x çysel
zadaçy L . Pust\ ϕ ( x , λ ) =
ϕ λ
ϕ λ
1
2
( , )
( , )
x
x
— reßenye uravnenyq (1), udovletvorqg-
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
606 R.=X.=AMYROV
wee naçal\nomu uslovyg ϕ ( 0 , λ ) =
0
1−
y uslovyqm (4). Tohda s pomow\g
vektor-reßenyj
F ( x , λ ) = Y ( x , λ ) ⋅
1
−
i
, f0 ( x , λ ) = Y0 ( x , λ ) ⋅
1
−
i
moΩno zapysat\
ϕ ( x , λ ) = 1
2i
F x F x( , ) ( , )λ λ−[ ], ϕ0 ( x , λ ) = 1
2 0 0i
f x f x( , ) ( , )λ λ−[ ].
Sledovatel\no, yspol\zuq predstavlenye (6), dlq reßenyq ϕ ( x , λ ) ymeem
ϕ ( x , λ ) = ϕ0 ( x , λ ) + K x t
t
t
dt
x
x
( , )
sin
cos
λ
λ−
−
∫ .
Tohda dlq ϕ1 ( x , λ ) — pervoj komponent¥ reßenyq ϕ ( x , λ ) — poluçaem
ϕ1 ( x , λ ) = ϕ01 ( x , λ ) + K x t t dt K x t t dt
x
x
x
x
11 12( , )sin ( , )cosλ λ
− −
∫ ∫− ,
yly
ϕ1 ( x , λ ) = ϕ01 ( x , λ ) + ˜ ( , )sin ˜ ( , )cosK x t t dt K x t t dt
x x
11
0
12
0
λ λ∫ ∫+ ,
hde
˜ ( , )K x t11 = K x t K x t11 11( , ) ( , )+ − ,
˜ ( , )K x t12 = − − −K x t K x t12 12( , ) ( , ) ,
˜ ( , ), ˜ ( , )K x K x11 12⋅ ⋅ ∈ L2 ( –x , x ) .
Takym obrazom, xarakterystyçeskoe uravnenye zadaçy L budet ymet\ vyd
∆ ( λ ) = ∆0 ( λ ) + ˜ ( , )sin ˜ ( , )cosK t t dt K t t dt11
0
12
0
π λ π λ
π π
∫ ∫+ = 0.
Lemma 2. Sobstvenn¥e znaçenyq zadaçy L prost¥e, t.$e. ∆̇( )λn ≠ 0.
Dokazatel\stvo. Poskol\ku
B ϕ′ ( x , λ ) + Ω ( x ) ϕ ( x , λ ) = λ ϕ ( x , λ ) ,
B ˙ ′ϕ ( x , λ ) + Ω ( x ) ϕ̇ ( x , λ ) = λ ϕ̇ ( x , λ ) + ϕ ( x , λ ) ,
umnoΩaq skalqrno v evklydovom prostranstve R
2
pervoe uravnenye na ϕ̇ ( x , λ ) ,
vtoroe uravnenye na ϕ ( x , λ ) y v¥çytaq yz vtoroho ravenstva pervoe, poluçaem
〈 ϕ , ϕ 〉 = B ˙ ,′ϕ ϕ – B ′ϕ ϕ, ˙ .
Yntehryruq poslednee ravenstvo na otrezke [ 0 , π ] y uçyt¥vaq, çto
αn = ϕ ϕ
π
n n dx,
0
∫ = ϕ λ ϕ λ
π
1
2
2
2
0
( , ) ( , )x x dxn n+[ ]∫ ,
posle yntehryrovanyq po çastqm poluçaem
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
O SYSTEME DYFFERENCYAL|NÁX URAVNENYJ DYRAKA … 607
αn = – ˙ ( )∆ λn ϕ2 ( π , λn ) .
Otsgda sleduet, çto ˙ ( )∆ λn ≠ 0.
Lemma dokazana.
Lemma 3. Dlq sobstvenn¥x znaçenyj zadaçy L spravedlyvo asymptoty-
çeskoe ravenstvo
λn = λn
0 + εn ,
hde { εn } ∈ l2 .
Dokazatel\stvo. Oboznaçym
Γn = λ λ λ β
: , , , , ,| | | |= + = ± ± …{ }n n0
2
0 1 2 ,
Gδ = λ λ λ δ: , , , , ,| |− ≥ = ± ± …{ }n n0 0 1 2 ,
hde δ — dostatoçno maloe poloΩytel\noe çyslo ( δ << β / 2 ) . Yz vyda ∆0 ( λ ) y
lemm¥ 1 sleduet, çto ∆0 ( λ ) qvlqetsq funkcyej typa „synusa”. Poπtomu [13,
s. 118, 119] dlq λ ∈ Gδ v¥polnqgtsq neravenstva
| ∆0 ( λ ) | > C eδ
λ π| |Im , ˙ ( )∆0
0λn ≥ γ > 0.
S druhoj storon¥ [2] (lemma 1.3.1),
lim ( ) ( )Im ( )
| |→∞
−| | −
λ
λ π λ λe ∆ ∆0 =
= lim ˜ ( , )sin ˜ ( , )cosIm Im
| |→∞
−| | −| |∫ ∫+
λ
λ π
π
λ π
π
π λ π λe K t t dt e K t t dt11
0
12
0
= 0,
t.=e. pry dostatoçno bol\ßyx n dlq λ ∈ Γn v¥polnqetsq neravenstvo
| ∆ ( λ ) – ∆0 ( λ ) | <
C
eδ λ π
2
| |Im .
Znaçyt, dlq λ ∈ Γn , hde n — dostatoçno bol\ßoe natural\noe çyslo, ymeem
| ∆0 ( λ ) | ≥ C eδ
λ π| |Im >
C
eδ λ π
2
| |Im > | ∆ ( λ ) – ∆0 ( λ ) | .
Tohda, yspol\zuq teoremu Ruße, poluçaem, çto vnutry kontura Γn pry dosta-
toçno bol\ßom n funkcyy ∆0 ( λ ) y ∆0 ( λ ) + { ∆ ( λ ) – ∆0 ( λ ) } = ∆ ( λ ) ymegt
odynakovoe çyslo nulej, t.=e. 2n + 1 nulej: λ – n , … , λ0, … , λn . Analohyçno,
yspol\zuq teoremu Ruße, moΩno dokazat\, çto pry dostatoçno bol\ßyx n v
kaΩdom kruhe | λ – λn
0
| < δ ymeetsq rovno odyn nul\ funkcyy ∆ ( λ ) . Poskol\-
ku δ — lgboe dostatoçno maloe çyslo, moΩno zapysat\ λn = λn
0 + εn , hde
lim
n n→∞
ε = 0, a tak kak çysla λ n qvlqgtsq kornqmy xarakterystyçeskoj funk-
cyy ∆ ( λ ) , to
∆ ( λn ) = ∆0 ( λn
0 + εn ) + ˜ ( , )sin( )K t t dtn n11
0
0
π λ ε
π
+∫ +
+ ˜ ( , )cos( )K t t dtn n12
0
0
π λ ε
π
+∫ = 0.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
608 R.=X.=AMYROV
S druhoj storon¥,
∆0
0( )λ εn n+ = ˙ ( )( )∆0
0 1 1λ εn no+[ ] .
Poπtomu dlq opredelenyq povedenyq εn poluçym
ε λ π λ ε
π
n n n no K t t dt˙ ( ) ( , )sin( ) ( )˜∆0
0
11
0
0
1 1+[ ] + +∫ +
+ ˜ ( , )cos( )K t t dtn n12
0
0
π λ ε
π
+∫ = 0. (10)
Dalee, sleduq rezul\tatam rabot [14, 15], ymeem
λn
0 = n + hn , sup | hn | ≤ µ .
Poπtomu [2, s. 67]
˜ ( , )sin( )K t t dtn n11
0
0
π λ ε
π
+
∫ ∈ l2 , ˜ ( , )cos( )K t t dtn n12
0
0
π λ ε
π
+
∫ ∈ l2 .
Sledovatel\no, yz (10) poluçaem, çto { εn } ∈ l2 .
Lemma dokazana.
Lemma 4. Dlq normyrovoçn¥x çysel zadaçy L spravedlyvo asymptotyçes-
koe ravenstvo
αn = αn
0 + δn ,
hde { δn } ∈ l2 .
Dokazatel\stvo. V sylu lemm¥ 3 yz predstavlenyj ∆ ( λ ) y ϕ2 ( π , λ ) ne-
trudno poluçyt\ sledugwye sootnoßenyq:
˙ ( )∆0 λn = ˙ ( )∆0
0λn + O ( εn ) ,
ϕ2 ( π , λn ) = ϕ π λ2 0
0
, ( ), n + δ̃n , δ̃n{ } ∈ l2 .
Poπtomu
αn = – ˙ ( )∆ λn ϕ2 ( π , λn ) = – ˙ ( ) , ˜( ) ( ),∆0
0
2 0
0λ ε ϕ π λ δn n n nO+[ ] +[ ] = αn
0 + δn ,
hde
δn = – ˙ ˜( )∆0
0λ δn n + O n n( ) ,, ( )ε ϕ π λ2 0
0 + O n n( ) ˜ε δ .
Oçevydno, çto { δn } ∈ l2 .
Lemma dokazana.
4. Reßenye Vejlq. Funkcyq Vejlq. Pust\ vektor-funkcyq Φ ( x , λ ) =
=
Φ
Φ
1
2
( , )
( , )
x
x
λ
λ
qvlqetsq reßenyem uravnenyq (1) y udovletvorqet uslovyqm
Φ1( 0, λ) = 1, Φ1 ( π , λ ) = 0, a takΩe uslovyqm sklejky (4). Funkcyg Φ ( x , λ )
budem naz¥vat\ reßenyem Vejlq dlq kraevoj zadaçy L .
Oboznaçym çerez ψ ( x , λ ) =
ψ λ
ψ λ
1
2
( , )
( , )
x
x
y C ( x , λ ) =
C x
C x
1
2
( , )
( , )
λ
λ
reßenyq urav-
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O SYSTEME DYFFERENCYAL|NÁX URAVNENYJ DYRAKA … 609
nenyq (1), udovletvorqgwye naçal\n¥m uslovyqm ψ ( π , λ ) =
0
1−
, C ( 0 , λ ) =
=
1
0
y uslovyqm sklejky (4). Qsno, çto funkcyy ψ ( x , λ ) y C ( x , λ ) qvlqgt-
sq cel¥my po λ . Tohda funkcyg ψ ( x , λ ) moΩno predstavyt\ v vyde
ψ ( x , λ ) = – ψ2 ( 0 , λ ) ϕ ( x , λ ) – ∆ ( λ ) C ( x , λ ) .
Oboznaçym
M ( λ ) = + ψ λ
λ
2( , )
( )
x
∆
−
ψ λ
ψ λ
2
1
0
0
( , )
( , )
. (11)
Qsno, çto
Φ ( x , λ ) = C ( x , λ ) + M ( λ ) ϕ ( x , λ ) . (12)
Funkcyg – Φ2 ( 0 , λ ) = M ( λ ) budem naz¥vat\ funkcyej Vejlq dlq zadaçy L .
Reßenye Vejlq y funkcyq Vejlq qvlqgtsq meromorfn¥my po λ funkcyqmy s
polgsamy na spektre zadaçy L . Yz (11) y (12) sleduet
Φ ( x , λ ) = − ψ λ
λ
( , )
( )
x
∆
. (13)
Teorema 2. Spravedlyvo predstavlenye
M ( λ ) = 1 1 1
0 0
0 0
0
α λ λ α λ λ α λ( ) ( ),−
+
−
+
=−∞
≠
+∞
∑
n n n nn
n
. (14)
Dokazatel\stvo. Analohyçno predstavlenyg reßenyq ϕ ( x , λ ) netrudno
poluçyt\ predstavlenye dlq reßenyq ψ ( x , λ ) :
ψ ( x , λ ) = ψ0 ( x , λ ) + N x t
t
t
dt
x
x
( , )
sin
cos( )
λ
λπ
π
−
− −
−
∫ =
= ψ0 ( x , λ ) + ˜( , )
sin
cos
N x t
t
t
dt
x λ
λ
π
−
−
∫
0
, (15)
hde πlement¥ ˜ ( , )N x tij matryc¥-funkcyy ˜( , )N x t = ˜ ( , )
,
N x tij i j
{ } =1
2
pry kaΩdom
fyksyrovannom x ∈ [ 0 , π ] prynadleΩat prostranstvu L 2 ( 0 , π ) po pere-
mennoj t ,
ψ0 ( x , λ ) =
−
−
−
−
−
×
×
− +
− +
< <
−
−
−
< <
+ −α
λ π
λ π
α
λ π
λ π
λ π
λ π
π
sin ( )
cos ( )
sin ( )
cos ( )
, ,
sin ( )
cos ( )
, .
x
x
x a
x a
x a
x
x
a x
1 0
0 1
2
2
0
Vvedem sledugwye oboznaçenyq:
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
610 R.=X.=AMYROV
M0 ( λ ) = − ψ λ
ψ λ
20
10
0
0
( , )
( , )
=
− + −
+ −
+ −
+ −
α λπ α λ π
α λπ α λ π
cos cos ( )
sin sin ( )
2
2
a
a
,
(16)
fi ( λ ) = ˜ ˜( , )sin ( , )cosN t t N t t dti i1 2
0
0 0λ λ
π
−{ }∫ , i = 1, 2.
Tohda yz (15) ymeem
ψi ( 0 , λ ) = ψi , 0 ( 0 , λ ) + fi ( λ ) .
Poskol\ku lim ( )Im
| |→∞
−| | | |
λ
λ π λe fi = 0, | ∆ ( λ ) | > C eδ
λ π| |Im
pry λ ∈ Gδ , yz ra-
venstva
M ( λ ) – M0 ( λ ) =
f f
M2 1
0
( )
( )
( )
( )
( )
λ
λ
λ
λ
λ
∆ ∆
−
sleduet
lim sup ( ) ( )
| |→∞ ∈
| |−
λ λ δ
λ λ
G
M M0 = 0. (17)
Dalee, vektor-funkcyy ϕ ( x , λn ) ( ϕ0 ( x , λn
0 ) ) y ψ ( x , λn ) ( ψ0 ( x , λn
0 ) ) qv-
lqgtsq sobstvenn¥my funkcyqmy zadaçy L ( L0 ) . Poπtomu suwestvugt kon-
stant¥ βn ( )βn
0
takye, çto
ψ ( x , λn ) = βn ϕ ( x , λn ) ψ λ β ϕ λ0
0 0
0
0( ) ( ), ,x xn n n=( ).
Otsgda poluçaem
βn = – ψ2 ( 0, λn ) = − 1
2ϕ π λ( , )n
, βn
0 = – ϕ2, 0 ( 0 , λn
0 ) = − 1
2 0
0ϕ π λ, ( , )n
.
Yspol\zuq ravenstva αn = – ˙ ( )∆ λn ϕ2 ( π , λn ) , αn
0 = − ˙ ,( ) ( ),∆0
0
2 0
0λ ϕ π λn n , ymeem
Res
λ λ
λ
= n
M( ) =
ψ λ
λ
2 0( , )
˙ ( )
n
n∆
= − 1
2
˙ ( , )( )∆ λ ϕ π λn n
= 1
αn
,
(18)
Res
λ λ
λ
= n
M
0 0( ) = 1
0αn
.
Teper\ rassmotrym konturn¥j yntehral
In ( λ ) = 1
2
0
π
µ µ
λ µ
µ
i
M M
d
n
( ) ( )−
−∫
Γ
, λ ∈ int Γn .
V sylu (17) ymeem lim ( )n nI→∞ λ = 0. S druhoj storon¥, sohlasno teoreme o
v¥çetax yz (18) ymeem
In ( λ ) = – M ( λ ) + M0 ( λ ) + 1 1
0 0
0α λ λ α λ λλ λn n n nn n n
( )int int ( )−
−
−∈ ∈
∑ ∑
Γ Γ
.
Otsgda pry n → ∞
M ( λ ) = M0 ( λ ) + 1 1
0 0α λ λ α λ λn n n nn ( ) ( )−
−
−
=−∞
+∞
∑ .
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O SYSTEME DYFFERENCYAL|NÁX URAVNENYJ DYRAKA … 611
Yz vyda (16) funkcyy M0 ( λ ) sleduet
M0 ( λ ) = 1 1 1 1
0
0 0 0α λ α λ λ λ
+
−
+
=−∞
+∞
∑
nn n n
.
Sravnyvaq dva poslednyx ravenstva, poluçaem
M ( λ ) = 1 1 1 1
0 0
0 0
0
α λ λ α λ λ α λ( ) ( ),−
+
−
+
=−∞
≠
+∞
∑
n n n nn
n
.
5. Obratnaq zadaça. V πtom punkte yssledugtsq obratn¥e zadaçy vossta-
novlenyq kraevoj zadaçy L po zadann¥m funkcyy Vejlq y dyskretn¥m spek-
tral\n¥m dann¥m. ∏ty obratn¥e zadaçy qvlqgtsq obobwenyqmy yzvestn¥x
obratn¥x zadaç dlq system¥ Dyraka (sm. [5, 6]).
Narqdu s L rassmotrym kraevug zadaçu L̃ toho Ωe vyda, no s potencyalom
˜ ( )Ω x . Uslovymsq, çto esly nekotor¥j symvol α oboznaçaet obæekt, otnosq-
wyjsq k zadaçe L , to α̃ budet oboznaçat\ obæekt, otnosqwyjsq k zadaçe L̃ .
Teorema 3. Esly M ( λ ) = ˜ ( )M λ , to L = L̃ . Takym obrazom, zadanye funk-
cyy Vejlq odnoznaçno opredelqet kraevug zadaçu L .
Dokazatel\stvo. Vvedem matrycu P ( x , λ ) = [ Pj k ( x , λ ) ]j, k =1, 2 po formule
P x( , )
˜
˜
˜
˜
λ
ϕ
ϕ
1 1
2 2
Φ
Φ
=
ϕ
ϕ
1 1
2 2
Φ
Φ
.
Yspol\zuq tot fakt, çto vronskyan reßenyj ϕ̃ =
˜
˜
ϕ
ϕ
1
2
y Φ̃ =
˜
˜
Φ
Φ
1
2
W x x˜ ( , ), ( , )˜ϕ λ λΦ{ } ≡ ˜ ( , ) ( , )˜ϕ λ λ1 2x xΦ – ˜ ( , ) ( , )˜ϕ λ λ2 1x xΦ = 1,
naxodym
P11 ( x , λ ) = ϕ λ λ1 2( , ) ,˜x xΦ ( ) – Φ1 2( , ) ˜ ( , )x xλ ϕ λ ,
P12 ( x , λ ) = Φ1 1( , ) ˜ ( , )x xλ ϕ λ – ϕ λ λ1 1( , ) ,˜x xΦ ( ),
(19)
P21 ( x , λ ) = ϕ λ λ2 2( , ) ,˜x xΦ ( ) – Φ2 2( , ) ˜ ( , )x xλ ϕ λ ,
P22 ( x , λ ) = Φ2 1( , ) ˜ ( , )x xλ ϕ λ – ϕ λ λ2 1( , ) ,˜x xΦ ( );
ϕ1 ( x , λ ) = P x x P x x11 1 12 2( , ) ˜ ( , ) ( , ) ˜ ( , )λ ϕ λ λ ϕ λ+ ,
ϕ2 ( x , λ ) = P x x P x x21 1 22 2( , ) ˜ ( , ) ( , ) ˜ ( , )λ ϕ λ λ ϕ λ+ ,
(20)
Φ1 ( x , λ ) = P x x11 1( , ) ,˜λ λΦ ( ) + P x x12 2( , ) ,˜λ λΦ ( ) ,
Φ2 ( x , λ ) = P x x21 1( , ) ,˜λ λΦ ( ) + P x x22 2( , ) ,˜λ λΦ ( ).
Yz (19), (12) y (18) sleduet
P11 ( x , λ ) = 1 + 1
1 2 2∆( )
( , ) ˜ ( , ) ( , )
λ
ϕ λ ψ λ ψ λx x x−( )( –
– ψ λ ϕ λ ϕ λ1 2 2( , ) ˜ ( , ) ( , )x x x−( )) ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
612 R.=X.=AMYROV
P12 ( x , λ ) = 1
1 1 1 1∆( )
( , ) ˜ ( , ) ( , ) ˜ ( , )
λ
ψ λ ϕ λ ϕ λ ψ λx x x x−( ) ,
P21 ( x , λ ) = 1
2 2 2 2∆( )
( , ) ˜ ( , ) ( , ) ˜ ( , )
λ
ϕ λ ψ λ ψ λ ϕ λx x x x−( ),
P22 ( x , λ ) = 1 + 1
2 1 1∆( )
( , ) ˜ ( , ) ( , )
λ
ψ λ ϕ λ ϕ λx x x−( )( –
– ϕ λ ψ λ ψ λ2 1 1( , ) ˜ ( , ) ( , )x x x−( )).
Yz predstavlenyj reßenyj ϕ ( x , λ ) y ψ ( x , λ ) qsno, çto
lim max ( , )
λ
λ
π
δ
λ
→∞
∈
≤ ≤
| |−
G
x
P x
0 11 1 = lim max ( , )
λ
λ
π
δ
λ
→∞
∈
≤ ≤
| |−
G
x
P x
0 22 1 =
= lim max ( , )
λ
λ
π
δ
λ
→∞
∈
≤ ≤
| |
G
x
P x
0 12 = lim max ( , )
λ
λ
π
δ
λ
→∞
∈
≤ ≤
| |
G
x
P x
0 21 = 0. (21)
Sohlasno (12) y (20) ymeem
P11 ( x , λ ) = ϕ λ λ λ ϕ λ1 2 1 2( , ) ( , ) ( , ) ˜ ( , )˜x C x C x x− +
+ ˜ ( ) ( ) ( , ) ˜ ( , )M M x xλ λ ϕ λ ϕ λ−( ) 1 2 ,
P12 ( x , λ ) = C x x x C x1 1 1 1( , ) ˜ ( , ) ( , ) ( , )˜λ ϕ λ ϕ λ λ− +
+ M M x x( ) ( ) ( , ) ˜ ( , )˜λ λ ϕ λ ϕ λ−( ) 1 1 ,
P21 ( x , λ ) = ϕ λ λ λ ϕ λ2 2 2 2( , ) ( , ) ( , ) ˜ ( , )˜x C x C x x− +
+ ˜ ( ) ( ) ( , ) ˜ ( , )M M x xλ λ ϕ λ ϕ λ−( ) 2 2 ,
P22 ( x , λ ) = C x x C x x2 1 1 2( , ) ˜ ( , ) ( , ) ( , )˜λ ϕ λ λ ϕ λ− +
+ M M x x( ) ( ) ( , ) ˜ ( , )˜λ λ ϕ λ ϕ λ−( ) 2 2 .
Takym obrazom, esly M ( λ ) ≡ ˜ ( )M λ , to pry kaΩdom fyksyrovannom x funkcyy
Pi j ( x , λ ) qvlqgtsq cel¥my po λ . Vmeste s (21) P11 ( x , λ ) ≡ 1, P12 ( x , λ ) ≡ 1,
P22 ( x , λ ) ≡ 1, P21 ( x , λ ) ≡ 0. Podstavlqq yx v (20), naxodym ϕ1 ( x , λ ) ≡ ˜ ( , )ϕ λ1 x ,
ϕ2 ( x , λ ) ≡ ˜ ( , )ϕ λ2 x , Φ1 ( x , λ ) ≡ ˜ ,Φ1( )x λ , Φ2 ( x , λ ) ≡ ˜ ,Φ2( )x λ pry vsex x y λ ,
y, sledovatel\no, L ≡ L̃ .
Teorema 3 dokazana.
Rassmotrym teper\ obratnug zadaçu vosstanovlenyq L po dyskretn¥m spek-
tral\n¥m dann¥m { } =−∞
∞λ αn n n, . Yz formul¥ (14) dlq funkcyy Vejlq y teo-
rem¥ 3 v¥tekaet sledugwaq teorema.
Teorema 4. Esly λn = λ̃n , αn = α̃n pry vsex n ∈ Z , to L ≡ L̃ . Takym ob-
razom, zadanye spektral\n¥x dann¥x odnoznaçno opredelqet zadaçu L .
Avtor v¥raΩaet blahodarnost\ Y.=M.=Husejnovu za vnymanye k rabote y kry-
tyçeskye zameçanyq.
1. Levytan$B.$M., Sarhsqn$Y.$S. Operator¥ Íturma – Lyuvyllq y Dyraka. – M.: Nauka, 1988. –
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Poluçeno 29.01.2004
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| id | umjimathkievua-article-3626 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:00Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/0ca9546ead9c4a5e428e4ac434eab7b0.pdf |
| spelling | umjimathkievua-article-36262020-03-18T20:00:32Z On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval O системе дифференциальных уравнений Дирака с условиями разрыва внутри интервала Amirov, R. Kh. Амиров, Р. Х. Амиров, Р. Х. We study representations of solutions of the Dirac equation, properties of spectral data, and inverse problems for the Dirac operator on a finite interval with discontinuity conditions inside the interval. Вивчаються зображення розв'язків рівняння Дірака, властивості спектральних даних та обернені задачі оператора Дірака на скінченному інтервалі з умовами розриву всередині інтервалу. Institute of Mathematics, NAS of Ukraine 2005-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3626 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 5 (2005); 601–613 Український математичний журнал; Том 57 № 5 (2005); 601–613 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3626/3982 https://umj.imath.kiev.ua/index.php/umj/article/view/3626/3983 Copyright (c) 2005 Amirov R. Kh. |
| spellingShingle | Amirov, R. Kh. Амиров, Р. Х. Амиров, Р. Х. On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval |
| title | On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval |
| title_alt | O системе дифференциальных уравнений Дирака с условиями разрыва внутри интервала |
| title_full | On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval |
| title_fullStr | On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval |
| title_full_unstemmed | On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval |
| title_short | On a System of Dirac Differential Equations with Discontinuity Conditions Inside an Interval |
| title_sort | on a system of dirac differential equations with discontinuity conditions inside an interval |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3626 |
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