CC BY
13TO'RT O'LCHAMLI QO'ZG'ALISHGA EGA IKKI KANALLI MOLEKULYAR-REZONANS MODELINING REZOLVENTASI
Dildora Erkinovna Ismoilova
Buxoro davlat universiteti
ANNOTATSIYA
Ushbu maqolada Fok fazosining qirqilgan ikki zarrachali qism fazosida 2-tartibli operatorli matritsa ko'rinishidagi A, panjaraviy ikki kanalli molekulyar-rezonans modeli qaraladi. A operatorga rezolventa operatori uchun aniq formula topiladi.
Kalit so'zlar: Fok fazo, operatorli matrisa, molekulyar-rezonans modeli, rezolventa operatori.
REZOLVENT OF THE TWO-CHANNEL MOLECULAR RESONANCE MODEL WITH FOUR DIMENSIONAL PERTURBATION
Dildora Erkinovna Ismoilova
Bukhara State University
ABSTRACT
In this paper a lattice two-channel molecular-resonance model A acting in the two-particle cut subspace of Fock space is considered as operator matrix of order 2. An exact form of the resolvent operator of A is found.
Key \words: Fock space, operator matrix, molecular-resonance model, resolvent operator.
MASALANING QO'YILISHI Td orqali d o'lchamli torni, H0 := C orqali bir o'lchamli kompleks fazoni (1-kanal) va Hx := L2 (Td) orqali Td to'plamda aniqlangan kvadrati bilan integrallanuvchi (umuman olganda kompleks qiymat qabul qiluvchi) funksiyalarning Gilbert fazosini (2-kanal) belgilaymiz. H0 va H fazolarning to'g'ri yig'indisini H orqali belgilaymiz, ya'ni H := H0 0 H. Odatda H0 fazoga Fok fazosining nol zarrachali qism fazosi, H fazoga Fok fazosining bir zarrachali qism fazosi va H Gilbert fazosiga esa Fok fazosining qirqilgan ikki zarrachali qism fazosi deyiladi.
Operatorlar nazariyasidan bizga yaxshi ma'lumki, H Gilbert fazosi ikkita Gilbert fazolar to'g'ri yig'indisidan iboratligi bois unda aniqlangan har qanday chiziqli chegaralangan operator hamisha 2-tartibli operatorli matrisa ko'rinishida tasvirlanadi. Mazkur maqolada H Gilbert fazosidagi quyidagi ikkinchi tartibli blok operatorli matritsani qaraymiz:
A:-
Bunda matritsa elementlari
A00 Mo A01
yMo A0*1 A11 -MlV1 -M2V2
4)0 /o = a/0, 4)1 /1 = j V0 (t^ (t)dt,
rpd
(A11/1 )(x)= u(x)/1(x), (VJx)(x) = V1(x) jjV1 (t)dt,
(V2/1)(x) = V2( x) j V2(t )f(t )dt
pd
tengliklar yordamida ta'sir qiladi. A operatornng parametrlari bo'lgan a va m (k = 0,1,2) sonlari hamda u(),v0 (•), v (') va v2 (•) funksiyalarga quyidagi shartlar qo'yiladi:
a - fiksirlangan haqiqiy son, ^ (k = 0,1,2)- fiksirlangan haqiqiy musbat sonlar (odatda ularga ta'sirlashish parametrlari deyiladi), u('), v0 (•),v (•) va v2 (•) funksiyalar esa
Td da aniqlangan haqiqiy qiymatli uzluksiz funksiyalardir.
A operator uning parametrlariga qo'yilgan yuqoridagi shartlarda H Gilbert fazosidagi chiziqli, chegaralangan va o'z-o'ziga qo'shma bo'ladi. Bu tasdiq Funksional analiz kursidan ma'lum bo'lgan ta'rif va metodlar yordamida isbotlanadi. Xususan, berilgan operatorni chegaralanganlikka tekshirishda / = (/, /) e H element normasi
r
. 1/2
I/0I2 + jl /1(t )f dt
V Td y
kabi aniqlanishidan hamda Koshi-Bunyakovskiy tengsizligidan foydalaniladi. O'z-o'ziga qo'shma ekanligini tekshirish uchun / = (/,/), g = (g0,g) e H elementlar skalyar ko'paytmasi
(/, g) = /0 'g0 + j /1(t)• g^fjdt
rpd
tenglik orqali aniqlanishidan foydalaniladi.
Zamonaviy matematik fizikada A operatorga ikki kanalli molekulyar-rezonans modeli deyiladi. Ikki va uch o'lchamli qo'z'g'alishga ega bo'lgan hollar [1 -6] ishlarda o'rganilgan.
Chekli o'lchamli qo'zg'alishlarda muhim spektrning o'zgarmasligi haqidagi G.Veyl teoremasiga ko'ra A operatorning muhim spektri uchun
^ (A) = [m; M] tenglik o'rinlidir, bu yerda m va M sonlari
m := min u(x), M := max u(x).
xeTd xeTd
formula orqali aniqlangan sonlardir. Ko'rinib turibdiki, A operatorning muhim spektri juk> 0, k = 0,1,2 ta'sirlashish parametrlaridan bog'liq emas.
Funksional analiz fanidan bizga yaxshi ma'lumki, A blok operatorli matritsa o'z-o'ziga qo'shma bo'lganligi bois, uning barcha xos qiymatlari haqiqiydir. A blok operatorli matritsaning diskret spektrini aniqlashda muhim bo'lgan hamda C\[m,M] sohada
v20(t)dt,
A0(rf0,z) = a — z— J V0
jrd u(t) z
Vo(t)vl(t)dt .
1 oi(Mo>z) = rfo J"
T
d u(t) - z
■V0(t )V0(t )dt
1 oo(ßo>z) = Mo J^^fvr-
jd u(t) — z
V(t)Vo(t)dt .
Iio(z) = J V
d u(t) — z
Ai(M, z) = 1—rf J ;
TdU(t) — z
Ao(rfo,z) = 1 — rf JV0F
jdU(t) — z
regulyar funksiyalarni qaraymiz. Odatda,
A(rfo,Mi, Mo,z) = A0(Mo,z)A1 (Mi,z)A0(Mo ,z) — 0MlMoI01(Mo,z) 100(Mo,z)I10(z)
— Ml1 h^ z )A 0 z) Mo1 oo (Mo, z) A1 (M^ z) — M1M0I10 (z)A 0 (M0, z) funksiyaga A operatorga mos Fredgolm determinant deyiladi. Sodda mulohazalar yordamida har bir fiksirlanganjuk>0,(k = 0,1,0) lar uchun zeC\[m;M] soni A operatorning xos qiymati bo'lishi uchun Arf0,, z) = 0 bo'lishi zarur va yetarli ekanligini isbotlash mumkin. O'z navbatida bu tasdiqdan A operatorning diskret spektri haqidagi quyidagi tenglik hosil qilinadi:
°dlsc(A) = e C\ [m;M]: A(rf,rf,Mo,z) = 0}-Endi A blok operatorli matritsaning rezolventasini topish uchun har bir fiksirlangan z e C \ [m, M] soni uchun
Af — zf = g (1)
tenglamani qaraymiz. Bu yerda
f = (f,, f1); g = (go, g1 )eH
(1) tenglamani quyidagi tenglamalar sistemasi ko'rinishida yozib olamiz:
af0 + Mo J V0 (t)f1 (t)dt — zf0 = go ;
j-d
MoVo( x)fo + U (x)f1( x) — (2)
— M1V1(x) J V1(t)f1(t)dt — M0V0 (x) J V1 (t)f1 (t)dt — zf1 (x) = g1(x)■
dd
(2)-tenglamalar sistemasining ikkinchi tenglamasida
C := Jvl(t)fl(t)dt
va
c2 := Jv2(t)fi(t)dt
kabi belgilashlar kiritib, (2)-tenglamalar sistemasidan f (x) funksiya uchun
gl ( X) - Mo V0 ( X)f0 + MlV1 ( x)C + M2 V2 ( X)C 2
f 1 ( x)
(3)
(4)
(5)
u (x) - z
ifodaga ega bo'lamiz. f ( x) uchun topilgan (5) ifodani (2)-tenglamalar sistemasining birinchi tenglamasiga va (3), (4) belgilashlarga olib borib qo'yamiz. Yuqoridagi belgilashlardan foydalanib
•Vo(t ) gi(t )dt.
a 0 (mo ? z)f0 + ml 101 (mo ? z)ci + m2102 (mo ? z)C2 = g0 - Mo J
101(Mo? z )f0 + A 1(Ml, z )C1 M2112 ( z )C2 = J
102(Mo? z)f0 M2112 (z)C1 +A2(M2. z)C2 = J
d u (t ) - z V1(t ) g1(t )dt
u (t ) - z v2(t ) g1(t )dt.
(6)
(7)
(8)
Jd u(t ) - z
f0,c va c2 noma'lumlarga nisbatan bir jinsli bo'lmagan tenglamalar sistemasiga ega bo'lamiz. (6), (7), (8)-tengliklarning koeffisiyentlari orqali tuzilgan ushbu
A f0 (Mo? M Ml,z ) :=
-Mo J Vo(t)g1(t )dt M1101(Mo? z) M2I02(M0, z)
u(t) - z
I* V1 (t)g1(t)dt
rJd u (t) - z
JV2(t ) g1(t )dt J,d u (t) - z
Ac (Mo? Mi, M z ):=
A1(M1, z) M2112(z) M2112(z) A2(M2? z)
Ao(Mo,z) go-Mo J V°(/)g|(/)dt) M2102 (Mo,z)
1 o1(Mo?z )
102 CM z)
u(t) - z vl(t ) gx(t )dt u(t) - z
V2(t ) gx(t )dt
u(t) - z
M2112( z ) A 2(M2? z)
d
1
d
T
T
d
1
AC2 fr,z) •=
Ao(Mo,z) MihiiMo,z) go -Mo j
hiiMo,z) Ai(Mi,z)
1 o2(Mo , z) -M2I i2(z)
Vo(t ) gi(t )dt
u (t ) - z
Vi(t ) gi(t )dt j<d U (t) z
v2(t ) g,(t )dt
j
j
u(t ) - z
determinantlarni qurib olamiz. Bizga yaxshi ma'lumki, agar z soni A operatoming spektriga tegishli bo'lmasa, u holda unga mos Fredgolm determinanti bu nuqtada nolmas qiymatni qabul qiladi. Shu sababli mazkur nuqtada berilgan sistema yagona yechimga ega bo'ladi. Yechimni chiziqli algebra kursidan yaxshi ma'lum bo'lgan Kramer usulini qo'llab f,c va c2 noma'lumlarni topib olamiz. Bizga yaxshi ma'lumki, Kramer usuliga ko'ra noma'lumlar quyidagicha topiladi;
fo =
ci =
C2 =
Af (Mo,^^z) . A(Mo,Mi,Mi,z) ' Ac (Mo, ^ M2, z) . A(ßo, M, M2, z) ' Ac2 (Mo,Mi, M2, z) A(^ jUi, H2.,z)
A0 (Mo ,M\,M2, z), A (.Mq ,M\,M2, z) va A (.Mq ,M\,M2, z) laming aniq ko'rinishidan foydalanib, f, c va c2 yechimlarni
Ai(Mi, z)A2(.Mi, z) - m!!212(z)
fo =
A(Mo, Mi-, Mi, z )
g o +
MoM2I12( z ) - Mo Ai (Mi, z )A 2 (.Mo , Mi-> Mi , z ) f vo(t ) gi(t )dt
u(t) - z
A(Mo, Mi-, M2, z )
j-
MÎi oi(Mo,z)Ii2(z) + MiAz)Iol(Mo,z) ivi(t)gi(t)dt _ A(^0 , ßx, ß2, z) rJd u (t ) - z
MiMiIoi^z)Ii2(z) + MiAi(Mi,z) I oi^z) îv2(t)gi(t)dt A(^0 , ßx, ß2, z ) jd u (t ) - z
Mi Io2(Mo, z )Ii2( z) + ^^ z) hi^ z)
(8)
ci = -
A(Mo,Mi,M2, z)
-go +
MoMiIoi(Mo,z)Ii2(z) + Mo^^z)Ioi(Mo,z) îvo(t)gi(t)dt
+ / o 2 1 o2(fro , z )I i 2(z ) ' H-oA2(H-2, z)Io i (W), z) f vo ( yu i y y +
,^,№2,z) Td u(t) - z
*o ' r*i ■> r*2? / Ta
A2 (Mi, z)Ao (Mo , z) - Mi^2 (Mo, z) rvi(t)gi(t)dt
A(Mo,Mi z)
u(t) - z
)
d
T
+
d
T
5z)I02(M0 5 z) + (Mo5 z)I12(z) rv2(t)&(t)dt . ^
--Tf-^-J-^-' (9)
A(m0,z) td u(t)— z
_ _ M2I01(M0, z)I12(z) + A1(Ml, z)I02(M0, z) ,
C2 _ w x g0 +
A(Mo, M> Mi,z)
, M0M2102 (M>5 z)I12(z) + Mo A1(M5 z)l02(Mo 5 z) rvo(t)g1(t)dt , A(m0 , M, M2, z) ^d u (t) - z
, MI01 (Mo 5 z) 102 (Mo 5z) + MlA0 (Mo ,z)I12 (z) rv2(t)g1(t)dt + A(m , M, Mi, z) yd u (t) - z
, A1(M1,z)A0 (M0 5z) — MI01 (M0 5z) rv1(t)g(t)dt.
J-
A(m0 , MM, m2, z) fd u (t) — z
ko'rinishda yozish mumkin. Topilgan (8)-(10)-ifodalarni (5)-ifodaga qo'yib
f1(x) = (—
M0V0(X) . A1 (M5 z)A2(M25 z) — M221122(z) _ u(x) — z A(m ,M ,M, z)
MV1(x) , MlI02(M05 z)I12(z) + A2(M25 z)l01(Mp5 z) _ u(x) — z A(m M >M 5 z)
M2V2(X) . M2101 (M0 5 z)I12 (z) + A1 (M5 z) 102 (M0 5 z^ + g1(X) _ u(x) — z A(m M >M 5 z) 0 u( x) — z
M0v0(x) ^m!I12(z) — M0A1(M5^CMMM^z) rv0(t)g1(t)dt _ u(x) — z A(mM5M25z) rJd u(t) — z
M2102(M05 z)I12(z) + MA2(M5 z)l01(M05 z) rV1(t)g1(t)dt _ A(mM5M25z) ^d u(t) — z
M1M2I01 (M 5 z)I12(z) + M2 A1 (M5 z)I02 (M 5 z) rv2(t)g1(t)dt) , A(m MMi 5 z) rJd u(t) — z
MV1( x) /M0M2I02(M05 z)l12( z ) + M0A2(M25 z)l01M05 z) fV0(t) g1(t )dt
! . 1 1v , (M0M21 02 (M0 5 z) I12(z ) + M0A2(M25 z)1 01(M05 z) f
u( x) — z A(m M5M2 5 z) rJd u(t) — z
| A2(M25 z)A0(M05 z) — M2102(M05 z) rV1(t)g1(t)dt _ A(mM>M5z) rJd u(t) — z
M2101(M05 z)I02(M0 5 z) + M2A0 (M)5 z)I12(z) rv2(t)g1(t)dt) + A(m M >M 5 z) rJd u(t) — z
, M2V2(x) (M0M2102(M0 5 z)l12(z) + M0 A1 (M15 z)l02(M05 z) V0 (t)&(t)dt + u( x) — z A(m MMi 5 z) rJd u(t) — z
, MI01(M0 5 z) 102 (M05 z) + M2 A0 (M0 5 z)I12 (z) j* V2 (t)g1(t)dt + A(m MMi 5 z) rJd u(t) — z
(10)
Ai (m, z)A0 (Mo,z) - M/c2i (Mo,z) f V (t)gl (t)dt
r VlC)glC)dt \ J , u(t\ _ V )
A(m,M,M,z) Td u(t) - z
tenglikga ega bo'lamiz.
(7)-tenglik va f (x) uchun hosil bo'lgan tenglikdan ko'rish mumkinki, A blok operatorli matritsaning rezolventasi H Gilbert fazosida 2-tartibli
R( z)
^ooCM Mi, M2;z) Roi(Mo, Mi, M2;z)
V R10(M0, M^ M2;z) R11(M0, Ml, M2;z) y blok operatorli matritsa ko'rinishda bo'ladi. Bu yerda
RooMo, M, M2; z)go = A1(M1'zA)A 2(M2' z)-MM 7122( z) go;
A(Mo, Ml, M2'z)
R (u u u-z)e = M0M227122(z)-MoA1 (M1,z)A2(Mo,M1,Mi,z) rVo(/)gl(t)dt
01 ^/^0' *1' /ö1 L / \ j /j\
A(Mo, Ml, M2 , z) Td u(t) - z
M221 02 (Mo, z) 1 12 ( z) + M1A 2 (M2 , z) 101 (Mo,z) fV1(i)g1(t)dt
A(Mo, Ml, M2 , z) Td u (t) - z
M1M21 01 (Mo, z) 1 12 (z) + M2 A1 (M1, z) 102 (Mo,z) j*V2(t)g1(t)dt A(M , M , M , z) d u(t) - z
r/? r» // „^WVrW MoV0 (x) MMp z)A2(M2, z)-M2112(z) (R10(M0 , M1, M2; z)g0)(x) - (-"
u( x) - z A(m ,M,M, z)
MV1(x) , M2102(M0, z)I12(z) + A2(M2, z)I01(M0, z) _ u(x) - z A(m ,M ,M, z)
m2V2(X) . m2701 (mo , z)I12 (z) + A1 (m1, z)102 (mo , z) u( x) - z A(m0 , m, m, z)
g1( x)
) g 0;
(Rll(Mo, M M2; z ) g1)(x)
VA*0 > < 1 > < 2 ' ✓ ^ 1 ✓ v ✓ /N
u (x) - z
MoVo(x) ^ M0M2 /12(z) - MoAl (M, z)A2 (Mo, Ml, M2, z) rv0(t)gl(t)dt u(x) - z A(m,M,M,z) J u(t) - z
M22/02(M0, z)/12(z) + MACM ^oiCM z) J V1(t)gl(t)dt
A(m,M,M,z) fd u(t) - z
M1M2/01(M0 , z)/12(z) + M2 A1 (Ml, z)/02 (Mo , z) |* V2 (t)gl(t)dt^
A(m ,M,M, z) rJd u(t) - z
, MV1(x) ( M0M2/02(Mo , z)/12(z) + Mo A2 (M2, z)/01 (Mo , z) f V0(t)gl(t)dt + u( x) - z A(m ,M,M, z) yd u(t) - z
+ A2(M2, z) A0 (Mo , z) - M2/02 (Mo , z) f V1(t)gl(t)dt
1-
A(m0 ,m ,m, z) Td u(t) - z
Ta'kidlash joizki, A operator ko'p hollarda umumlashgan Fridrixs modeli deb ham yuritiladi. Bunday turdagi modellarning muhim spektri, xos qiymatlari, rezolventa operatori, bo'sag'aviy xos qiymatlari, virtual sathlari, mos Fredgolm determinanti va uning analitik xossalari, sonli va kvadratik sonli tasvirlari bilan bog'liq tadqiqot ishlari [1-10] kabi maqolalarda olib borilgan. [11-23] maqolalarda esa umumlashgan Fridrixs modeli deb ataluvchi operator xossalari yordamida soni saqlanmaydigan va uchtadan oshmaydigan zarrachalar sistemasiga mos blok operatorli matrisalarning muhim spektri va uning joylashuv o'rni, diskret spektrning chekli yoki cheksizligi hamda diskret spektr asimptotikasi bilan bog'liq natijalar olingan. Maqolada o'rganilgan ikki kanalli molekulyar-rezonans modeli ham ikkinchi tartibli blok operatorli matrisa ko'rinishga ega. Uning diagonal elementlaridan biri An - цУх - цу2 bo'lib, zamonaviy matematik fizikada bu operator Fridrixs modeli nomi bilan mashhur. Ko'rinib turibdiki, bu model ikki o'lchamli qo'zg'alishga ega. Uning spektral xossalari [24-25] va yoyiluvchi operatorlarning spektri haqidagi teoremadan foydalanib panjaradagi uchta zarrachalar sistemasiga mos model operatorning muhim va diskret spektrlarini o'rganish mumkin [26-30]. Xususan, Fridrixs modeli muhim spektrining chap chegarasiga virtual sathga ega bo'lsa, uch zarrachali model operator diskret spektr cheksiz to'plam bo'lishini isbotlash mumkin. Agar muhim spektrning chap chegarasi Fridrixs modeli uchun bo'sag'aviy xos qiymatga ega bo'lsa, bu holda uch zarrachali model operator diskret spektrining chekliligi bilan bog'liq natijalarni olish mumkin.
REFERENCES
1. Исмоилова Д.Э. (2021). О свойствах определителя Фредгольма, ассоциированного с обобщенной модели Фридрихса. НТО, 1(60), 21-24.
2. Тошева Н.А., Исмоилова Д.Э. (2021). Явный вид резольвенты обобщенной модели Фридрихса. Наука, техника и образование, 2-2(77), 39-43.
3. Tosheva N.A., Ismoilova D.E. (2021). Ikki kanalli molekulyar-rezonans modeli xos qiymatlarining soni va joylashuv o'rni. Scientific progress. 2:1, 61-69.
4. Tosheva N.A., Ismoilova D.E. (2021). Ikki kanalli molekulyar-rezonans modelining sonli tasviri. Scientific progress. 2:1, 1421-1428.
5. Tosheva N.A., Ismoilova D.E. (2021). Ikki kanalli molekulyar-rezonans modelining rezolventasi. Scientific progress. 2:2, 580-586.
6. Тошева Н.А., Исмоилова Д.Э. (2021). Икки каналли молекуляр-резонанс модели хос кийматларининг мавжудлиги. Scientific progress. 2:1, 111-120.
7. Исмоилова Д.Э. (2020). Метод формирования в преподовании темы евклидовых пространств. Проблемы педагогики. 51:6, C. 87-89.
8. Ismoilova D.E. (2021). To'rt o'lchamli qo'zg'alishga ega ikki kanalli molekulyar-rezonans modelining muhim va diskret spektrlari. Scientific progress. 2:3, 44-50 b.
9. Ismoilova D.E. (2021). Ba'zi xususiy hollarda to'rt o'lchamli qo'zg'alishga ega ikki kanalli molekulyar-rezonans modelining Fredgolm deterninanti. Scientific progress. 2:3, 69-76 b.
10. Rasulov T.H., Dilmurodov E.B. (2020). Eigenvalues and virtual levels of a family of 2x2 operator matrices. Methods Func. Anal. Topology, 1(25), 273-281.
11. Muminov M., Rasulov T., Tosheva N. (2019). Analysis of the discrete spectrum of the family of 3x3 operator matrices. Comm. in Math. Analysis, 1(11), 17-37.
12. Rasulov T.H., Tosheva N.A. (2019). Analytic description of the essential spectrum of a family of 3x3 operator matrices. Nanosystems: Phys., Chem., Math., 5(10), 511519.
13. Rasulov T.H., Dilmurodov E.B. (2020). Analysis of the spectrum of a 2x2 operator matrix. Discrete spectrum asymptotics. Nanosystems: Phys., Chem., Math., 2(11), 138144.
14. Rasulov T.H., Dilmurodov E.B. (2019). Threshold analysis for a family of 2x2 operator matrices. Nanosystems: Phys., Chem., Math., 6(10), 616-622.
15. Расулов Т.Х., Дилмуродов Э.Б. (2020). Бесконечность числа собственных значений операторных (2х2)-матриц. Асимптотика дискретного спектра. ТМФ. 3(205), 368-390.
16. Dilmurodov E.B., Rasulov T.H. (2020). Essential spectrum of a 2x2 operator matrix and the Faddeev equation. European science, 2(51), 7-10.
17. Латипов Х.М. (2021). О собственных числах трехдиагональной матрицы порядка 4. Academy, 3(66), 4-7.
18. Латипов Х,.М. (2021). 4-тартибли матрица хос сонларининг таснифи. Scientific progress. 2:1, 1380-1388 бетлар.
19. Латипов Х.М., Пармонов Х.Ф. (2021). Некоторые задачи, сводимые к операторным уравнениям. ВНО, 113:10, часть 3, С. 15-21.
20. Латипов Х,.М., Хдйитова М.А. (2021). Компакт тупламда узлуксиз функция хоссалари ёрдамида ечиладиган айрим масалалар. Scientific progress. 2:3, 77-85-betlar.
21. Лакаев С.Н., Расулов Т.Х. (2003). Модель в теории возмущений существенного спектра многочастичных операторов. Математические заметки. 73:4, С. 556-564.
22. Лакаев С.Н., Расулов Т.Х. (2003). Об эффекте Ефимова в модели теории возмущений существенного спектра. Функциональный анализ и его приложения, 37:1, С. 81-84.
23. Albeverio S., Lakaev S.N., Rasulov T.H. (2007). On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics. Journal of Statistical Physics, 127:2, pp. 191-220.
24. Albeverio S., Lakaev S.N., Rasulov T.H. (2007). The Efimov Effect for a Model Operator Associated with the Hamiltonian of non Conserved Number of Particles.
Methods of Functional Analysis and Topology, 13:1, pp. 1-16.
25. Bahronov B.I., Rasulov T.H. (2020). Structure of the numerical range of Friedrichs model with rank two perturbation. European science. 51:2, pp. 15-18.
26. Umirkulova G.H., Rasulov T.H. (2020) Characteristic property of the Faddeev equation for three-particle model operator on a one-dimensional lattice. European science. 51:2, Part II, pp. 19-22.
27. Rasulova Z.D. (2014). Investigations of the essential spectrum of a model operator associated to a system of three particles on a lattice. J. Pure andApp. Math.: Adv. Appl., 11:1, 37-41.
28. Rasulova Z.D. (2014). On the spectrum of a three-particle model operator. Journal of Mathematical Sciences: Advances and Applications, 25, pp. 57-61.
29. Rasulov T.H., Rasulova Z.D. (2014). Essential and discrete spectrum of a three-particle lattice Hamiltonian with non-local potentials. Nanosystems: Physics, Chemistry, Mathematics, 5:3, pp. 327-342.
30. Расулов Т.Х., Расулова З.Д. (2015). Спектр одного трехчастичного модельного оператора на решетке с нелокальными потенциалами. Сибирские электронные математические известия. 12 (2015), С. 168-184.
