MAVZU: PANJARADAGI IKKI ZARRACHALI SHREDINGER OPERATORIGA MOS MODEL OPERATOR SPEKTRI

Masalaning qo’yilishi – butun sonlar to‘plami,   ) (2       da aniqlangan kvadrati bilan jamlanuvchi funksiyalarning Hilbert fazosi bo‘lsin. Bir o‘lchamli panjaradagi ikkita ixtiyoriy kvant zarralari sistema gamiltoniani hˆ ga mos, model operatori koordinata ko‘rinishida ,   ) (2       Hilbert fazosida chegaralangan o‘z -o‘ziga qo‘shma operator sifatida quyidagicha aniqlanadi: , ˆ ˆ = ˆ 0   v h h  bunda )], , (ˆ) ( ˆ ) , (ˆ) (ˆ[ =) , )(ˆ ˆ( 2 1 2 2 1 1 2 1 0 s n n s n s n s n n h s           Z ). , (ˆ ) ( ˆ =) , )(ˆ ˆ( 2 1 2 1 2 1 n n n n v n n v     

Masalaning qo’yilishi Bu yerda )( ˆ1   ,  )( ˆ2 bir o ‘ lchamli panjaradagi zarrachaning bir tugundan boshqa ( ba ’ zi qo ‘ shni ) tugunlariga ko ‘ chishini tavsiflovchi dispersion funksiyalar va  )( ˆv  da aniqlangan zarrachalarning o ‘ zaro ta ’ sir potensiali bo ‘ lib , ular quyidagi formulalar bilan aniqlanadi            , hollarda boshqa 0, 2, = agar , 2 1 0, = agar , 1 =) (ˆ s m s m s i i i va      , hollarda boshqa 0, 2, = agar , 0, = agar , 2 =) ( ˆ 1 0 s s s v    bunda  0 >i m i - zarrachaning massasi, 1,2 =i va 0 0   , 0 1  .

Masalaning qo’yilishi,] ; ( =    ) (2    L –    da aniqlangan kvadrati bilan integrallanuvchi funksiyalarning Hilbert fazosi bo‘lsin. Bu fazoda h operatorning spektral xossalarini keltiramiz , ) ( =) ( 0   v  k h k h bunda )( 0 kh )( k  funksiyaga ko‘paytirish operatori: 2 2 2 1 2 1 2 1 1 2 cos 2 1 =) ( , 2 cos) ( 1 1 =) ( m k m m m k a p k a m m p k      va  v yadrosi ) 2( cos =) ( 1 0 s p s p v       ko‘rinishga ega bo‘lgan integral operator, ya’ni   ). ( , ) ( ) 2( cos =) )( ( 2 1 0   L f ds s f s p p f       v

Masalaning qo’yilishi Masalaning qo’yilishiTa ’kidlab o‘tish joizki , muhim spektr turg ’unligi haqidagi Ve yl teoremasiga ko ‘ra [13] , ) (k h operatorning spektri )) ( ( k h ess   ,  v kompakt qo ‘zg ‘alishlarda qo ‘zg ‘almas ) (0 k h operatorning spektri bilan ustma -ust tushadi . Bundan quyidagi tenglik o‘rinli bo‘ladi )], ( ), ( [ = )) ( ( = )) ( ( 0 k M k m k h k h ess    bunda ). ( 1 1 =) ( max =) ( ), ( 1 1 =) ( min =) ( 2 1 2 1 k a m m p k M k a m m p k m k p k p          