Panjaradagi bir zarrachali Schr¨odinger operatori spektri va unga mos giperbolalar oilasi
![Kirish
Masalaning qo'yilishi. Bir o'lchamliZpanjarada ^ v
; ;
2R;
potensial yordamida tasirlashuvchi ikki zarrachali sistema energiya oper-
atoriga mos diskret b
H (
k ) = b
H 0(
k ) + b
V Schrodinger operatorlarining
keng sin ni tadqiq qilish. Bir o'lchamli panjaradagi ikki zarrachali b
H (
k ) Schrodinger operatori
tashqi ta'sir energiyasi ; 2R parametrlar orqali berilganda har bir
k 2 T uchun uning
ess (b
H (
k )) muhim spektridan tashqaridagi xos qiy-
matlarining aniq sonini va joylashgan o'rnini topishga doir teoremalarni
isbotlash. Mavzuning dolzarbligi. Ikki zarrachali diskret Schrodinger operator-
larining spektral xossalari keyingi yillarda faol o'rganilmoqda (masalan [1],
[7], [8], [14], [21]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 ([3]-[5]).
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 [20] 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 bir o'lchamli panjaradagi ikki zarrachali b
H (
k ) Schrodinger
operatori tashqi ta'sir energiyasi ; 2R parametrlar orqali berilganda
2](https://docx.uz/documents/48ef83b5-44e3-43df-b717-735f993818a2/page_2.png?v=1)
![Teorema 1.7.2.
kA 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 1.7.3. (A ) [m; M ]. Shuningdek, m; M2(A ) .
Natija 1.7.1. Har qanday chegaralangan o'z-o'ziga qo'shma operatorning
spektri bo'sh emas.
Teorema 1.7.4. O'z-o'ziga qo'shma operatorning har xil xos qiymatlariga
mos keluvchi xos vektorlari ortogonal.
Teorema 1.7.5. Ao'z-o'ziga qo'shma operator bo'lsin. zsoni Aoperator
uchun xos qiymat bo'lishi uchun R
(A](https://docx.uz/documents/48ef83b5-44e3-43df-b717-735f993818a2/page_34.png?v=1)
![BOB 3
H (
k ) operatorning xos qiymatlari
soni va ularning joylashgan o'rni
3.1 H
(
k ); k 2T operatorning muhim spektri.
H 0(
k ); k 2T operator Tda aniqlangan uzluksiz haqiqiy qiymatli E
k(
p )
funksiyaga ko'paytirish operatori bo'lganligi uchun uning spektri faqat
muhim spektrdan iborat bo'ladi:
(H
0(
k )) =
ess (
H
0(
k )) = [ E
min (
k ); E
max (
k )] ; (3.1.1)
bu yerda Emin (
k ) := min
q2 T d E
k(
q ) = 2(1](https://docx.uz/documents/48ef83b5-44e3-43df-b717-735f993818a2/page_45.png?v=1)
![Proof.Zaruriyligi.
Faraz qilaylik, ixtiyoriy ; 2R va k2 T uchun
z 2 Cn[E
min (
k ); E
max (
k )] soni H
(
k ) operatorning xos qiymati bo'lsin,
ya'ni
H (
k )f = zf
tenglama L2
;e
(T ) fazoda noldan farqli yechimga ega bo'lsin.
U holda quyidagi munosabatlar o'rinli:
"k(
p )f (p ) + 1 2
Z
T(
+ cos pcos q)f (q )dq =zf (p );
( "
k(
p )](https://docx.uz/documents/48ef83b5-44e3-43df-b717-735f993818a2/page_48.png?v=1)
![FOYDALANILGAN
ADABIYOTLAR
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62](https://docx.uz/documents/48ef83b5-44e3-43df-b717-735f993818a2/page_62.png?v=1)
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63](https://docx.uz/documents/48ef83b5-44e3-43df-b717-735f993818a2/page_63.png?v=1)
Panjaradagi bir zarrachali Schr¨odinger operatori spektri va unga mos giperbolalar oilasi
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