Anotaciones de Un Curso de Física Del Estado Sólido

Rgfvbrsfhmh Gmkfegmc hb Bhukmkfûg m Hfstmgkfm. Amkuctmh hb Kfbgkfms. Hbpmrtm`bgte hb Aàsfkm Aughm`bgtmc.

Jrmhe bg Aàsfkm

Mgetmkfegbs hb ug kurse hb

AÀ[FKM HBC B[XMHE [ÛCFHE

Justmve Mhecae ^ærbz [ígkobz.

Kurse <>=2/<>=1

Mc bqufpe hekbgtb. êr tmg jrmtm y pmrtfkucmr bxkbpkfûg.

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«Gugkm sulbstf`bs bc pcmkbr qub cb preperkfegms mc pôlcfke kegtíghecbs mcje qub ym ke`prbghbg¹

A.Lceko.

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Àghfkb jbgbrmc jbgbrmc [elrb bstms Getms

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= Cm Bstrukturm Bstrukturm ^brfûhf ^brfûhfkm km hb ces Krfstm Krfstmcbs cbs

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< Hfa Hfarmk rmkkfû kfûg g y cm \bh \bkàpr \bkàprekm ekm

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; Bcbk Bcbktregb tregbss bg ug êtbgkfmc êtbgkfmc ^brfûhfke ^brfûhfke.. Xberb` Xberb`mm hb Lcek Lceko o

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8 Bc @ehbce @ehbce hb Hruhb-[e`` Hruhb-[e``brabc brabch h hb ces @btmcbs @btmcbs

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2 Lmghm Lmghmss hb Bgbrj Bgbrjàm. àm. Bcbktregbs Bcbktregbs Kumsfcflrb Kumsfcflrbss y Aubrtb ubrtb`bg `bgtb tb Cfjmhes

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1 Hfgí` Hfgí`fkm fkm hb ces ces Bcbktre Bcbktregbs gbs hb Lceko Lceko

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: Hfg Hfgí`f í`fkm km hb hb cm cm \bh \bh

? ^repf ^repfbhmhb bhmhbss Xær`fk Xær`fkms ms hb ces [ûcfhe [ûcfhess

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6 Hba Hbabkt bktes es ûgtum ûgtumcbs cbs hb cm cm \bh

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=> Kmrmktbràstfkms Aughm`bgtmcbs hb ces @mtbrfmcbs [b`fkeghukterbs y @mjgætfkes

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M Kegs Kegstmg tmgtbs tbs Aàsfkms Aàsfkms

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L Xmlcm ^brfûhfkm ^brfûhfkm hb ces Bcb`bgtes Bcb`bgtes y ^mrí ^mrí`btr `btres es hb Fgt Fgtbræs bræs

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\babrbgkfms

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Àghfkb jbgbrmc

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[elrb bstms Getms Mgetmkfegbs hb ug Kurse hb Aàsfkm hbc Bstmhe [ûcfhe, qufzís bsb sbm bc tàtuce `ís mkbrtmhe qub vfstb mc tbxte m kegtfgumkfûg. Rgms getms, uges mpugtbs sf sb qufbrb, ug rbsu`bg hb ces kegtbgfhes hb cm msfjgmturm Aàsfkm hbc Bstmhe [ûcfhe, msfjgmturm qub sb f`pmrtb bg bc kumrte kurse hbc jrmhe bg aàsfkm hb cm Rgfvbrsfhmh Gmkfegmc hb Bhukmkfûg m Hfstmgkfm. Kegstm hb hfbz kmpàtuces e tb`ms bg ces qub sb om fgtbgtmhe pcms`mr gbjre selrb lcmgke ce qub bg tmg sûce cm ou`fchb epfgfûg hbc muter, supegb ug kegtbgfhe hb rbcbvmgkfm qub ge hblb sbr pmsmhe per mcte, mc `bges kcmre, sf cm fgtbgkfûg hbc cbkter, pubhb qub ug bstuhfmgtb, sbm cm hb mhqufrfr ug `àgfè hb kegekf`fbgtes rbspbkte hb bstm rm`m hb cm aàsfkm. Hfbz kmpàtuces tmc y keè bstí bstrukturmhm cm msfjgmturm f`pmrtfhm, ke`bgzmghe keg cm bstrukturm pbrfûhfkm pbrfû hfkm hb ces krfstmcbs, pfcmr aughm`bgtmc qub sestfbgb bg jrmg `bhfhm bc rbste hb kegekf`fbgtes, pmsmghe per bc bstuhfe hb cm hfarmkkfûg hb rmyes _, cm tberàm hb lmghms (pesflcb`bgtb bc `myer cejre hb cm aàsfkm hbc bstmhe sûcfhe), cms vflrmkfegbs hb cm rbh krfstmcfgm, cms prepfbhmhbs tær`fkms y bcæktrfkms hb ces sûcfhes, y tbr`fgmghe keg ces hbabktes pugtumcbs hb ces krfstmcbs y cms prepfbhmhbs aughm`bgtmcbs hb ces krfstmcbs sb`fkeghukterbs, y hbc `mjgbtfsè bg ces sûcfhes. Bc cbkter hblb tbgbr bg kubgtm, qub ge sfbghe cms prbsbgtbs `ís qub ug rbsu`bg, y mugqub bg cm `bhfhm hb ce pesflcb om sfhe kegkblfhe keg ug `àgfè hb bxpcfkmkfegbs y hb hbsmrrecces `mtb`ítfkes, ge bs hb gfgjugm `mgbrm ug tbxte qub pubhm sbrvfr pmrm bc bstuhfe ke`pcbte y preaughe hb ce qub sb trmtm mquà, pmrm bcce, sb om fgkcufhe mc gmc bg cms rbabrbgkfms, ugm cfstm hb elrms hb cbkturm kmsf elcfjmhm, qub bg lubgm `bhfhm, omg hmhe m cuz m bstb... hfjmès `mgumc hb rbabrbgkfm, kmdûg hb smstrb hbc qub qufzís ugfkm`bgtb pehrí smkmrsb pmrtfhe, mkuhfbghe m æc keg uges kegekf`fbgtes sukfbgtbs y msbgtmhes hb aàsfkm bg jbgbrmc y hb aàsfkm hbc bstmhe sûcfhe bg pmrtfkucmr. Bstms getms, hb ug èhe bxkbpkfegmc, bspbrmg ku`pcfr keg bc rbqubrf`fbgte pmrm ug g, ug g qub tmc vbz per p er mgtedmhms, cms mlfjmrrmhms sbghms hbc hbstfge omg qubrfhe kegvbrtfr bg hes, cm supbrmkfûg hb cm msfjgmturm bg kubstfûg, y keg bcce cm kuc`fgmkfûg hbc pcmg hb bstuhfes. ûbs qub msà sbm. Ce kfbrte bs tm`lfæg, qub hurmgtb su krbmkfûg, cms prbsbgtbs sb omg fhe kegvfrtfbghe bg ug pcmkbr amskfgmgtb, pcmkbr qub qufzís sûce pubhbg bgtbghbr y sbjurm`bgtb sbgtfr, bses kurfeses hbc `bkmgfsè ôctfè hb Hfes. Edmcí sfrvm bstb tbxte m su ke`btfhe y sbm rbkflfhe keg mkbptmkfûg per bc cbkter, sf bs msà, lubgm pmrtb hbc eldbtfve bstmrí ku`pcfhe.

RGBH.

Mgetmkfegbs hb ug kurse hb Aàsfkm hbc Bstmhe [ûcfhe.

= Cm Bstrukturm ^brfûhfkm hb ces Krfstmcbs

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= Cm Bstru Bstruktu kturm rm ^br ^brfûhfk fûhfkmm hb ces ces Krfstmc Krfstmcbs bs Xehe sûcfhe krfstmcfge, e krfstmc sfg `ís, sb bgkubgtrm bg cm gmturmcbzm aer`mhe per lcequbs fhægtfkes rbpbtfhes hb aer`m pbrfûhfkm (rbh) , tmcbs lcequbs bstíg kegaer`mhes per íteès e jrupes hb íteès hfspubstes bg ugm bstrukturm trfhf`bgsfegmc pbrfûhfkm hb cmrje mckmgkb. =

Rgm rbh krfstmcfg krfstmcfgmm bs per tmgte tmgte ug kegdugte kegdugte hb pugtes pugtes hfskrbtes hfskrbtes hbc bspmkfe keg vbkter vbkter pesfkfûg hmhe per4 0 g = m= + g + g < m< + g + g ; m; , r 0 g (=.=) heghb ces gf lmrrbg tehes ces bgtbres y ces m ces mf seg ces ccm`mhes vbkterbs ccm`mhes vbkterbs lmsb prf`ftfves. prf`ftfves. Hbgetmrbès per \ m tehe vbkter qub ugb kumcbsqufbrm hes pugtes hb rbh. Bc kegdugte hb íteès qub sb msekfm m tehes y kmhm uge hb ces pugtes hb rbh bs kegekfhe keè lmsb mtû`fkm (m vbkbs tm`lfæg ètfve), msà, tehm bstrukturm krfstmcfgm bs cm ke`lfgmkfûg hb rbh krfstmcfgm + lmsb mtû`fkm.

Krfstmc lfhf`bgsfeg lfhf`bgsfegmc mc hbskrfte hbskrfte keè rbh krfstmcfgm `ís lmsb mtû`fkm. Aubgtb4 Aubgtb4 Y2 Y2P. Afjurm =.=4 Krfstmc

-Kbchm ugftmrfm y kbchm prf`ftfvm4 Bs prf`ftfvm4 Bs bc vecu`bg hb bspmkfe qub per trmscmkfegbs bg vbkterbs hb rbh kulrb tehe bc bspmkfe y sfg secmpm`fbgtes. [b ccm`m kbchm ugftmrfm prf`ftfvm sf mhb`ís kegtfbgb tmg sûce ug pugte hb rbh, m vbkb bs ôtfc hbgfr kbchms ge prf`ftfvms ccm`mhms kbchms kegvbgkfegmcbs qub prbsbrvbg cms sf`btràms hb cm rbh, sf`btràm trmscmkfegmc, retmkfegmc, bspbkucmr, hb fgvbrsûg, btk. Bs bvfhbgtb qub cm kbchm prf`ftfvm bgkfbrrm bc `bger vecu`bg pesflcb. -Kbchm hb Zfjgbr-[bftz4 Rgm Zfjgbr-[bftz4 Rgm `mgbrm hb bskejbr ugm kbchm prf`ftfvm prf`ftfvm kegsfstb bg trmzmr càgbms qub ugmg ug pugte hb rbh keg sus vbkfges `ís prûxfès, prûxfès, pestbrfer`bgtb trmzmr ces pcmges `bhfmtfkbs m hfkoms càgbms. Cm rbjfûg bgkbrrmhm per ces pcmges ( ug pecfbhre bg ;H e ug pecàjege bg
Afjurm =.<4 Kbchm ugftmrfm prf`ftfvm hb ugm rbh krfstmcfgm trfhf`bgsfegmc. Kmhm pugte hb rbh bs ke`pmrtfhe per ? kbchms, omy per tmgte ug pugte hb rbh per kbchm. Aubgtb4 ottp4//www.upv.bs =

Ge tbgbès bg kubgtm mquà kfbrtes sûcfhes ccm`mhes mèraes keè per bdb`pce bc vfhrfe, kuym bstrukturm, sf lfbg prbsbgtm kfbrte erhbg, æstb bs tmg sûce hb kerte mckmgkb.

RGBH.



;

Afjurm =.;4 Bdb`pces hb kbchms hb Zfjgbr-[bftz, lfhf`bgsfegmc y trfhf`bgsfegmc. Aubgtb4 Zfifpbhfm -\bhbs hb Lrmvmfs4 Bxfstbg =8 tfpes hb rbhbs mtbghfbghe m sus sf`btràms, seg cms ccm`mhms rbhbs hb Lrmvmfs èstrmhms bg cm jurm =.8 jurm =.8 . . Bstms rbhbs sb mjrupmg m su vbz bg : sfstb`ms krfstmcfges.

ugftmrfms hb cms =8 rbhbs hb Lrmvmfs Lrmvmfs pesflcbs trfhf`bgsfegmcbs. trfhf`bgsfegmcbs. Afjurm =.84 [fstb`ms krfstmcfges y kbchms ugftmrfms Cm btfqubtm btfqubtm ^, sb kerrbspeghb kerrbspeghb keg kbchm prf`ftfvm, prf`ftfvm, cms F, A y K seg kbchms kegvbgkfegmcbs, cm kbgtrmhm bg bc kubrpe, cm kbgtrmhm bg cms kmrms y cm kbgtrmhm bg hes kmrms rbspbktfvm`bgtb. Aubgtb4 scfhbsomrb.gbt

-^mrí`btre hb rbh4 [b rbh4 [b ge`lrm msà m cm cegjftuh «m¹ hb cm mrfstm hb cm kbchm kegvbgkfegmc (e prf`ftfvm) kôlfkm (ôgfkm`bgtb bg sfstb`ms kôlfkes heghb bs m 0 l 0 k. Afjurm =.8 Afjurm =.8). ). -Àghfkb hb keerhfgmkfûg FK4 Bs FK4 Bs bc gô`bre hb vbkfges `ís prûxfès m kmhm pugte hb rbh. Keè pubhb aíkfc`bgtb ke`prelmrsb, per bdb`pce bg cm rbh kôlfkm sf`pcb e sk, bstb gô`bre bs fjumc m 1, bg cm kôlfkm kbgtrmhm bg bc kubrpe e lkk bs ?, bg cm kôlfkm kbgtrmhm bg cms kmrms e akk bs =<, y bg cm rbh obxmjegmc ke`pmktm û okp bs=<. -Hfrbkkfegbs y pcmges krfstmcfges4 Xehe krfstmcfges4 Xehe pugte hb ugm rbh qubhm hbtbr`fgmhe per cms ; keerhbgmhms (x , y , z) rbabrfhms mc sfstb`m hb bdbs qub hbgbg cm lmsb vbkterfmc bcbjfhm. Bs f`pertmgtb cm aer`m hb hbsfjgmr hfrbkkfegbs e pcmges hbgtre hb ug krfstmc, perqub ùkoms hb cms prepfbhmhbs hb ces sûcfhes krfstmcfges hbpbghbg hbc pcmge e hfrbkkfûg qub sb kegsfhbrb. êr bcce, rbsuctm hb bspbkfmc fgtbræs bgkegtrmr ugm aer`m kûèhm y rípfhm hb fhbgtfkmr cms hfrbkkfegbs y pcmges krfstmcfges. Cm getmkfûg b`pcbmhm sb hbge`fgm getmkfûg hb àghfkbs hb @fccbr. RGBH.



8

Cms tbrgms Yo i cP hb ke`pegbgtbs bgtbrms `àgf`ms rbabrfhms m cm lmsb bcbjfhm, hbgbg ug vbkter bg ugm hfrbkkfûg krfstmcfgm hmhm, tmcbs ke`pegbgtbs seg ces ccm`mhes àghfkbs hb @fccbr hbc krfstmc, per kegvbgfe, kegvbgfe, ces vmcerbs gbjmtfves seg hbsfjgmhes kecekmghe mc àghfkb bg kubstfûg ugm càgbm selrb bc `fsè. Rg pcmge krfstmcfge bs hbgetmhe per àghfkbs hb @fccbr bg cm aer`m (o i c) y tmc pcmge bs pbrp bghfkucmr m cm hfrbkkfûg Yo i cP. Cm `mgbrm hb hbtbr`fgmr ces àghfkbs hb @fccbr hb ug pcmge kumcqufbrm, kegsfstb bg bgkegtrmr ces rbkàprekes hb cms fgtbrsbkkfegbs fgtbrsbkkfegbs hbc pcmge keg ces bdbs keerhbgmhes hb cm rbh y rbhukfrces m bgtbres `àgfès.

Afjurm =.24 Mcjuges pcmges krfstmcfges y sus àghfkbs hb @fccbr hb ugm rbh kôlfkm sf`pcb (sk). [ûce sb ùbstrm ùbstrm cm fgtbrsbkk fgtbrsbkkfûg fûg hb ces pcmges keg cm kbchm prf`ftfvm. prf`ftfvm. Aubgtb4 ubgtb4 Zfifpbhfm

-B`pmqubtm`fbgte -B`pmqubtm`fbgte Ke`pmkte4 Cms Ke`pmkte4 Cms bstrukturms krfstmcfgms ègebspbkfb hb `myer ke`pmkfhmh, bste bs, mqubccms qub mprevbkomg mc `íxfè bc bspmkfe hfspegflcb, seg cm akk y cm okp < . Bste pubhb vbrsb aíkfc`bgtb bg cm jurm =.1 jurm =.1 , , bg cm qub sb mpfcmg, bg kmpms hb cm aer`m `ís ke`pmktm pesflcb, bsabrms ràjfhms m èhe hb pugtes hb rbh. < Auæ Jmuss qufbg hbèstrû `mtb`ítfkm`bgtb bstb obkoe `bhfmgtb bc mpfcm`fbgte hb bsabrms ràjfhms, ∔ eltuve bc vmcer ώ/; < ≏ > , :8.

RGBH.



2

Afjurm =.14 B`pmqubtm`fbgte ke`pmkte hb bsabrms ràjfhms. Bg m ) ugm kmpm M hb bsabrms sb mkeèhmg hb cm aer`m `ís ke`pmktm pesflcb. Bg l ) ugm sbjughm kmpm L hb bsabrms ekupmg cm `ftmh hb ces fgtbrstfkfes hbdmhes per cm kmpm M. Rgm tbrkbrm kmpm, bg k ) y h) , pubhb mkeèhmrsb hb hes aer`ms hfstfgtms rbspbktfvm`bgtb, ugm kmpm M ulfkmghe cms bsabrms bg cms `fs`ms pesfkfegbs hb cm prf`brm kmpm, bstm bs cm bstrukturm okp9 e lfbg, ugm kmpm K ulfkmghe cms bsabrms bg bc rbste hb ces fgtbrstfkfes hbdmhes per cm prf`brm prf`brm kmpm, bs bstm cm bstrukturm bstrukturm akk. Aubgtb4 ubgtb4 scfhbsomrb.gbt scfhbsomrb.gbt

-Amkter hb b`pmqubtm`fbgte4 [b b`pmqubtm`fbgte4 [b hbgb bc amkter hb b`pmqubtm`fbgte a keè4 a 0

Q bsabrms , Q kbchm

(=.<)

keè bc kekfbgtb bgtrb bc vecu`bg ekupmhe per bsabrms ràjfhms bg ugm kbchm ugftmrfm, Q bsabrms , y bc prepfe vecu`bg hb cm kbchm, Q kbchm . ûbhb hbèstrmrsb keg amkfcfhmh, m pmrtfr hb su jbe`btràm, qub pmrm cms rbhbs sk, lkk, akk y okp, bstb vmcer sb kerrbspeghb rbspbktfvm`bgtb keg >, 2<, >, 1?, >, :8 y > , :8 mprexf`mhm`bgtb9 mprexf`mhm`bgtb9 bste bs, ces íteès bg ugm rbh akk û okp ekupmg ug bspmkfe hbc :8 % hbc tetmc hfspegflcb. -Fgtbrstfkfes e oubkes4 Bc oubkes4 Bc bspmkfe cflrb u equbhmhbs hbdmhms bgtrb íteès hb ugm rbh krfstmcfgm sb kegekb keè fgtbrstfkfes e oubkes hb bstrukturm. Kmhm fgtbrstfkfe ùbstrm ugm jbe`btràm pmrtfkucmr bg augkfûg hbc gu`bre hb vbkfges `ís prûxfès bg kegtmkte keg æc9 msà, bg cm bstrukturm kôlfkm sf`pcb, ug oubke obxmæhrfke (? vbkfges bg kegtmkte) sb bgkubgtrm bg bc kbgtre hb cm kbchm prf`ftfvm. Bg cms bstrukturms ke`pmktms akk y okp, sb hbùbstrm qub bxfstbg tmgtes oubkes ektmæhrfkes (1 vbkfges bg kegtmkte) y bc helcb hb oubkes tbtrmæhrfkes (8 vbkfges bg kegtmkte) qub bc gô`bre hb pugtes hb rbh per kbchm kegvbgkfegmc. ^mrm cm akk bg pmrtfkucmr, ces oubkes ektmæhrfkes sb bgkubgtrmg ulfkmhes, uge hb bcces bg bc kbgtre hb cm kbchm kegvbgkfegmc kôlfkm, y ces ; rbstmgtbs bg bc kbgtre hb cms mrfstms hbc kule, m ug =/8 hb oubke per mrfstm9 ces ? oubkes tbtrmæhrfkes sb ulfkmg bg cms pesfkfegbs ( =/8, =/8, =/8) rbspbkte hb kmhm værtfkb hb cm kbchm kegvbgkfegmc. Afgmc`bgtb, bg cm lkk bxfstbg 1 fgtbrstfkfes ektmæhrfkes hbaer`mhes y =< tbtrmæhrfkes tm`lfæg hbaer`mhes. RGBH.



1

Y2P. Afjurm =.:4 Mcjuges fgtbrstfkfes bg bstrukturms krfstmcfgms kôlfkms. Aubgtb4 Y2

Afjurm =.?4 Mcjuges bdb`pces hb bstrukturms krfstmcfgms. Elsærvbsb keè bg bc GmKc ces fegbs Gm +

sb

ulfkmg bg tehes ces oubkes ektmæhrfkes `fbgtrms ces Kc ∜ bg cms pesfkfegbs hb rbh akk. Aubgtb4 Y 2P

RGBH.


< Hfarmkkfûg y cm \bh \bkàprekm

:

< Hfar Hfarmk mkkf kfûg ûg y cm cm \bh \bh \bkà \bkàpr prekm ekm -\bh Hfrbktm4 Bg Hfrbktm4 Bg mhbcmgtb \H, bs bc kegdugte hb tehes ces { \g } keg vbkterbs lmsb prf`ftfves {mf } sbjôg (=.= (=.=), ), bste bs, bc kegdugte hb vbkterbs qub «cekmcfzm¹ bg bc bspmkfe m tehe pugte hb ugm rbh krfstmcfgm. -\bh \bkfprekm4 Bg \bkfprekm4 Bg mhbcmgtb \\, bs bc kegdugte hb vbkterbs hb eghm i kuyms eghms `mgtfbgbg cm pbrfehfkfhmh hb cm \H, bs hbkfr, tmcbs qub bf iº(r+\) 0 bf iºr. Xmcbs vbkterbs per tmgte omg hb ku`pcfr i º \ 0 <ώg, g ∏ T . Bs pubs bc kegdugte hb vbkterbs bskrftes keè4 J 0 ΰ = l= + ΰ < l< + ΰ < l; ΰ f ∏ T, 9 (<.=) heghb ces {l d } tmcbs qub m qub m f º ld 0 < ώθ fd fd , keg θ fd fd cms hbctms hb Iregbkibr, seg ces vbkterbs lmsb prf`ftfves hb cm \\ qub bg tær`fges hb ces vbkterbs lmsb prf`ftfves hb cm \H tfbgbg cm aer`m4 m< Ú m; m; Ú m= m= Ú m< , l< 0 <ώ , l; 0 <ώ , (<.<) l= 0 <ώ m= º (m< Ú m; ) m= º (m< Ú m; ) m= º (m< Ú m; ) y bvfhbgtb`bgtb ku`pcbg cm keghfkfûg bxfjfhm dugte keg ces {mf }. Bc cbkter omlrí getmhe qub bc bspmkfe rbkàpreke hb cm \\, mcjblrmfkm`bgtb per hbgfkfûg (smcve amkter hb < ώ), ge bs `ís qub bc humc hbc bspmkfe hfrbkte hb cm \H, e sf sb qufbrb, bs bc bspmkfe kevmrfmgtb kevmrfmgtb msekfmhe mc bspmkfe kegtrmvmrfmgtb kegtrmvmrfmgtb hb cm \H. -^rf`brm Tegm hb Lrfcceufg (^TL )4 [b )4 [b ge`lrm msà mc vecu`bg kegtbgfhe bg cm kbchm hb Zfjgbr-[bftz kegstrufhm hbc èhe qub sb bxpuse bg cm sbkkfûg mgtbrfer. Cm ^TL bs bs hb jrmg f`pertmgkfm bg bc bstuhfe hb cm hfarmkkfûg hb rmyes _ per krfstmcbs, bg cm Xberàm hb Lmghms, y bg jbgbrmc, bg bc bstuhfe hb cms prepfbhmhbs hb ces sûcfhes krfstmcfges. [b hbgb fjumc`bgtb fjumc`bgtb ugm [bjughm Tegm hb Lrfcceufg ( [TL ), ), ugm Xbrkbrm Tegm hb Lrfcceufg ( XTL ), ), ... kegstrufhms bg lmsb m ces sbjughes, tbrkbres, ... vbkfges prûxfès.

Afjurm Afjurm <.=4 <.=4

^rf` ^rf`br brms ms 8 zegm zegmss hb Lrfc Lrfcce ceuf ufg g hb ugm ugm rbh rbh lfhf lfhf`b `bgs gsfe fegm gmcc kumh kumhrm rmhm hm.. Aubg ubgtb4 tb4 ottp4//www.um`.bs

Elsærvbsb Elsærvbsb bg cm jurm mgtbrfer mgtbrfer,, qub tehms cms zegms hb Lrfcceufg Lrfcceufg kulrbg cm `fs`m `fs`m írbm (vecu`bg bg ugm rbh trfhf`bgsfegmc). Cms rbcmkfegbs (<.< (<.<)) bxfjbg trmlmdmr keg vbkterbs lmsb prf`ftfves hb cm \H pmrm hbgfr lmsbs prf`ftfvms prf`ftfvms hb cm \\. ^mrm bc sfstb`m kôlfke, sfbghe f sfbghe f, d, i ces vbrserbs bg cm hfrbkkfûg RGBH.



?

hb kmhm mrfstm hb cm kbchm kôlfkm kegvbgkfegmc hb pmrí`btre m, qubhmg hbgfhes ces vbkterbs lmsb prf`ftfves keè4 sk 4 lkk 4

m= 0

akk 4

m

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m= 0 m f , m< 0 m d , m; 0 m i,

∜f + d + i) , m 0 < (f ∜ d + i) , m m

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m= 0

d + i) ,

m

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m

m< 0

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f+i) ,

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m; 0

0 m< (f + d m

<(

∜ i) ,

(<.;)

f + d).

Rsmghe cms (<.< (<.<)) keg cms mgtbrferbs, rbsuctm qub cm \\ hb ugm sk, bs etrm sk hb pmrí`btre <ώ hb rbh m 9 cm \\ hb ugm lkk, bs ugm akk hb pmrí`btre 8mώ 9 y cm hb ugm akk, akk, bs ugm lkk hb 8ώ pmrí`btre m . -Augkfegbs pbrfûhfkms bg cm rbh4 [upûgjmsb rbh4 [upûgjmsb ugm augkfûg g (r) keg cm pbrfehfkfhmh hb cm rbh (per bdb`pce cm kegkbgtrmkfûg bcbktrûgfkm). Hbsmrreccíghecm bg sbrfb hb Aeurfbr, g (r) 0

�

gI bfIºr ,

(<.8)

I

y bxfjfbghe bg vfrtuh hb cm pbrfehfkfhmh qub4 g (r + \) 0

�

gI bfIº(r+\) 0

I

�

gI bfIºr 0 g (r) 0

fI

⇜b

I

º\ 0 = 9 ∎ \ ∏ \H, (<.2)

ces I ces I seg seg bgtegkbs vbkterbs hb cm \\, cubje bs I 0 J , J , y bg hbgftfvm, g (r) 0

�

fJ

gJ b ºr

9

gJ 0

J

= Q kbchm

ˁ

g (r) b∜fJºr hQ.

(<.1)

kbchm

-Hfarmkkfûg hb rmyes _4 [bm _4 [bm ugm eghm bcbktre`mjgætfkm keg vbkter hb eghm i eghm i fgkfhbgtb, fgkfhbgtb, qubs bs hfspbrsmhm per ug krfstmc keg kbchms bg pesfkfegbs hb rbh {\g } y lmsb mtû`fkm hb d íteès bg pesfkfegbs {rd }, cm m`pcftuh M hb cm eghm hfspbrsmhm bg cm hfrbkkfûg i hfrbkkfûg i ‼ bs4 lmsb

M

� ∐

a d b∜fr º(i ∜i) ‼

d

rbh

�

b∜f\g º(i ∜i) , ‼

(<.:)

g

d

keg a d bc ccm`mhe amkter hb aer`m mtû`fke hmhe per4 a d 0

ˁ

gd (ύ)b∜fύºJ h; ύ,

(<.?)

krfstmc

heghb gd (ύ) bs kegkbgtrmkfûg bcbktrûgfkm hbc íteè d hb cm lmsb bg bc vecu`bg h; ύ . [b hbsprbghb hb fg`bhfmte hb (<.: ( <.:)) qub bc `íxfè hb M sb mckmgzm kumghe i‼ ∜ i 0 ▻i 0 J,

(<.6)

pubs bc sbjughe su`mterfe hb (<.: ( <.:)) , qub sb bxtfbghb m tehm cm \H arbgtb mc prf`bre qub bs ugm su`m hb uges pekes, bs qufbg `mrkm hfkoe vmcer. vmcer. Cm keghfkfûg mgtbrfer bs kegekfhm keè keghfkfûg hb Cmub. RGBH.



6

Msà pubs, ces `íxfès hb hfarmkkfûg hb ugm eghm bcbktre`mjgætfkm per ug krfstmc, e kurrbg mc ku`pcfrsb cm keghfkfûg mgtbrfer9 keghfkfûg qub lmde hfspbrsfûg bcístfkm ( i‼ 0 i ), bcbvmghe mc kumhrmhe m`les `fb`lres hb ▻i 0 J , J , y km`lfmghe J km`lfmghe J per per ∜J (ce qub ge èhfkm cm keghfkfûg), pubhb bskrflfrsb tm`lfæg keè4 = <

i º J 0 J<

(<.=>)

-Bkumkfegbs hb Cmub4 @uctfpcfkmghe Cmub4 @uctfpcfkmghe bskmcmr`bgtb m`les `fb`lres hb (<.6 ( <.6)) per ces vbkterbs lmsb prf`ftfves hb cm \H, sb eltfbgbg, bg vfrtuh hb ( <.= <.=)4 )4

º ▻i 0 <ώΰ =, m< º ▻i 0 <ώΰ < , m; º ▻i 0 <ώΰ ; , m=

(<.==)

kegekfhms keè bkumkfegbs hb Cmub, jbe`ætrfkm`bgtb kmhm ugm hb bccms rbprbsbgtm ug kege hb jbgbrmtrfz ▻i bg terge mc bdb hfrfjfhe per mf 9 msà, cm keghfkfûg hb `íxfè hb hfarmkkfûg sb smtfsamkb kumghe bc vbkter ▻i kefgkfhb keg cm càgbm hb fgtbrsbkkfûg hb ces trbs keges, ugm keghfkfûg aubrtb qub sûce sb ku`pcb `bhfmgtb bc lmrrfhe, lfbg hb cegjftuhbs hb eghm ι hb cm rmhfmkfûg utfcfzmhm (`ætehe hb Cmub) e lfbg hb erfbgtmkfegbs hbc krfstmc (`ætehe hbc krfstmc retmterfe), e `bhfmgtb bc use hb pecve hb krfstmc mprevbkomghe cm erfbgtmkfûg mzmresm hb ces jrmges krfstmcfges (`ætehe hb Hblyb-[kobrrbr). -^cmges krfstmcfges y vbkterbs hb cm \\4 Hmhe ug vbkter kumcqufbrm J hb cm \\ bg hfrbkkfûg Yo i cP rbabrfhm m ces vbkterbs lmsb prf`ftfves, bs aíkfc prelmr qub æstb, bs pbrpbghfkucmr mc kegdugte hb pcmges hbc krfstmc pmrmcbces hb cm am`fcfm {o i c}. Xmcbs pcmges bstíg hfstmgkfmhes ugm cegjftuh hoic (ccm`mhm hfstmgkfm fgtbrpcmgmr) . [b hbùbstrm sfg hfkuctmh qub sf J sf J `¸Ľg Ľg bs bc vbkter `ís kerte pbrpbghfkucmr m hfkoes pcmges, bgtegkbs4 hoic 0

<ώ J`¸Ľg Ľg

.

(<.=<)

-Cby hb Lrmjj4 \bcmkfegmghe Lrmjj4 \bcmkfegmghe (<.=< (<.=<)) keg (<.=> (<.=>), ), y tbgfbghe bg kubgtm qub i 0 cm cegjftuh hb eghm hb cm rmhfmkfûg utfcfzmhm, sb hbhukb qub4

<ώ ι

keg ι (<.=;)

heghb bc bgtbre g mpmrbkb mc tbgbr bg kubgtm qub bg (<.=> ( <.=>)) bc vbkter J vbkter J 0 gJ`¸Ľg Ľg bs ug vbkter hb cm \\ `ís jbgbrmc, y οoic bs bc ígjuce bgtrb bc vbkter hb eghm i y bc pcmge (o i c). Cm rbcmkfûg mgtbrfer bs cm lfbg kegekfhm Cby hb Lrmjj pmrm cm hfarmkkfûg hb rmyes _ per ug krfstmc. ^mrm cms rbhbs sk, sustftuybghe h oic bg cm cby hb Lrmjj, per su bxprbsfûg bg augkfûg hb ces vbkterbs lmsb prf`ftfves, sb tfbgb qub4

∔ o< +
oic 0 gι,

(<.=8)

ce qub pbr`ftb p br`ftb bxpbrf`bgtmc`bgtb bxpbrf`bgtmc`bgtb pmrm ugm ùbstrm krfstmcfgm, hbtbr`fgmr ces àghfkbs hb @fccbr hb ces pcmges fgvecukrmhes bg ces `íxfès hb hfarmkkfûg keg ígjuce ο oic kegekfhe (pcmges hb Lrmjj) , pmrm ugm rmhfmkfûg hb cegjftuh hb eghm ι hbtbr`fgmhm. RGBH.



=>

Afjurm <.<4 @íxfè hb hfarmkkfûg (bg hfspbrsfûg bcístfkm ) hb Cmub y bqufvmcbgkfm keg cm cby hb Lrmjj bg hfspbrsfûg bcístfkm. Aubgtb4 ottp4//www.s.puk.k

-Amkter hb bstrukturm hb cm lmsb4 Lmde lmsb4 Lmde cm keghfkfûg hb `íxfè hb hfarmkkfûg hb Cmub, bc sbjughe su`mterfe hb (<.: ( <.:)) sb rbhukb elvfm`bgtb mc gô`bre hb kbchms G hb cm rbh, Cm m`pcftuh hfspbrsmhm rbsuctm bgtegkbs4 lmsb

M

∐ G [

J

9

[ J 0

�

a d b∜fr d º(i ∜i) . ‼

(<.=2)

d

M [ J sb cb hbge`fgm amkter hb bstrukturm hb cm lmsb y rbprbsbgtm ugm su`m fgtbrabrbgkfmc fgtbrabrbgkfmc hbgtre hb kmhm kbchm hbc krfstmc. Bg lmsbs pecfmtû`fkms, mcjuges `íxfès hb hfarmkkfûg pbr`ftfhes pubhbg rbsuctmr preoflfhes e bxtfgjufhes per cm prepfm lmsb mtû`fkm ([ J 0 >). Bstb obkoe su`fgfstrm ugm rbcbvmgtb fgaer`mkfûg selrb cm bstrukturm hb cm `fs`m, msà pubs, kegsfhbrmghe per bdb`pce m cm rbh lkk keè ugm sk keg lmsb ègemtû`fkm (ug íteè bg bc kbgtre hb cm kbchm), sb bxtfgjufríg cms rbbxfegbs hb pcmges ( o i c) tmcbs qub o + i + c 0 f`pmr 9 bg bc kmse hb ugm akk (sk `ís lmsb trfmtû`fkm), sb bxtfgjufríg mqubccms mqubccms o, i , c tfbgbg hfstfgtm pmrfhmh. rbbxfegbs heghb o, Afgmc`bgtb,bc Afgmc`bgtb,bc amkter hb aer`m mtû`fke a d (lmde (lm de keghfkfûg hb `íxfè hb hfarmkkfûg) su`fgfstrm fgaer`mkfûg mkbrkm hb cm hfstrflukfûg bcbktrûgfkm hbc íteè d hb cm lmsb mtû`fkm, eltbgfæghesb hfabrbgkfms pbqubýms rbspbkte hb ces vmcerbs tbûrfkes hb ces íteès cflrbs. - Hfarmkkfûg hb gbutregbs y bcbktregbs4 Ge bcbktregbs4 Ge sûce ces rmyes _ seg utfcfzmhes per cms tækgfkms hb hfarmkkfûg per krfstmcbs, etrms rmhfmkfegbs seghm keè ces gbutregbs y ces bcbktregbs pubhbg sbr usmhms tm`lfæg keg bsb aàg, su utfcfhmh kcmre bstí vfbgb hbtbr`fgmhm per cms kmrmktbràstfkms hbc bxpbrf`bgte y hb cms prepfbhmhbs keè cm kmrjm, bgbrjàm, btk hb cms pmrtàkucms seghm . Hb bstb èhe, cms eghms bcbktre`mjgætfkms fgtbrmkkfegmg keg ces bcbktregbs hbc krfstmc, ces gbutregbs ce omkbg keg ces gôkcbes mtû`fkes y bcbktregbs `mjgætfkes sfbghe hb utfcfhmh bg bc bstuhfe hb krfstmcbs `mjgætfkes, y per ôctfè ces bcbktregbs, qub fgtbrmkkfegmg aubrtb`bgtb pbgbtrmghe peke, y per bcce, rbsuctmg ôtfcbs bg bc bstuhfe hb prepfbhmhbs supbrkfmcbs hb sûcfhes y krfstmcbs hbcjmhes.

RGBH.


; Bc Bcbktregbs bg ug êtbgkfmc ^brfûhfke. Xberb`m hb Lceko

==

; Bcbktr Bcbktregb egbss bg ug êtbgkf etbgkfmc mc ^brfûhfk ^brfûhfke. e. Xber Xberb`m b`m hb Lceko Bstuhfmrbès mc sûcfhe krfstmcfge kegsfhbríghece hb aer`m sbpmrmhm keè ug kegdugte hb fegbs fg`ûvfcbs bg sus p esfkfegbs hb bqufcflrfe, y ug kegdugte hb bcbktregbs keg èvf`fbgte fghbpbghfbgtb (mprexf`mkfûg hb bcbktregbs fghbpbghfbgtbs) hblfhe mc petbgkfmc pbrfûhfke krbmhe per ces fegbs, R ( ( r + \) 0 R ( ( r)

9

∎\ ∏ \H.

(;.=)

Cm prf`brm kegsfhbrmkfûg qubhm dustfkmhm per cm hfabrbgkfm hb `msm bgtrb ces fegbs y bcbktregbs, ce qub pbr`ftb p br`ftb omkbr fghbpbghfbgtbs sus èvf`fbgtes, bs bstm cm ccm`mhm mprexf`mkfûg hb Lerg-Epbgobf`br. [elrb cm sbjughm, cm cegjftuh hb eghm hb Hb Lrejcfb hb ces bcbktregbs (bg bc èhbce hb bcbktregbs cflrbs) bs hbc `fsè erhbg hb `mjgftuh qub cm pbrfehfkfhmh hb cm rbh. Xehe bste rbhukb gubstre bstuhfe m rbsecvbr cm bkumkfûg hb [korñhfgjbr hb ug bcbktrûg (mprexf`mkfûg ègebcbktrûgfkm) se`btfhe m ug petbgkfmc R ( ( r) keè bc mgtbrfer, pmrm ce kumc tbgbès bc sfjufbgtb4 -Xberb`m. (hb Lceko)4 Cms augkfegbs hb eghm hb Lceko, secukfegbs hb cm bkumkfûg hb [korñhfgjbr fghbpbghfbgtb hbc tfb`pe pmrm ug bcbktrûg lmde bc petbgkfmc ( ;.= ( ;.=)4 )4 <

∜ < ` ∊<ί (r) + R ( (r)

(r) 0 β ί (r) ,

(;.<)

seg tmcbs qub, ίgi (r) 0 u gi (r) bfiºr ,

(;.;)

∎ ∏ \H y g ∏ T.

keg ugi (r + \) 0 u gi (r) 9 \

Ces bcbktr bcbktregb egbss msà hbskrf hbskrftes tes seg ccm`mccm`mhes bcbktregbs hb Lceko, pubhb bgkegtrmrsb ugm hbèstrmkfûg hbc tberb`m bg per bdb`pceY= bdb`pceY=P. Rgm aer`m bqufvmcbgtb hb bgugkfmr bc tberb`m hb Lceko, bs hbkfr qub ces mutebstmhes ˁ , pubhbg bshbc epbrmher om`fctegfmge O \bprbsbgtmk tmkfûg fûg hb ugm augkfû augkfûg g hb Afjurm ;.=4 \bprbsbg kejbrsb hb èhe qub kmhm augkfûg hb eghm Afjurm Lceko. Bs ugm eghm pcmgm èhucmhm per ugm augkfûg ίgi (r) msekfmhm m kmhm vbkter hb eghm i ug keg cm pbrfehfkfhmh hb cm \H. Aubgtb4 Y= Y=P. smtfsamkb4 ίgi (r + \) 0 b fiºr ίgi (r) 9 ∎\ ∏ \H. (;.8) i

Omy qub getmr qub mquà, cm kmgtfhmh  i ge bs bc è`bgte cfgbmc hbc bcbktrûg (cflrb) W ίgi (r) 0 ∜ f ∊ίgi (r)̿ 0  iίgi (r), cubje ίgi (r) ge bs muteaugkfûg hbc hmhe qub p epbrmher è`bgte W è`bgte Wp, rmzûg per cm qub  i bs ccm`mhe kumsff`pucse e è`bgte krfstmcfge hbc bcbktrûg. Cms secukfegbs hb cm aer`m (;.; (;.;)) qub ku`pcbg cms keghfkfegbs hb kegterge ugi (r + \) 0 ugi (r) , surjbg hb cm rbsecukfûg hb cm bkumkfûg hb [korñhfgjbr bg ugm ôgfkm kbchm prf`ftfvm prf`ftfvm y qub bs cm bkumkfûg hb mutevmcerbs sfjufbgtb4 RGBH.



W O (r) ui (r) 0

�

< 

<`

�

∜ ∊ + i) + R ( (r)

( f

=<

ui (r) 0 β i ui (r) .

(;.2)

êr tmgte, pmrm kmhm i kmhm i omlrí omlrí ug kegdugte fggfte y hfskrbte hb mutevmcerbs β gi fghbxmhes per g . Hblbès Hblbès bspbrmr mhb`ís mhb`ís qub bc bspbktre bspbktre hb secukfegbs secukfegbs sbm kegtfg kegtfgue, ue, pubs i bs pmrí`btre hbc om`fctegfmge, cubje sf`pcb`bgtb bskrflfrbès βg (i) . Ces vbkterbs hb eghm i sfb`prb pubhbg rbhukfrsb m cm ^TL , pubs tmgte cms augkfegbs hb eghm keè ces mutevmcerbs mutevmcerbs rbsuctmg fjumcbs tmgte pmrm i keè i keè pmrm i pmrm i + J, ∎J ∏ \\. -Hfmjrm`m y rbcmkfegbs hb hfspbrsfûg hb bcbktregbs4 M tmc kegdugte βg (i) cb ccm`mrbès hfmjrm`m hb hfspbrsfûg hb bcbktregbs, ccm`mghe rbcmkfûg hb hfspbrsfûg m kmhm β (i) pmrm kmhm g. Bs usumc hbkfr, mugqub bste kegstftuyb ug mluse hb cbgjumdb, qub bc hfmjrm`m hb hfspbrsfûg hb bcbktregbs aer`m ugm bstrukturm hb lmghms, y qub kmhm rbcmkfûg hb hfspbrsfûg aer`m cm lmghm hb bgbrjàm g . -Qbkterbs i pbr`ftfhes4 [fg rbstrfgjfr cm jbgbrmcfhmh, supegjmès ugm rbh krfstmcfgm sk hb vecu`bg gfte Q keg G f kbchms m ce cmrje hb kmhm hfrbkkfûg mf (hb ces vbkterbs lmsb prf`ftfves). Bc krfstmc tfbgb bg tetmc G 0 G = G < G ; kbchms prf`ftfvms, bxtbghfbghe bc vecu`bg Q m m tehe bc bspmkfe f`pegfbghe cms ccm`mhms keghfkfegbs hb kegterge hb LergQeg Imr`mg sfjufbgtbs m ces bcbktregbs hb Lceko, ίgi (r + G f f mf ) 0 ίgi (r)

r∏Q

9

,

f 0 =, <, ;,

qub mpcfkmhms m cms augkfegbs hb Lceko, ίgi (r + G f f mf ) 0 b fiºG m ίgi (r) 0⇜ bfiºG m 0 = 9 f 0 =, <, ;, f f f

f f f

(;.1)

(;.:)

hb heghb sb bxtrmb qub4 `= ` ` l= + < l< + ; l; G = G < G ;

i 0

9

`f

∏ T,

(;.?)

keg { lf } ces vbkterbs lmsb prf`ftfves hb cm \\. Bs bstm keghfkfûg pubs, cm qub hblbg smtfsamkbr ces vbkterbs i . Bc vecu`bg hbc bspmkfe rbkàpreke msekfmhe m ug i pbr`ftfhe = = Úvecu`bg hb cm kbchm prf`ftfvm hb Yl= º (l< Ú l; )P 0 G bs trfvfmc`bgtb trfvfmc`bgtb |▻i| 0 G cm \\, ; y tbgfbghe bg kubgtm qub bc vecu`bg hb cm kbchm prf`ftfvm hb cm \\ bs (
|▻i| 0

(<ώ ); Q

.

(;.6)

-Hbgsfhmh hb bstmhes4 ^mrm bstmhes4 ^mrm bc kíckuce hb mcjugms prepfbhmhbs aàsfkms sb rbmcfzm ugm su`m hb ugm augkfûg A g (i) bxtbghfhm mc kegdugte hb vbkterbs i vbkterbs i pbr`ftfhes. pbr`ftfhes. [f A g (i) bs mprexf`mhm`bgtb kegstmgtb bg bc vecu`bg ▻i, y sf mhb`ís prbsbgtm cm pbrfehfkfhmh hb cm \\, sb ccbvm m kmle cm mprexf`mkfûg4

�

A g (i)

<

g,i

∜↜

Q ↜∗

(<ώ )

ˁ

A g (i) hi,

(;.=>)

K^

heghb bc < prevfbgb hb cm hbjbgbrmkfûg hb bspàg bcbktrûgfke y K ^ bs cm kbchm prf`ftfvm prf`ftfvm hb cm \\. RGBH.



=;

[f A g (i) hbpbghb mhb`ís hb g y i y i m m trmvæs hb βg (i) , sb hbgb cm hbgsfhmh hb bstmhes e hb gfvbcbs Hg (β) pmrm kmhm lmghm g , hb èhe qub4

�

A g (i) 0

<

g,i

�ˁ g

A g (β)

K^

H (β) hβ, g

(;.==)

cubje per hbgfkfûg bs Hg (β) hβ 0 <Úgô`bre hb vbkterbs i vbkterbs i pbr`ftfhes pbr`ftfhes bg cm lmghm g keg bgbrjàm bgtrb β y β + hβ. [f sb kegsfhbrmg [ g (β) y [ g (β + hβ) supbrkfbs bqufbgbrjætfkms keg bgbrjàm β y β + hβ rbspbktfvm`bgtb rbspbktfvm`bgtb b fg`brsms bg cm K ^, bc kíckuce fgtbjrmc keghukb m4

H

ˁ

[ g (β)

h[ . βg (i)

|∊

|

(;.=<)

-[upbrkfb hb Abr`f y bstrukturm hb lmghms4 Rg lmghms4 Rg sfstb`m hb G bcbktregbs hb Lceko bg bc bstmhe aughm`bgtmc, aughm`bgtmc, sb kegstruyb ekupmghe e kupmghe ces gfvbcbs ègebcbktrûgfkes β g (i) pmrm tehe g omstm ke`pcbtmr bc gô`bre hb bcbktregbs hbc sfstb`m, m æste subcb ccm`mrsb keghfkfûg hb ekupmkfûg e hb ccbgmhe. Bg ug kegtfgue hb bgbrjàm, bc vmcer βA hb bgbrjàm, tmc qub, βg (i) 3 βA pmrm tehes ces gfvbcbs ekupmhes sb hbge`fgm bgbrjàm hb Abr`f, bste bs, mqubccm qub sbpmrm (bg ug kegtfgue) bc kegdugte hb gfvbcbs ekupmhes hb ces hbsekupmhes. Cm supbrkfb bqufbgbrjætfkm βg (i) 0 β A hbc bspmkfe rbkàpreke sb hbge`fgm supbrkfb hb Abr`f, y sb ccm`m rm`m g mqubccm kerrbspeghfbgtb m ug vmcer hb g hbtbr`fgmhe. ^mrm kfbrtes sûcfhes ekurrb qub mc ekupmr ces gfvbcbs βg (i) , bc vmcer hb bgbrjàm `àgfè hb mqubcces qub qubhmg hbsekupmhes, bs `myer qub bc vmcer `íxfè hb ces gfvbcbs ekupmhes, omy per tmgte ug fgtbrvmce hb bgbrjàm preoflfhe kegekfhe keè jmp ( hbc fgjcæs) hb bgbrjàm, zmgdm, e lrbkom hb bgbrjætfkm. Mc preybktmr cm hbgsfhmh hb bstmhes selrb bc bdb hb bgbrjàm, sb elsbrvmg zegms e fgtbrvmces hb bgbrjàm, lmghms qub kerrbspeghbg m ugm e vmrfms rbcmkfegbs hb hfspbrsfûg (elsærvbsb bg cm jurm ;.< keè cms rbcmkfegbs hb hfspbrsfûg pubhbg Afjurm ;.<4 Lmghms hb bgbrjàm. Cm hb `bger bgbrjàm secmpmrsb), y etres fgtbrvmces qub ge ke- pmrbkb prevbgfr hb hes rbcmkfegbs hb hfspbrsfûg hfsrrbspeghbg m gfgjugm, ces prepfes jmps hb tfgtms keg hbgsfhmhbs hb bstmhe M y L9 cm hb bgbrjàm bgbrjàm (bs bste ce qub hm sbgtfhe mc ge`- supbrfer bg km`lfe, prevfbgb hb ugm ôgfkm rbcmkfûg lrb hb lmghms). hb hfspbrs hfspbrsfûg. fûg. Elsbrvb Elsbrvb bc jmp hb bgbrjàm bgbrjàm bgtrb bgtrb cms Y =P. [b ccm`m lmghm hb vmcbgkfm, vmcbgkfm, LQ, m mqubccm hbgsfhmhbs L y K. Aubgtb4 Y= hb gfvbcbs hb bgbrjàm `ís mctes ekupmhm per bcbktregbs bg bc kbre mlsecute. Bg ces sûcfhes mfscmgtbs y sb`fkeghukterbs omy ug jmp hb bgbrjàm duste per bgkf`m hb cm LQ y m bgbrjàms supbrferbs ugm lmghm hb keghukkfûg, LK. êr ce hfkoe, bvfhbgtb`bgtb bstes sûcfhes ge prbsbgtmg bgbrjàm hb Abr`f lfbg hbgfhm gf per tmgte supbrkfb hb Abr`f. Cm mgkourm hbc jmp hb bgbrjàm βj , bgtrb cm LQ y cm LK, hbtbr`fgm hb ug èhe jbgbrmc sf bc sûcfhe bs ug sb`fkeghukter e mfscmgtb ; , sf βj ≏ = bQ sbrí ug sb`fkeghukter ym ;

[b vbrí bg bc Kmpàtuce 1 Kmpàtuce 1 qub bste bs msà ôgfkm`bgtb sf cm LQ sb bgkubgtrm pmrkfmc`bgtb ekupmhm.

RGBH.



=8

qub m tb`pbrmturms hfstfgtms hb kbre, bcbktregbs hb cm LQ pubhbg per preèkfûg tær`fkm smctmr m cm LK, per bc kegtrmrfe, keg βj ≏ => bQ ces bcbktregbs ge pubhbg supbrmr cm lrbkom bgbrjætfkm y bc sûcfhe sbrí ug mfscmgtb. Bg bc kmse hb ces `btmcbs qub seg sûcfhes keg supbrkfb hb Abr`f, ge tfbgbg LQ sfge ôgfkm`bgtb LK, ge omy jmp hb bgbrjàm y per kegsfjufbgtb kegsfjufbgtb seg keghukterbs.

RGBH.


8 Bc @ehbce hb Hruhb-[e``brabch hb ces @btmcbs

=2

8 Bc @ehbce @ehbce hb Hruhb Hruhb-[e -[e``b ``brab rabch ch hb ces @btm @btmcbs cbs Bc èhbce hb Hruhb mpcfkm cm bxftesm tberàm kfgætfkm hb ces jmsbs pmrm bxpcfkmr cms prepfbhmhbs hb ces `btmcbs. ^mrm bcce sb kegsfhbrm qub ces bcbktregbs hb vmcbgkfm, qub seg ces hb keghukkfûg, sb ùbvbg cflrb`bgtb bg bc `btmc y per tmgte pubhbg sbr èhbcmhes keè ug jms kcísfke, ces fegbs hbc `btmc kegsfhbrmhes qufbtes bg su pesfkfûg hb bqufcflrfe (rbh aràm), vfstes keè ug aeghe hb kmrjm pesftfvm kuye petbgkfmc kegstmgtb pubhb te`mrsb keè kbre (ccm`mhe m vbkbs èhbce hb dmcbm). Bc èhbce bxpcfkm mcjugms hb cms prepfbhmhbs aàsfkms hb ces `btmcbs keè cm keghuktfvfhmh bcæktrfkm, keghuktfvfhmh tær`fkm, bc babkte Omcc, prepfbhmhbs ûptfkms, bgtrb etrms. @bhfmgtb mrju`bgtes kuígtfkes sb kegkcuyb, qub sf lfbg bc bstuhfe hb cm hfgí`fkm hb ces bcbktregbs pubhb sbr mlerhmhe hb ug èhe purm`bgtb kcísfke, bc bstuhfe bstmhàstfke hbshb ugm pbrspbktfvm kcísfkm ge rbsuctm mhbkumhe, auæ [e``brabch qufbg rbkurrfû m cm bstmhàstfkm kuígtfkm pmrm trmtmr m ces bcbktregbs keè ug kegdugte hb abr`fegbs, keè ug jms hb Abr`f (`bhfmgtb cm bstmhàstfkm hb Abr`f-Hfrmk bg cujmr hb cm hb @mxwbcc-Lectz`mg). Bc èhbce, lísfkm`bgtb supegb ce sfjufbgtb4 =. [b hbsprbkfm cm fgtbrmkkfûg bcbktrûg-bcbktrûg (mprexf`mkfûg hb bcbktregbs fghbpbghfbgtb) y cm fgtbrmkkfûg fgtbrmkkfûg fûg-bcbktrûg (mprexf`mkfûg hb bcbktregbs cflrbs). <. Bxfstb mcjôg `bkmgfsè `bkmgfsè hb kecfsfûg (e skmttbrfgj) keg ces fegbs fegbs hbc `btmc qub tbr`mcfzm bc sfstb`m. Cm prelmlfcfhmh hb kecfsfûg hbgtre hb ug fgtbrvmce hb tfb`pe ht bs ht/ψ , sfbghe ψ bc tfb`pe hb rbcmdmkfûg, tfb`pe hb kecfsfûg e tfb`pe cflrb `bhfe. Bstb tfb`pe tfb`pe ge hbpbghb gf hb cm pesfkfûg r pesfkfûg r hbc hbc bcbktrûg gf hb su vbcekfhmh v vbcekfhmh v.. Xrms ugm kecfsfûg bc bcbktrûg «ecvfhm¹ su ofsterfm mgtbrfer per ce qub su vbcekfhmh hb smcfhm sbrí mcbmterfm y sece hbpbghbrí bg `mjgftuh hb cm tb`pbrmturm. -Bkumkfûg hfgí`fkm hb ces bcbktregbs4 [upegjmès bcbktregbs4 [upegjmès ug bcbktrûg keg è`bgte cfgbmc t p ( ) se`btfhe m ugm aubrzm A (t). Hb cm sbjughm supesfkfûg, bg ug tfb`pe ht , bc bcbktrûg tbghrí ugm prelmlfcfhmh = ∜ htψ hb ge kecfsfegmr. [f ge kecfsfegm, su è`bgte trms bc tfb`pe ht sbrí p (t + ht) 0 p (t) + A ( t) ht (bg prf`br erhbg hb mprexf`mkfûg). Cm kegtrflukfûg bg pre`bhfe hb ces bcbktregbs qub ge omg kecfsfegmhe (sulfghfkb gk ) sbrí4

 

⟯p

�∜ �⟯

(t + ht) 0 =

gk

⟦

ht h t ψ

⟦ ∜ htψ ⟯p (t)⟦ . (8.=)

( p (t) + A (t) ht) 0 p (t) + A (t) ht

⟦ ⟯

⟦

⟯

⟦ ⟯

Cm kegtrflukfûg mc è`bgte, è`bgte, hb ces bcbktregbs qub sà omg kecfsfegmhe, pubste qub cubje hb cm kecfsfûg su hfrbkkfûg bs mcbmterfm y bc gô`bre hb æstes bs ht/ψ Úgô`bre tetmc hb bcbktregbs, bs kcmrm`bgtb hb erhbg hes bg ht y pehbès hbsprbkfmrcm (sfjufbghe bg prf`br erhbg hb mprexf`mkfûg). mprexf`mkfûg). Bg hbgftfvm, hbgftfvm, omkfbghe ht ↜ > bg (8.= ( 8.=)) tbgbès4 hp (t) ht

� �

⟦ ∜ ψ = ⟯p (t)⟦ ,

0 A (t)

⟯

(8.<)

bxprbsfûg kegekfhm keè bkumkfûg hfgí`fkm, y kuym fgtbrprbtmkfûg bs fg`bhfmtm, bc babkte gbte hb cms kecfsfegbs fgtrehukbg ug tær`fge hb mèrtfjum`fbgte ψ = ⟯p (t)⟦ keè sf hb ugm aubrzm hb rezm`fbgte e hb rbsfstbgkfm preperkfegmc m cm vbcekfhmh hb ces bcbktregbs sb trmtmsb. RGBH.



=1

-Keghukkfûg hb kerrfbgtb kegtfgum. Cby hb Eo`4 [e`btfhe Eo`4 [e`btfhe ug `btmc m ug km`pe bcæktrfke kegstmgtb B , hb cm bkumkfûg hfgí`fkm (mckmgzmhe bc ræjf`bg bstmkfegmrfe) y tbgfbghe bg kubgtm qub bc pre`bhfe tb`permc bgtrb kecfsfegbs (hb ces bcbktregbs keg ces fegbs hbc `btmc) bs ψ , rbsuctm cm vbcekfhmh cà`ftb e hb mrrmstrb bg pre`bhfe, ∜bBψ . v 0 (8.;) `b

Hb cm cby hb Eo`, d 0 ϋB, keg ϋ cm keghuktfvfhmh bcæktrfkm y D bc vbkter hbgsfhmh hb kerrfbgtb, ùctfpcfkmghe cm rbcmkfûg mgtbrfer per bc gô`bre hb bcbktregbs per ugfhmh hb vecu`bg g , y per cm kmrjm ∜b hbc bcbktrûg, sb ccbjm m qub d 0 ∜gbv 0

gb< Bψ `b

0⇜ ϋ 0

gb< ψ , `b

(8.8)

ce qub bxprbsm cm rbcmkfûg cfgbmc hb cm cby hb Eo` supubste qub ces bcbktregbs kecfsfegmg keg ces fegbs hbc `btmc. Bg jbgbrmc, bxpbrf`bgtmc`bgtb bxpbrf`bgtmc`bgtb sb ke`prublm qub cm keghuktfvfhmh hbpbghb hb cm tb`pbrmturm, rbspbkte hb æste, pmrm `ís hbtmccbs y mkcmrmkfegbs, pubhb hfrfjfrsb per bdb`pce m Y8P e m Y= Y=P -Babkte Omcc y `mjgbterrbsfstfvfhmh4 Mgmcfk Mgmcfkbè bèss cm fgubg fgubgkfm kfm hb ug km`pe km`pe `mjgætfke bxtbrge selrb cm keghuktfvfhmh bcæktrfkm. Bc ègtmdb bxpbrf`bgtmc tàpfke (Omcc =?:6) sb ùbstrm bg cm jurm 8.= 8.=.. Cm ke`pegbgtb `mjgætfkm hb cm aubrzm hb Cerbgtz prehukb ugm hbbkkfûg hb cms trmybkterf ybkterfms ms bcbktrûgf bcbktrûgfkms kms qub hm cujmr m ugm mkuùcmkfûg hb kmrjm bg bc lerhb hb cm ùbstrm. Keè kegsbkubgkfm mpmrbkb bg- Afjurm 8.=4 Bsqub`m hbc bxpbrf`bgte ccbvmhe m kmtegkbs ug km`pe bcæktrfke trmgsvbrsmc B y , le per Omcc. Aubgtb4 Y2 Y2P. qub krbkb mc mkuùcmrsb cm kmrjm omstm qub kegtrmrrbstm cm aubrzm hb Cerbgtz `mjgætfkm mckmgzíghesb ug ræjf`bg bstmkfegmrfe. y Ges fgtbrbsm cm kmgtfhmh ccm`mhm kebkfbgtb Omcc, \ O ≭ dB x L , y cm kmgtfhmh ccm`mhm `mj`mjgbterrbsfstfvfhmh, ύ ≭ B dyx . Mpcfkmghe cm bkumkfûg hfgí`fkm, bg ke`pegbgtbs pre`bhfmhms rbsuctm4 B bψ vx 0

∜

vy 0

∜

∜ φ ψv

x

k

`b B y bψ `b

vz 0

∜

y

∜ φ ψv k

x

(8.2)

B z bψ , `b

bL heghb φk ≭ ∜ ` bs cm ccm`mhm arbkubgkfm kfkcetrûg e jfrearbkubgkfm hbc bcbktrûg. Bg bc k ræjf`bg bstmkfegmrfe bs v y 0 >, cubje keg d keg d 0 ∜gbv sustftuybghe cms rbcmkfegbs mgtbrferbs bg cms hbgfkfegbs hb kebkfbgtb Omcc y `mjgbterbsfstfvfhmh, sb eltfbgb4

\O 0 ύ 0

RGBH.

∜=, gb

(8.1)

`b . gb< ψ



=:

Bxpbrf`bgtmc`bgtb Bxpbrf`bgtmc`bgtb sb bgkubgtrm qub cm `mjgbterrbsfstfvfhmh `mjgbterrbsfstfvfhmh ùbstrm ugm jrmg vmrfbhmh hb ke`pertm`fbgtes bg augkfûg hbc km`pe `mjgætfke mpcfkmhe, rbspbkte hbc kebkfbgtb Omcc, pmrm ces `btmcbs gelcbs y mckmcfges, rbsuctm mkbptmlcb pmrm ces prf`bres y lubge pmrm ces sbjughes, sfg b`lmrje, pmrm bc rbste hb `btmcbs bs fgmkbptmlcb, bg kegkrbte perqub bc sfjge bs kegtrmrfe (smcve (smcve qub ces pertmherbs p ertmherbs hb kmrjm aubrmg pesftfves, pbre bste bs fgke`pmtflcb keg ug èhbce hb bcbktregbs cflrbs). Bstms hfskrbpmgkfms cms rbsubcvb cm Xberàm hb Lmghms qub tfbgb bg kubgtm cm fgtbrmkkfûg hbc petbgkfmc pbrfûhfke hb ces fegbs hbc krfstmc keg ces bcbktregbs. \bærmsb m Y =P per bdb`pce pmrm preaughfzmr. -Kmcer -Kmcer bspbkàke bspbkàke bcbktrûgf bcbktrûgfke4 ke4 Hbshb bc pugte hb vfstm hb cm bstmhàstfkm kcísfkm hb @mxwbcc-Lectz`mg, mpcfkmghe bc tberb`m hb bqufpmrtfkfûg hb cm bgbrjàm, sb eltfbgb, pmrm cm kegtrflukfûg hb ces bcbktregbs mc kmcer bspbkàke hb ug `btmc m vecu`bg kegstmgtb y ; per ugfhmh hb vecu`bg4 k bc v 0 < giL , qub bs kcmrm`bgtb kegstmgtb. Hbshb ugm pbrsp bktfvm bktfvm kuígtfkm, rbsecvfbghe cm bkumkfûg hb [korñhfgjbr fghbpbghfbgtb hbc tfb`pe pmrm ug kegdugte hb G bcbktregbs cflrbs bg ug kule hb mrfstm C , kuyms secukfegbs seg hb cm aer`m ί i (r) ∐ bfiºr , f`pegfbghe m æstms keghfkfegbs hb kegterge hb Lerg-veg Imr`mg ίi (x + C , y , z ) 0 ίi (x, y + C, z) 0 ίi (x , y , z + C) 0 ίi (x , y , z ), sb hbhukb pmrm ces vbkterbs hb eghm cm keghfkfûg i keghfkfûg i 0
8 ; ; ώi A < ώ; ? /Q

0⇜ iA 0 ;ώ< g <

 

=/;

9

g 0

G , Q

(8.:)

keg bgbrjàm hb Abr`f keè βA 0 <ìA b , tb`pbrmturm hb Abr`f hmhm per iL X A A 0 βA , y = < vbcekfhmh hb Abr`f < `b vA 0 β A . Cm hbgsfhmh hb bstmhes H (β) pubhb eltbgbrsb m pmrtfr hb cm rbcmkfûg hb hfspbrsfûg mpcfkmghe (;.=< (;.=<)) , hb `mgbrm qub bxprbsmhm bg augkfûg hb cm bgbrjàm hb Abr`f, bs4

H

ˁ

[

h[ hβ/hi

; G 0 < βA

�

β . βA

(8.?)

^mrm cm bgbrjàm pre`bhfe hbc sfstb`m hb G bcbktregbs cflrbs, ⟯R ( ( X )⟦ , omkfbghe use hb cm hfstrflukfûg hb Abr`f-Hfrmk (jurm 8.< (jurm 8.<)) = a A H (β, X ) 0 β ² , heghb ² bs bc petbg∜

b iL X +=

kfmc quà`fke y pmrm cm kumc bg bc kbre mlsecute bs a A H (², >) 0 <= , a A H (β 3 ², >) 0 = y a A H (β 7 ², >) 0 >, per ce qub pubhb hbkfrsb qub βA 0 ² (X 0 > I). Cm hfstrflukfûg rbprbsbgtm cm prelmlfcfhmh hb qub bc bstmhe β (i) bstæ ekupmhe, per tmgte, kegtmghe keg H (β) , y keg β (i) pmrm bcbktre- Afjurm 8.<4 Augkfûg hb Hfstrflukfûg hb Abr`f-Hfrmk gbs cflrbs, bgkegtrmès m X 0 > I qub bg m X 0 > I (càgbm pugtbmhm) y m X 7 > I (càgbm kegtfgum). Aubgtb4 Y2 Y2P.

RGBH.


8 Bc @ehbce hb Hruhb-[e``brabch hb ces @btmcbs bc bstmhe aughm`bgtmc bs4 ∗

⟯R ( (X 0 >)⟦ 0

ˁ

hβ

>

βA

H (β) βa

A H ( β, >)

0

ˁ

=?

H (β) β 0 ;2 G β

hβ

>

A ,

(8.6) ̿ > I , cm fgtbjrmc qub bs hbc vmcer eltbgfhe `bhfmgtb cm bstmhàstfkm kcísfkm. M X 0 mgtbrfer ge rbsuctm trfvfmc, sfg b`lmrje, pubhb mprexf`mrsb `bhfmgtb `bhfmgtb cm ccm`mhm bxpmgsfûg hb [e``brabch, ce qub ccbvm sfg bgtrmr bg hbtmccbs m4 ∗ ∌ ώ< X ώ < < X = bc kv 0 hβH (β) βa A H (β, X ) ≋ giL 0 giL , (8.=>)
Q

�

∌X

ˁ >

�

Q

<

X A A

<

βA

heghb sb om tbgfhe bg kubgtm cm bxprbsfûg hb cm tb`pbrmturm hb Abr`f X A A hmhm mgtbrfer`bgtb. Cm rbcmkfûg cfgbmc keg X , bg bc rbsuctmhe mgtbrfer, sb ke`prublm bxkbcbgtb`bgtb bg bc cmlermterfe, y keè kmlb bspbrmr sb mgucm pmrm X 0 > I , sfg b`lmrje, bc amkter hb preperkfegmcfhmh rbsuctm ùy vmrfmlcb vmrfmlcb pmrm ces hfstfgtes `btmcbs, fgkcuse pmrm mcjuges ùy mcbdmhe hbc vmcer bxpbrf`bgtmc. Cm `bhfhm bxpbrf`bgtmc hb kbc v hblb omkbrsb m ùy lmdms tb`pbrmturms pmrm ge vbrsb bg`mskmrmhm per cm kegtrflukfûg hb cm rbh krfstmcfgm, qub keè sb vbrí `ís mhbcmgtb, hbpbghb hb X ; kumghe X ↜ > I.

-Etres \bsuctmhes hbc èhbce4 Bc èhbce4 Bc èhbce mrredm mrre dm etres rbsuctmhes qub mc `bges `brbkb cm pbgm ke`bgtmr. \bspbkte hb cm keghuktfvfhmh bg kerrfbgtb mctbrgm, sb bgkubgtrm qub > ϋ (φ ) 0 =∜ϋfφψ , heghb mquà ϋ> bs cm keghuktfvfhmh hb kerrfbgtb kegtfgum ym eltbgfhm y keè bs hb bspbrmr ϋ (φ) ↜ ϋ> kumghe φ ↜ >.

^mrm cm kegtrflukfûg bcbktrûgfkm m cm keghuktfvfhmh tær`fkm, sb eltfbgb δ bc 0 ;= v< ψ kbc v , < < β heghb v bs cm vbcekfhmh hb ces bcbktregbs. Omkfbghe v ≋ v A 0 A /<`b y keg (8.=> (8.=>), ), sb < < ώ δ bc 0 ;b
⟯⟦

⟯⟦

Keg rbcmkfûg m cms prepfbhmhbs ûptfkms, pmrm cm fgtbrmkkfûg hb ugm eghm bcbktre`mjgætfkm hb arbkubgkfm φ keg ug `btmc, sb eltfbgb pmrm cm pbr`ftfvfhmh bcæktrfkm rbcmtfvm β (φ) 0 fϋ β φ > = + =∜>/fφψ . M lmdms arbkubgkfms sb mprexf`m keè β (φ ) ≋ βfϋ> φ> , qub mc sbr f`mjfgmrfm f`pcfkm f`pcfkm qub cm eghm sb mtbgôm mtbgôm bg bc `btmc `btmc mckmgzmgh mckmgzmghee ugm preaughfhmh preaughfhmh hb pbgbtrmkfûg pbgbtrmkfûg < φp <β> k< θ 0 β (φ ) ≋ = ∜ φ< heghb φ p< ≭ ϋ>/β> ψ ϋ> φ . M mctms arbkubgkfms sb mprexf`m per bs cm ccm`mhm arbkubgkfm hb pcms`m, qub kerrbspeghb m cm arbkubgkfm kmrmktbràstfkm hb eskfcmkfegbs cegjftuhfgmcbs hb cm hbgsfhmh hbc jms hb bcbktregbs kuye kumgte sb ccm`m pcms`ûg. ^mrm φ 3 φ p bs β (φ ) 3 > y per bghb cm eghm sb mtbgôm bg bc `btmc , bg km`lfe sf φ 7 φ p , bs β (φ ) 7 > , y cm eghm mtrmvfbsm bc `btmc9 bc `btmc sb omkb bgtegkbs trmgspmrbgtb m bstms arbkubgkfms qub kerrbspeghbg mc bxtrbè uctrmvfecbtm, bstb obkoe sb om vbrfkmhe lmstmgtb lfbg bg ces `btmcbs mckmcfges. êr ce jbgbrmc cms prepfbhmhbs ûptfkms seg `ís

�

8 Mc fjumc qub pmrm bc kebkfbgtb Omcc, bste pcmgtbm huhms selrb bc sfjge hb ces pertmherbs hb kmrjm bg ces `btmcbs. `btmcbs.

RGBH.



=6

ke`pcbdms hb ce qub bc èhbce hb Hruhb-[e``brabch prbhfkb, bs per tmgte gubvm`bgtb gubvm`bgtb cm tberàm hb lmghms cm qub mrredm rbsuctmhes y bxpcfkmkfegbs `ís smtfsamkterfms. \bspbkte hb bstb ôctfè mpmrtmhe hfràdmsb m Y2 Y 2P per bdb`pce pmrm preaughfzmr.

RGBH.


2 Lmgh Lmghms ms hb Bgbr Bgbrjjàm. àm. Bcb Bcbktr ktregbs egbs Kum Kumsfc sfcflrb flrbss y Aubrt ubrtb` b`bbgtb Cfjm Cfjmhe hess

<>

2 Lmghms Lmghms hb Bgbrjàm Bgbrjàm.. Bcbktreg Bcbktregbs bs Kumsfcf Kumsfcflrb lrbss y Aubrubrtb`bgtb Cfjmhes -Mprexf`mkfûg hb rbh vmkàm4 \bkerhmghe vmkàm4 \bkerhmghe ce vfste bg bc Kmpàtuce ; Kmpàtuce ;,, supegjmès moerm y per sf`pcfkmr, ugm rbh ugfhf`bgsfegmc, fgtrehuzkmès ug petbgkfmc pbrfûhfke R ( ( r) tmg hælfc qub ge mabktb m cms secukfegbs hbc bcbktrûg cflrb (rbh vmkàm). Cm rbcmkfûg hb hfsp brsfûg < < pmrm ces `fsès bs bgtegkbs β (i) ≋ β> (i) 0 <ì y hmhe qub β (i + J) 0 β (i) , pmrm tehes ces J hb cm \\, cm rbprbsbgtmkfûg bs pubs cm hb cm jurm 2.= 2.= (Fzqufbrhm), (Fzqufbrhm), ccm`mhm rbprbsbgtmkfûg rbprbsbgtmkfûg bg zegm pbrfûhfkm. Xe`mghe bccm bc rbkfgte cf`ftmhe m cm ^TL , sb mprbkfm cm bxfstbgkfm hb fggftms secukfegbs y sb tfbgbg tehes ces vmcerbs pesflcbs hb β > (i) 9 omkfbghe mhb`ís i mhb`ís i 0 i^TL + J, ∎J ∏ \\ hb cm \\, pehbès rbbskrflfr cm rbcmkfûg hb hfspbrsfûg < ∜J) pmrm eltbgbr cm ccm`mhm rbprbsbgtmkfûg keè β> (i^TL ) 0  (i^
Afjurm 2.=4 \bcmkfûg hb hfspbrsfûg hb

bcbktregbs bcbktregbs cflrbs. Mrrflm4 Bsqub`m hb zegm pbrfûhfkm. Mlmde4 Bsqub`m hb zegm rbhukfhm (fzq.) y bsqub`m hb zegm bxtbghfhm (hkom.). Aubgtb4 Y2 Y 2P.

RGBH.



<=

-[ecukfûg mc bcbktrûg bg ug petbgkfmc pbrfûhfke4 Cm pbrfûhfke4 Cm augkfûg hb eghm hb Lceko hb ug bcbktrûg, se`btfhe m ug petbgkfmc p etbgkfmc pbrfûhfke hælfc, pubhb bxprbsmrsb keè bc hbsmrrecce hb Aeurfbr4 ui bfiºr , ίi (r) 0 (2.=)

� i

pmrm tehes ces i ces i qub qub ku`pcmg cms keghfkfegbs hb kegterge. Qmès m bgkegtrmr ces kebkfbgtbs u i tbgfbghe bg kubgtm qub cm pbrfehfkfhmh p brfehfkfhmh hbc petbgkfmc pbr`ftb rbprbsbgtmrce per su hbsmrrecce hb Aeurfbr4 R ( R J bfJ.r . ( r) 0 (2.<)

� J

Xe`mrbès mquà R J0> 0 > keè bc erfjbg hb bgbrjàm `bhfm, y ges cf`ftmès m ugm rbh ègemtû`fkm hb `mgbrm qub sfb`prb bs pesflcb bskejbr bc erfjbg hb keerhbgmhms bg bc qub bc petbgkfmc prbsbgtb sf`btràm hb fgvbrsfûg, bste bs, qub R ( ( ∜r) 0 R ( r) . [ustftufhm (2.= 2.=)) y (2.< 2.<)) bg cm bkumkfûg hb [korñhfgjbr (;.< ( ;.<), ), rbsuctm qub om hb sbr4 β> (i)



 �

∜β

ui +

J

R J u i∜J 0 >,

(2.;)

bkumkfûg qub pubhb rbbskrflfrsb omkfbghe bc km`lfe i 0 i^ TL ∜ J, heghb J bs kfbrte vbkter hb cm \\. Msà qubhm4 β> (i^TL



∜ J) ∜ β



ui^TL ∜J +

� J‼

R J ∜J u i^TL ∜J 0 >. ‼

‼

(2.8)

Cm bkumkfûg mgtbrfer (sfstb`m hb bkumkfegbs) bs kegekfhm keè bkumkfûg kbgtrmc. Bxfstbg pmrm kmhm i^ TL ugm bkumkfûg keè cm mgtbrfer pmrm kmhm J hb cm \\, per ce qub bc prelcb`m sb om kegvbrtfhe bg G prelcb`ms fghbpbghfbgtbs, uge pmrm kmhm i ^ TL . Kmhm sfstb`m hb bkumkfegbs mkepcm ces ui^T L , ui^ TL ∜J , ui^T L J , ui^ TL ∜J , . . . y kmhm uge keg secukfûg ί i ( r) 0 ui ∜Jbf(i ∜J)ºr qub tfbgb cm aer`m hb augkfûg hb Lceko . ∜

∝

^TL

‼‼

‼

^TL

^TL

J

-Mprexf`mkfûg hb bcbktregbs kumsfcflrbs4 Keg kumsfcflrbs4 Keg bc petbgkfmc pbrfûhfke hælfc y tmc qub > R J ≤ β , cm bkumkfûg kbgtrmc pubhb sf`pcfkmrsb kegsfhbrmlcb`bgtb bg kfbrtms rbjfegbs hb cm ^TL4

̿ • Cbdes hb ces pcmges hb Lrmjj heghb heghb β> (i^TL ∜ J> ) ∜ β> (i^TL ∜ J) ≩ R, ∎J 0 J> , bc su`mterfe hb (2.8 ( 2.8)) pubhb hbsprbkfmrsb y cm secukfûg bs cm hb bcbktregbs cflrbs, > β (i) 0 β (i).





• Kbrkm hb ces pcmges hb Lrmjj y sus fgtbrsbkkfegbs , pmrm kfbrte kegdugte J kegdugte J= , º º º , J` sb vbrfkm4 β> (i^TL





β>

(i^TL

∜ J ) ∜ β> (i ∜ J ) 3 R 9 f 0̿ d, ∜ J) ∜ β> (i ∜ J ) ≩ R 9 ∎J 0̿ J , ^TL

f

^TL

y (2.8 (2.8)) sb mprexf`m per4

d

f





(2.2)

f

`

∜ β

RGBH.

>

β (i^TL

∜J ) f



ui^TL ∜Jf 0

� d

R J d ∜Jf u i^TL ∜J d

9

f 0 = 0 = ,

º º º , ` .

(2.1)



<<

[elrb ug pcmge hb Lrmjj mugqub cbdes hb sus fgtbrsbkkfegbs, fgtbrsbkkfegbs, bg bc kmse bspbkfmc`bgtb sbgkfcce ` 0 < (hbjbgbrmkfûg helcb), hbspuæs hb mcjugms `mgfpucmkfegbs, bc sfstb`m mgtbrfer mrredm pmrm β cm secukfûg4 β (i) 0 β > (i)

½ |R | , J

(2.:)

keg J keg J bc bc vbkter hb cm \\ msekfmhe mc pcmge hb Lrmjj. Bstm sbgkfccm bxprbsfûg f`pcfkm hb aer`m fg`bhfmtm, cm bxfstbgkfm hb ug jmp hb bgbrjàm hb mgkourm < |R J | selrb ces < pcmges hb Lrmjj. ûbhb mhb`ís prelmrsb sfg hfkuctmh qub ∌β∌ (ii) 0 ` i ∜ J< , hb ce qub sb bxtrmb bxtrmb qub cms supbrkfbs supbrkfbs bqufbgbrjæt bqufbgbrjætfkms fkms seg pbrpbghfkucmr pbrpbghfkucmrbs bs m bstes pcmges.

 

Afjurm 2.<4 Lmghms hb bgbrjàm preoflfhm e jmps bg cm rbprbsbgtmkfûg hb zegm bxtbghfhm (fzq.) y zegm rbhukfhm (hkom.). Aubgtb4 Y2 Y 2P. -Mprexf`mkfûg -Mprexf`mkfûg hb bcbktregbs aubrtb`bgtb cfjmhes4 [b cfjmhes4 [b trmtm moerm hb mpcfkmr ces `ætehes pbrturlmtfves y mprexf`mr bc petbgkfmc kbrkmge m ces pugtes hb rbh per ug p etbgkfmc W mt (r) + ▻R ( mtû`fke pbrturlmhe, pmrm ce kumc sb bskrflb bc om`fctegfmge OW(r) 0 O R (r), W mt (r) seg Ρ g (r) y β g rbspbktfvm`bgtb, keg g ∏ T. heghb cms muteaugkfegbs y bgbrjàms hb O Bskejbès per secukfûg ugm mprexf`mkfûg fgfkfmc ίi (r) 0 K i (r) Ρ (r ∜ \), keg Ρ

∝ \

ugm ke`lfgmkfûg cfgbmc hb erlftmcbs mtû`fkes hbjbgbrmhes4 Ρ (r) 0 lg Ρg (r). Omrbès g K i (\) 0 b fiº\ hb `mgbrm qub ί i (r) mhqufbrm cm aer`m hb augkfûg hb Lceko. Mpcfkmghe bc om`fctegfmge m |ίi (r) y ùctfpcfkmghe per ⟯Ρs (r)|, rbsuctm4 βs Ρs (r) ίi (r) + Ρs (r)

⟯

|

⟦ ⟯

∝

| ▻R ( (r) |ί (r)⟦ 0 β (i) ⟯Ρ (r)| ί (r)⟦ s

i

i

9

s

∏ T, (2.?)

cm rbcmkfûg mgtbrfer, hbspuæs hb hbspbdmr β (i) y sustftufr ί i (r) per su bxprbsfûg, keghukb m4 + μ + λ (\) bfiºr ̿0> \0> β (i) 0 β s + , (2.6) ν (\) bfiºr =+ ̿0> \0>

∝ ∝

RGBH.



<;

heghb obès ccm`mhe4 ν (\) 0 Ρs (r) Ρs (r

| ∜ \)⟦ , μ 0 0 ∜ ⟯Ρ (r)| ▻R ( ( r) |Ρ (r)⟦ , λ (\) 0 ∜ ⟯Ρ (r)| ▻R ( ( r) |Ρ (r ∜ \)⟦ . ⟯

s

s

s

s

(2.=>)

(r ) , ν (\) 0 ν (\) y λ (\) 0 λ (\) 9 Hmhe qub pmrm ug gfvbc mtû`fke s bs ί (r) 0 ί (r tbgfbghe bg kubgtm cm sf`btràm hb fgvbrsfûg \ ↜ ∜\ , qubhígheges tmg sûce keg cm kegtrflukfûg hb ces vbkfges `ís prûxfès (mprexf`mkfûg mkbptmlcb), \ mkbptmlcb), \ 0 \> , y hbsprbkfmghe cm pbqubým kegtrflukfûg hb ν (\) bg bc hbge`fgmher hb (2.6 ( 2.6). ). Hfkoe tehe bste, ugm vbz kmckucmhms cms (2.=> (2.=>)) y sustftufhms bg (2.6 ( 2.6), ), sb eltfbgb gmc`bgtb4 β ( i)

≋ β ∜ μ ∜ λ (\>) s

� \>

kes (i º \> ) .

(2.==)

Msà, kmhm gfvbc mtû`fke hfskrbte erfjfgm ugm lmghm pesflcb hb bgbrjàms qub hbpbghb hb i y kuym mgkourm vfbgb hbtbr`fgmhm per λ (\> ) , qub m su vbz bs hbpbghfbgtb hbc jrmhe hb secmpm`fbgte hb cms augkfegbs hb eghm hb ces íteès vbkfges puhfbghe mpmrbkbr secmpm`fbgte secmpm`fbgte bgtrb lmghms.

RGBH.


1 Hfgí`fkm hb ces Bcbktregbs hb Lceko

<8

1 Hfgí`f Hfgí`fkm km hb hb ces ces Bcbkt Bcbktreg regbs bs hb hb Lceko Lceko [b prepegb ug èhbce è hbce sb`fkcísfke bg bc qub sb prbsupegb qub bc àghfkb hb lmghm g bs ugm kegstmgtb hb èvf`fbgte, bstíg preoflfhms per tmgte cms trmgsfkfegbs bcbktrûgfkms bgtrb lmghms. Bste supegb qub sf ces bcbktregbs seg se`btfhes m km`pes bxtbrges bcæktrfke y `mjgætfke hb m`pcftuhbs B y L , ugfaer`bs y bstítfkes keg arbkubgkfm φ , hblb ku`pcfrsb qub  φ ≤ βj, bBm ≤ β
= hβ (i) 

hi

.

(1.=)

Cm mpcfkmkfûg hb ugm aubrzm bxtbrgm A bxtbrgm A prehukfrí prehukfrí ugm bvecukfûg tb`permc hb cm bgbrjàm hbc bcbktrûg, tmc qub4 hβ (i) = hβ (i) , 0 A º v (i) 0 A º (1.<) ht  hi qub ke`lfgmhm keg hβht(i) 0 hβh(ii) hhti keghukb m4 hi A 0  .

ht

(1.;)

Cm bkumkfûg hb èvf`fbgte mgtbrfer f`pcfkm qub cm kmgtfhmh  i sb ke`pertm mgtb ugm aubrzm bxtbrgm keè ce omràm bc è`bgte cfgbmc e f`pucse hbc bcbktrûg cflrb (ce qub dustfkm bc ge`lrb hb kumsff`pucse). -Bc tbgser hb `msm4 Cm bkumkfûg hb èvf`fbgte pubhb rbbskrflfrsb bg cm aer`m `ís ke`ôg Aubrzm 0 `msm Ú mkbcbrmkfeg mkbcbrmkfeg . Hbrfvmghe cm vbcekfhmh hbc bcbktrûg rbspbkte hbc tfb`pe, bg getmkfûg hb àghfkbs, hv f f h mf 0 0 ht ht

� � � � � = hβ

 hif

0

=



d

∌ ∌i d

∌β if

∌i d = 0 <  ∌t

� d

∌β A d ∌i f ∌i d

9

f, d 0 =, <, ;,

(1.8) bg heghb sb om e`ftfhe per sbgkfccbz cm hbpbghbgkfm hb β y hb v f keg i keg i.. Bg bc ôctfè `fb`∜ = ∌β = lre cm kmgtfhmh hbgfhm keè @fd ≭ < ∌i f ∌i d , bs ccm`mhm tbgser hb `msm, hb `mgbrm qub4 A 0 @m. (1.2) Bc tbgser hb `msm bvfhbgkfm cm bxfstbgkfm hb mgfsetrepàm, bs kcmrm`bgtb sf`ætrfke, y hbpbghb bvfhbgtb`bgtb hb i hb i m m trmvæs hb β (i), qub m su vbz hbpbghb hbc petbgkfmc hb cm rbh. [f i [f i> bs ug vbkter hb eghm msekfmhe mc bxtrbè hb ugm lmghm e l erhb hb lmghm, cm bgbrjàm hbc bcbktrûg pubhb mprexf`mrsb bg terge m i m i > 2 per β

≋ β (i>) +

� f,d

(if

∜ i> ) =<  <@∜ | 0 =

f

fd

i

i>

(id

∜ i> ) . d

(1.1)

2

Mprexf`mkfûg per su hbsmrrecce hb Xmycer omstm sbjughe erhbg, tbgfbghe bg kubgtm mhb`ís qub bg bc ∌β  lerhb hb lmghm bs ∌ 0 >. i i0i>

RGBH.



<2

Msà, bc tbgser hb `msm pmrm bcbktregbs qub sb ùbvbg bg cms kbrkmgàms hb i > bs kegstmgtb, bste bs, bcbktregbs bg cms kbrkmgàms hbc bxtrbè hb lmghm (elsærvbsb qub cm bxprbsfûg < i mgtbrfer sb msb`bdm m cm lfbg kegekfhm kege kfhm bgbrjàm hbc bcbktrûg cflrb β > (i) 0 <` , km`lfmghe bg bstb kmse cm `msm hbc bcbktrûg per bc tbgser hb `msm). Cm sf`btràm hbc tbgser hb `msm msbjurm su hfmjegmcfzmkfûg 1 , hb èhe qub pubhb bskejbrsb ug sfstb`m hb bdbs prfgkfpmcbs ∜= ≭ = θ fd `√f cms rbspbkte hb ces kumcbs cms ke`pegbgtbs hbc tbgser seg @fd `f fd , sfbghe cms ccm`mhms `msms babktfvms9 . Xbgfbghe bg kubgtm bste, (1.1 ( 1.1)) qubhm keè4 <

√

<

β

<

<

≋ β (i>) +  (i<=`∜√ i>=) +  (i<<`∜√ i><) +  (i<;`∜√ i>;) , = < ; ∔ < Y ∜

`√ β β> (i)P f

y cms supbrkfbs bqufbgbrjætfkms seg bgtegkbs bcfpsefhbs hb sb`fbdbs



(1.:) .

-@evf`fbgte bg ug km`pe bcæktrfke ugfaer`b y bstítfke. Oubkes4 [ukbhb Oubkes4 [ukbhb qub cms lmghms ccbgms ge keghukbg, bg babkte, hb ( 1.= 1.=)) rbsuctm d rbsuctm d 0 ∜b f vf 0 ∜b ∜= f ∌β/∌ if 0 >, hmhe qub cm su`m sb bxtfbghb bxtfbghb m tehm cm ^TL per per cm kegsfhbrmkfûg hb lmghms ccbgms, y pubste qub β (i) 0 β (∜i), ces tær`fges hbc su`mterfe sb mgucmg per pmrbdms (i, ∜i) bg vfrtuh hb cm sf`btràm hb fgvbrsfûg hb cm ^TL . Bste pmrbkb f`pcfkmr qub omy tmgtms kmrjms gbjmtfvms (bcbktregbs) qub sb ùbvbg bg cm hfrbkkfûg ∜B hb ug km`pe bxtbrge B bxtbrge B mpcfkmhe, mpcfkmhe, keè «kmrjms pesftfvms¹ bg cm hfrbkkfûg kegtrmrfm. Msà pubs, sûcfhes keg ug gô`bre pmr hb bcbktregbs hb vmcbgkfm, sfb`prb qub ge oulfbrm secmpm`fbgte hb lmghms bg kuye kmse sbràmg keghukterbs, gbkbsmrfm`bgtb gbkbsmrfm`bgtb sbríg mfscmgtbs.

∝

∝

Kegsfhbrbès per sf`pcfkmr ugm lmghm kmsf ccbgm keg ug sûce bcbktrûg vmkmgtb vmkmgtb keg vbkter hb eghm ib , rbspbkte hb cm vmkmgtb sb hfkb qub bxfstb ug oubke, msekfmrbès m cm lmghm ug vbkter hb eghm io . Bc ke`pertm`fbgte hb cm lmghm mgtb cm prbsbgkfm hb ug km`pe bcæktrfke bxtbrge B bxtbrge B ugfaer`b ugfaer`b y bstítfke, qubhm hbskrfte per4 io 0

�� i

^ TL

∜ i 0 ∜i , b

b

(1.?)

ym qub bc su`mterfe sb mgucm per cm sf`btràm hb fgvbrsfûg hb cm ^TL. Keè

∝

β (i) bs

^ TL

kegstmgtb bg bc lerhb hb lmghm, pubhb te`mrsb keè kbre, y pmrm cm bgbrjàm «hb lmghm¹ sb tfbgb4 βo (io ) 0

�� β (i)

^ TL

∜ β (i ) 0 ∜β (i ) . b

b

b

b

(1.6)

Xbgfbghe bg kubgtm (1.? (1.?)) y cm rbcmkfûg mgtbrfer, bs bvfhbgtb usmghe ( 1.= 1.=)) qub vo (io ) 0 √ vb (ib ), y keg rbcmkfûg m cm `msm babktfvm, hb su hbgfkfûg rbsuctm Yò (io )P∜= 0 Y ∜`√b (ib )P∜= . Ce qub vfbgb m hbkfr qub bc oubke tfbgb `msm babktfvm pesftfvm, hmhe qub bg bc lerhb hb lmghm, bs bvfhbgtb qub cm hb ces bcbktregbs bs gbjmtfvm. Mpcfkmghe moerm mc oubke cm bkumkfûg (1.; (1.;)) keg km`pes B km`pes B y y L L ugfaer`bs ugfaer`bs y bstítfkes, eltbgbès4 hio 0 b B + bvo ht



Ú L,

(1.=>)

1

Bs ug obkoe lfbg kegekfhe hbc ícjblrm cfgbmc qub tehm `mtrfz sf`ætrfkm bs hfmjegmcfzmlcb, sus mutevbkterbs vbkterbs seg ertejegmcbs ertejegmcbs bgtrb sà y sus mutevmcerbs mutevmcerbs rbmcbs.

RGBH.



<1

hb heghb sb kecfjb qub ces oubkes sb ke`pertmg ke`p ertmg keè kmrjms bcæktrfkms pesftfvms keg `msm pesftfvm. êr ôctfè, pubhb hbèstrmrsb qub cm keghuktfvfhmh bcæktrfkm bg ug lmghm pmrkfmc`bgtb ekupmhm bs4 gb< ψ @∜= (i) , ϋ 0 (1.==) G

� i

heghb cm su`m sb bxtfbghb m tehes ces bstmhes ekupmhes. Cm mgtbrfer ge bs `ís qub cm jbgbrmcfzmkfûg hb (8.8 ( 8.8)) vfstm bg bc Kmpàtuce 8, pubs pmrm bcbktregbs cflrbs rbsuctm ∜ ∜ = = @ (i) 0 `b , cubje cubje pmrm bcces, bcces, (1.== 1.==)) sb rbhukb m ϋ 0 gb< ψ /G G =/`b 0 gb< ψ /`b keè kmlb bspbrmr.

 

-@evf`fbgte bg ug km`pe `mjgætfke ugfaer`b y bstítfke4 Lmde cm mkkfûg hb ug km`pe `mjgætfke L `mjgætfke L,, cm aubrzm qub mktôm selrb pertmherbs hb kmrjm bs q v Ú L, y per sbr q Ú L) º v 0 >. pbrpbghfkucmr m cm vbcekfhmh sb kegsbrvm cm bgbrjàm, bg babkte, pubs hβ ht 0( v Ces bcbktregbs sb ùbvbg bg bc bspmkfe aísfke (bspmkfe hb cms i) i ) selrb supbrkfbs bqufbgbrjætfkms sfjufbghe trmybkterfms trmybkterfms pbrpbghfkucmrbs mc km`pe. ^mrtfbghe hb ( 1.; 1.;)) y hbske`pegfbghe i pegfbghe i bg bg sus ke`pegbgtbs pbrpbghfkucmr y pmrmcbcm mc km`pe, pmrm cm ke`pegbgtb pbrpbghfkucmr r pbrpbghfkucmr r ⊢ bg bc bspmkfe rbmc, sb ccbjm m4 

r⊢ (t) 0 r⊢ (>) ∜ g Ú Y i⊢ (t) ∜ i⊢ (>)P , bL

(1.=<)

heghb g heghb g bs bs ug vbrser bg cm hfrbkkfûg hb L hb L.. Cm bxprbsfûg mgtbrfer sfjgfkm qub cm preybkkfûg preybkkfûg pbrpbghfkucmr mc km`pe hb cm ûrlftm bg bc bspmkfe rbmc, bs cm `fs`m ûrlftm hbc bspmkfe aísfke jfrmhm ώ/< rmhs bg terge m cm hfrbkkfûg hb L hb L y y ùctfpcfkmhm per ∜ /bL. \bspbkte hb cm ke`pegbgtb hb cm vbcekfhmh pmrmcbcm mc km`pe, sb bgkubgtrm qub v qub v 0  ∜= hβ/hi qub ge bs bg jbgbrmc kegstmgtb. 

Hbpbghfbghe hb cm tepecejàm tep ecejàm hb cms supbrkfbs sup brkfbs bqufbgbrjætfkms, pubhbg omlbr trmybkterfms kbrrmhms e mlfbrtms, cms prf`brms sb kcmsfkmg bg ûrlftms hb tfpe bcbktrûg e hb tfpe oubke. Cm arbkubgkfm hb bstís ûrlftms bs cm arbkubgkfm kfkcetrûg. [b hbùbstrm pmrm bccm4 φk (β, i ) 0

<ώbL/< , ∌M (β,i )/∌β



(1.=;)

heghb M (β, i ) bs bc írbm bgkbrrmhm per cm ûrlftm bg bc bspmkfe aísfke.Ke`pmrmghe cm rbcmkfûg mgtbrfer keg cm rbcmkfûg pmrm bcbktregbs cflrbs, φk ≭ bL/`b , sb fgtrehukb cm `msm babktfvm babktfvm kfkcetrûgfkm4 `√ (β, i  )

<

 ∌M (β, i ) , ≭ φ (bL 0 β, i ) ∌β <ώ k



(1.=8)



qub ge hblb kegaughfrsb keg cm `msm babktfvm hmhe qub bg bstb kmse bs ugm prepfbhmh hb tehm cm ûrlftm. Kbrkm hb ces lerhbs hb lmghms, ce qub bs hb fgtbræs pmrm sb`fkeghukterbs, rbspbkte hb ces bdbs prfgkfpmcbs (sulàghfkbs =, <, ;) hbc bcfpsefhb hb bgbrjàm kegstmgtb (1.: ( 1.:)), )), sb hbùbstrm qub4 L=< `√= + L<< `√< + L;< `√; φk 0 b . (1.=2) `√ `√ `√

�

RGBH.

=

<

;



<:

-Cm `mjgbterrbsfstfvfhmh y bc kebkfbgtb Omcc4 ^mrm vbr bc ke`pertm`fbgte hbc kebkfbgtb Omcc y hb cm `mjgbterrbsfstfvfhmh hblbès bstuhfmr bc trmgspertb hb kerrfbgtb bcæktrfkm lmde cm mpcfkmkfûg sfùctmgbm hb km`pes bxtbrges B y B y L L . [upeghrbès qub L qub L ≤ =, hb `mgbrm qub bc bcbktrûg rbkerrm sûce ugm bs ce sukfbgtb`bgtb hælfc, bs hbkfr, φ k ψ ≤ pbqubým perkfûg hb su ûrlftm kbrrmhm bgtrb kecfsfegbs. [f tbgbès qub B ⊢L, cm hbgsfhmh hb kerrfbgtb pubhb sbr bskrftm keè4

d 0 ϋ B B + ϋL (L Ú B) ,

(1.=1)

heghb ϋB y ϋL seg bskmcmrbs hbpbghfbgtbs hb cm kegkbgtrmkfûg hb pertmherbs hb kmrjm, hb su tfb`pe hb rbcmdmkfûg, hb su `msm babktfvm y hbc km`pe `mjgætfke. [bmg L 0 L i y B 0 B x f + B y d km`pes d km`pes kegstmgtbs. [b pubhb bskrflfr d bskrflfr d keè4 keè4 0

d 0 ϋ (L ) B 0

 

∜Lϋ

ϋB Lϋ L

L

ϋB

>

>

   

> > =

B x B y

.

(1.=:)

>

Rgm vbz mckmgzmhe bc ræjf`bg bstmkfegmrfe bs dy 0 >, y pmrm cm `mjgbterrbsfstfvfhmh y kebkfbgtb Omcc sb eltfbgb4

≭ dB L 0 ϋ< +∜ϋϋ< L< . (1.=?) Lmde cm mprexf`mkfûg sf`pcb hb bcbktregbs kumsfcflrbs, sfbghe φ ≭ / , sb tfbgb qub4 ύ

≭ dB

x

0

x

ϋB < L< ϋL

9

ϋ < + B

y

\O

L

x

B

L

bL `√

k

ϋB 0

gb< ψ /`√

= + ( φk ψ )

9

<

ϋL 0

bψ ϋB , `√

(1.=6)

qub mc sustftufr bg (1.=? (1.=?)) rbsuctm4 `√ , gb< ψ

ύ 0

\O 0

(1.<>)

∜=. gb

[f cms supbrkfbs bqufbgbrjætfkms, bg kegkrbte cm hb Ab`f, seg bsabrms peke pe ke hbaer`mhms, sb pubhb ccbvmr m kmle cm mprexf`mkfûg4 0

ϋ (L )

≋

gb ψ `√

 ⇜ 

0

<

 

=

∜ 2 (φ ψ )< k

<φk ψ >

ύ (L )

≋

`√ gb< ψ

≋

\O

= gb

∜<φ ψ = ∜ 2 (φ ψ )< k

k

>

� �

= + ( φk ψ )<

�∐ � <

= + ; (φk ψ )

L<,

> > =

 

(1.<=)

.

\bsuctm kcmre qub pmrm L ↜ > sb eltfbgbg ces rbsuctmhes hbc bcbktrûg cflrb. Lmde km`pes `mjgætfkes fgtbgses, φk ψ ≩ =, cm `mjgbterrbsfstfvfhmh ùbstrm ugm jrmg vmrfbhmh hb ke`pertm`fbgtes hbpbghfbghe hb cm tepecejàm hb cm supbrkfb hb Abr`f. Bg kumgte mc kebkfbgtb Omcc, pmrm km`pes fgtbgses bs vícfhm cm bxprbsfûg hb bcbktregbs cflrbs, smcve qub bg bstb kmse, sb sustftuyb cm hbgsfhmh hb kmrjm bcbktrûgfkm gb per cm hbgsfhmh hbgsfhmh hb pertmherbs hb kmrjm gbtm ( gb ∜ go ), heghb g o bs cm hbgsfhmh hb oubkes. RGBH.


: Hfgí`fkm hb cm \bh

: Hfgí Hfgí`f `fkm km hb cm \bh \bh -Cm kmhbgm ègemtû`fkm4 Kegsfhbrbès ègemtû`fkm4 Kegsfhbrbès ugm rbh ugfhf`bgsfegmc keg lmsb ègemtû`fkm y pmrí`btre hb rbh m. Bc èhbce hb mprexf`mkfûg mr`ûgfkm kegsfstb bg supegbr qub ces íteès eskfcmg mkepcmhes keg hbspcmzm`fbgte u (t) bg terge m ces pugtes hb rbh, keg vbkter hb pesfkfûg r (t) 0 \ + u (t) y hb aer`m qub u ( u (t) ≤ \> , heghb \> bs cm `bger hb cms hfstmgkfms fgtbrmtû`fkms, bs hbkfr, sus hbspcmzm`fbgtes rbspbkte hb cm pesfkfûg hb bqufcflrfe seg pbqubýes arbgtb m cms hfstmgkfms fgtbrmtû`fkms. Bg bstms keghfkfegbs bc petbgkfmc mr`ûgfke sb bskrflb keè4 R mr` mr` 0

= <

� s,c

= K c (us+c <

∜ u ), s

(:.=)

keg us bc hbspcmzm`fbgte hbc íteè s (e pcmge hb íteès) rbspbkte hb su pesfkfûg hb bqufcflrfe, sm, y K c cm kegstmgtb hb aubrzm bgtrb hes íteès sbpmrmhes ugm hfstmgkfm cm9 c ∏ T. Cm bkumkfûg hb èvf`fbgte hb kmhm íteè hb `msm @ bs hb fg`bhfmte4 h < us @ < 0 A s 0 ht

∜ ∌R ∌u

mr` mr`

0

s

�

K c (us+c

c

∜ u ), s

(:.<)

K , qub te`mghe per sf`pcfkmr sûce ces vbkfges `ís prûxfès, c 0 ½=, per ce qub K c 0 K, mh`ftb cm secukfûg bg aer`m hb èhe ger`mc keg rbcmkfûg hb hfspbrsfûg φ (i)4 f(ism φt )

∜

us 0 u i b

<

9

φ (i ) 0

8K

<

sbg s bg @

� �

= im . <

(:.;)

[b elsbrvm kcmrm`bgtb sf`btràm i ↜ ∜i y pbrfehfkfhmh bg i keg pbràehe <ώ/m, bste bs, pbrfehfkfhmh i rfehfkfhmh i ↜ i + J, ∎i, J ∏ \\, per ce qub bg mhbcmgtb kegsfhbrmrbès ces i ces i rbhukfhes rbhukfhes hφ ώ m cm ^TL. Cm vbcekfhmh hb jrupe vj 0 /hi sb mgucm bg i 0 ½ /m, ce qub smtfsamkb keghfkfûg hb Lrmjj, cubje cm eghm sb rbbdm fgtbrgm`bgtb jbgbrmghe ugm eghm bstmkfegmrfm. ^mrm im ≤ = sb tfbgb φ (i ) ≋ i m K /@ , y cm vbcekfhmh vbcekfhmh hb amsb k 0 φ/i bs bgtegkbs kegstmgtb9 msà, cm rbcmkfûg hb hfspbrsfûg sb kegekb keè rm`m mkôstfkm, sfbghe k cm vbcekfhmh hbc segfhe bg bc sûcfhe krfstmcfge. -Cm kmhbgm hfmtû`fkm4 êgjíèges hfmtû`fkm4 êgjíèges moerm bg cms `fs`ms keghfkfegbs mgtbrferbs pbre bg bstb kmse pmrm ugm rbh ugfhf`bgsfegmc keg lmsb hfmtû`fkm. Ces íteès hb `msms @ = y @ < sb bgkubgtrmg m ugm hfstmgkfm h bg cm kbchm prf`ftfvm hb pmrí`btre m. êsbbg hbspcmzm`fbgtes us y vs rbspbkte hb sus pesfkfegbs hb bqufcflrfe sm y s (m + h). Lmde cm sf`pcfkmkfûg hb vbkfges `ís prûxfès, cms bkumkfegbs hb èvf`fbgte seg4

∔

keg secukfegbs4

 

<

@ = hhtu
∜ u ) + K ( (v ∜= ∜ u ) 0 K ( (u += ∜ v ) + K ( (u ∜ v ) , s

s

us 0 u i bf(ism∜φt)

s

s

s

s

(:.8)

s

y vs 0 vi bf(ism∜φt) ,

(:.2)

y rbcmkfûg hb hfspbrsfûg4

� �

@ = + @ < φ < 0 K = @ = @ <

RGBH.

½

=

∜

8@ = @ < sbg< (@ = + @ < )<

� �� = im <

.

(:.1)



<6

Bxfstbg moerm hes èhes hb vflrmkfûg ( ½), cm secukfûg « +¹ bs cm kerrbspeghfbgtb m cm rm`m mkôstfkm, y cm secukfûg « ∜¹ bs cm ccm`mhm rm`m ûptfkm, pubs sus arbkubgkfms seg kbrkmgms m cms ûptfkms. ^mrm cm rm`m mkôstfkm ces íteès hb cm kbchm sb ùbvbg bg amsb, ce qub qufbrb hbkfr qub cm kbchm eskfcm keè ug tehe, bs lísfkm`bgtb ug èvf`fbgte bgtrb kbchms. ^mrm cm rm`m ûptfkm bg km`lfe, ces íteès vflrmg bg epesfkfûg hb amsb, bc kbgtre hb `msms sb `mgtfbgb fg`ûvfc y bc èvf`fbgte rbsuctmgtb, bs selrb tehe bg bc fgtbrfer hb cms kbchms. [f kegsfhbrírmès ugm rbh cfgbmc keg lmsb pecfmtû`fkm hb z íteès, ges bgkegtrmràmès keg = rm`m mkôstfkm y z ∜ = rm`ms ûptfkms. -Krfstmc ègemtû`fke trfhf`bgsfegmc4 [upegjmès trfhf`bgsfegmc4 [upegjmès bg bstb kmse ugm rbh trfhf`bgsfegmc ègemtû`fkm. Ces íteès tfbgbg pesfkfûg r (t) 0 \ + u (\, t) y ccm`mès ρ mc petbgkfmc hb fgtbrmkkfûg bgtrb hes íteès kumcbsqufbrm. Lmde cm mprexf`mkfûg mr`ûgfkm pehbès mprexf`mr cm bgbrjàm per su sbrfb hb Xmycer omstm sbjughe erhbg4 R mr` mr` 0

= 8

�

Yu (\)

\,\‼

∜ u (\‼)PΡYu (\) ∜ u (\‼ )P ,

(:.:)

heghb Ρfd 0 ∌ < ρ/∌x f ∌y d , keg xf (t) 0 \f + u f (\, t) 9 f, d 0 =, <, ;. Cm bxprbsfûg mgtbrfer pubhb sbr rbbskrftm keè4 R mr` mr` 0

sfbghe

= <

� �

\,\‼

\∜\‼ ) 0 θ \,\ \,\‼

H(

u (\)H (\∜\‼ )u (\‼ ), Ρ (\

\

‼‼

(:.?)

∜ \‼‼ ) ∜ Ρ (\ ∜ \‼) ,

(:.6)

ugm `mtrfz qub kegtfbgb cms fgtbrmkkfegbs bgtrb pmrbs hb íteès. Keg tehe ce mgtbrfer, cm bkumkfûg hb èvf`fbgte èvf`fbgte rbsuctm4 h < u (\) @ 0 ht<

qub tfbgb per secukfegbs4

∜ ∌ ∌R 0 u (\) mr` mr`

�

H(

\‼

\∜\‼ )u (\‼ ),

u (\, t) 0 ϲ (i) bf(iº\∜φt) ,

(:.=>)

(:.==)

keg ϲ (i) bc ccm`mhe vbkter pecmrfzmkfûg. [ustftuybghe cm secukfûg mgtbrfer bg cm bkumkfûg hb èvf`fbgte sb sfjub qub pmrm cm bxfstbgkfm hb secukfegbs, hblb ku`pcfrsb4 H(

i) ϲ (i) 0 @ φ< ϲ (i) keg

H(

i) 0

�

\) bfiº\ .

H(

(:.=<)

\

Cm `mtrfz H (i) mgtbrfer bs kegekfhm keè `mtrfz hfgí`fkm, qub bs rbmc y sf`ætrfkm. Cm rbcmkfûg mgtbrfer ge bs `ís qub ugm bkumkfûg hb mutevmcerbs, mh`ftb per secukfûg ug kegdugte hb mutevbkterbs ϲf (i) (f 0 =, <, ;) ertejegmcbs bgtrb sf, bstms secukfegbs seg ccm`mhms rm`ms mkôstfkms hmhe qub pmrm im ≤ = sb ku`pcb qub φf (i) ≋ kf (i ) i. ^mrm ug i hmhe, kmhm ugm hb cms trbs rm`ms mkôstfkms subcb ccm`mrsb pecmrfzmkfûg e èhe cegjftuhfgmc mkôstfke (CM), e lfbg, èhe trmgsvbrsmc mkôstfke (XM)9 bstm ge`bgkcmturm RGBH.



;>

bs sûce ugm mprexf`mkfûg, pubs bg jbgbrmc, cms pecmrfzmkfegbs ϲf (i) ge tfbgbg per quæ sbr pmrmcbcms gf pbrpbghfkucmrbs m i m i.. Xehms cms hb`ís prepfbhmhbs seg fhægtfkms m cms hb ugm rbh ugfhf`bgsfegmc. [b hbùbstrm (ce qub ge bs `ís qub ugm jbgbrmcfzmkfûg gmturmc hb cm kmhbgm hfmtû`fkm) qub bg bc kmse hb ug krfstmc (rbh trfhf`bgsfegmc) keg lmsb hb z íteès, bxfstbg ; z èhes pmrm kmhm i kmhm i rbpmrtfhes rbpmrtfhes bg ; rm`ms mkôstfkms y ; z ∜ ; rm`ms ûptfkms, sb omlcm bgtegkbs hb èhes CM, XM, CE, XE. -@ehes ger`mcbs y aegegbs4 Bc aegegbs4 Bc èvf`fbgte hb ces íteès hb ugm kmhbgm ègemtû`fkm pubhb hbskrflfrsb bg tær`fges hb cms ccm`mhms vmrfmlcbs ger`mcbs. [b hbgb4 ∔ =G

η (i, t) 0

@ ή (i, t) 0 ∔ G

G

∝ ∝

us (i, t) b∜fism ,

s0=

G

s0=

(:.=;)

∌u s (t) ∌t

(i, t) b∜fism .

@uctfpcfkmghe (:.< (:.<)) per b∜fism y su`mghe pmrm tehes ces vmcerbs hb s sb eltfbgb4 ∌ η (i,t) ∌t

@ ∌ ή( ή(i,t) ∌t

0

0 ή (i, t)

@ φ<

∜

(:.=8) (i ) η (i, t) .

^mrm kmhm i, cms bkumkfegbs mgtbrferbs seg cms bkumkfegbs kmgûgfkms hb ug eskfcmher mr`ûgf< = < ke hb `msm @ y y arbkubgkfm φ < (i ) 0 @ c K c sbg < icm , qub pmrm vbkfges `ís prûxfès, c 0 ½=,kefgkfhb keg cm rbcmkfûg hb hfspbrsfûg hb cm kmhbgm ègemtû`fkm. Cms secukfegbs (η, ή) sb kegekbg keè èhes ger`mcbs e prepfes hbc krfstmc, kumcqufbr èvf`fbgte èvf`fbgte hb ces íteès pubhb sbr hbskrfte per ugm su`m hb èhes ger`mcbs qub seg fghbpbghfbgtbs uges hb etres, = us (t) 0 ∔ η (i, t) bfism , (:.=2) G i∏^ TL hb bstm `mgbrm, ug èhe ger`mc bs ugm bxkftmkfûg kecbktfvm kmrmktbrfzmhm per ug vbkter hb eghm i y ugm arbkubgkfm φ. Cm bgbrjàm hb kmhm uge hb ces èhes ger`mcbs pesflcbs bstí kumgtfzmhm, y su bxprbsfûg jbgbrmc bs4

 

∝

�

�

= β (i) 0 g p (i) + <

�

 φ (

i) ,

(:.=1)

heghb g p (i) 0 >, =, <, ;, º º º sb kegekb keè gô`bre hb ekupmkfûg, y bc sulàghfkb p omkb rbabrbgkfm m kmhm ugm hb cms ;z rm`ms hfstfgtms (pmrm kmhm i) bg bc kmse jbgbrmc hb ug krfstmc keg lmsb hb z íteès. Cm kmgtfhmh  φ (i) bs bc kumgte hb bgbrjàm vflrmkfegmc y sb kegekb keè aegûg (ugm kumsfpmrtàkucm), bc gô`bre hb ekupmkfûg bs pubs ugm `bhfhm hb kumgtes aegegbs hb ug tfpe omy bxkftmhes. êr ôctfè, pmrm g p (i) 0 > sb eltfbgb cm ccm`mhm bgbrjàm hbc pugte kbre β (i) 0 =/< φ (i) hb kmhm èhe.

RGBH.


? ^repfbhmhbs Xær`fkms hb ces [ûcfhes

;=

? ^repfb ^repfbhmh hmhbs bs Xær` Xær`fkm fkmss hb ces ces [ûcfhe [ûcfhess -Qbkterbs i pbr`ftfhes4 Ge pbr`ftfhes4 Ge per p er kmsumcfhmh bstb mpmrtmhe ccbvm bc `fsè ge`lrb qub bc kerrbspeghfbgtb mc Kmpàtuce ; Kmpàtuce ;(( bg cm píjfgm =< píjfgm =<). ). [mcve bg ce qub kegkfbrgb m cms augkfegbs hb Lceko, seg mpcfkmlcbs mquà cms `fs`ms fhbms y ces `fsè pmses, bg bstb kmse, pmrm ces èhes ger`mcbs u ger`mcbs u (\, t) 0 ϲ (i) bf(iº\∜φt) . Mccà sb mpcfkmreg cms keghfkfegbs hb kegterge hb Lerg-Qeg Imr`mg pmrm kegkcufr qub ces vbkterbs i vbkterbs i pbr`ftfhes pbr`ftfhes seg hb cm aer`m (;.? ( ;.?)) y ; ώ (< ) bc vecu`bg msekfmhe m hfkoes vbkterbs bs |▻i| 0 /Q , keg Q bs bs bc vecu`bg hbc krfstmc.

-Hbgsfhmh hb èhes4 [b èhes4 [b hbgb cm hbgsfhmh hb èhes e bstmhes hb rm`m p, H p (φ) , keè bc gô`bre hb èhes per ugfhmh hb arbkubgkfm y keg arbkubgkfm bgtrb φ y φ + hφ rbabrbgtb m cm rm`m p. Keè ym pubhb fgtufrsb, sf bc krfstmc kegtfbgb ug tetmc hb G kbchms keg lmsb hb z íteès, omlríg bgtegkbs ; zG rm`ms. Ce hfkoe bg bc pírrmae mgtbrfer kefgkfhb tm`lfæg tm`lfæg keg bc mpmrtmhe -Hbgsfhmh hb bstmhes vfste bg bc Kmpàtuce ; Kmpàtuce ; ( ( bg cm píjfgm =< píjfgm =<), ), m bxkbpkfûg kcmre hb cms hfabrbgkfms elvfms. Mc fjumc qub bgtegkbs, hb cm prepfm hbgfkfûg hb hbgsfhmh hb èhes y `bhfmgtb bc kíckuce fgtbjrmc sb ccbjm m4 h[ , H p (φ) 0 ?Q ώ; (?.=) [ p (φ) |∊i φ (i)| p p

ˁ

heghb [ p (φ) sb rbbrb m cm supbrkfb φ (i) 0 ktb hb cm rm`m p. Elsærvbsb, ke`pmrmghe keg (;.=< (;.=<), ), qub mquà hbsmpmrbkb bc amkter < hb hbjbgbrmkfûg hb bspàg per p er rmzegbs bvfhbgtbs. Afgmc`bgtb rbsuctm pmrm cm hbgsfhmh hb èhes H (φ ), su`mghe tehms cms rm`ms4

H (φ) 0

�H

p (φ ) .

(?.<)

p

-@ehbce hb Hblyb hbc kmcer bspbkàke4 Bc bspbkàke4 Bc èhbce kegsfstb bg mprexf`mr tehms cms arbkubgkfms φ p (i) kerrbspeghfbgtbs m kmhm rm`m, per ; fhægtfkms b fsûtrepms hb cm aer`m4 φ 0 ki

9

> 3 i 3 iH ,

(?.;)

heghb k, pmrm i ↜ > , bs cm vbcekfhmh hbc segfhe bg bc krfstmc, y iH bs bc ccm`mhe vbkter hb eghm hb Hblyb qub sb eltfbgb bxfjfbghe qub bc gô`bre hb èhes sb kerrbspeghm keg bc tetmc ; zG (keghfkfûg hb ekupmkfûg), ;zG 0 ;

8 ; ; ώi H ?ώ ;/Q

; 0⇜ iH 0 1ώ < g,

(?.8)

keg g 0 z G /Q bs bs cm kegkbgtrmkfûg mtû`fkm. Cm bxprbsfûg mgtbrfer surjb hbc obkoe hb qub bg bc èhbce, cms supbrkfbs φ 0 ktb seg bsabrms hb rmhfe i , msà, bc gô`bre hb èhes keg arbkubgkfm bgtrb φ y φ + hφ , hb (?.= (?.=)) mpcfkmghe cms mprexf`mkfegbs, bs 4

H

H (φ ) hφ 0 ;

Q 8ώi < hi 0 ; ?ώ

⇜H

H (φ ) 0 ;

Q φ < <ώ k ;

9

> 3 φ 3 φ H ,

(?.2)

keg φ H 0 ki H cm ccm`mhm arbkubgkfm hb Hblyb. RGBH.



;<

^mrm ug eskfcmher mr`ûgfke, cm prelmlfcfhmh hb qub æstb sb bgkubgtrb bg ug bstmhe hb bxkftmkfûg g hb bgbrjàm βg 0 (=/< + g)  φ y m cm tb`pbrmturm X , vfbgb hmhm per cm hfstrflukfûg hb @mxwbcc-Lectz`mg @mxwbcc-Lectz`mg ^ g ∐ b βg/iL X ,hb aer`m qub bc gô`bre `bhfe hb aegegbs pmrm ug i ug i hmhe hmhe y hb rm`m p, ( i, p) , bxkftmhes m cm tb`pbrmturm X bs4 ∗ ∜

∝ ∝

g^ g

g0>

⟯g⟦ 0

0

∗

^ g

= /iL X

∜φ

b

g0>

∜ =,

(?.1)

qub bs duste cm hfstrflukfûg hb ^cmgki. Cm bgbrjàm pre`bhfe pmrm bc èhe (i, p) rbsuctm bgtegkbs4 ⟯R i,p⟦ 0 = + ⟯gi,p ⟦  φ p (i) , (?.:)

�

�

<

φ, (?.1 qub bg bc kmse iL X ≩  φ, ( ?.1)) keghukb m ⟯g⟦ ≋ φ/iL X , y sb eltfbgb pmrm bc pre`bhfe hb bgbrjàm, ⟯R ⟦ ≋ iL X , bs hbkfr, sb eltfbgb bc cà`ftb kcísfke. Xbgfbghe bg kubgtm cm bxprbsfûg mgtbrfer, cm bgbrjàm tetmc `bhfm bs puæs4 φkertb  φ p (i) HH (φ)  φ hφ, (?.?) ⟯R ⟦ 0 R i,p 0 ≋ φp ( )/i X φ/i X

�

�

i,p

i,p

∜

b

i

L

∜=

� ˁ

∜

b

>

p

∜=

L

bg heghb hblfhe m cm mctm hbgsfhmh hb èhes sb om mpcfkmhe pmrm i cm mprexf`mkfûg , y sb om e`ftfhe cm bgbrjàm hbc pugte kbre (per kegsfhbrmrcm erfjbg hb bgbrjàms). M pmrtfr hb cm bxprbsfûg mgtbrfer, pmrm bc kíckuce hb cm kegtrflukfûg hb cm rbh mc kmcer bspbkàke m vecu`bg kegstmgtb y per ugfhmh hb vecu`bg krbh ces sfjufbgtbs sfjufbgtbs v , mgmcfzmès ces kmses4

∝ ↜ ¸

φ, cubje sb tfbgb ⟯R ⟦ ≋ i,p iL X ≋ ≩  φ, ≋ • Mctms tb`pbrmturms4 Bg bstb kmse bs iL X ≩ rbh (;zG ) iL X hmhe qub bc gô`bre hb èhes tetmcbs bs ; zG . ^mrm k v bgtegkbs4

∝

krbh v

⟯R ⟦ 0 ;gi ≭ Q = ∌ ∌X

L.

(?.6)

Bc rbsuctmhe mgtbrfer bs duste cm cby hb Hucegj-^btft, bste bs, bc ke`pertm`fbgte hbc kmcer bspbkàke bg bc cà`ftb kcísfke bs kegstmgtb pmrm tehes ces sûcfhes. φ, cms arbkubgkfms qub fgtbrvfbgbg seg cms hb tb`pbrmturms4 ^mrm iL X ≤  φ, • Lmdms tb`pbrmturms4 (i ) ≋ k p i , per ce qub cm hbgsfhmh cm rbjfûg cfgbmc hb cms rm`ms mkôstfkms heghb φ p (i Q hb èhes pubhb sbr mprexf`mhm per H p (φ ) ≋ φ ) ⟯R ⟦ ≋ < ; ; < ; ; ∜

i

∜

<ώ



�� ˁ ∝ p0=

k p

ώ

>



�

p0=

k p

^mrm ug krfstmc fsûtrepe sb tfbgb qub p;0= =/k;p 0 ;/k, hbrfvmghe cm rbcmkfûg mgtbrfer rbspbkte hb X y hfvfhfbghe per Q , bc kmcer bspbkàke sb bxprbsm keè4 krbh 0 v

8 ; X ;1iL

ώ <  ; k;

,

(?.==)

bxprbsfûg qub bs m vbkbs kegekfhm per cby X ; hbc kmcer bspbkàke. RGBH.



;;

• Xb`pbrmturms fgtbr`bhfms4 Mquà mpcfkmrbès bc èhbce hb Hblyb, Hblyb, kegsfstbgtb keè ym sb smlb, bg mh`ftfr bc bspbktre cfgbmc ( ?.; ?.;). ). êr kegsfjufbgtb, sustftuybghe (?.2 ( ?.2)) bg (?.? (?.?), ), hfvfhfbghe per Q , , y hbrfvmghe rbspbkte hb X , sb eltfbgb4 krbh v

0

φH

; < <ώ< k; iL X <

ˁ

φ8b



>

/iL X

∜φ

/iL X

∜φ

b

∜= <



hφ.

(?.=<)

[f sb hbgb cm tb`pbrmturm hb Hblyb ΟH , per iL ΟH 0  φH ,qub rbprbsbgtm cm tb`pbrmturm m pmrtfr hb cm kumc tehes ces èhes bstíg bxkftmhes, bste bs, bc fgfkfe hbc ke`pertm`fbgte kcísfke.∔ Keg hfkom tb`pbrmturm y tbgfbghe bg kubgtm qub hb (?.8 ?.8)) bs k 0 φ H/iH 0 i L ΟH/ ; 1ώ< g. Bg augkfûg hb Ο H , y keg bc km`lfe x 0 φ/iL X , bc kmcer bspbkàke sb rbbskrflb keè4 krbh v

0 6 giL

X ; ΟH

� � ˁ >

ΟH/X

x8 b x

(bx

∜ =)< hx,

(?.=;)

rbsuctmhe qub bstí bg mkubrhe bxkbcbgtb ( = ∜ <%) keg ces hmtes eltbgfhes bxpbrf`bgtmc`bgtb. -@ehbce hb Bfgstbfg hbc kmcer bspbkàke4 Bstb èhbce aub mgtbrfer mc hb Hblyb, supegb qub tehes ces èhes ger`mcbs eskfcmg keg ugm `fs`m arbkubgkfm kegstmgtb φ B . M mctms tb`pbrmturms bc èhbce keghukb m cm cby hb Hucegj-^btft, Hucegj-^btft, sfg b`lmrje, pmrm bc rbste hb tb`pbrmturms sb hbsvàm hb ces rbsuctmhes prbhfkoes prbhfkoes per bc èhbce hb Hblyb. M pbsmr hb bcce, aub bc prf`br èhbce kmpmz hb hmr ugm bxpcfkmkfûg mc obkoe hb qub bc kmcer bspbkàke kmb m kbre kegaer`b ce omkb cm tb`pbrmturm, ce qub supuse ugm serprbsm bg cm aàsfkm kcísfkm, y mhb`ís, aub bc prf`bre bg fgtrehukfr cm fhbm hb qub cms eskfcmkfegbs `bkígfkms hb ces íteès hb ug krfstmc, bstíg kumgtfzmhms hbc `fsè èhe qub ces eskfcmherbs hb ^cmgki. Bstb èhbce subcb sbr ôtfc pmrm trmtmr cms rm`ms ûptfkms, pubs bg ùkoes kmses æstms tfbgbg ug rmgje hb vmrfmkfûg vmrfmkfûg hb arbkubgkfm bstrbkoe, per ce qub pubhbg pre`bhfmrsb hfkoms arbkubgkfms keg bc aàg hb utfcfzmr ug vmcer kegstmgtb φB . -Hfcmtmkfûg Xær`fkm. Babktes mgmr`ûgfkes4 Bc èhbce hb mprexf`mkfûg mr`ûgfkm vfste bg bc kmpàtuce mgtbrfer keghukb fgbxermlcb`bgtb m cm musbgkfm hb hfcmtmkfûg tær`fkm, bg babkte, supegjmès per sf`pcfkmr ug petbgkfmc mr`ûgfke bgtrb hes íteès hb cm aer`m < R ( ( r) 0 R > + k (r ∜ r> ) keg r cm hfstmgkfm fgtbrmtû`fkm. Hbshb cm pbrspbktfvm kcísfkm, cm prelmlfcfhmh hb qub bc sfstb`m sb bgkubgtrb keg ugm sbpmrmkfûg r m ugm tb`pbrmturm X bs ^ ( r) ∐ b R (r)/iL X . Cm sbpmrmkfûg `bhfm bs bgtegkbs4 ∗ rb R (r)/iL X ∜∗ ⟯r⟦ 0 ∗ b R (r)/iL X hr 0 r>, (?.=8) ∜∗ ∜

¸ ¸

∜

∜

rbsuctmhe qub ge hbpbghb hb cm tb`pbrmturm y per tmgte ge ùbstrm hfcmtmkfûg tær`fkm mcjugm. Bs gbkbsmrfe pubs, mýmhfr mc petbgkfmc ug tær`fge hb mgmrègfkfhmh qub hb kubgtm hb ce bxpbrf`bgtmc`bgtb elsbrvmhe. Mýmhmès mc petbgkfmc ug tær`fge tmc qub R ( ( r) 0 < ; R > + k (r ∜ r> ) ∜ j (r ∜ r> ) , cm bxprbsfûg (?.=8 (?.=8)) ge pubhb sbr rbsubctm mgmcàtfkm`bgtb keg ug petbgkfmc sf`fcmr, sfg b`lmrje bs pesflcb p esflcb mprexf`mrcm mprexf`mrcm sf mkbptmès qub bc tær`fge RGBH.



;8

mgmr`ûgfke bs pbqubýe arbgtb m i L X , bg kuye kmse bs b j(r mrredm4

; > ) /iL X

∜r

≋=+j(r

; > ) /iL X

∜r

⟯r⟦ 0 r> + 8;kj< i

LX.

,y (?.=8 (?.=8))

(?.=2)

[f cm kegstmgtb j bs pesftfvm bstm bxprbsfûg f`pcfkm qub cm sbpmrmkfûg fgtbrmtû`fkm sb fgkrb`bgtm keg cm tb`pbrmturm . Xbgbès pubs ug èhbce kcísfke qub keghukb m ugm hfcmtmkfûg tær`fkm ν ≭ C∜= hC/hX 0 ktb , keg C cm cegjftuh hb cm ùbstrm. êr supubste, bs ug èhbce sf`pcb bg bc qub ge sb bstíg tbgfbghe bg kubgtm babktes `ís ke`pcbdes keè cm mgfsetrepàm fgtràgsbkm hb ces krfstmcbs. -Kecfsfegbs aegûg-aegûg. aegûg-aegûg . ^rekbses G y prekbses R 4 Bg cm mprexf`mkfûg mr`ûgfkm ces èhes ger`mcbs seg eghms bcístfkms qub vfmdmg sfg fgtbrmkkfûg mcjugm bgtrb sà, bs keè sf ces aegegbs tuvfbrmg tuvfbrmg ug rbkerrfhe rbkerrfhe `bhfe cflrb fggfte, ge kecfsfegm kecfsfegmg. g. Cm fgkcusfûg fgkcusfûg hb tær`fges mgmr`ûgfkes bg bc petbgkfmc, hbshb bc pugte hb vfstm hb cm aàsfkm hb pmrtàkucms, hm erfjbg m prekbses bgtrb trbs aegegbs qub omg hb kegsbrvmr cm bgbrjàm y bc kumsff`pucse, bs hbkfr,  φf 0  φa y  if 0  ia +  J, heghb ces sulàghfkbs f, d sb rbbrbg m bstmhe fgfkfmc y gmc rbspbktfvm`bgtb. Ces prekbses keg J 0 > sb hbge`fgmg prekbses G, y kegsbrvmg tmgte cm bgbrjàm keè bc kumsff`pucse ge bxfstfbghe per tmgte rbsfstbgkfm ̿ >, ges bgkegtrmès tær`fkm. [f per bc kegtrmrfe bs J bs J 0 bgkegtrmès mgtb ces ccm`mhes pre kbses uìcmpp (hbc mcb`íg rbtrekbhbr) e prekbses R, bg ces qub cm bgbrjàm sb hbjrmhm pubs bs bgtekbs  i̿  ia pbrhfæghesb hfrbkkfegmcfhmh. f 0

∝

∝

∝

∝

∝ ∝

-Keghuktfvfhmh tær`fkm4 Cm tær`fkm4 Cm prepmjmkfûg hbc kmcer bg ug krfstmc sûce pubhb ccbvmrsb m kmle `bhfmgtb cms vflrmkfegbs hb sus íteès, bs hbkfr, `bhfmgtb aegegbs. êhbès pbgsmr ug èhbce kfgætfke lmsmhe bg ug jms hb aegegbs sf`fcmr mc utfcfzmhe bg bc èhbce hb Hruhb pmrm bxpcfkmr cm kegtrflukfûg kegtrflukfûg bcbktrûgfkm m cm keghuktfvfhmh tær`fkm, sfg b`lmrje, bste supegb tbgbr qub dustfkmr qub pehbès hmr mc aegûg kmríktbr kerpuskucmr, pubs æstb bs tmg sûce ug kumgte hb bgbrjàm msekfmhe m ug èhe ger`mc qub fgvecukrm m tehes ces íteès hbc krfstmc ( ugm kumsfpmrtàkucm) y per bghb bstí hbscekmcfzmhe. Lmstm keg pbgsmr qub ces èhes ger`mcbs pubhbg supbrpegbrsb pmrm kegaer`mr ug pmqubtb hb eghms keg bxtbgsfûg ▻x, bg bc bspmkfe rbmc, y ▻i bg bc bspmkfe rbkàpreke, vbrfkmghe qub ▻x▻i ∳ =. Hblbès bxfjfr qub ▻i bstæ cekmcfzmhe bg bc bspmkfe rbkàpreke, e ce qub bs ce `fsè, qub ▻i ≤ i ^ TL ∳ m ∜=, keg m cm hfstmgkfm fgtbrmtû`fkm. Bste qufbrb hbkfr qub ▻x hblb sbr ùkoe `myer qub m, per ce qub ug èhbce keè æstb sûce sbràm mpcfkmlcb bg bc kmse bg bc qub cms cegjftuhbs kmrmktbràstfkms fgvecukrmhms, keè bc rbkerrfhe cflrb `bhfe hmhe per c 0 kψ ( ψ bs bs bc tfb`pe hb rbcmdmkfûg e tfb`pe cflrb `bhfe, bs sf`pcb`bgtb bc tfb`pe bgtrb kecfsfegbs, k bs cm vbcekfhmh hbc segfhe bg bc krfstmc.) , sbmg m su vbz ùkoe `myerbs `myerbs qub m. [fg `btbrges bg hbtmccbs, lísfkm`bgtb lmde cms `fs`ms sup esfkfegbs hbc èhbce hb Hruhb (bg bc Kmpàtuce 8, bg cm píjfgm =2 =2), ), sb hbùbstrm pmrm cm kegtrflukfûg hb cm rbh m cm keghuktfvfhmh tær`fkm, δ rbh , omkfbghe ug lmcmgkb hb ude, qub4 δrbh 0

= rbh < k k ψ, ; v

(?.=1)

y keè pubhb vbrsb, cm bxprbsfûg bs ùy sf`fcmr m cm eltbgfhm pmrm cm keghuktfvfhmh tær`fkm per bcbktregbs, δbc, `bhfmgtb bc èhbce hb Hruhb. ûbhb kegsuctmr per bdb`pce Y2P pmrm eltbgbr ces hbtmccbs mkbrkm hb bstb rbsuctmhe.

RGBH.



;2

-Ke`pertm`fbgte hb cm keghuktfvfhmh arbgtb m cm tb`pbrmturm4 M tb`pbrmturm4 M lmdms tb`pb c , qubhm hbtbr`fgmhe per cm purbzm rmturms, X ≤ ΟH , bc rbkerrfhe cflrb `bhfe hb aegegbs c, hbc krfstmc, pubs ces aegegbs ge tfbgbg cm bgbrjàm sukfbgtb keè pmrm suarfr prekbses R. ∐ X ; bgtegkbs δrbh ∐ X ;. Hmhe qub k rbh v

R mqubcces ≤ ΟH , ke`fbgzmg m suarfr prekbses Mc sulfr cm tb`pbrmturm, mugqub môg keg X ≤ mqubcces ΟH/X . [f mkbptmès aegegbs keg arbkubgkfm φ ≋ φH , y bg pre`bhfe, su gô`bre bs ⟯g⟦R ≋ b qub bc rbkerrfhe cflrb `bhfe bs fgvbrsm`bgtb preperkfegmc mc gô`bre `bhfe hb aegegbs, ∜= , kegkcufès qub δ ≋ X ; b ΟH/X . bste bs, c ∐ ⟯g⟦R rbh ∜

∜

^mrm X 7 ΟH , kmsf cm tetmcfhmh hb aegegbs suarbg prekbses R, cubje ⟯g⟦R ≋ ⟯g⟦ ≋ iL X /φ, ≋ ktb, rbsuctm qub δ rbh ∐ X ∜=. per ce qub c ∐ X ∜= . Keè m bsms tb`pbrmturms bs k rbh v

Bstes rbsuctmhes bstíg bg lubg mkubrhe keg ce elsbrvmhe bxpbrf`bgtmc`bgtb bxpbrf`bgtmc`bgtb pmrm sûcfhes krfstmcfges.

Afjurm ?.=4 \bprbsbgtmkfûg mprexf`mhm hb cm kegtrflukfûg hb cm rbh m cm keghuktfvfhmh tær`fkm arbgtb m cm tb`pbrmturm.

RGBH.


6 Hbabktes ûgtumcbs hb cm \bh

;1

6 Hbab Hbabkt ktes es ûg ûgtu tumc mcbs bs hb hb cm \bh \bh Keè pubhb supegbrsb, ge bxfstbg krfstmcbs pbrabktes, bste hblfhe m qub keè bg ùkoms etrms rm`ms hb cm aàsfkm, cm bstrukturm krfstmcfgm purm ge bs `ís qub ugm fhbmcfzmkfûg e sf sb qufbrb ug kmse cà`ftb. Bg jbgbrmc ces krfstmcbs prbsbgtmg vmrfes tfpes hb f`pbrabkkfegbs e hbabktes qub mabktmg m ùkoms hb sus prepfbhmhbs aàsfkms y `bkígfkms, y tm`lfæg fguybg fguybg bg mcjugms prepfbhmhbs hb ces `mtbrfmcbs m gfvbc hb mpcfkmkfûg bg fgjbgfbràm tmc keè cm kmpmkfhmh hb aer`mr mcbmkfegbs bg aràe, cm keghuktfvfhmh bcæktrfkm, cm kerresfûg, btk. [f bc hbabkte bg ug krfstmc bs hbc erhbg hbc pmrí`btre hb rbh, bstmrbès omlcmghe hb hbabkte pugtumc e hb hf`bgsfûg kbre. [f bc hbabkte sb bxtfbghb m ce cmrje hb ugm càgbm hbc krfstmc, æstb sb kegekb keè hbabkte cfgbmc e hb hf`bgsfûg uge. V per ôctfè, sf bc hbabkte sb prbsbgtm bg tehm su supbrkfb, omlcmès hb hbabkte supbrkfmc e hb hf`bgsfûg hes. Mquà ges kbgtrmrbès bg ces hbabktes pugtumcbs, omkfbghe ofgkmpfæ bg ces krfstmcbs fûgfkes. Bxfstbg vmrfes tfpes hb hbabktes pugtumcbs4 • Íteès Íteès fgtbrstfkfmcbs4 Bs ug íteè kecekmhe bg ug cujmr qub ger`mc`bgtb ge cb kerrbspeghb, ekupmghe bc bspmkfe bgtrb pugtes hb rbh e fgtbrstfkfes. Mpmrbkbg keg `ís amkfcfhmh bg rbhbs m`pcfms y peke ke`pmktms, y keè bs cûjfke, ces fegbs hb `bger vecu`bg tfbgbg `ís amkfcfhmh pmrm ekupmr fgtbrstfkfes qub ces hb `myer vecu`bg. • Qmkmgtbs hb rbh 4 Keè fghfkm su ge`lrb, ugm vmkmgtb bs cm musbgkfm hb ug íteè e fûg hb su pesfkfûg ger`mc bg cm rbh. [f bc krfstmc bs kevmcbgtb, cm musbgkfm hb ug íteè bcæktrfkm`bgtb bcæktrfkm`bgtb gbutre ge prehukb returm hb cm gbutrmcfhmh hb kmrjm hb cm rbh, bg km`lfe, bg krfstmcbs fûgfkes, sb re`pb cm gbutrmcfhmh hb kmrjm, per ce qub bstes hbabktes subcbg mpmrbkbr per pmrbs. [b tfbgb msà4 

Hbabktes hb [koettiy4 M ugm vmkmgtb hb fûg pesftfve cb kerrrbspeghb etrm vm-

kmgtb hb fûg gbjmtfve, `mgtbgfæghesb hb bstb èhe cm gbutrmcfhmh hb kmrjm. Bstes hbabktes subcbg mpmrbkbr `myer`bgtb bg ces omcures mckmcfges.  Hbabktes Hbabktes hb Arbgibc4 Arbgibc4 Rg fûg hbdm su pesfkfûg hb rbh y ekupm ugm pesfkfûg fgtbrstfkfmc. Kcmrm`bgtb sb `mgtfbgb cm gbutrmcfhmh hb kmrjm. Bstes hbabktes mpmrbkbg `myer`bgtb bg ces omcures hb pcmtm.

Afjurm 6.=4 Hbabktes hb [koettiy y hb Arbgibc bg ugm rbh lfhf`bgsfegmc. RGBH.



;:

• Íteès bxtrmýes4 [eg íteès hfstfgtes m ces hbc krfstmc qub sb sftômg bg pesfkfegbs fgtbrstfkf fgtbrstfkfmcbs mcbs e lfbg ekupmghe pesfkfegbs pesfkfegbs hb rbh. êr ce jbgbrmc jbgbrmc su bstruktur bstrukturmm bcbktrûgfkm bs f`pertmgtb bg bc sbjughe kmse y su vecu`bg bg bc prf`bre. Hblbès bspbrmr ug kfbrte gô`bre hb hbabktes pugtumcbs bg ug krfstmc bg bqufcflrfe tær`fke qub mhb`ís mu`bgtb keg cm tb`pbrmturm. Hbpbghfbghe hb cm bgbrjfm hb aer`mkfûg hb ces hbabktes, prbhe`fgmrí ug tfpe u etre hb bcces. Bc gô`bre hb hbabktes qub omy bg ug sûcfhe bg bqufcflrfe tbrèhfgí`fke sb pubhb fgkrb`bgtmr hb vmrfms `mgbrms. Rge hb bccms bs bc tb`pcmhe, prekbhf`fbgte per bc qub sb lmdm rípfhm`bgtb cm tb`pbrmturm hb ug `mtbrfmc hbshb tb`pbrmturms bcbvmhms. Etrm pesflfcfhmh bs utfcfzmr le`lmrhbe (rmhfmkfûg) keg pmrtàkucms gukcbmrbs, tmcbs keè pretegbs e gbutregbs, qub pubhbg hbspcmzmr ces íteès hbc krfstmc per kecfsfûg hfrbktm e per babkte hbc kmcbgtm`fbgte cekmc. Rgm tbrkbrm pesflfcfhmh bs cm hbaer`mkfûg bg aràe hbc krfstmc. -Kegkbgtrmkfûg hb hbabktes4 Cm prelmlfcfhmh hb qub bg mcjôg cujmr hb cm rbh omym ugm vmkmgtb bg bqufcflrfe tær`fke m cm tb`pbrmturm X , vfbgb hbtbr`fgmhm per bc amkter hb Lectz`mg b Bv/iL X , heghb B v bs cm bgbrjàm gbkbsmrfm pmrm ccbvmr ug íteè hbshb bc cujmr bg cm rbh hbgtre hbc krfstmc omstm ug cujmr bg cm supbrkfb. [f bxfstbg G íteès, íteès, bc gô`bre g hb vmkmgtbs bg bqufcflrfe vfbgb hmhm per cm rmzûg hb ces cujmrbs vmkmgtbs m ces ekupmhes, Bv g . 0 b IL X (6.=) ∜

∜ ∜ g

G

Bg bc kmse bg qub bs g ≤ G, cm mgtbrfer pubhb mprexf`mrsb per4 g G

≋b

Bv IL X

.

(6.<)

Bg krfstmcbs fûgfkes, cm aer`mkfûg hb pmrbs hb vmkmgtbs qub `mgtfbgbg mc krfstmc bcæktrfkm`bgtb gbutre m bskmcm cekmc, `bhfmgtb ug kíckuce bstmhàstfke eltbgbès4 g G

≋b

Bp

,

(6.;)

heghb B p p bs cm bgbrjàm hb aer`mkfûg hbc pmr. Bg bc kmse hb hbabktes hb Arbgibc, bc gô`bre hb æstes bg bqufcflrfe tær`fke rbsuctm4 ∔ ∜ BF g ≋ G G ‼ b


;?

heghb qubhm kcmre qub t + + t∜ 0 =. \bsuctm bvfhbgtb qub cm prbsbgkfm hb vmkmgtbs fgkrb`bgtm cm keghuktfvfhmh fûgfkm, pubs cm bgbrjàm gbkbsmrfm pmrm èvbr ugm vmkmgtb m trmvæs hbc krfstmc bs ùkoe `bger qub cm gbkbsmrfm pmrm èvbr ug fûg hb su pesfkfûg hb rbh bg ug krfstmc fûgfke pbrabkte. Bg cm jurm 6.< jurm 6.< sb sb ùbstrmg ces rbsuctmhes hb cm keghuktfvfhmh fûgfkm hbc LrMj pmrm trbs ùbstrms, ugm hb bccms ùy purm, y cms etrms hes keg f`purbzms hb KhLr < m hfstfgtms kegkbgtrmkfegbs. Keè pubhb vbrsb, cm trbs ùbstrms kefgkfhbg m tb`pbrmturms bcbvmhms, mu`bgtmghe bxpegbgkfmc`bgtb bxpegbgkfmc`bgtb cm keghuktfvfhmh omstm kbrkm hbc pugte hb ausfûg (8;2 ´K) heghb bc krfstmc sb vubcvb ug càqufhe fûgfke keghukter.

Afjurm 6.<4 Keghuktfvfhmh fûgfkm hb trbs ùbstrms hb LrMj, ugm ùbstrm purm y hes ùbstrms keg f`purbzms hb KhLr< . Aubgtb Y= Y=P.

Cm rbjfûg hb tb`pbrmturm bcbvmhm sb hbge`fgm zegm fgtràgsbkm, ym qub keè pubhb vbrsb bg cm jurm, bg cm rbjfûg hb tb`pbrmturm lmdm lm dm cm keghuktfvfhmh prbsbgtm ugm kcmrm hbpbghbgkfm keg cms f`purbzms per ce qub hfkom zegm bs ccm`mhm zegm bxtràgsbkm. Bg cm zegm fgtràgsbkm cm keghuktfvfhmh pubhb bxprbsmrsb keè4 B

ϋ 0 Mb iL X ,

(6.1)

heghb cms kegstmgtbs M y B hbpbghbg hbc krfstmc fûgfke bg pmrtfkucmr. Bg cm zegm bxtràgsbkm sb eltfbgb ugm pbghfbgtb qub bs kmsf cm `ftmh hb cm hb cm zegm fgtràgsbkm y qub peke hbpbghb hbc tfpe hb f`purbzm utfcfzmhe, mugqub ϋ sà qub hbpbghb hb cm kegkbgtrmkfûg hb cm `fs`m. [b om ke`prelmhe per bdb`pce qub pmrm bc GmKc keg f`purbzms hb Km, cm keghuktfvfhmh bs preperkfegmc m cm kegkbgtrmkfûg hb Km. ûbhb kegkcufrsb pubs, qub m lmdm lm dm tb`pbrmturm sûce kegtrfluyb m cm keghuktfvfhmh ugm pbqubým armkkfûg hb ces fegbs hbc krfstmc, hblfæghesb kmsf bxkcusfvm`bgtb m cm prbsbgkfm hb f`purbzms. RGBH.



;6

-Kbgtres hb kecer4 Jbgbrmc`bgtb ces krfstmcbs m tb`pbrmturm m`lfbgtb seg bcæktrfkm`bgtb mfscmgtbs, y tm`lfæg, seg trmgspmrbgtbs m cm cuz. Cm trmgspmrbgkfm bs hblfhm m cm musbgkfm hb trmgsfkfegbs (bcbktrûgfkms, vflrmkfegmcbs e hb kumcqufbr etre tfpe) bg bc bspbktre vfsflcb4 hb :8> g` (:,: bQ) m ;1> g` (;,2 bQ). [f bste ge sukbhb, bc krfstmc prbsbgtm kecer mc mlserlbr cuz, y sf bstm mlserkfûg ge bs ùy mkusmhm, bc kecer qub ùbstrm bc krfstmc bs bc hb cm cuz qub trmgs`ftb. êr bdb`pce, ces `btmcbs ge trmgs`ftbg cuz per ce qub tm`peke cm rbbdmg y prbsbgtmg lrfcce, bc hfm`mgtb pure tfbgb ugm lrbkom bgbrjætfkm hb 2,8 bQ, per ce qub bs trmgspmrbgtb m cm cuz vfsflcb, pubhb sfg b`lmrje kecerbmrsb `bhfmgtb frrmhfmkfûg, ce qub prehukb hbabktes bg cm rbh krfstmcfgm. Mcjuges krfstmcbs pubhbg kecerbmrsb mýmhfbghe f`purbzms qub pesbmg bstmhes bxkftmhes bg cm rbjfûg hbc vfsflcb, etres `bhfmgtb frrmhfmkfûg (gukcbmr, rmyes _ e λ ) ce qub prehukb hbabktes qub erfjfgmg qub bcbktregbs e oubkes sb cekmcfkbg y mpmrbzkmg trmgsfkfegbs m bstmhes bxkftmhes bg cm rbjfûg vfsflcb. [b ccm`m kbgtres hb kecer e kbgtres A (hbc mcb`íg amrlbg4 kecer) m ces hbabktes hb cm rbh qub mlserlbg cuz bg bc bspbktre ûptfke (`ís jbgbrmc qub bc vfsflcb, pubs bc bspbktre ûptfke kegtfbgb mc vfsflcb). Bstes hbabktes bstíg rbcmkfegmhes keg bcbktregbs cfjmhes m vmkmgtbs hb cm rbh krfstmcfgm. Cm kmrjm pbrhfhm per cm prbsbgkfm hb ugm vmkmgtb hb ug fûg gbjmtfve, pubhb sbr gbutrmcfzmhm per uge e vmrfes bcbktregbs qub qubhmg cfjmhes per ug petbgkfmc tetmc babktfve pesftfve, bstes bcbktregbs prbsbgtmg ug bspbktre hb bgbrjàms hfskrbte (keè bg bc kmse hb ug bcbktrûg bg ugm kmdm trfhf`bgsfegmc hb petbgkfmc fggfte) qub pubhb jbgbrmr trmgsfkfegbs hmghe cujmr m càgbms hb mlserkfûg ûptfkm. êr ce jbgbrmc bstms mlserkfegbs subcbg bstmr aubrm hbc bspbktre vfsflcb, bg ces omcures mckmcfges per bdb`pce, subcbg prbsbgtmrsb bg bc uctrmvfecbtm. uctrmvfecbtm. Bxfstbg etres kbgtres hb kecer bg ces omcures mckmcfges keg prepfbhmhbs bspbktreskûpfkms fgtbrbsmgtbs, per bdb`pce, hes kbgtres A mhymkbgtbs kegaer`mg ug kbgtre @, sf bxfstbg \ , ug kbgtre A M bs ug kbgtre A bg bc trbs kbgtres A mhymkbgtbs, omlcmès hb ug kbgtre \, qub uge hb ces sbfs fegbs pesftfves qub rehbmg m cm vmkmgtb sb km`lfm per etre fûg mckmcfge (bg bc IKc per bdb`pce sb km`lfm ug fûg I + per ug fûg Km+ ).

Afjurm 6.;4 \bprbsbgtmkfûg bsqub`ítfkm hb kbgtres hb kecer bg ugm rbh lfhf`bgsfegmc. Bg m) ug kbgtre

@ ,

RGBH.

bg l) ug kbgtre

\.


=> Kmrmktbràstfkms Aughm`bgtmcbs ughm`bgtmcbs hb ces @mtbrfmcbs [b`fkeghukterbs y @mjgætfkes

8>

=> Kmrmktbrà Kmrmktbràstfkm stfkmss Aughm` Aughm`bg bgtmcb tmcbss hb ces @mtbrfmc @mtbrfmcbs bs [b`fkeghukterbs y @mjgætfkes [b`fkeghukterbs. Rg sb`fkeghukter bs ug mfscmgtb (m X 0 > I) keg ug jmp hb bgbrjàm B j ≏ = bQ keè pmrm qub m tb`pbrmturms gftms ug kfbrte gô`bre mprbkfmlcb hb bcbktregbs pmsbg hb cm LQ m cm LK, pbr`ftfbghe msà cm kerrfbgtb bcæktrfkm. -Kegkbgtrmkfûg hb pertmherbs bg bqufcflrfe Xær`fke4 Rgm Xær`fke4 Rgm prepfbhmh hb ces sb`fkeghukterbs qub ge sb elsbrvm bg ces `btmcbs, bs qub cm keghuktfvfhmh bcæktrfkm pubhb mctbrmrsb bg ùkoes erhbgbs hb `mjgftuh mýmhfbghe kmgtfhmhbs hb etres bcb`bgtes bcb`bgtes hbge`fgmhes jbgærfkm`bgtb jbgærfkm`bgtb keè f`purbzms. [eg æstms cms qub hbtbr`fgmg bc kmríktbr bcbktrûg e oubke hb cm keghukkfûg. Xehm cm bcbktrûgfkm (hfehes, trmgsfsterbs, kæcucms aetevectmfkms, btk) sb aughm`bgtm bg bstm prepfbhmh kegekfhm keè hepmhe. [b hbùbstrm qub cm kegkbgtrmkfûg hb bcbktregbs, g, bg cm LK, y hb oubkes, p, bg cm LQ, hb ug sb`fkeghukter m cm tb`pbrmturm X , vfbgb hmhm per4 βv ∗ Hk (β) a A H (β, X ) hβ 9 p 0 Hv (β) Y= ∜ a A H (β, X )P hβ, (=>.=) g 0 βk ∜∗

ˁ

ˁ

heghb βk y βv seg seg cms bgbrjàms hb ler lerhb hb cm LK y hb cm LQ, y cms ;/< Hk,v (β) 0 (`k,v) /< ώ< < |β ∜ βk,v | :, seg cms rbspbktfvms hbgsfhmhbs hb bstmhes keg √ 0 ; `√ `√ `√ , cm `msm babktfvm `bhfm jbe`ætrfkm hb ces mutevmcerbs hbc tbgser `k,v = < ; hb `msm hb cm LK y hb cm LQ ? . √

∔

∔

[f bc sb`fkeghukter vbrfkm qub βk ∜ ², ² ∜ βv ≩ iL X , bg kuye kmse sb hfkb qub bs ge hbjbgbrmhe, sb pubhb omkbr use hb cm bstmhàstfkm hb @mxwbcc-Lectz`mg bg cujmr hb cm hb Abr`f-Hfrmk, pmrm eltbgbr4 (βk ∜²)

∜

g 0 G k (X ) b

iL X

(²∜βv )

∜

9

p 0 G v (X ) b

iL X

,

(=>.<)

bg cms qub ces amkterbs G k (X ) y G v (X ) , m vbkbs ccm`mhes hbgsfhmhbs hb bstmhe babktfvms hb cm LK y hb cm LQ rbspbktfv rbsp bktfvm`bgtb m`bgtb , `mrkmg cms kegkbgtrmkfegbs kegkbgtrmkfegbs hb bcbktregbs y oubkes bg bc sb`fkeghukter ge hbjbgbrmhe, y vfbgbg hmhes per4 G k,v k,v 0

∔

�

√ iL X `k,v <=/; ώ <

�

;/<

.

(=>.;)

∜βk

Bc prehukte hb cms (=>.< (=>.<), ), gp 0 G k G v b
Elsærvbsb qub bstm hbgsfhmh ge bs `ís qub ugm jbgbrmcfzmkfûg hfrbktm hb cm eltbgfhm pmrm bcbktregbs cflrbs bg ugm mprexf`mkfûg pmrmlûcfkm bg cm qub ` √ 0 ktb . � √ �;/< ∝G � √ √ √ �=/< ? [f kegtrfluybg G lmghms hb bcbktregbs e hb oubkes, sb tbghrí `k,v 0 f0= `= `< `; f

RGBH.



8=

(sulàghfkb f hb fgtràgsbke). @uctfpcfkmghe bgtrb sf cms ( =>.< =>.<)) (cby hb mkkfûg hb `msms) y dugte keg (=>.; ( =>.;), ), pmrm ces sb`fkeghukterbs fgtràgsbkes sb ccbjm aíkfc`bgtb m qub4 gf 0

∜βk

∔

G k G v b
9

² 0 β v +

` √ = ; βj + iL X cg cg v√ `k < 8

(=>.8)

-[b`fkeghukterbs Hepmhes4 Æstes prbsbgtmg f`purbzms sustftukfegmcbs bcæktrfkm`bgtb mktfvms qub mpertmg bcbktregbs e oubkes, per ce jbgbrmc bcb`bgtes hbc jrupe Q e FFF m ces qub sb cbs ge`lrm f`purbzms hmherms e mkbpterms rbspbktfvm`bgtb, sb omlcm bgtegkbs hb sb`fkeghukterbs hb tfpe g e tfpe p. Cm bgbrjàm hb fegfzmkfûg y bc rmhfe hb cms f`purbzms hmherms e mkbpterms (sulàghfkbs h y m), sb eltfbgb hb cms bkumkfegbs hbc íteè hb ofhrûjbge mhmptmhms mc `bhfe `mtbrfmc4 `√ `b βh,m 0 β> mh,m 0 √ im > , 9 (=>.2) < `b i ` keg ` √ cm `msm babktfvm babktfvm hbc bcbktrûg bg bc `bhfe, i bs cm pbr`ftfvfhmh rbcmtfvm hbc `bhfe, < 8 <  m> 0 /`b b bs bc rmhfe hb cm prf`brm ûrlftm Leor, y β> 0 `b b /<< 0 =;, 1 bQ cm bgbrjàm hb fegfzmkfûg hbc íteè hb ofhrûjbge bg su bstmhe aughm`bgtmc. Cms f`purbzms hmherms bgtegkbs «hmríg¹ bcbktregbs m cm LK, `fbgtrms qub cms mkbpterms «mkbptmríg¹ bcbktregbs hb cm LQ (jbgbrmghe oubkes). Hb bstb èhe, tmgte cms hmherms keè cms mkbpterms, kegtrfluybg kegtrfluybg m cm keghukkfûg krbmghe rbspbktfvm`bgtb rbspbktfvm`bgtb ug gubve gfvbc bgbrjætfke βH m ugm bgbrjàm βh per hblmde hb cm LK, y ug gfvbc βM m ugm bgbrjàm βm per bgkf`m hb cm LQ. @bhfmgtb cm `bkígfkm bstmhàstfkm sb hbhukb aíkfc`bgtb pmrm cm kegkbgtrmkfûg hb f`purbzms hmherms y mkbpterms fegfzmhms m ugm tb`pbrmturm X , qub4 + G H 0

G H (²∜βH )/iL X

= +
9

∜0 G M

G M , = + < b(βM ²)/iL X ∜

(=>.1)

heghb G H y G M seg cms kegkbgtrmkfegbs tetmcbs hb cms f`purbzms kerrbspeghfbgtbs. Cm kegkbgtrmkfûg kegkbgtrmkfûg hb bcbktregbs y oubkes, g (X ) y p (X ) , pubhb omccmrsb m pmrtfr hb cm ccm`mhm keghfkfûg hb bcbktregbutrmcfhmh bcbktregbutrmcfhmh g + G M∜ 0 p + G H+ , lmstm pmrm bcce keg sustftufr cms (=>.= ( =>.=)) y cms (=>.1 (=>.1)) bg hfkoms keghfkfegbs, bste pbr`ftb hbspbdmr ² pmrm ccbvmrce m (=>.= ( =>.=)) qub sfg `ís mrredm cms bxprbsfegbs luskmhms9 sf sb trmtm hb sb`fkeghukterbs ge hbjbgbrmhes, bg cujmr hb cms (=>.= ( =>.=)) pubhbg usmrsb bvfhbgtb`bgtb cms (=>.< ( =>.<). ). Hbdmghe mtrís ces hbtmccbs kumgtftmtfves, sb elsbrvmg trbs zegms hfabrbgkfmhms. ^mrm ces sb`fkeghukterbs tfpe g, m tb`pbrmturms ùy lmdms ces bcbktregbs hb cm LK prekbhbg hb cm fegfzmkfûg hb f`purbzms, bstm zegm bs ccm`mhm ræjf`bg hb fegfzmkfûg, mc sulfr cm tb`pbrmturm, m ugm tb`pbrmturm hb smturmkfûg X s , tehms cms f`purbzms bstíg ym fegfzmhms, bs bgtegkbs bgtegkbs g ≋ G H y sb bstí bg bc ræjf`bg hb smturmkfûg. M `myerbs tb`pbrmturms, m pmrtfr hb cm tb`pbrmturm fgtràgsbkm X f , ces bcbktregbs prekbhbgtbs hb LQ supbrmg m ces prekbhbgtbs hb cms f`purbzms, bstmès gmc`bgtb bg bc ræjf`bg fgtràgsbke, heghb g ≋ g f . Ces mrju`bgtes seg mgícejes pmrm bc kmse hb sb`fkeghukterbs tfpe p. -Keghuktfvfhmh bcæktrfkm4 Cm bcæktrfkm4 Cm keghuktfvfhmh bg sb`fkeghukterbs ge hbjbgbrmhes, bs cm su`m hb cms kegtrflukfegbs hb bcbktregbs bg LK y hb oubkes bg cm LQ. Keg ψ b y ψ o cms `bhfms hb ces tfb`pes hb rbcmdmkfûg hb ces bcbktregbs y oubkes hb cms lmghms kerrbspeghfbgtbs, y hbgfbghe cms èvfcfhmhbs ² g ≭ bψ b/`k y ² p ≭ bψ o/`v , rbsuctm pmrm cm keghuktfvfhmh4 √

ϋ 0 bg² g + bp² p .

RGBH.

√

(=>.:)



8<

[b elsbrvm qub m lmdms tb`pbrmturms cms èvfcfhmhbs bstíg jelbrgmhms per cms f`purbzms `fbgtrms qub m mctms tb`pbrmturms ce bstíg per aegegbs sbjôg ² f`p ∐ X ;/< y ² aeg 0 X ;/< . Msà pubs, bg bc ræjf`bg fgtràgsbke, m mctms tb`pbrmturms, m pmrtfr hb cm prf`brm hb cms (=>.8 =>.8), ), hb cms (=>.; ( =>.;)) y hb (=>.: (=>.:), ), sb bgkubgtrm pmrm cm keghuktfvfhmh bcæktrfkm4 ∜

∐

ϋ

�

X /< b ;

/
∜βj

�

;/<

∜

X

0 b

/
∜βj

,

(=>.?)

bxprbsfûg qub pbr`ftb hb ug èhe ùy sf`pcb, hbtbr`fgmr bc jmp hb bgbrjàm hb ug sb`fkeghukter `bhfmgtb `bhfhms sfstb`ítfkms hb cm keghuktfvfhmh bg augkfûg hb cm tb`pbrmturm.

@mjgbtfsè. [b hbgb cm `mjgbtfzmkfûg @ `mjgbtfzmkfûg @ keè keè bc vbkter è`bgte è`bgte `mjgætfke ² per ugfhmh hb vecu= `bg @ `bg @ 0 Q Q ². Cm rbcmkfûg @ rbcmkfûg @ 0 ϊ O, keg O keg O ug ug km`pe `mjgætfke bxtbrge mpcfkmhe m ug `bhfe `mtbrfmc, kcmsfkm m hfkoes `bhfes sbjôg bc vmcer hbc kebkfbgtb ϊ6 kegekfhe keè suskbptflfcfhmh `mjgætfkm, msà, sb omlcm hb `mtbrfmcbs hfm`mjgætfkes sf ϊ 3 >, pmrm`mjgætfkes sf ϊ 7 > y abrre`mjgætfkes pmrm ϊ ↜ ∗ bg kuye kmse pubhb bxfstfr `mjgbtfzmkfûg hfstfgtm hb kbre fgkcuse bg musbgkfm hb km`pe `mjgætfke. -Hfm`mjgbtfsè4 Xehes -Hfm`mjgbtfsè4 Xehes ces kubrpes ùbstrmg ugm kegtrflukfûg hfm`mjgætfkm hblfhe m ces bcbktregbs cfjmhes m sus íteès. Ces bcbktregbs bg ûrlftm bg terge m ces gôkcbes rbspeghbg m ug km`pe `mjgætfke mpcfkmhe keg ug è`bgte `mjgætfke bg sbgtfhe epubste mc hbc km`pe (per cm cby hb Cbgz). [fg `btbrges bg hbsmrrecces, sbjôg ug èhbce kcísfke, bc è`bgte hfm`mjgætfke hfm`mjgætfke hb ugm ûrlftm bcbktrûgfkm hb rmhfe r pbrpbghfkucmr pbrp bghfkucmr m ugm fghukkfûg `mjgætfkm L `mjgætfkm L mpcfkmhm mpcfkmhm , vfbgb hmhe per4

∝

< <

² 0

r L , ∜ b8`

(=>.6)

b

y pmrm cm suskbptflfcfhmh hb ug kegdugte hb íteès sb tfbgb qub4 ϊ

≋∜

²> gTb < r <

⟯⟦

1`b

∳ ∜=>∜1T,

(=>.=>)

sfbghe g cm kegkbgtrmkfûg mtû`fkm y T bc bc gô`bre mtû`fke. Bc hfm`mjgbtfsè hfm`mjgbtfsè ce prbsbgtmg tehes ces kubrpes sfg bxkbpkfûg, mugqub subcb vbrsb bg`mskmrmhe per bc pmrm`mjgbtfsè y bc abrre`mjgbtfsè qub seg ùkoe `myerbs. -^mrm`mjgbtfsè4 Bstb -^mrm`mjgbtfsè4 Bstb bs hblfhe mc è`bgte `mjgætfke pbr`mgbgtb qub ùbstrmg mcjuges íteès keè kegsbkubgkfm hbc è`bgte mgjucmr hb bspàg  [ y hbc è`bgte mgjucmr erlftmc  C hb sus bcbktregbs. Bc è`bgte `mjgætfke vfbgb hmhe per4 ² 0

∜j²

L

D,

heghb  D 0  (C + [) bs bc è`bgte mgjucmr tetmc bcbktrûgfke, ²L 0 =>∜<8 D /X bs bc `mjgbtûg hb Leor y j bs cb amkter hb Cmghæ.

(=>.==) b/<`b

0 6, ;

Ú

6

Bg rbmcfhmh ϊ bs ug tbgser qub hm kubgtm hb cm mgfsetrepàm mgfsetrepàm hb ces `mtbrfmcbs `mtbrfmcbs rbspbkte hbc `mjgbtfsè, `mjgbtfsè, mquà sfg b`lmrje, cm kegsfhbrmrbès ug bskmcmr keè sf keg `mtbrfmcbs fsûtrepes trmtírmès.

RGBH.



8;

Bg musbgkfm hb km`pe `mjgætfke bxtbrge, ces è`bgtes mtû`fkes sb bgkubgtrmg erfbgtmerfbgtmhes mc mzmr rbsuctmghe ugm `mjgbtfzmkfûg gucm. Bg prbsbgkfm hb ugm fghukkfûg `mjgætfkm L pbrpbghfkucmr m cm ûrlftm bcbktrûgfkm, bc è`bgte `mjgætfke ² tfbgb ugm bgbrjàm4 R 0

∜² º L 0 ∜j² D º L 0 ∜j² L

z L L D z

9

D z 0

∜D, D + + =, º º º , D,

(=>.=<)

heghb D z bs cm ke`pegbgtb hb D bg cm hfrbkkfûg hb L. L . Keè pubhb vbrsb, cm bgbrjàm bs `àgf`m kumghe bstí erfbgtmhe sbjôg L sbjôg L.. Mpcfkmghe cm `bkígfkm bstmhàstfkm sb bgkubgtrm qub m cm tb`pbrmturm X cm `mjgbtfzmkfûg rbsuctm4 j² L D L @ 0 gj² gj ²L D LD ( x) x 0 , 9 (=>.=;) iL X

sfbghe LD ( x) cm ccm`mhm augkfûg hb Lrfcceufg (rbærmsb m Y 2P per bdb`pce) qub m tb`pbrmturms mctms e km`pes lmdes ( x ≤ =) , pubhb mprexf`mrsb bg prf`br erhbg per L D ( x) ≋ (D +=) +=)x/;D , y rbsuctm pmrm cm suskbptflfcfhmh4 ϊ 0

@ O

< <

≋ ²>L@ 0 gp;i² X ²> 0 K X ,

L

(=>.=8)

L

heghb p 0 j D ( ( D + + =) bs bc gô`bre babktfve hb `mjgbtegbs hb Leor y K bs cm kegstmgtb hb Kurfb. Cm rbcmkfûg mgtbrfer bxprbsm cm ccm`mhm Cby hb Kurfb, qub per bdb`pce, m X 0 ;>> I, mrredm ug vmcer ϊ ∳ => ∜; . M tb`pbrmturms lmdms e km`pes mctes ( x ≩ =) , sukbhb qub LD ( x) ↜ =, sb tfbgb bgtegkbs pmrm cm `mjgbtfzmkfûg qub @ 0 gj² L D ccm`mhm `mjgbtfzmkfûg hb smturmkfûg.

∔

-Abrre`mjgbtfsè4 [bjôg bc èhbce hb Obfsb`lbrj, cm fgtbrmkkfûg bgtrb hes íteès f y d hb è`bgtes mgjucmrbs tetmcbs  [f y  [d => vfbgb hmhm per cm ccm`mhm bgbrjàm hb fgtbrkm`lfe4 R f,d (=>.=2) f,d 0 ∜D f,d f,d [f º [d , heghb D f,d fgtbrkm`lfe, sf D f,d mcfgbm`fbgte f,d bs kegekfhm keè fgtbjrmc hb fgtbrkm`lfe, f,d 7 > sb tfbghb mc mcfgbm`fbgte hb bspfgbs (abrre`mjgbtfsè), per bc kegtrmrfe, sf bs D f,d f,d 3 > sb tfbghb mc mgtfmcfgbm`fbgte hb ces `fsès (mgtfabrre`mjgbtfsè). (mgtfabrre`mjgbtfsè). Bg prbsbgkfm hb ugm fghukkfûg L fghukkfûg L cm cm bgbrjàm hb ug kegdugte hb bspfgbs rbsuctm4 R 0

�

∜ D f,d

f,d f,d

�

[f º [d + j² L L

[f .

(=>.=1)

f

Kegsfhbrmghe sûce cm fgtbrmkkfûg hb ces z vbkfges `ís prûxfès, bs d D f,d f,d 0 z D , ce qub dugte keg cm mprexf`mkfûg mprexf`mkfûg hb km`pe `bhfe, bg cm qub cm fgtbrmkkfûg fgtbrmkkfûg bgtrb [f y [d bs mprexf`mhm sustftuybghe [ sustftuybghe [d per p er su vmcer `bhfe ⟯[d ⟦ , rbsuctm qub bc `mtbrfmc sb ke`pertm keè pmrm`mjgætfke se`btfhe m ug km`pe babktfve L babktfve L ba ,

∝

Lba 0 L + ι²@ =>

9

ι

D ≭ gj z ²D ²

< < L >

Bs ug km`lfe hb getmkfûg mkestu`lrmhe bg bc `mjgbtfsè bc bskrflfr tetmc D, msà, sb omlcm hb bspàg sfg `ís.

RGBH.

. [ pmrm

(=>.=:)

bc è`bgte mgjucmr



88

Cm `mjgbtfzmkfûg rbsuctm bgtegkbs cm (=>.=; ( =>.=;)) keg x 0 j²L DL ba /iL X , y pmrm mctms tb`pbrmturms, (x ≤ =), cm suskbptflfcfhmh pubhb bgtegkbs mprexf`mrsb per4 ϊ 0

@ O

K ≋ ²> @ 0 L X ∜ ∜ Kι .

(=>.=?)

Bxprbsfûg kegekfhm keè cby hb Kurfb-Zbfss, vícfhm pmrm X 7 X k ≭ K ι , heghb X k bs cm tb`pbrmturm hb Kurfb. ^mrm X 3 X k sb ccbjm m ugm bkumkfûg f`pcàkftm pmrm @ , qub pmrm bc kmse hb bspàg =/< y bg musbgkfm hb km`pe rbsuctm4 @ 0 g² > tmgo X k hbtbr`fgm

� � @ /g²L X /X k

.

(=>.=6)

pubs, cm tb`pbrmturm per hblmde hb cm kumc pubhb omlbr `mjgbtfzmkfûg

bspegtígbm.

-He`fgfes abrre`mjgætfkes4 M abrre`mjgætfkes4 M tb`pbrmturm tb`pbrmturmss `bgerbs `bgerbs qub cm tb`pbrmturm tb`pbrmturm hb Kurfb, Kurfb, tehes ces bspfgbs hblbràmg bstmr mcfgbmhes, sfg b`lmrje, sb elsbrvm qub æstes sb mcfgbmg p er zegms `mkreskûpfkms ccm`mhms he`fgfes keg erfbgtmkfegbs hfstfgtms. Bste bs msà aughm`bgtmc`bgtb pmrm `fgf`fzmr cm bgbrjàm hb mgfsetrepàm (msekfmhm m cm bxfstbgkfm hb hfrbkkfegbs krfstmcfgms hb «aíkfc¹ f`mgmkfûg) y m cm bgbrjàm `mjgbtestítfkm msekfmhm m cm fgtbrmkkfûg `mjgætfkm kcísfkm hb cmrje mckmgkb bgtrb ces bspfgbs.

RGBH.


M Kegstmgtbs Aàsfkms

82

Mpæghfkb M M Kegs Kegstm tmg gtbs tbs Aàsf Aàsfkm kmss

Kumhre M.=4 Kegstmgtbs aàsfkms. Aubgtb4 Y8 Y8P.

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L Xm Xmlcm ^brfûhfkm hb ces Bcb`bgtes y ^mrí`btres hb Fgtbræs

81

Mpæghfkb L

L Xmlcm mlcm ^brfûhfk ^brfûhfkmm hb ces Bcb`bg Bcb`bgtes tes y ^mrí`b ^mrí`btre tress hb Fgtbræs

Xmlcm Afjurm L.=4 Xmlcm

pbrfûhfk pbrfûhfkm m hb ces bcb`bgte bcb`bgtess y pmrí`bt pmrí`btre re hb fgtbr fgtbræs. æs. Mpcàqubsb Mpcàqubsb bc zee` pmrm ugm vfsumcfzmkfûg mhbkumhm. Aubgtb4 ottp4//sfkm.kfbgkfms.ukofcb.kc Aubgtb4 ottp4//sfkm.kfbgkfms.ukofcb.kc

RGBH.


\babrbgkfms

8:

\babrbgkfms Y=P Mcvmrbcces. @mtbrfmc Ke`pcb`bgtmrfe hb Aàsfkm hbc Bstmhe [ûcfhe. Kumrte kurse hbc Jrmhe bg Aàsfkm (RGBH), RGBH, Bgbre <>=1. Y
>=. Y;P Lrewg. Aàsfkm hb ces [ûcfhes. Bhfterfmc \bvbrtæ, Lmrkbcegm, =6:>. Y8P Ifttbc. Fgtrehukkfûg m cm Aàsfkm hbc Bstmhe [ûcfhe. Bhfterfmc \bvbrtæ, [.M., [.M., Lmrkbcegm, =66:. Y2P @mzm, @esqufrm Qbfrm. Qbfrm. Aàsfkm hbc Bstmhe [ûcfhe. Rgfvbrsfhmh Rgfvbrsfhmh hb [mgtfmje hb Ke`pestbcm, [mgtfmje hb Ke`pestbcm, <>>?. Y1P ^mvcev. Aàsfkm hbc Bstmhe [ûcfhe. Bhfterfmc @F\, =6?6. Y:P \ejmg, Jutfærrbz, @bgæghbz. Fgtrehukkfûg m cm Aàsfkm hb ces [ûcfhes. Rgfvbrsfhmh hb Kofcb.

RGBH.


8?

RGBH.


Anotaciones de Un Curso de Física Del Estado Sólido

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