Hilbert fazolarida o‘z-o‘ziga qo‘shma
![Kirish
Masalaning qo'yilishi. d-o'lchamli Zd
; d 3; kubik panjarada ^ v
tashqi maydonidagi ikkita zarrachalarning o'zaro ta'sirini tasvirlovchi b
H b
V (
k ) =
b
H 0(
k ) + b
V Schrodinger operatorlarning keng sin ni tadqiq qilish. Ikki
zarrachali b
H b
V (0) Schrodinger operatorining xos qiymatlarini mavjudligi
yoki yo'qligi ^ vpotentsialga va Zd
; d 3; panjaraning o'lchamiga bog'liq
holda o'rganish hamda b
H b
V (0) Schrodinger operatorining
ess (b
H b
V (0)) muhim
spektridan quyida xos qiymatlari mavjudligi yoki yo'qligi uchun teore-
malarni isbotlash. Mavzuning dolzarbligi. Ikki zarrachali diskret Schrodinger operator-
larining spektral xossalari keyingi yillarda faol o'rganilmoqda (masalan [1]{
[5] va [6]{[11] larni qarang).
Uzluksiz Schrodinger operatori muhim spektrning chapida yotuvchi disk-
ret spektri (xos qiymatlari)ning mavjudligi faqat zarrachalar ta'sir ener-
giyasi b
V ; > 0 ga bog'liq bo'ladi ([6],[7],[8],[13]).
Ammo, ikki zarrachali diskret Schrodinger operatorlari uchun xos qiy-
matlarning mavjudligi va soni faqatgina ta'sir potentsiali b
V ga bog'liq
bo'lmasdan balki sistema kvazi-impulsiga ham bog'liqdir. O'lchami d 3 bo'lgan panjarada ikkita ixtiyoriy zarrachali sistemaga
mos diskret Schrodinger operator b
H (k ) ning diskret spektri mavjudligi sis-
tema kvazi-impulsining barcha qiymatlari uchun [1] da o'rganilgan. Uch o'lchamli panjaradagi Schrodinger operatorlari uchun xos qiymat-
larning mavjudligi va xos qiymatlar o'rnining kontakt va qo'shni tugun-
larda juft-jufti bilan o'zaro ta'sirlashuvchi potentsialga bog'liqligi [10] da
keltirilgan. Schrodinger operatori b
H (k ) ning xos qiymatlari soni uchun chegaralar
[3] va [4] maqolalarda keltirilgan. Xususan, b
H (k ) operatorning muhim
spektridan chapda xos qiymatlarning mavjudligi ko'rsatilgan.
Dissertatsiya ishining maqsadi va vazifalari. Ushbu magistrlik dis-
sertatsiya ishida d-o'lchamli Zd
; d 3; kubik panjarada ^ vtashqi may-
donidagi ikkita zarrachalarning o'zaro ta'sirini tasvirlovchi b
H b
V (
k ) = b
H 0(
k )+ b
V Schrodinger operatorining muhim spektri o'rni, Zd
; d 2](https://docx.uz/documents/7807a716-6f8a-4ffc-85cb-f7fc8bf24c89/page_2.png?v=1)
![Teorema 1.10.1. k
A k = max fjm j; jM jg.
Ma'lumki, (A ) kA k radiusli doira ichida saqlanar edi. O'z-o'ziga qo'shma
operatorlar uchun esa bu baholash yanada aniqroq. Teorema 1.10.2.
(A ) [m; M ]. Shuningdek, m; M2(A ) . Natija 1.10.1. Har qanday chegaralangan o'z-o'ziga qo'shma operatorning
spektri bo'sh emas. Teorema 1.10.3. O'z-o'ziga qo'shma operatorning har xil xos qiymatlariga
mos keluvchi xos vektorlari ortogonal. Teorema 1.10.4. A
o'z-o'ziga qo'shma operator bo'lsin. zsoni Aoper-
ator uchun xos qiymat bo'lishi uchun R
(A](https://docx.uz/documents/7807a716-6f8a-4ffc-85cb-f7fc8bf24c89/page_34.png?v=1)
![BOB 3
b
H
b
V (
k ) Schrodinger operatorining
spektral xossalari
b
H b
V (
k ) operatorning muhim spektri
H 0(
k ) k2 Td
operator, Td
da aniqlangan uzliksiz haqiqiy qiymatli
E k(
) funksiyaga ko'paytirish operatori bo'lganligi uchun H
0(
k ) operatorn-
ing spektri faqat muhim spektrdan iborat bo'ladi va quyidagi munosabatlar
o'rinli bo'ladi:
(H
0(
k )) =
ess (
H
0(
k )) = [ E
min (
k ); E
max (
k )] ; (3.0.1)
bu yerda
Emin (
k ) min
p 2 Td E
k(
p ); E
max (
k ) max
p 2 Td E
k(
p ):
H 0(
k ) operator b
H 0(
k ) operatorga unitar ekvivalent operator bo'lganligi
uchun ( b
H 0(
k )) = (H
0(
k ))
bo'ladi. b
H 0(
k ) operator o'z-o'ziga qo'shma operator va qo'zg'alish operatori b
V
kompakt operator ekanligidan muhim spektri turg'unliga haqidagi Weyl
teoremasiga ko'ra b
H b
V (
k ) operatorning
ess (b
H b
V (
k )) muhim spektri b
H 0(
k )
operatorning
ess (b
H 0(
k )) muhim spektri bilan ustma-ust tushadi, ya'ni
ess (b
H b
V (
k )) =
ess (b
H 0(
k )) = (H
0(
k )) : (3.0.2)
Min-max prinsipiga ko'ra va ^ v 0 bo'lganligi uchun b
H b
V operatorn-
ing barcha chekli karrali xos qiymatlari
ess (b
H b
V ) muhim spektrning
E
min
bo'sag'asidan quyida yotadi.
Biz shuningdek potensialning ishorasini o'zgarganda Rd
dagi Schrodinger
operatorlardan farqli o'laroq, diskret Schrodinger operatorlarining muhim
spektrdan o'ngda yotuvchi xos qiymatlari mavjud bo'lishini ta'kidlaymiz. 45](https://docx.uz/documents/7807a716-6f8a-4ffc-85cb-f7fc8bf24c89/page_45.png?v=1)
![ya'ni
^
f 2 `
0(
Z d
): bo'ladi.
^
2`2
(Z d
) funksiya
[
b
V
1 2
b
R 0(
E
min (0))
b
V
1 2
] ^
= ^
: (3.1.26)
tenglikning yechimi bo'lsin. ^
f = b
R 0(
E
min (0))
b
V
1 2
^
(3.1.27)
belgilash orqali biz ^
f 2 `
0(
Z d
) egamiz. ( ??) tenglama
b
V
1 2
^
f = ^
;
b
V
^
f =
b
V
1 2
^
: (3.1.28)
tengliklarni keltirib chiqaradi. ( ??) va ( ??) tengliklar ko'rsatadiki,
b
H 0(0)](https://docx.uz/documents/7807a716-6f8a-4ffc-85cb-f7fc8bf24c89/page_53.png?v=1)
![FOYDALANILGAN
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The Threshold E ects
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Number of Eigenvalues of the
Three-Particle Schrodinger Operators on Lattices, 18(2012), 18pp. 387-420 [3] V. Bach, W. de Siqueira Pedra, S. Lakaev
:Bounds on the Discrete Spectrum of
Lattice Schrodinger Operators Preprint mp-arc 10-143. [4] V. Bach, W. de Siqueira Pedra, S. Lakaev:
Bounds on the Pure Point Spectrum of
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Energy-momentum spectrum
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Phys. Theor. 67, 91{107 (1997). [6] Klaus M.
andB.Simon: Coupling constants thresholds in non-relativistic quantum me-
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andB. Simon: Coupling constant thresholds in nonrelativistic quantum me-
chanics. I. Short-range Two-body case. Annals of Physics, 108,251-281(1977). [8] S.N.Lakaev:
The E mov e ect in a system of three identical quantum particles. Funct.
Anal. Appl. 27, 166{175 (1993). [9] S.N.Lakaev, Sh.M. Tilavova: Merging of eigenvalues and resonances of a two-particle
Schrodinger operator. Theoret.and Math. Phys., 101,1320 -1331,(1994) [10] S. N. Lakaev, I.N. Bozorov:
The number of bound states of a one-particle Hamiltonian
on a three-dimensional lattice, Theoretical and Mathematical Physics, 1582009, pp. 360-
376. [11] A.A.Pankov:
Lecture notice operators on Hilbert space. New York: Nova Science Publish-
ers, 2007. [12] M.Reed
andB.Simon: Methods of modern mathematical physics. IV: Analysis of Oper-
ators, Academic Press, New York, 1979. [13] S.N.Lakaev, S.X.Abduxakimov: Panjaradagi ikki fermionli sistemaga mos diskret
Schrodinger operatori xos qiymatlari mavjudligi, SamDU ilmiy axborotnomasi.3, (2017). [14] S.N.Lakaev, I.N.Bozorov: The number of bound states of a one-particle Hamiltonian on a
three-dimensional lattice, Theoretical and Mathematical Physics, . 158, 2009, pp.360-376. 59](https://docx.uz/documents/7807a716-6f8a-4ffc-85cb-f7fc8bf24c89/page_59.png?v=1)
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The low energy expansion in nonrelativistic
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A class of exactly solvable three-body quantum
mechanical problems and universal low energy behavior // Phys. Lett. A.83:1, 25{27 (1971). [19] S. Albeverio, S.N. Lakaev, Z.I. Muminov.
Schrodinger operators on lattices. The E mov
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Number of eigenvalues of the three-particle
Schrodinger operators on lattices // Markov Proc. Relat. Fields.18:3, 387{420 (2012). [21] V. Bach, W. de Siqueira Pedra, S.N. Lakaev.
Bounds on the discrete spectrum of lattice
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Energy-momentum spectrum of some two-
particle Hamiltonians // Phys. Rev. E.66:3, 016130 (2002). [23] M. Klaus.
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The number of bound states of a one-particle Hamiltonian on a
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The few-body problem on a lattice // Rev. Modern Phys.58, 361{379 (1986). [31] R.A. Minlos, Y.M. Suhov.
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Hamiltonians in solid state physics as multi-particle discrete Schrodinger op-
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Number of bound states of three-particle systems and E -
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![[34] M. Reed, B. Simon.
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(1968). 61](https://docx.uz/documents/7807a716-6f8a-4ffc-85cb-f7fc8bf24c89/page_61.png?v=1)
Hilbert fazolarida o‘z-o‘ziga qo‘shma