REGULAR STURM-LIOUVILLE OPERATORS WITH TRANSMISSION CONDITIONS AT FINITE INTERIOR DISCONTINUOUS POINTS

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1 ournl o Mthetl enes: dvnes nd ppltons Volue Nuer Pes REGULR TURM-LIOUVILLE OPERTOR WITH TRNMIION CONDITION T INITE INTERIOR DICONTINUOU POINT XIOLING HO nd IONG UN hool o Mthetl enes Inner Monol Unverst Hohhot P. R. Chn e-l: sun@u.edu.n strt In ths pper we nvestte lss o tur-louvlle proles wth trnssson ondtons t nte nteror dsontnuous ponts. We dene the l nd nl opertors ssoted wth ondtons t the nteror ponts n n pproprte Hlert spe. onstruton o the l nd nl opertors we estlsh neessr nd suent ondtons whh urntee seldjontness o the ordnr derentl opertors wth oupled oundr ondtons nd rel ntere ondtons t the nteror ponts. Introduton The tur-louvlle theor s ver portnt or solvn proles n thets phss. In ths pper we stud reulr tur-louvlle proles whh hve dsontnutes t nte nteror ponts. Ths reserh s otvted the theor o het nd ss trnser vred ssortent o phsl trnser proles [] nd eletrostt potentl Mthets ujet Clsston:. Kewords nd phrses: tur-louvlle opertors dsontnuous ponts trnssson ondtons sel-djontness. Ths work ws supported the Ntonl Nture ene oundton o Chn (868). Reeved Otoer 9 ent dvnes Pulshers

2 66 XIOLING HO nd IONG UN proles o pont hre []. To del wth nteror dsontnutes soe ondtons re posed on the dsontnuous ponts nd suh ondtons nvolve let nd rht lts o solutons nd ther qus-dervtves t the dsontnuous ponts whh re oten lled trnssson ondtons or ntere ondtons. ollown Mukhtrov nd Ykuov n [] we use n pproprte Hlert spe H wth n nner produt whh depends on the trnssson ondtons nd ve neessr nd suent ondtons or sel-djontness o LP s wth rel trnssson ondtons n H. We dene the l nd nl opertors LMT LT n eton whh depends on the trnssson ondtons t the nteror ponts o the ntervl ( ). It s the ke pont or solvn the prole. We prove tht the nl opertor L s setr nd L re n djont pr. Thus the hrtertons o sel-djontness o the prole s equvlent to onsdern ll sel-djont etensons o L T. Note tht our rel trnssson ondtons re not requred to e sel-djont.e. the oeent tres o trnssson ondtons re rtrr rel oupln tres whose deternnt s postve. Ths ontrsts wth the usul theor whh requres the oupln tr to hve deternnt one. wth. The Ml nd Mnl Opertors ssoted wth the Opertor T Consder the reulr setr derentl equton l : ( p ) q λ on I λ C (.) T T I ( ) ( ) ( ) < ( ) p q L ( I ( R) ); oundr ondtons Y Y ( ) Y ; p (.) nd ntere ondtons

3 REGULR TURM-LIOUVILLE OPERTOR 67 [] [ ] : l (.) [] [ ] : l (.) where [] s the -th qus-dervtves o j j ; ( j ) re ople tres suh tht the tr hs ull rnk; nd the oeents j j j re rel nuers stsn. > > ollown Mukhtrov nd Ykuov [] n ths pper we dene the slr nner produt n H s ollows: d d d d where I L. Then H s Hlert spe wth ths nner produt. Denton.. The l opertor M L enerted the derentl epresson l n H s dened s { [ ] [ ] lo lo M C C H L D [ ] } nd H l C lo. M M L D l L Denton.. The nl opertor L enerted l n H s dened s

4 68 XIOLING HO nd IONG UN { [ ] [ D L D L ] ( ) ( ) M [] [ ] [ ] ( ) } L l D( L ). Rerk. Here ( ) ( ) l ( ). Theore.5. n [7] these lts est nd re nte or n l don unton enerted l restrted to ( ) ( ) nd ( )( ). lrl ( ) ( ) nd [] ( )( ) dened rht lts est nd re nte. [ ] ( ) Denton.. We dene the opertor T enerted l n H s ollows: D ( T ) { D Y Y ( ) l l } M T l or D( T ). In order to nvestte the queston o sel-djontness or ll LP (.)- (.) we ntrodue the new l nd nl opertors ssoted wth T. Denton.. The l opertor dened s ollows: ssoted wth T s D ( L ) { D( L ) l l } MT M LMT l or D( LMT ). Denton.5. The nl opertor dened L T ssoted wth T s { [ ] [ D L D L ] ( ) } T MT LT l or D( LT ). Ovousl we hve L L T L L. T MT M

5 REGULR TURM-LIOUVILLE OPERTOR 69 Le.. The nl opertor L s losed nd dense-dened M M L setr derentl opertor nd L L L. propert o the djont opertor we hve L LMT T LT LM. (.5) or ever u v D( LMT ) we hve ( lu v) ( u lv) W ( u ) W ( u ) (.6) where W ( u ) u v [ ] u [ ] v. Hene or ever D( L ) nd v D we hve u T ( LT u v ) ( u LMTv ). (.7) It s ler tht opertor. T LMT L nd L T s dense-dened setr Theore.. or n ople nuers α β α there ests β u D suh tht [ ] [ u α u β u α u ] ( ). β Proo. Let nd e lnerl ndependent solutons o the equton ( p u ) q u ( ( )) wth the ntl-vlue [] [ nd ] respetvel. Let ( ) nd ( ) e lnerl ndependent solutons o the equton ( p u ) q u dened on the ntervl ( )( ) ( )( ) stsn ( ) [ [] ( )] [] [ [] ( )]

6 XIOLING HO nd IONG UN 7 nd [ [] ] [] [ [] ]. It s es to ver tht nd re lnerl ndependent. Put.. We see tht nd re n D nd the re lnerl ndependent solutons o. lu Let where sts the ollown ondtons: β. α (.8) ne nd re lnerl ndependent the Gr deternnt. Hene nd re unquel deterned.

7 REGULR TURM-LIOUVILLE OPERTOR 7 Let w D e the soluton o the ntl-vlue prole lw I w w [] ( ). In ter o (.8) we n otn tht w stses [ w w ] [] w α nd w ( ). In the se w we hve v D β suh tht v α [] nd [ ] v β v v ( ). Put u w v. Then u s the requred unton. Theore.. Let H. Then the equton l hs soluton v/ D( L T ) nd onl s orthoonl to ll solutons o l whh re n D ( ). Proo. Neesst. or H where ( ) ( ) l hs soluton v/ D( L T ) then or ever D( ) suh tht l we hve ( ) ( lv/ ) ( l v/ ) ( v/ L ) ( v/ L ). T T MT uen. Consder the ntl-vlue prole ( p ) q ( ) []. the theor o estene nd unqueness o solutons the prole hs unque soluton v/. lrl let v/ ( ) e the unque soluton o the ollown ntl-vlue prole;

8 7 XIOLING HO nd IONG UN ( p ) q ( ) nd ( ) [( )/ ( )/ [] v v ] [] [( ) ( ) [] ] v/ v/. v/ ( ) v/ ( ) Then v/ v/ ( ) s soluton o l stsn the trnssson ondtons (.) nd (.) nd v [ / v/ ]. Ovousl v/ D. Let D suh tht l. Then ( [] [] LMT v/ v/ LMT v/ v/ ( ) ( )). Note tht ( LMT v/ ) ( v/ LMT ) ( LMTv/ ) ( ). Thereore or eh D( ) suh tht l we hve [ ] [ v / v/ ] ( ) ( ). Theore. we hoose D suh tht ( ) ut [] ( ). o v /( ). lrl we et [ v / ] ( ). ove soe prelnr theores we hve the ollown theore: Theore.. The l nd nl opertors T MT L T LMT ssoted wth T re n djont pr n the sense tht L L nd L T. T Proo. () (.7) L L. In the ollown we prove tht. Let H MT T L nd or ever D ( L ) ( l ) ( ). Denote v/ D e soluton o l. Then or ever D( L T ) ro (.7) we hve T ( l ) ( ) ( lv/ ) ( l v/ ).

9 REGULR TURM-LIOUVILLE OPERTOR 7 Thereore ( l v/ ). ne D( L ) s rtrr the ove LT T equton shows tht v/ R. Theore. we otn v/ N. o v/ D.e. v/ D nd l lv/ l lv/. It ples tht LT LMT. Hene L T LMT. () L MT L T LMT LT LMT. Thereore MT s derentl opertor enerted l nd D( L ) D. or n u v D( LMT ) (.6) we know tht v D ( LMT ) nd onl or n u D ( u( ) v [] ( ) u [] ( ) v( )) ( u v [] u [] v). (.9) Theore. we hoose [ u u ] ( ) u nd u [] ndependentl. We ssert tht or ll u D (.9) holds nd onl v ( ) v [] ( ) v v [ ].e. v DL T. o D ( L MT ) D ( L T ).. Theore.. The djont opertor T o T stses L T T Proo. Ths ollows ro (.5) nd Theore... The Neessr nd uent Condtons or el-djontness Theore.. Let v D ( T ) nd onl v D nd or ever u D( T ). T denote the djont opertor o T. Then W ( u ) W ( u )

10 7 XIOLING HO nd IONG UN Proo. Let v D T. Theore. v D. or ever u D( T ) we hve ( ) ( T Tu v u v). Hene W ( u ) W ( u ). or the onverse v D( ) nd W ( u ) W ( u ) or eh u D( T ) then ( lu v) ( u lv).e. v D ( T ). Theore.. The opertor T s sel-djont nd onl D stses () D( T ) D; () or n u v D ( T ) W ( u ) w( u ) ; () I v D( ) nd the eqult W ( u ) w( u ) holds or n u D( T ) then v D( T ). Proo. The proo s slr to Theore.. We ot the detls. elow we wll ve the dretl nlt desrpton o the sel-djont oundr ondtons. or ternolo suh s oundr ors utull djont oundr ondtons et. see [5]. We use the notton to denote the ople onjute trnspose o tr. Theores. nd. we otn the net onluson whh spees the ondtons on the djont oundr ondtons o opertor T. Theore.. Let D nd V Y Y ( ) e twodensonl oundr ors. Then the oundr ondtons U Y () Y ( ) o T nd V re utull djont nd onl.

11 REGULR TURM-LIOUVILLE OPERTOR 75 Proo. et Y Y C then C Y Y U where rnk. Let e tres suh tht N s nonsnulr tr. Then Y Y U C re oundr ors whh re opleentr to the ornl oundr ors. U Thus. NC U U U Let [] Z then ; Y Z Z Y W where.. denotes the nner produt o ople Eulden spe. Put. Then. ; ; C N U C C W W Ovousl N s nonsnulr. et C N V V V where. ) ( Z Z V Z Z V Then we hve. N C V Hene N we hve

12 XIOLING HO nd IONG UN 76. E Thereore. E We otn. Conversel ssue tht there est tres nd suh tht nd rnk (.) rnk let denote the spe enerted two lnerl ndependent row vetors o. ro (.) we hve. Ths shows tht ll olun vetors o the tr R elon to. However. R It s ovous tht rnk R. rnk o two lnerl ndependent olun vetors o R or o se o. lrl we hve rnk rnk R nd ts two olun vetors onsst o se o the spe. Thereore there ests

13 REGULR TURM-LIOUVILLE OPERTOR 77 nonsnulr tr P suh tht RP R.e. ( ) P. o ( ) P( ). We hve P nd P. Hene V Z( ) Z( ) PV nd V re the djont oundr ondtons o U. Le.. The derentl opertor T s sel-djont nd onl U Y Y ( ) re sel-djont oundr ondtons. The net theore s our n result n ths seton; t ves the suent nd neessr ondton on the oundr ondtons or seldjontness o the opertor T. Theore.. The opertor T s sel-djont nd onl. Proo. ro Le. nd Theore. we n otn the dret onluson. Reerenes [] D. ushnn G. tol nd. Wednn One-densonl hrod ner wth lol pont ntertons. Rene new. Mth. 67 (995) []. V. Lkov nd Y.. Mkhllov The Theor o Het nd Mss Trnser Qosenerdt 96 (Russn). [] C.. Londs nd D. P. Chrssoulds Two-ntervl tur-louvlle theor or the eletrostt potentl o pont hre ner deletr one. ppl. Mth. 9 (to pper). [] O. h. Mukhtrov nd. Ykuov Proles or derentl equtons wth trnssson ondtons pplle nlss 8 () -6. [5] M.. Nrk Lner Derentl Opertors Prt II Hrrp London 968. [6]. P. Wn. un nd. Zettl Two-ntervl tur-louvlle opertors n oded Hlert spes. Mth. nl. ppl. 8 (7) [7]. Zettl tur-louvlle Theor Mthetl urves nd Monorphs Provdene R.I.: er. Mth. o. (5).