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VDZghgbq_kdb_fh^_eb < gZklhys_c ]eZ\_ Zjl_nZdlu h[hehqdb iZjZ^b]fu jZkkfZljb\Zxlky im l_fZgZebaZj_r_gbcijhkl_crbofgh]hdjZlgh\klj_qZxsbokyaZ^ZqDhrbb djZ_\uoaZ^Zq^eyh^ghf_jgh]hebg_cgh]hmjZ\g_gbyl_iehijh\h^ghklb±dZ ghgbq_kdbofh^_e_c[42–49,71,72] Fh^_ev  ± AZ^ZqZ Dhrb ^ey h[eZklb 6 5 – 5  ijb G \b^ghf gZ qZevghfjZkij_^_e_gbbl_fi_jZlmju

7  [

M [

$G [  $

'4 6F9

7 / 



A^_kv ∆4 = 6F9 7 /  − dhebq_kl\h l_ieZ ih^\_^_ggh]h d lhqd_ [   ijb W   F9 ±h[t_fgZyl_ieh_fdhklvkj_^u\h[eZklb 5  6±iehsZ^vihi_ j_qgh]hk_q_gbyh^ghf_jghch[eZklb 7  /±l_fi_jZlmjZb©rbjbgZªwd\b \Ze_glgh]h G bklhqgbdmdhg_qgh]hl_ieh\h]hbfimevkZJ_r_gb_aZ^ZqbDhrb kmkeh\b_f 

7  [ W =

 [  7 /   [ ° 5 W ° 5  H[S −  DW   πDW  



Fh^_ev±AZ^ZqZDhrbijbgZqZevghfjZkij_^_e_gbbl_fi_jZlmjulb iZ©l_ieh\hcbfimevkª

7  [ 

M [

Î7  [ °>  /  / @ Ï [ °  /  /   >   @ Ð



J_r_gb_aZ^Zqbbf__l\b^

7  [ W

[  /  Ø Û 7 Ë È [  /  Ø  [ ° 5  W ° 5   HUI È HUI Ì  Í Ê  DW Ú Ê  DW Ú ÜÝ

Fh^_ev±AZ^ZqZDhrbijbgZqZevghfjZkij_^_e_gbbl_fi_jZlmjulb iZ©klmi_gvdZª

7  [

M [

Î7 [ !  7 ! 7  Ï7  [     Ð 



J_r_gb_bf__l\b^

7  7 7  7 [ Ø   HUI È Ê  DW Ú  

7  [ W



Fh^_evDjZ_\ZyaZ^ZqZkihklhygguf]jZgbqgufmkeh\b_fI-]hjh^Z 

^eyh[eZklb Σ = 5 + × 5+ ijbihklhygghcgZqZevghcl_fi_jZlmj_

7  [

7 7  W = 7F > 7 

M [

J_r_gb_bf__l\b^

[ Ø 7F   7F  7 HUI È  Ê  DW Ú

7  [ W





Fh^_ev  ± DjZ_\Zy aZ^ZqZ k ]jZgbqgufb mkeh\byfb IV]h jh^Z ^ey ^\mokehcghckbkl_fu 5  5 kihklhyggufbghjZaebqgufbgZqZevgufb

^

l_fi_jZlmjZfb\h[eZklyo 5L

7 [

M  [

`

L

 

7  [ ° 5 7  [

M   [

7  7  [ ° 5 

J_r_gb_aZ^Zqb^ey 7 [ W ^ey 7  [ W ZgZeh]bqgh bf__l\b^

7 + . ε7 7 − 7  [   7  [ W =  + HUI    + .ε  + .ε D W    



]^_ .H H H   HL O L F9L DL O L F9L L   Fh^_ev±DjZ_\ZyaZ^ZqZk]jZgbqgufbmkeh\byfbI-]hjh^Z^eyh[eZk [ °   /   /  kehcdhg_qghclhesbgu =jZgbq lb 6  : – 5  : gu_bgZqZevgZyl_fi_jZlmjuihklhyggughjZaebqgu

^

`

M [

7

FRQVW  [ °:



7  /   W

7F

FRQVW W ° 5 7F  7 



7  [ 7   /   W J_r_gb_lZdh\h



- [ W ]^_

‡

  Q  [ Ø FRVÈ P Q H[S  PQ )  /  P Ê Ú Q  Q 

7  [ W  7F 7  7F PQ



Ç

 Q   S  )

DW

 /  





Fh^_ev  ± =jZgbqgZy klZpbhgZjgZy  aZ^ZqZ k pbdebq_kdbfb ]jZgbq gufbmkeh\byfbIV]hjh^Z ^ey^\mokehcghckbkl_fu >: : @ L_fi_jZlm ju\kehyo± 7L  [L  [L ∈ ΩL = {[L ∈ ( /L )}L  >eyi_j\hcdjZ_\hcaZ^Zqb

kbkl_fZ {Ω Ω  }±g_pbdebq_kdZy 

7  [ [ =  = 7  [ [ =  = 7  7 [ [ / 7F7  [ [ /      

7F 

L_fi_jZlmju 7L  [L \h[eZklyo :L m^h\e_l\hjyxlmjZ\g_gbyf

λL

G  7L

G [L 

+ I L  [L =  L

 



]^_ I L  [L ±nmgdpbbiehlghkl_cbklhqgbdh\l_ieZ\ :L  Fh^_ev  ± G_ebg_cgZy djZ_\Zy aZ^ZqZ lbiZ Kl_nZgZ  JZkkfZljb\Z xlkymjZ\g_gby\b^Z ^ey^\monZa©l\_j^hcªb©`b^dhcª jZa^_e_gguo ih^\b`ghc]jZgbp_cnZah\h]hi_j_oh^Z [ I [W kaZ^ZggufgZg_cmkeh\b _f

∂7 ∂7 G ξW   − λ  = γσ ∂[ [ = ξW ∂[ [ =ξW GW ]^_ O L  ± l_iehijh\h^ghklv nZa J  ± iehlghklv l\_j^hc nZau V  ± m^_evgZy kdjulZyl_iehlZnZah\h]hi_j_oh^Z [W ±dhhj^bgZlZnjhglZnZah\h]hi_j_ λ

oh^Z^eydhlhjhcbaj_r_gbyaZ^Zqbke_^m_l[43]

[ W

E W E

FRQVW 



Nhjfmeu lbiZ   oZjZdl_jgu b ^ey j_r_gbc [he__ keh`guo aZ^Zq lbiZ Kl_nZgZ[55,73] VAZ^ZqbDhrb Bkke_^h\Zgb_ Fh^_e_c ± kljhbf lZd Ih j_r_gbyf       ± nmgdpbyf 7L  [ W L b TL  [ W = − Ì 

±ghf_jZfh^_e_c gZoh^bfnmgdpbb 3L  [ W

˜7L ˜W

∂7L Ih\_^_gb_nmgdpbc 7L  3L  TL bamqZ_fgZ]jZgbpZoh[eZklb ∂[

6 Z\gmljb__±ih\_^_gb_ 7L  [ W <i_j\hfkemqZ_\uqbkey_fh^ghdjZlgu_ ij_^_eu [    [  ‡  W    W  ‡  Z aZl_f ± ^\hcgu_ ij_^_eu dhf[bgZpbb  h^ghdjZlguo  Ihke_^gb_ gZoh^bf ihhq_j_^gh gZijbf_j \gZ qZe_ OLP 7L  W  Z aZl_f OLP 7L  [  b h^gh\j_f_ggh dh]^Z W→ 

[→

OLP 7L  [ W = OLP 7L  [ W  W = 7L  
[ → W →

wdklj_fmfiheZ]Zy [ ±iZjZf_ljhfZ W ±g_aZ\bkbfhcdhhj^bgZlhc bgZoh^bf nhjfmeu±hp_gdbwnn_dlb\ghcehdZebaZpbb 7L  [ W H^ghdjZlgu_b^\hcgu_ ij_^_eu^ey [    b [ → −∞ g_\uqbkey_f\\b^mbg\ZjbZglghklbj_r_ gbchlghkbl_evghij_h[jZah\Zgby [   [  GZoh^bfnmgdpbb 3L  [  W

˜7L  ˜W

  [   7 /   [  3  [ W =     ⋅  −  H[S −    DW     DW   πDW      3  [ W



  È 7 Ø Ë È / [ Ø  / FKÈ / [ Ø Û H[SÈ  [   /  Ø  [ VK É Ù É Ù Ê DW Ú  Ê DW Ú ÜÝ DW Ê  SDW  Ú ÌÍ Ê Ú

3  [ W
O

È 7  7 Ø È [ Ø [ H[SÉ  Ù É Ù Ê DW Ú Ê  SDW  Ú



˜7L  ˜W

 [   Ì7 /   [  −   T  [ W =     ⋅  H[S   DW    Ñ DW   DW    T  [  W

  È O7 Ø VKÈ / [ Ø H[SÈ  [   /  Ø  É Ù DW Ê SDW Ú Ê DW Ú Ê Ú

 [  Ì(7 − 7 )  T  [ W = ⋅ H[S −  DW   Ñ DW  





>eynmgdpbc 7L  [ W bf__f

7 [ W [ 

7 /  7  [ W [ →∞ = 7  [ W W → = 7  [ W W →∞ =    SDW 

 /  7  [ W [ → = 7 HUI    7  [ W [ ‡   DW  7  [ W W 

7  [ W W ‡

Î7  [  /  Ñ7 Ï   [ /   Ñ [ ! /   Ð



7 +7 7  [ W [ → = 7  [ W W →∞ =   7  [ W [ ‡  7  [ W W 



7

[ Î7  Ñ 7  7 Ï   [   Ñ7  [! Ð



>eynmgdpbc 3L  [ W ba ± ke_^m_l

3  [ W [ → = −

7 /  3  [ W [ →∞ = 3  [ W W → = 3  [ W W →∞ =    πDW 

 (/ )   7 /    3  [ W = 3  [ W W →∞ =   3  [ W [ → =     H[S −  [ →∞      DW    πDW     [ ≠ /   3  [ W [ → =  Ψε  [ = / 



3  [ W [ → = 3  [ W [ →∞ = 3  [ W W → = 3  [ W W →∞ =    >eynmgdpbc TL  [  W ba ± ke_^m_l

T  [ W [ → = T  [ W [ →∞ = T  [ W W → = T  [ W W →∞ =   T  [ W [ → = T  [ W [ →∞ = T  [ W W →∞ =   T  [ W T  [ W [ → =

W

→



 [ ≠ /   = :ε  [ = /  

λ(7 − 7 )  T  [ W [ →∞ = T  [ W = T  [ W W →∞ =    πDW W →

<ihke_^gbobakhhlghr_gbc b nmgdpbb



:H


:H  [ W

7 È [H  /  Ø  ÉÊ SDW  ÙÚ

 W 

ÎÑÈ O7 Ø Ë [  /  Ø  ÛÞÑ  H[SÌÈ H  Üß ÏÊ ÑÐ  SDW Ú ÌÍ Ê  DW Ú ÜÝÑàW 



/   1  SDW  H  1  FRQVW  H !  >ey:H  [ W iheh`bf [H /    1  DW  H  1  FRQVW  H !  Ihkdhevdmijb [ H   /     1    Zijb [ H   /     1  !  bf_ >ey
_f

<

OLP
H °   · Ñ 7 Ï  1   H  Ñ H !  Ð = −Ψ−  

È 7 Ø OLP Ê   1  W H  Ú W  1    Ψ+ =

OLP Ψε  [ W W →  [ε →(/  )+ 

>ey:H  [ W g_aZ\bkbfhhlagZdZ [ − /  ihemqZ_f

ε ∈ (−  )  λ7   ε :± = OLP Ψε  [ W = OLP   H[S − 1  W =  ∞ε ≥  W →_ [ε _→(/  ) W →    πDW 

(

)



LZdbf h[jZahf m[_`^Z_fky \ lhf qlh   GZqZevgu_ mkeh\by ^ey nmgdpbc 7L  [ W g_\uihegyxlkym 7 [ W ©ijhiZ^Z_lª G nmgdpbym 7  [ W b 7 [ W h[gZjm`b\Zxlky jZaju\u I]h jh^Z ijb _ [_ /   b [   khhl\_lkl\_ggh    =jZgbqgu_ agZq_gby ijb [   nmgdpbc 7 [ W  b 7 [ W  \ gZqZevguc fhf_gl \j_f_gb  W    bf_xl jZaju\u II]h b I]h jh^h\ khhl\_lkl\_ggh  7 W ijb W   h[jZsZ_lky\∞Z 7  W ijb W   bf__lkdZqhdijbgb fZy agZq_gb_ 7 + 7     GZqZevgZy kdhjhklv baf_g_gby l_fi_jZlmju 7 W   3 W  ijb W    [_kdhg_qgZ Z l_fi_jZlmju 7  /   W  3  /   W  ijb W    aZ\bkbl hl kihkh[Z ij_^_evgh]h i_j_oh^Z [H   /  ijbq_f\aZ\bkbfhklbhl\u[hjZiZjZf_ljh\ 1  b H \ hgZ fh`_lijbgbfZlvihfh^mex ex[u_agZq_gbyhl^h∞ Ihke_^g__hlgh kblkybdnmgdpbb T  /   W ijb W   kl_fhlebqb_fqlhagZd [ H  /  jhebg_b]jZ_lZ\aZ\bkbfhklbhl\u[hjZiZjZf_ljZ H nmgdpbyfh`_lijbgb fZlv ^\Z agZq_gby ±  b ∞   =jZgbqgh_ agZq_gb_ ijb [ =   nmgdpbb T  [ W  \ gZqZevguc fhf_gl \j_f_gb W     bkiulu\Z_l jZaju\ II]h jh^Z h[jZsZ_lky\∞  

ey 7 [ W bf__f

/    =   /   = ∞  /  ∞  = /    ∞ = /  ∞ =   / ∞   = / ∞ = / ∞ =   / ∞ ∞  = / ∞   ∞ = / ∞ ∞ =  



< hlebqb_ hl ^jm]bo ij_^_eu ijb [   W   g_ kh\iZ^Zxl / › /   Ihwlhfm /  \uqbkey_lky k ihfhsvx

[H  W

 1  DW H  H !  

/

Î H °    χ H •  Ð



>ey 7  [ W  gZoh^bf iheZ]Zy [ε W = /  −  1  DW H !  

/ 

/

/

7  /‡

/ ‡ /  ‡ /‡ ‡ /‡  ‡ / /  

+ ε

/‡ 

 _ 1 _ ‡ 

/ ‡

/ ‡  /‡ ‡ 

7  [ →  /  −  7 7  /  /    =    [ = /     [ →  /  + 

/ /  







Î( H  1  !  [H   /    Ï   Ð( H  1    [H   /    

7  Æ <      Æ <  7 7 ( ε+ =  ( + HUI _ 1  _) Æ =   ( ε− =  ( − HUI _ 1  _) Æ =     7   Æ> 7  Æ >      

>ey 7  [ W ihemqZ_f

/

7 [ →  +  7  7  /    =   / ‡ ‡   7  [ →  − 

7 +7 / ∞   ∞ =    /‡  / ‡  / 

/  ‡

/‡   / ‡

/‡ 7  7  

7 7



 H °   7  7 È 7  7 Ø ÎÑ  – ÏHUI_ 1  _ H  Ê  Ú Ñ  Ð H ! 

ε ∈ −  7 + 7  7 − 7   / ∞ ∞ = +  × HUI _ 1  _ ε =       ε >  



Ba ± ke_^m_lqlh_klv^\_]jmiiu\_ebqbg /LM D E \i_j\hc]jmi i_©ghjfZevghcª  /LM DE /LM D E /LM D E l_^\hcgu_ihhq_j_^gu_ ij_^_eu g_ aZ\bkyl hl ihjy^dZ bo \uqbke_gby b kh\iZ^Zxl k ^\hcgufb \u qbke_ggufb ijb h^gh\j_f_gghf i_j_oh^_ [  D  W  E   [ [ H W  \h \lh jhc ]jmii_ ©ZghfZevghcª  ^\hcgu_ ij_^_eu aZ\bkyl hl ihjy^dZ i_j_oh^h\ jZaebqguf_`^mkh[hcbg_kh\iZ^Zxlk^\hcgufbh^gh\j_f_ggufb >ey nmgdpbc 3L  [ W  b TL  [  W  ©ghjfZevgu_ª ij_^_eu himkdZ_f Z ©ZghfZevgu_ª^Ze__ijb\h^bf

/ 

‡  /  

/  /  

/ 

 / 

 /  /   

/  

ÎÑ H °    χ H  ÑЇ H ! 

‡  /  /   = Ψ+ 

 ε ∈ −  ε >     ε ∈     /  =  ∞   1  7 − 7  ε =    π 

/ 

/  

/  /  

/ 

   ε ∈ −  ε >    ε ∈ −   /  =  ∞   λ1  / 7  ε =    π D    

‡  /  /   

/  

 /  /   = :± 

  / 

Î H °   χ H •  Ð



A^_kv <  b :“  khhl\_lkl\mxl   b   Iheh\bgZ ^\hcguo ij_^_eh\    ijbihhq_j_^ghf\uqbke_gbbkh\iZ^Zxlh^gZdhg_kh\ iZ^Zxlk^\hcgufbh^gh\j_f_ggufbM^jm]hciheh\bgu   ihhq_j_^gu_ ij_^_eu g_ kh\iZ^Zxl gb ^jm] k ^jm]hf gb k h^gh\j_f_ggufb < ihke_^gbo \u[bjZy khhl\_lkl\mxsb_ H  b 1   ^ey /LM   b /  /   fh`gh ihemqblv ex[u_ gZi_j_^ aZ^Zggu_ agZq_gby \ bgl_j\Ze_ ∞  beb ±∞∞ LZdbfh[jZahfmjZkkfhlj_gguonmgdpbc\lhqdZo [  b [ /  bf_xlkyjZaju\u JZkkfhljbfih\_^_gb_nmgdpbc 7L  [ W \h\gmlj_ggbolhqdZoh[eZklb 6  Bamkeh\bywdklj_fmfZnmgdpbc 7L  [ W ih W  [ b]jZ_ljheviZjZf_ljZ 

˜7L ˜W ihemqZ_fiheZ]Zy [

3L  [ W





[ W W

 7 /   [   [  3  [  W =     ⋅   −  H[S −   =   [ !  W !    DW      DW      πDW    AgZq_gb_ W W _klvlhqdZfZdkbfmfZnmgdpbb 7 [ W b k\yaZgh k dhhj^bgZ lhc [ [ °  ‡ ‡ ke_^mxs_cba aZ\bkbfhklvx

[

DW 



FZdkbfZevgh_\lhqd_ [

[ agZq_gb_7 [ W ^hklb]Z_lkyijbW W bjZ\gh  È 7 / Ø   PD[ 7 [ W 7P  [ [ W °6 SH Ê [ Ú

Ijb W  W l_fi_jZlmjZ\lhqd_ [ [ \hajZklZ_lZijb W ! W m[u\Z_llZdqlh  kh\iZ^Zxs__khp_gdhcwnn_dlb\ghcehdZebaZpbbl_fi_jZlmjgh]hihey 

 ^Z_ldhhj^bgZlm [ ±ehdZevgh]hfZdkbfmfZl_fi_jZlmju>ey 7  [ W bamkeh\by 3  [  W  ke_^m_l

/ [ / / [ [ VKÈ   Ø   FKÈ   Ø Ê DW Ú  Ê DW Ú Iheh`b\ [ /  K !   

K  DW  /  

  [ !  W ! 



)  ba   ihemqZ_f ^ey [ ! / 

K K WKÈ Ø Ê ) Ú





?kebj_rblvmjZ\g_gb_ lhihemqbf K K K  ) beb\ha\jZsZykv di_j\hgZqZevgufi_j_f_gguf [ = [ W Ij_h[jZam_f d\b^m

K )

È   Ø K Ù   OQÉ É  Ù K É  Ù Ê K Ú

]

/   ]   [



bj_rZ_f_]hijb[eb`_gghjZaeZ]Zyeh]Zjbnf

 )



È ] ] ] Ø  ]  É     Ù     Ê Ú





Ba ijb ]    ke_^m_lmjZ\g_gb_]hijb[eb`_gby

 )

 ] 



Imklv D   hagZqZ_lqlh D …  Lh]^Z kijZ\_^eb\h^ey

[ •   /  



([ ) = DW 



J_r_gb_ 

l_kh\iZ^Z_lk Ba b ihemqZ_f

/ W •    D MjZ\g_gb_]hijb[eb`_gby

 )

È ] Ø  ] É   Ù Ú Ê







ke_^m_lba ijb\uiheg_gbbmkeh\by

]    ] 



  ] ≤  



hldm^Zke_^m_lmlhqg_gb_hp_gdb 

[ •   /  






È / Ø DW     / DW # DW É   Ù  Ê DW Ú

>

@



Ihke_^g__ khhlghr_gb_ \uihegy_lky k lhqghklvx \  K lhqghklvx \  fh`ghaZibkZlv

 [

ijb



# DW



 /  W ≥       D   



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[[7  [ W ]∞−∞ =  [  ∂7  

∞

∂[  − ∞

= 



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 DW

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DW \7 [ W  bihemqbf 7 W

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7 δ W 7 / =     SDW 7  W



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7 G W 7

  W ≥ W∗ =  /  D 



Hp_gdZ δσ W = /  +  DW ijbih^klZgh\d_\ ^Z_l

7 G V  W …  ¹   W !  7



>ey 7  [ W ihemqbf

7 G W

7  7  7 7 G V  W

7   ¹  7  7 

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= /  πD ≅  /  D GZoh^bf

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  [δ   W  W  [   =  −  OQ   È G Ø   Ê / Ú W  W /  W /   /   / Ijb  W W / …  W W / I  nhjfmeZ   hibku\Z_l ^\b`_gb_ l_fi_jZlmjgh]h

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[G Ijbmkeh\bb 7  [ G  W



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 [   7F − 7     3  [ W = [ H[S −   DW     πDW   



 [   λ(7 − 7 )   T  [ W =  F   H[S −  DW  πDW    



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a a 7  [ W ≅ 7F + (7 − 7k )θ  [ W  T [ W

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˜ ˜7 Ø ˜ ˜7 Ø ˜ ˜7 Ø ˜7  È O  [  È O 7  È O  [  7   ˜[ Ú ˜[ Ê ˜[ Ú ˜[ Ê ˜[ Ú ˜[  ˜[ Ê

<ihke_^g_fbakemqZ_\ \klj_qZxlkyhrb[dbijbij_h[jZah\Zgbyo\[86] \qZklghklbaZibkZgh

˜ È ˜7 O [ 7 Ø ˜[ Ê ˜[ Ú

O



˜  7 ˜O ˜7 ˜O È ˜7 Ø    ˜[  ˜[ ˜[ ˜7 Ê ˜[ Ú



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˜O  [ 7 ˜7 ˜[ ˜[

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˜O ˜7 ˜O È ˜7 Ø    ˜[ ˜[ ˜7 Ê ˜[ Ú

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GF G F  ∂7 ∂7 ∂X ∂ F9 7   = = F9 + 7 9 =  F9 + 7 9  GW G 7  ∂W ∂W ∂W ∂W 



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D 7  [ W = λ π Ijb T W

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 a q0  dτ 2q a  T (0, t ) =  t+∞ = 0  λ π  λ π −∞ t − τ

∫

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1 T ( x,0) = 2a π

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G δW = ϑn < ∞  GW W =  LZdbf h[jZahf njhgl ^\b`msbcky kh kdhjhklvx - n  kms_kl\m_l gh \ ijb [eb`_gbb W W U  Ijb W  ‡ ba ke_^m_lZkbfilhlbq_kdZynhjfmeZ G  W

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K fZeufb yq_cdZfb ©dm[bdZfbª  kj_^u \ dhlhjhc ijhl_dZ_l ^bkkbiZ lb\gucijhp_kkgZdhlhju_jZa[b\Z_lkykbkl_fZijbqbke_gguojZkq_lZoZk khpbbjm_lkyN;FH\[33,85] AZ\_jrZy jZkkfhlj_gb_ N;FH ijb\h^bf lhqdm aj_gby gZ ihe_\h_ dhglbgmZevgh_ hibkZgb_Z\lhjh\[28]©lZdhcij_^_evguc i_j_oh^g_\ha fh`_gm`_baaZd\Zglh\uobfhe_dmeyjguok\hckl\\kbemdhlhjuojZkkfZl jb\Zlv nbabq_kdb_ \_ebqbgu mf_gvr_ggu_ k\_jo g_dhlhjuo ]jZgbp ebr_gh kfukeZIhwlhfmnbabdb\\h^yl©nbabq_kdb[_kdhg_qghfZeu_ª\_ebqbgug_ ^Z\Zywlbfihgylbyfhij_^_e_gbyª Ijbgpbi ehdZevgh]h d\ZabjZ\gh\_kby IEDJ  b]jZxsbc \ iZjZ^b]f_ dexq_\mxjhevhibku\Z_ldbg_lbq_kdh_ih\_^_gb_N;FHgZhkgh\_ke_^mx s_c]bihl_au[145]<ex[hckbkl_f_kms_kl\mxl^\ZjZaebqguofZkrlZ[Z^ey \j_f_gj_eZdkZpbb±mklZgh\e_gbyjZ\gh\_kgh]hkhklhygbykbkl_fu\p_ehf± W /  b j_eZdkZpbb \ N;FH ± W U  ijbq_f W U  W /  EhdZevghjZ\gh\_kgh_ kh klhygb_kbkl_fu\gZqZe_mklZgZ\eb\Z_lkyaZ\j_fy W U \N;FHZaZl_fkbkl_ fZklj_fblkydjZ\gh\_kbx\p_ehfqlhhij_^_ey_lky\j_f_g_f W / LZdbfh[ jZahf kh]eZkgh IEDJ \j_f_ggZy w\hexpby \ N;FH ij_^klZ\ey_l kh[hc ih ke_^h\Zl_evghklvd\ZabjZ\gh\_kguokhklhygbc^eydhlhjuoh[hkgh\Zghkms_ kl\h\Zgb_ l_jfhklZlbq_kdbo iZjZf_ljh\ < i_j\uc i_jbh^ jZa\blby g_jZ\gh \_kghcl_jfh^bgZfbdbqZklhih^q_jdb\Zehkvqlhkbkl_fZkeZ[hg_jZ\gh\_kgZ l_ ]jZ^b_glu iZjZf_ljh\ b ihlhdb \ gbo fZeu d\ZabklZpbhgZjgu  K_cqZk kqblZxl [] qlh IEDJ kijZ\_^eb\ b ^ey ^Ze_dbo hl jZ\gh\_kby kbkl_f \iehlv^h]jZ^b_glh\≤Dkf  Ke_^kl\b_fbaIEDJy\ey_lkyp_gljZevgh_iheh`_gb_y^jZl_jfh^bgZfb q_kdhciZjZ^b]fukhhlghr_gb_=b[[kZbal_jfhklZlbdbkijZ\_^eb\hb\l_j fh^bgZfbd_l_ijbf_gbfhdg_h[jZlbfufg_klZpbhgZjgufijhp_kkZfi_j_ ghkZ Ij_^iheZ]Z_lky qlh kijZ\_^eb\Z ^ey N;FH b wj]h^bq_kdZy ]bihl_aZ ± \hafh`ghklv aZf_gu mkj_^g_gby ih \j_f_gb mkj_^g_gb_f ih nZah\hfm ijh kljZgkl\mIjbwlhf\j_fykdZqdhh[jZagh]hbaf_g_gbyl_jfh^bgZfbq_kdboiZ jZf_ljh\\N;FH^he`gh[ulvfgh]hf_gvr_bgl_j\ZeZ\j_f_gbf_`^mi_j_ oh^hf ba h^gh]h d\ZabjZ\gh\_kgh]h khklhygby \ ihke_^mxs__ [6] Wlb ©kdZqdbª\fZdjhkdhibq_kdbomjZ\g_gbyoi_j_ghkZg_mqblu\Zxlky
6 = 6 [7 U  W  3 U W  & U W ]



<nbabq_kdhcdbg_lbd_ZgZeh]IEDJ±ijbgpbi^_lZevgh]hjZ\gh\_kby± nhjfmebjmxl \ \b^_ mkeh\by jZ\_gkl\Z \_jhylghkl_c ijyfuo b h[jZlguo i_ j_oh^h\[8,147] 

ZΓ′ Γ′  Γ Γ = ZΓ l  Γl  Γ′ l  Γ′l 



]^_ * l  ± agZq_gb_ iZjZf_ljh\ *  ijb aZf_g_ W  W  BgZq_ ]h\hjy dZ`^uc fbdjhkdhibq_kdbcijhp_kkklhedgh\_gbc khijh\h`^Z_lkyijhlb\hiheh`guf _fmJZ\_gkl\h g_fh`_lkem`blvh[hkgh\Zgb_fIEDJlZddZdihemq_gh ba mkeh\by klZpbhgZjghklb nmgdpbb jZkij_^_e_gby e_\Zy qZklv \ mjZ\g_gbb dbg_lbq_kdh]h[ZeZgkZIZmebjZ\gZgmex[8] Kljh]bcZgZeba[6,9]ihdZau\Z_l ke_^mxs__eyihemq_gbyM;W±©]eZ\gh]hªmjZ\g_gbyl_jfh^bgZfbdbgZhkgh\_ba eh`_gguoij_^klZ\e_gbchNF;HbIEDJkmq_lhf bkihevam_lkykhhl ghr_gb_=b[[kZl_jfhklZlbdb

7 G 6 = GX + 3Gϑ −

∑ µ N G &N 



∑ µk



N

dhlhjh_aZf_gy_lkyl_jfh^bgZfbq_kdbf

T

dS du dϑ = +P − dt dt dt

k

d Ck  dt

I_j_oh^hl d ±h[yaZl_evgucwe_f_glbaeh`_gby\ex[hfmq_[gb d_bebfhgh]jZnbbihl_jfh^bgZfbd_F_`^ml_fykghklb\_]hh[hkgh\Zgbb g_l :gZeba bklhqgbdh\ ihdZau\Z_l qlh \klj_qZ_lky q_luj_ ih^oh^Z d Zj]m f_glZpbb ©kgZ[`_gby \j_f_g_fª khhlghr_gby =b[[kZ < i_j\hf ± ©ih mfhe qZgbxª±ibrml©hlkx^ZihemqZ_f ª±gbdZdg_h[tykgyy©ihemq_gb_ª [2,6,7,9,17,110,151]
^bnn_j_gpbjm_lkydZdkeh`gZy\gZqZe_±ihl_jfh^bgZfbq_kdbfiZjZf_ljZf Z aZl_f ihke_^gb_ ± ih \j_f_gb [20,77,122] < q_l\_jlhf ijb\h^ylky ©d\Zabh[hkgh\Zgbyª±Zj]mf_glukdhj__©dhgljZªq_f©ijhª[3,6,8,9] >\Zi_j\uoih^oh^Z ih ihgylguf ijbqbgZf g_ dhff_glbjmxlky Lj_lbc ih^oh^ \u]ey^bl dhjj_dlguf h^gZdh \hagbdZ_l \hijhk hldm^Z m iZjZf_ljh\ l_jfhklZlbdb aZ\bkbfhklv hl \j_f_gb ^Z _s_ b g_ij_ju\gZy " <_^v \j_fy \\h^blky\l_jfhklZlbdmbl_fkZfufhkms_kl\ey_lkyi_j_oh^dl_jfh^bgZ fbd_ bf_ggh\khhlghr_gbb "<[ZeZgkh\uomjZ\g_gbyolbiZ \j_fy ijbkmlkl\m_lgZZijbhjghchkgh\_±hgh©^Zghªijb[ZeZgkh\uo\u\h^Zowlbo mjZ\g_gbcdhlhju_[uebba\_klgukh\j_f_gNmjv_h^gZdhkqblZebkvq_flh ©ihklhjhggbfª^eyl_jfhklZlbdbG_jZ\gh\_kgZyl_jfh^bgZfbdZdZdjZab[u eZ ijba\ZgZ ©e_]Zebah\Zlvª wlb mjZ\g_gby h[hkgh\Z\ bo i_j_oh^hf  → gZhkgh\_IEDJIhwlhfm\lj_lv_fih^oh^_bf__feh]bq_kdbc djm] \j_fy \\h^blky \ nhjfmem l_jfhklZlbdb   ^ey ihemq_gby nhjfmeu l_jfh^bgZfbdb dhlhjZykh^_j`bl\j_fylhevdh ihlhfm qlh _]h kh^_j `bl[ZeZgkh\h_mjZ\g_gb_ gZebqb_\j_f_gb\dhlhjhfg_h[hkgh\u\Z_l kyZihklmebjm_lky^eyq_]hIEDJg_bkihevam_lky Ijbq_l\_jlhfih^oh^_bf__fihkmlbd\Zabbeb\_jg__Zglbh[hkgh\Z gby GZqg_f k deZkkbdh\ [3] ©Ke_^m_l \ hkh[_gghklb h[jZlblv \gbfZgb_ gZ jhev \j_f_gb \ hij_^_e_gbb wgljhibb Wgljhiby _klv \_ebqbgZ oZjZdl_jb amxsZy kj_^gb_ k\hckl\Z l_eZ aZ g_dhlhjuc hlebqguc hl gmey ijhf_`mlhd \j_f_gb 'W AZ^Z\ 'W fu^he`gu^eyhij_^_e_gby 6 fuke_gghjZa^_eblvl_ ehgZqZklbgZklhevdhfZeu_qlh[ubokh[kl\_ggu_\j_f_gZj_eZdkZpbb[ueb fZeu ih kjZ\g_gbx k 'W  Ihkdhevdm \ lh `_ \j_fy wlb qZklb b kZfb ^he`gu [ulvfZdjhkdhibq_kdbfblhykghqlh^eykebrdhffZeuobgl_j\Zeh\ 'W ih gylb_wgljhibb\hh[s_ihl_jy_lkfuke\qZklghklbg_evay]h\hjblvhf]gh \_gghf__agZq_gbbªLhqdZaj_gbyDI=mjh\Z[6]©i_j_oh^hlh^gh]hnb abq_kdh]hwe_f_glZjgh]hh[t_fZl_±N;FH±B< dkhk_^g_fmbebi_j_ oh^hlh^gh]hbgl_j\ZeZ\j_f_gb]b^jh^bgZfbq_kdh]hfZkrlZ[Zdke_^mxs_ fm _klv ^bkdj_lgZy hi_jZpby b khhl\_lkl\mxsb_ mjZ\g_gby ^he`gu [ulv kljh]h ]h\hjy aZibkZgu dZd mjZ\g_gby \ dhg_qguo jZaghklyoª Khfg_gby \ ^bnn_j_gpbjm_fhklb g_ij_ju\ghklb  wgljhibb jZa^_ey_l b D; Lheiu]h [60]©\ijhp_kk_i_j_^Zqbl_ieZwgljhibyfh`_lg_bf_lvkfukeZ_kebgZ ijbf_jl_iehijh\h^ghklvijhbkoh^blhq_gv[ukljhlZddZdwgljhiby_klvoZ jZdl_jbklbdZ kbkl_fu gZ g_dhlhjhf bgl_j\Ze_ \j_f_gb \ l_q_gb_ dhlhjh]h fZdjhkdhibq_kdb_oZjZdl_jbklbdbkbkl_fugZijbf_jwg_j]byl_l_fi_jZlm jZ±B< ij_^iheZ]Zxlkyg_baf_ggufbªB:D\Zkgbdh\dZkZykvdhgdj_lgh khhlghr_gby =b[[kZ aZdexqZ_l [8] ©I_j\h_ b \lhjh_ gZqZeZ ^ey d\ZabklZlb q_kdboijhp_kkh\

7G6

G X  3 G -  P G 1  $ G D \dhlhjhfagZd^bnn_j_gpbZeZ G ) mnb]mjbjmxsbo\_ebqbgihgbfZ_lkydZd

fZdjhkdhibq_kdh_ ©[_kdhg_qgh fZeh_ª l_ nbdkbjm_fh_ k ihfhsvx fZdjh kdhibq_kdbo ijb[hjh\ baf_g_gb_ wlhc \_ebqbgu )  ijb i_j_oh^_ kbkl_fu ba 

jZ\gh\_kgh]h khklhygby  \ ©[ebae_`Zs__ª jZ\gh\_kgh_ khklhygb_  G ) œ )  )ªDwlhfm^h[Z\bfke_^mxs__
∂ϑ  ∂6  ρ7  + (9 ∇6 ) = σ′LN L + GLY(λ∇7 ) ∂[N  ∂W 



]^_ 9  -L  ± \_dlhj b dhfihg_glu kdhjhklb `b^dhklb V LN  ± ©\yadbcª l_gahj gZijy`_gbc <   e_\Zy qZklv ihke_ ^_e_gby gZ U7  _klv ihegZy km[klZgpbhgZevgZy ijhba\h^gZym^_evghcwgljhibbih\j_f_gbijZ\hcqZk lb fh`_l [ulv ijb^Zg \b^ kmffu ^b\_j]_gpbb ihlhdZ wgljhibb k h[jZlguf agZdhf bnmgdpbbiehlghklbbklhqgbdh\wgljhibblZdqlh ijbf_l\b^  <[17]bamjZ\g_gby ^Ze__ihemq_ghmjZ\g_gb_l_iehi_j_ghkZ\g_ k`bfZ_fhc `b^dhklb Nmjv_Dbjo]hnZ  Kh\iZ^_gb_ M;W k mjZ\g_gb_f l_i ehi_j_ghkZ h[gZjm`b\Z_lky b ^ey [_a^bkkbiZlb\ghc k\_jol_dmq_c `b^dhklb [17]

∂ (ρ6 ) + GLY(ρ69Q ) =   ∂W



]^_ 9Q ±kdhjhklv©ghjfZevgh]hªl_q_gby <nbabq_kdhcdbg_lbd_[145]ba\_klgZkbkl_fZmjZ\g_gbci_j_ghkZ ± wg_j]bbl_ieZ  ±bfimevkZmjZ\g_gb_GZ\v_±KlhdkZ  ±M;WIh^q_j db\Z_lkyqlh _klvke_^kl\b_ b \kbemaZdhgZkhojZg_gbyfZkkul_ M;Wwd\b\Ze_glghnZdlbq_kdbmjZ\g_gbxl_iehi_j_ghkZ[145, (22.69)] ©DZghgbq_kdbcª\u\h^[71,112,151]M;W\l_jfh^bgZfbd_[Zabjm_lkygZ ZdkbhfZlbq_kdbijbgbfZ_fhfmjZ\g_gbb ijb D = ρ6  

∂ (ρ6 ) + GLY , 6 = σ 6  ∂W



AZl_f \   ih^klZ\eyxlky qZklgu_ [ZeZgku lbiZ   b hkms_kl\ey_lky kjZ\g_gb_e_\uobijZ\uoqZkl_cwlbo^\momjZ\g_gbchlghkbl_evghwgljhibb <j_amevlZl_ihemqZxl

,6 =

   ,T − µ N , N +   V 6  7 N  

∑

   ,T ³7   7



H^ghagZqghklv ij_^klZ\e_gbc   h[_ki_qb\Z_lky bo bg\ZjbZglghklvx hl ghkbl_evgh ]jmiiu =Zebe_y b jZ\_gkl\hf V 6  \ khklhygbb jZ\gh\_kby MjZ\g_gb_ kqe_gZfb kh^_j`blihlhdbjZaebqghcijbjh^u\h[ s_f kemqZ_ ± \aZbfhk\yaZggu_ Ihwlhfm ^ey j_r_gby   g_h[oh^bfZ ^h ihegbl_evgZybgnhjfZpbyh\b^_k\ya_cihlhdh\kh[mkeh\eb\Zxsbfbbol_j fh^bgZfbq_kdbfbkbeZfbKh]eZkghl_hjbbEHgaZ]_jZwlZk\yavihklmebjm _lky\\b^_dhgklblmlb\guomjZ\g_gbcDM 

,L =

∑ /LN ;N 



N

h[h[sZxsbo qZklgu_ wfibjbq_kdb_ aZdhgu i_j_ghkZ Nmjv_ NbdZ HfZ b ^j <  ,L ±iehlghklvihlhdh\^Ze__±©ihlhdbª  ; N ±l_jfh^bgZfb q_kdb_ kbeu  ijhihjpbhgZevgu_ ]jZ^b_glZf khhl\_lkl\mxsbo l_jfh^bgZfb q_kdbo nmgdpbc ± ihl_gpbZeh\  ; T a ∇7  /LN  ± dbg_lbq_kdb_ dhwnnbpb_g luKmq_lhf ^eykdhjhklbwgljhih]_g_aZgZc^_gh

σ6 =

∑ ,L ;L = ∑ /LN ;L ;N ≥   L



LN

GZ hkgh\_ iheh`_gby h[ bg\ZjbZglghklb mjZ\g_gbc f_oZgbdb hlghkbl_evgh ©h[jZs_gby»\j_f_gbl_aZf_gu W  W EHgaZ]_jhf[uehihemq_ghkhhl ghr_gb_ \aZbfghklb mf_gvrZxs__ dhebq_kl\h g_aZ\bkbfuo dbg_lbq_kdbo dhwnnbpb_glh\

/LN

/NL 



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fh^bgZfbdbdhlhjucfh`_l[ulvbkihevah\Zgijbj_r_gbbkbkl_fu[ZeZgkh \uomjZ\g_gbc b\g_  vyjfZlb;ZoZj_\hc<mygh\bqZ;_clf_gZO_ckZQ_f [_jkZ FboZceh\Z±=eZamgh\Z AZjm[bgZ vyjfZlb [113] EhdZevgZynhjfZij_^eh`_ggh]h
δ20 = δ[σ − (Ψ + Φ )] =  



]^_ V khhl\_lkl\m_l   G  ± kbf\he \ZjbZpbb ih ihlhdZf b kbeZf 2 0  ± nmgdpbyHgaZ]_jZ±FZoemiZ < ≥b ) ≥±ehdZevgu_ihl_gpbZeujZkk_y gbyhij_^_ey_fu_nhjfmeZfb

<; ;

 /LN ;L ; N ), ,  LÇ N

 5LN ,L , N   LÇ N



FZljbpZ 5LN y\ey_lkyh[jZlghcihhlghr_gbxdfZljbp_ /LN 

1,i = k ,

∑ Rim Lmk = δik = 0,i ≠ k.



m

Mkeh\b_ehdZevgh]hwdklj_fmfZ wd\b\Ze_glghjZ\_gkl\m[113, c. 153]

V   <  )

PD[ 



>Ze__ba b Z\lhjhf\u\h^ylkyDMmjZ\g_gb_l_iehijh\h^ghklbb ijhq Wlh h[klhyl_evkl\h \_kvfZ bgljb]mxs_ ihkdhevdm ba  ±  b   ke_^m_l



<

 V ) 

 V  20 = σ − (Ψ + Φ ) = σ − σ =    

DZdba\_klgh
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Q  D  Q  PEQ ∆4

+

Q E  Q L_iehijblhdba;hj]ZjlhgZd^bk

τ =  6 T 6+ τ  T 6+ T 6  W



 = τ

τ

∫ T6 

+

τ G τ 

O  W a 7  70  W  O  W  6

>

@



Ihke_g_kdhevdh]jhfha^dboij_h[jZah\ZgbcgZoh^bf

O '7  ¹ )   D J  Q  O 




]^_

Q J Q

Q 

 J    J    Q

BamjZ\g_gby^bkkbiZlhjZ '4 W  qZ_f

 Q DQ

Q 

 D    D    Q



'4    W  kmq_lhf  ±  ihem

τ = aτ Q = ΨQWΦ  (α β γ Q )W

O  D 

)  D  E J  )   D  E J

)   D  E  J  Q



>ey i_j_oh^Z \   d kemqZx klZpbhgZjguo iZjZf_ljh\ ^hklZlhqgh \ nhj fmeZo^ey ) L L  iheh`blv H O H F H O  Lh]^Z )  ) )  b  ^Z_ljZg__ihemq_ggh_\ujZ`_gb_^ey W Q  GZb[hevrbcbgl_j_kij_^klZ\eyxlke_^mxsb_qZklgu_kemqZbZ gZqZev guckdZqhdiZjZf_ljh\dh]^Zbodhg_qgh_kbg^_dkhf©Sª agZq_gb_mklZgZ\ eb\Z_lkykjZam`_ijb W  [ kdZqhdiZjZf_ljh\hlgZqZevguoagZq_gbc^h dhg_qguo\dhgp_ ' i_jbh^Zijb τ = τ \ ebg_cgh_ih\j_f_gbbaf_g_gb_ iZjZf_ljh\Bkihevamyihemq_ggu_nhjfmeugZoh^bf KemqZcZ  D E J   PD Q PE Q  Q J  Q QD Q b

)   HO  HF )  KemqZc[  D

) KemqZc\  D

E

)

= ΨQ ΨQ −  b

J

‡ PD Q

) 

E

J

) )

PE Q

Q J  Q

  aWQ

  PD Q

PE Q

  HO  HF    HO  HO

) )

  H O  H O ) 

QD Q

WQ

Ijb


 Ψ (ε − εO ) Φ  = [ + Ψ Q− (εO + εF )] + Q λ  Ψ −  

 b



Q

−





Q  ba ke_^m_l )



È  H O  H F Ø È  H O  H O Ø  Ê  ÚÊ  Ú



>eyijhba\hevgh]hkhq_lZgbyiZjZf_ljh\ D E J  

PD Q °>@PE Q °>@Q J  Q °>@QD Q °>@ ) °>) ) @ )  °>)  )  @ ) ) )    H O ) 

  HO  HF 

  H O )  °>  aH    aH@  aε = ελ +  εO + εF 

AZ^ZqZG_ebg_cguc^bkkbiZlhjLZdh\ufgZah\_f^bkkbiZlhjiZjZ f_lju dhlhjh]h aZ\bkyl hl l_fi_jZlmju O = O 7  F9 = F9 7  Ì = Ì7  WlbaZ\bkbfhklbhibku\Z_fnhjfmeZfb ′ ′ ′ O 7 = O + ∆O Éα   F9 7 = F9  + ∆F9  Éβ  Ì7 = Ì + ∆ Ì É γ 



A^_kv D „ E „ J „ ±ijhba\hevgu_iheh`bl_evgu_ ihklhyggu_ Z T jZaf_jgZyl_fi_jZlmjZ^bkkbiZlhjZ

T W  ± [_a

a 70 τ − 7 Éτ =  T W °>@  76 − 7

3

W  W



Ba ke_^m_lqlh

T W

M Q K

KQ K

Ih^klZ\b\ \ gZoh^bf

O 7 = O + ∆Oηα  F9 7 = F9  + ∆F9 ηβ  Ì7 = Ì + ∆ Ì ηγ  ]^_ D QD „  β = Qβ′  γ = Qγ′ LZdbfh[jZahfaZ^ZqZk\_^_gZdaZ^Zq_ qlhiha\hey_lkjZam\uibkZlvj_r_gb_±nhjfmem^ey ' i_jbh^Zg_ebg_cgh]h ^bkkbiZlhjZ

W

WQQ


)D „ E „ Q  )  D „ J „ Q

  hij_^_e_gu ih   b   Ijb Q  Â′ = ÂÃ′ = ′ = ÃÄ = Ä  b j_r_gby aZ^Zq  b  iheghklvx kh\iZ^Zxl l_ ojhgh]_g_jZpby

]^_ ) L L

g_ebg_cgh]h^bkkbiZlhjZlh`^_kl\_ggZlZdh\hc^eyg_klZpbhgZjgh]h^bkkbiZ lhjZWlhy\ey_lkylj_lvbfZj]mf_glhf\u^_e_gghklbkemqZy Q  AZ^ZqZ  Wgljhih]_g_jZpby ^\mo g_\aZbfh^_ckl\mxsbo ^bkkbiZlh jh\Ihf_klbf\;hj]Zjlhg^\Zh^bgZdh\uo^bkkbiZlhjZkjZaebqgufbl_fi_ jZlmjZfb 7 76  '7 7 76  '7 JZkkfhljbfi_j_oh^u 7  76 b 7  76  \ ^bkkbiZlhjZo 0 b 0   Wgljhiby 0 [m^_l \hajZklZlv 

 'V  !  Z 0  ±mf_gvrZlvky 'V    GZkbgl_j_km_lihegh_baf_g_ gb_ wgljhibb \ kbkl_f_ ©;hj]Zjlhg±^bkkbiZlhjª Hij_^_ey_f l_jfhklZlbq_ kdmxZddmfmebjm_fmx wgljhibx

∆σ$

76

7  G7 =6OF9 OQ 6   7  7  7

∫

= 6OF9



∆σ$

76

= 6OF9

7  G7 = − 6OF9 OQ   7  76  7

∫



Ihkdhevdmbaf_g_gb_wgljhibb;hj]ZjlhgZjZ\ghgmexihhij_^_e_gbxihe gZywgljhibykbkl_fu[m^_l

'V$6

'V$  'V$

È 7 Ø 6OF9 OQÉ 6 Ù  Ê 77 Ú







Mkeh\b_ 'V$6 •  [m^_lkh[ex^_gh_keb 7 6 77 • Ijh\_jbfihke_^ g__g_jZ\_gkl\h

7 6 77

7 6 76  '7 76  '7

7 6

7 6

 '7



!   'V$6 !  

Ih\lhjy_fjZkq_l^ey^bkdj_lghcihlhdh\hcl_jfh^bgZfbq_kdhc wgljhibb

'V$

'7 6OF9 È  Ø  'V$ Ê 7 Ú

'V$6

'V$  'V$

 '7 Ø  6OF9 È Ê 7 Ú

  Ø !   6OF9 '7 È Ê 7 7 Ú



ihkdhevdm 7 76  '7 7  '7  7   '7 !  LZdbf h[jZahf h[Z f_lh^ZjZkq_lZijb\h^yldh^ghfmj_amevlZlmh[s__baf_g_gb_wgljhibbkbk l_fu ©;hj]Zjlhg±^bkkbiZlhjuª ± iheh`bl_evgh qlh y\ey_lky nZdlbq_kdb \u\h^hf\lhjh]h gZqZeZ l_jfh^bgZfbdb \ deZkkbq_kdhc nhjfmebjh\d_ gZ hk gh\_jZg__ijbgylhcwd\b\Ze_glghc_c <Kbevguc^bkkbiZlhj AZ^ZqZOjhgh]_g_jZpby^bkkbiZlhjZImklv\;hj]Zjlhgkl_fi_jZlm jhc 76 !! 7  ihf_s_g ^bkkbiZlhj k gZqZevghc l_fi_jZlmjhc 7  dhlhjZy aZ l_fih^gbfZ_lky 7  76 Hij_^_ebf ' i_jbh^u^eydZ`^h]hpbdeZih^gy 

lby l_fi_jZlmju ^bkkbiZlhjZ gZ '7  l_ bgl_j\Zeu \j_f_gb W N  g_h[oh^b fu_^eyi_j_oh^Z 7N   7N  N  1 Bf__f

7  '77

7

7  N'771

7N a 7N W

7  '7

7  '7 

7  1'7

76 



7N   '7M Q  W W N  ϕQ τ τN = ϕQ η = ηQ  

E_\ZyqZklvM>kf kh\iZ^Z_lklZdh\hc^eykeZ[h]h^bkkbiZlhjZ

'4N

6OF9 '7

'4



6 T N W N 



IjZ\ZyqZklvaZibku\Z_lkyZgZeh]bqgh

'4N

]^_

T N+

 = τN

τN

∫ Ta N

+

(τ)G τ TaN W



O a 7  7N W  O  6

>

@



Ih^klZ\b\ \ gZoh^bf

T N+



∫[

(

)]

[

]

 λ  λ∆7 (1 − N ) + Ψ −Q  =  ∆7  (1 − N ) +  − ηQ G η =   O O     Q 
IjbjZ\gb\Zy b kmq_lhf ihemqbf

W N Q

O  1  N

WQ  

NhjfmeZ hibku\Z_lojhgh]_g_jZpbxkbevgh]h^bkkbiZlhjZijbq_fiZ jZf_lju
τN.Σ = τN.− Σ + τN.  N  1  



A^_kv\_jogbcbg^_dk© . ªh[hagZqZ_l . fh^_ev^eydhlhjhckh]eZkgh 

7N

>

@

   7  76  7   H[S  WN. 6 W  τN. = W OQ +    1 −N 



>ey ' fh^_eb ij_h[jZamy   ihemqZ_f h[hagZqZy lbi fh^_eb \_jogbf bg^_dkhf

τN'

  ΨQW W  Ψ −Q  Ψ −Q   = = + − −  (1 − N )ΨQ +  1 − N  1 − N  1 − N    







Mkeh\b_f jZaeh`_gby \   y\ey_lky lj_[h\Zgb_
τN. =

   W   + −  −  1 − N  (1 − N ) (1 − N ) 



>eyijb[eb`_ggh]hjZ\_gkl\Z\j_f_g N ]hi_jbh^Z\ ' b . fh^_eyol_^ey \uiheg_gbymkeh\by WN. # WN' g_h[oh^bfh  \jy^Zo b m^_j `Zlvlhevdhi_j\u_^\Zqe_gZ  iheh`blv
((

\b_ [m^_l kh[ex^_gh _keb  (1 − N )

( ((1 − N ) ))≤  

−



))<<  qlh dhgdj_lbabjm_f \ \b^_

 Wlh ^Z_l  1  N •   ≈  BlZd ^ey \k_o

N ≤ 1 −  b Q  WN. # WN' qlh ^Z_l q_l\_jluc Zj]mf_gl \ ihevam \u [hjZagZq_gby Q  H[jZlguc i_j_oh^ hl ' fh^_eb b . fh^_eb we_f_glZj_g \   ^_eZ a a a xlkyaZf_gu W N  G W  '7  G 7   1  N  76  7  W G 7  7N  7  W  qlh^Z_l

GW W

a G7 a 76  7  W



\ khhl\_lkl\bb k   Hij_^_ebf ihegu_ \j_f_gZ j_eZdkZpbb aZiheg_gby ©yfuª 76  7  \ ' fh^_eyo b . fh^_eyo LZdh_ \j_fy i_j_oh^Z 7  76  \ . fh^_ebdZdb\kemqZ_keZ[h]h^bkkbiZlhjZjZ\gh∞
ijhp_^mjhc ©nbgblbaZpbbª iheh`b\ 76 → 7 ′ 6 = 76 − δ7 = 76 − ∆7 H  Lh]^Zba ihemqbf

(

)

τ∞ = τΣ. = W OQ 1H  = W  + OQ 1 



Ijb 1   W6. # W  Ihegh_ \j_fy i_j_oh^Z 7  76  \ ' fh^_eb fh`ghgZclbkmffbjh\Zgb_fih N \ ghwlh^hklZlhqgh]jhfha^dhIh 

wlhfm h]jZgbqbfky kmffbjh\Zgb_f WN'  ijb ey k ≤ N − 19 wlhkhhl\_lkl\b_ ijb[eb`_ggh_klvihwlhfm\hkihevam_fky qlh^Z_l

1 W OQÈ Ø  N Ê  Ú

W6'„ Ijb 1 ]^_

 1   

 ba ke_^m_l W6'„ W Ihegh_\j_fy W6' W6'„„

1



W6'„  W6'„„ 

   W È    Ø # W Ê   Ú 

Ç WN'

N 1 

HdhgqZl_evghihemqbf

W6'

W

W6.  W 



Mqblu\Zyqlh©g_\yadZª WfZeZfh`ghkqblZlvqlhkmffZjgu_\j_f_gZ ijhp_kkZ\ ' b . fh^_eyoijb[ebabl_evghkh\iZ^Zxlijb Q l_ihem qZ_fiylucZj]mf_gl\ihevam©\u^_e_gghklbªkemqZy Q  \mol_fi_jZlmjguc;hj]ZjlhgJZkkfhljbfh^ghf_jguc^bk kbiZlhj \ ;hj]Zjlhg_ ih^^_j`b\Zxs_f \ h^ghf l_iehh[f_gghf k_q_gbb _]h l_fi_jZlmjm 76 Z \ ^jm]hf ± 76  GZqZevgZy l_fi_jZlmjZ ^bkkbiZlhjZ 7  ijbq_f 76   7  76 76  76  1'7b

76  7

1'77  76 

1  '7 

76  76   1  1  1    a L_fi_jZlmjZ^bkkbiZlhjZ 7  W [m^_l\hajZklZlvhl 7 ^h 7 6 ihke_q_]hklZ 7  7 6

[bebabjm_lkylZddZdijblhdl_ieZd^bkkbiZlhjmhl;hj]ZjlhgZq_j_aih\_jo ghklvkl_fi_jZlmjhc 76kjZ\gy_lkykhllhdhfl_ieZhl^bkkbiZlhjZq_j_aih \_joghklvkl_fi_jZlmjhc 76 :gZeh]bqghij_^u^ms_fmihemqZ_f

T N+ = W N

>

[

[

]

]

Ì ∆7 Ì (1 − N ) + ¹Q−  T N− = ∆7 (1  + N ) − ¹−Q  O  O  

@

W  1 „  N 
 N

  1 „  1 „

  1  1     

I_j_oh^ d . fh^_eb ^ey   hq_\b^_g hgZ kh\iZ^Z_l k jZg__ jZkkfhlj_g ghc k aZf_ghc 76  7 6  Ihke_^gbc ' i_jbh^ W N  ijb N 1 „  kh\iZ^Z_l k W Q ^eykeZ[h]h^bkkbiZlhjZJZkkfhljbfkhhlghr_gbyf_`^mkhhl\_lkl\mx sbfb  N fb  ' i_jbh^Zfb ^ey ^\mo b h^ghl_fi_jZlmjgh]h ;hj]Zjlhgh\ Ba  b ijb
1  N    1 „  N   

W N WN



1   1   1  1    N  



Ba   \b^gh qlh ojhgh]_g_jZpby ^bkkbiZlhjZ \ ^\mol_fi_jZlmjghf ;hj ]Zjlhg_ ij_\urZ_l lZdh\mx ^ey h^ghl_fi_jZlmjgh]h l_ ijhp_kk ijhl_dZ_l [he__f_^e_gghbl_f\[hevr_ckl_i_gbq_f[eb`_l_fi_jZlmjZ^bkkbiZlhjZ dklZpbhgZjghc

W  W



1   1  W

W

W

 N ! N   1 „ W N W N  W1„ 1  1   

  1   1  



AZ^ZqZ>bkkbiZlhjki_j_f_ggufbiZjZf_ljZfbJZkkfhljbfi_j_ oh^ 7N   7N \kbevghf^bkkbiZlhj_Bf__f

a 7N  W

W 7N   '7 È Ø Ê WN Ú

Q

7   N   '7  '7KQ K

W  WN

Baf_g_gb_iZjZf_ljh\kh\j_f_g_fhibku\Z_lkynhjfmeZfb





O N    H O  N KD F9  N  W

O N  W

O N  W







O N    H O  N K J 

E_\ZybijZ\ZyqZklbM>khhl\_lkl\_ggh



τN

a ∆4 N = 6 O N τ F9  N  1 − N G 7N  1 − N  '4N

∫

]^_



F9  N    H F N KE 

6 T N ¹ W N 



T N+ = =

(

[

]

λN  1 − N a 76 − 7N τ = O N  1 − N 

)(

)

λ N − a  + ελ N ηγ  − εO N ηα (76 − 7N  1 − N ) O N −



Ih^klZ\b\ \ nhjfmeu   dhlhju_ ijbjZ\gb\Z_f ^jm] ^jm]m  ± ihke_g_kdhevdh]jhfha^dbo\udeZ^hdgZoh^bf

aτ = N

ΨQW N − Φ O Φ N  W N − = N −  Φ  N =  N [(1 − N )ΨQ + ] DN − Φ  N Φ N α β Q =  + µα Q εO N + µβ Q εO N  Φ  N α  γ  Q =  + ν N  γ  Q ε λ  N + ν N  α  Q ε O  N  PD Q

 1 − N Q + Q γ +  + Q − Q  ν N  γ Q =  DQ (1 − N ) + Ψ −Q (γ + )

[

]







GZebphZgZeh]bykhkeZ[uf^bkkbiZlhjhfijb N 1 \k_nhjfmeui_j_oh^yl \khhl\_lkl\mxsb_bfnhjfmeu^eykeZ[h]h^bkkbiZlhjZJZkkfhljbf\Z`gu_ qZklgu_ kemqZb   kdZqhd \k_o iZjZf_ljh\ ijb W     kdZqhd iZjZf_ljh\ ijb τ = a τN  ebg_cgh_ih\j_f_gbbaf_g_gb_jhkl iZjZf_ljh\Bf__f   α = β = γ =   PD Q

PE Q  ν N  γ Q = ν N α Q =  Q … lZddZdQ …   Q  1  N
) N

  D

E

J ‡ PD Q )  N 

  D

E

J

)  N

PE Q

 
Q > 1  N    Q @   1  N 
>

 ν N  γ Q = ν N α Q =  

PE Q 

@


Q  ÎÑ Ï 1  N     Q  ÑÐ 1  N  

   baf_gy_lky\ij_^_eZo    
IZjZf_lju Q N  J  Q b Q N D Q ijb Q

q_gby[ebadb_d AZ^ZqZ>bkkbiZlhjkg_ebg_cgufbiZjZf_ljZfbA^_kvlZd`_bf_ _lkyZgZeh]bykg_ebg_cgufkeZ[uf^bkkbiZlhjhfIjb Q bf__lkyihegZy wd\b\Ze_glghklv g_klZpbhgZjghfm ^bkkbiZlhjm JZkkfhljbf qZklgu_ kemqZb   ebg_cguc jhkl iZjZf_ljh\ k l_fi_jZlmjhc   kdZqhd iZjZf_ljh\ \ dhgp_ pbdeZgZijhly`_gbbdhlhjh]hhgb©aZfhjh`_guªBf__f   D E J     Q  
Q N  J  Q

Q N D Q

Q 1 N

1  N       Q 1  N ° È  Û  Ê   ÜÝ  1  N  

  H O  N  H F N )  N

) N

  H O  N  H O N Q 1  N )  N

)  N °>  aHN    aHN @  aHN


  D

) N  )  N

   H O  N   H F N   H   ON

E J ‡ PD Q PE Q Q N  J  Q Q ) N )  N )  N 

Q N D Q



LZdbfh[jZahfkemqZc ±kdZqdhh[jZagh]hbaf_g_gbyiZjZf_ljh\^bkkbiZlh jZ \ dhgp_ pbdeZ τ ∈ [ a τN ] ijb Q  ijb\h^bl d kh\iZ^_gbx ' i_jbh^h\ g_ebg_cgh]h b ebg_cgh]h ^bkkbiZlhjh\ Wlh ^Z_l r_klhc Zj]mf_gl ©\u^_e_gghklbªkemqZy Q  

AZ^ZqZ  Mkeh\b_ Z^^blb\ghklb ojhghrdZeu  Ijhp_kk i_j_oh^Z

7  76  \ kbevghf ^bkkbiZlhj_ ihjh`^Z_l ^\_ rdZeu l_fi_jZlmj ± 4 7 ^7  7 7   7N   76 ` b \j_f_gguo bgl_j\Zeh\ 4 τ = {τ Q  τ Q  τN  Q  τ 1  Q } Khhl\_lkl\b_ f_`^m wlbfb ^\mfy rdZeZfb mklZgZ\eb\Z_lky

nhjfmeZfbojhgh]_g_jZpbbBah[s_cl_hjbbrdZebbaf_j_gbc[40171]ke_ ^m_lqlh\kydZyrdZeZbgl_j\Zeh\_^bgkl\_ggZklhqghklvx^h]jmiiuebg_c guo ij_h[jZah\Zgbc \b^Z *DE  [  D[  E  Ihkdhevdm f_`^m 4 7  b 4 W bf__lky h^ghagZqgh_ khhl\_lkl\b_ Z^^blb\ghklb h^ghc ba gbo ^he`gZ khhl \_lkl\h\ZlvZ^^blb\ghklv^jm]hcBgZq_]h\hjy_keb_klv^\Zk^\b]ZihrdZe_ 4 7  7N   7N b 7N  7N wd\b\Ze_glgu_k^\b]m 7N   7N lhbgl_j \ZeZf W N Q b W N Q khhl\_lkl\mxsbfwlbfk^\b]Zf^he`_gkhhl\_lkl\h\Zlv bgl_j\Ze WNQ N bfwd\b\Ze_glguc

W N Q  W N Q

WNQ N  N

 1 



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XW

6W W XW XW

6W  W XW

6W  W XW  6W  W

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31,90

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28,28

26,86

32,22

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30,16

28,55

26,94

12,46

16,27

18,44

18,96

19,25

11,85

16,27

18,27

19,20

19,35



Ba lZ[e  ke_^m_l qlh ijb W •  W U  ih]j_rghklv aZf_gu ' fh^_eb . fh^_evx g_kms_kl\_ggZ l_ ^ey \j_f_g jZ\guo mq_l\_j_gghfm kh[kl\_g

ghfm \j_f_gb keZ[h]h ^bkkbiZlhjZ \ebygb_ ^bkdj_lghklb g_kms_kl\_ggh b fh`ghbkihevah\Zlv . fh^_evKjZ\g_gb_ihl_ieh\ufihlhdZf[uehijh\_ ^_gh^eylhq_d [ OdhglZdl 0b 0  b [ OdhglZdl 0 b 0  b l_o`_fhf_glh\\j_f_gbIjZdlbq_kdh_kh\iZ^_gb_ihlhdh\jZkkqblZgguoih ' b . fh^_eyf[uehihemq_ghk H    lZd`_m`_^ey W •  WU 



=E:<:P?IHQDB>BKKBI:LHJH< Fu g_ gZ[ex^Z_f g_ihkj_^kl\_ggh ^bnn_ j_gpbZevguo mjZ\g_gbc fu gZ[ex^Z_f ebrv dh g_qgu_mjZ\g_gbydhlhju_y\eyxlkyg_ihkj_^kl \_gguf\ujZ`_gb_fgZ[ex^Z_fuoy\e_gbc :ImZgdZj_ Ml\_j`^Zxlqlhmki_of_lh^Z=_ca_g[_j]Z fh`_l [ulv ijb\_^_g d qbklh Ze]_[jZbq_kdhfm f_lh^mhibkZgbyijbjh^ul_bkdexq_gbxbanb abdbg_ij_ju\guonmgdpbcGhlh]^Zgm`gh[m^_l \ ijbgpbi_ hldZaZlvky hl ijhkljZgkl\_ggh \j_f_ggh]hdhglbgmmfZ :Wgrl_cg

†Ebg_cgu_h^ghjh^gu_p_ihqdb <gZklhys_c]eZ\_jZkkfZljb\Zxlkykbkl_fu\aZbfh^_ckl\mxsbo^bkkb iZlhjh\±p_ihqdbba 1h^ghf_jguoiehlghdhglZdlbjmxsbo^jm]k^jm]hf h^ghjh^guo ^bkkbiZlhjh\ 0 N   N  1  =jZgbqgu_ ^bkkbiZlhju 0 b 0 1 bf_xll_fi_jZlmjm;hj]ZjlhgZ± 76 <gZklhys_fiZjZ]jZn_jZkkfZljb \Zxlky h^ghjh^gu_ ebg_cgu_ k ihklhyggufb b h^bgZdh\ufb iZjZf_ljZfb p_ihqdb^bkkbiZlhjh\Z^Ze__±boh[h[s_gbyg_h^ghjh^gu_p_ihqdb†  g_klZpbhgZjgu_† bg_ebg_cgu_p_ihqdb†  >ey\u\h^ZmjZ\g_gby^bkkbiZlhjghcp_ihqdbM>P \u^_ebf\g_cljb ijhba\hevguo ihke_^h\Zl_evguo ^bkkbiZlhjZ ^ 0 N  0 N  0 N ` kmq_lhf\aZbfh^_ckl\byk 0 N b 0 N  ^ey M ]hpbdeZ ' i_jbh^Z dh]^Z W ° W M Ijbjhklm[uev dhebq_kl\Zl_ ieZ\ 0 N 

∆4N  M = 6OF9 ∆7N  M  '7N  M

7N  M  7N  M 



IehlghklvihlhdZl_ieZba 0 N  b 0 N  beb \ h[jZlghf gZijZ\e_gbb  \ M f pbde_

T  NM − − N Õ =

[

]

λ a a 7N − M τ − 7N  M τ  N O

 1 

>ey]jZgbqguo^bkkbiZlhjh\ 0 b 0 1  h[f_gb\Zxsboky l_iehf k ;hj]Zj 

lhghf

T %M   W

a O 7  7  W T 1M  %  W  O  6  M

>

@

a O  W  76   7 O  1 M

>

@

]^_© % ªh[hagZqZ_l;hj]Zjlhg 76 ±l_fi_jZlmjuih^^_j`b\Z_fu_;hj]Zjlh ghfgZ\g_rgbo]jZgbpZop_ihqdbL   L_fi_jZlmju\ 0 N \gZqZe_ M ]h L

pbdeZ± 7N  M  N  1  M  -  7N 7 \dhgp_ M ]hpbdeZ± 7N  M Baf_ g_gb_l_fi_jZlmju^bkkbiZlhjZ 0 N \oh^_ M ]hpbdeZ

È WØ 7N  M   '7N  M É Ù  ÊW M Ú

a 7N  M  W



Kj_^gyyaZ M cpbdel_fi_jZlmjZ\ 0 N 

 7N  M = ÕM

τM

a

∫ 7N  M (Õ)GÕ = 7N  M − + 

∆7N  M








 T %M−  = M ÕM T  NM − − N = M

 ÕM

T N M−  N + = M

 ÕM

τM

∫ T % −(Õ)GÕ = O  (76 Ì

 M



)

− 7 M 



τM

∫ TN − −N (Õ)GÕ = O (7N − M − 7N  M ) N Ì

 M

 1 





τM

∫

T N M−  N + (Õ)GÕ =



T 1M − % =  M

 ÕM

(

)

Ì 7N  M − 7N + M  N O

' 4N M

 1  

τM

Ì  M ∫ T 1 − % (Õ)GÕ = O  (71  M − 76 ) 





J_amevlbjmxsbcl_iehijblhd\ 0 N 





Ë Û 6 W M T  NM   N  TN M  N  ÌÍ M M ÜÝ





6 W M Ijb N

O 7  7N  M  7N  M  N O  N  M

bf__f ' 4M 6 W M

Ijb N

1 

 1  



Ë Û 6 W M T%M   T M  ÌÍ M M ÜÝ O  7  7  7  7 M  O  6  M   M

>

@



  ∆ 4 1+  M = 6τ M  T  1M − − 1 − T 1M − 6  =    M  M  O 6 W M  7  71 M  71 M  71  M  O  6

>

@



Bamkeh\byl_ieh\h]h[ZeZgkZbkmq_lhf b ihemqZ_f

∆7N  M τM

(

)

D =  7N − M − 7N  M + 7N + M  N O

 1  



Ihemq_ggh_ mjZ\g_gb_   hlebqZ_lky hl \klj_qZxsboky \ ebl_jZlmj_ dh g_qghjZaghklguoZiijhdkbfZpbcmjZ\g_gbyl_iehijh\h^ghklbl_fqlh\_eb qbgu 7Q M  Q N   N  N   \g_f±mkj_^g_ggu_ih M fm\j_f_gghfmpbdem Zg_hlg_k_ggu_d M fmfhf_glm\j_f_gbWlhb]jZ_lj_rZxsmxjhevlZddZd iha\hey_lba ihemqblvp_eucdeZkkmjZ\g_gbc\qZklghklbbke_^mxs__ ba ijb W M  G W  ∆7N  M → G 7  O  G [ h[uqgh_mjZ\g_gb_Nmjv_

∂7 ∂ 7 =D   ∂τ ∂[



>Ze__ [m^_l ihdZaZgh qlh   ± g_ [he__ q_f gme_\h_ ijb[eb`_gb_ .  fh^_e_cke_^mxsboba ' fh^_eb  >ey ijZdlbq_kdbo jZkq_lh\ mjZ\g_gbx   p_e_khh[jZagh ijb^Zlv bgmx g_ kh^_j`Zsmx kj_^gbo l_fi_jZlmj nhjfm >ey wlh]h gZ hkgh\_  aZibr_f

7N − M = 7N − M − +

∆7N − M

7N  M



 7N  M

7N  M  

7N  M  

'7N  M  

'7N  M  



Ih^klZgh\dZ \ ihke_ijhkluoij_h[jZah\Zgbc^Z_lM>P

− ; N − M + 5 M ; N  M − ; N + M = EN  M  N

 1  

 

]^_

;N M

'7N  M  5 M '7

Ø È W  É  U  Ù EN  M 7  '7  N'   N  M  Ê WM Ú  ∆ 7N  M − ≡ 7N − M − − 7N  M − + 7N + M − 

N 

(

)

:gZeh]bqgu_   mjZ\g_gby ^ey ]jZgbqguo ^bkkbiZlhjh\ p_ihqdb  0 b 0 1 ke_^mxlba    

 5 M   ; M  ; M

E M  E M =





(

 76 − 7 M − + 7 M − ∆7

  ; 1  M  5 M   ; 1 M

E1 M 

 7  71 M   71  M   '7  6

E1 M

)



M>P kh\f_klghk  h[jZamxlkbkl_fmba 1g_ba\_klguo ; N  M  N  1 b 5 M Wlmkbkl_fmfh`ghaZibkZlv\fZljbqghf\b^_

$M ; M ]^_ ; M

^; N  M `N

 1

%M



^EN  M `N

\_dlhjklhe[_pg_ba\_klguo % M

 1

±\_d

lhjklhe[_pijZ\uoqZkl_chij_^_ey_fuokh]eZkgh ± FZljbpZ $ ± lj_o^bZ]hgZevgZybkbff_ljbqgZy

$M

−  ν = N −  N +    ν < N −  ν > N +    M  M  D N ν =  D N Q N  1  5 M δ Nν  N =  1 −   5 M +   δ Nν  N =  1

(

D N MQ

)

 N = ν  D QM N  δ Nν =   N ≠ ν



JZkkfhljbfkihkh[uj_r_gbykbkl_fu I_j\ucaZdexqZ_lky\aZ^Z gbb^eydZ`^h]h M ]hpbdeZ\_ebqbgu W M Zke_^h\Zl_evghb 5 M Qbkehg_ba \_klguolh]^Zkh\iZ^Z_lkqbkehfmjZ\g_gbcJ_r_gb_ fh`_l[ulvgZc ^_ghf_lh^hfh[jZlghcfZljbpu

;M



$ M % M  ; N  M

=

1

∑ (DN M6 ) 6 =

−

E6  M 





 ]^_ D N M 6 ±we_f_gluh[jZlghcfZljbpu $ M Ijb\lhjhfkihkh[_aZ^Zxlky \k_ \_ebqbgu ; 1 M  qlh iha\hey_l ba ihke_^g_]h mjZ\g_gby kbkl_fu gZclb

; 1   M bl^BkdexqZylZd ^; N  M `ijboh^bfdmjZ\g_gbyfkl_i_gb 1hl

ghkbl_evgh 5 M  Imklv wlh mjZ\g_gb_ j_r_gh Lh]^Z ih^klZgh\dhc 5 M  \h \k_

^

`

mjZ\g_gbygZqbgZykihke_^g_]hgZoh^bf\k_ ; N  M  Lj_lbc kihkh[ aZdex qZ_lky \ hlg_k_gbb 5 M  d g_ba\_klguf b \ aZ^Zgbb h^gh]h ba ©\gmlj_ggboª

^; N  M ` Imklv aZ^Zgh ; 6  M

;6 M  Lh]^Z kbkl_fZ   jZkiZ^Z_lky gZ ^\_ kbkl_fumjZ\g_gbclZdh]h`_\b^Zihjy^hdh^ghcbaih^kbkl_f± 6  Z^jm ]hc± 1   6   DZ`^mxba^\moihemq_gguo©mdhjhq_gguoªkbkl_ffh`gh

j_rZlv\lhjufkihkh[hfQ_l\_jluckihkh[ij_^klZ\ey_lkh[hc\lhjhcj_Zeb am_fuc©k\_jom\gbaªaZ^Z_lky ; M b^Ze__ZgZeh]bqgh AZ^ZqZ>\mf_jgu_kbkl_fuKbkl_fZ e_]dhh[h[sZ_lkygZkem qZc ^\mf_jguo d\Z^jZlguo ^bkkbiZlhjh\ ©dm[bdh\ª  aZihegyxsbo ijyfh m]hevgmx h[eZklv lhesbghc O  Ijhba\hevguc \gmlj_ggbc ^bkkbiZlhj kbkl_ fu fh`gh jZkkfZljb\Zlv dZd h[sbc we_f_gl ^\mo i_j_k_dZxsboky p_ihq_d \aZbfh^_ckl\mxsbckq_lujvfy©khk_^yfbªImklvlZdhc^bkkbiZlhj± 0 N P \oh^bl\dZq_kl\_ N ]h\i_j\mxp_ihqdmb\ dZq_kl\_ P]h ± \h \lhjmx _c i_ji_g^bdmeyjgmx  N  1  P  1   ?]h ©khk_^yfbª [m^ml ih i_j\hc p_ihqd_±^bkkbiZlhju 0 N Pb 0 N Pih\lhjhc± 0 N P b 0 N P  Dhebq_kl\Zl_ieZi_j_r_^r_]h\^bkkbiZlhj 0 N P hl©khk_^_cªgZ M fpbde_

'4N P M

O F9 '7N P M  '7N P M

7N P M  7N P M 



7H¡žR¡S›¢R›– 0 N P bl_iehhllhdbbag_]hh[mkeh\e_ggu_\aZbfh^_ckl\b_f k©khk_^yfbªfh`ghij_^klZ\blv\\b^_

' 4N M ' 4N M ]^_

 

+

+

O W M T N  M  ∆ 4 P M = O  τ M T P M   

 

O W M T N  M  ' 4P M

 

O W M T P M 



T N M

O   7  7N P M T P M O  N P M

O 7  7N P M  O  N P  M

T N M

Ì O 7  7N P M  T P− M = 7N P M − 7N  P − M   O  N P M O

(

)

KhklZ\eyy[ZeZgkl_ieZ\ 0 N P gZ M fpbde_gZhkgh\_ ± ihemqZ _f



'7N P M WM ]^_

Û DË 7 7    ' ' Ì Ü N  P  M N  P  M O ÌÍ N  ÜÝ P 



∆ 7N  P M = 7N − P M − 7N  P M + 7N + P M

N 

∆ 7N  P M = 7N  P − M − 7N P M + 7N  P + M 

 P 



MjZ\g_gb_   y\ey_lky ^\mf_jguf ZgZeh]hf mjZ\g_gby   b ijb mijh s_gghf j_^mpbjh\Zgghf  ij_^_evghf i_j_oh^_ W M  G W  O  G [G \  '7N P M  G 7  [ \ W ^Z_l^\mf_jgh_mjZ\g_gb_Nmjv_

˜7 ˜W

È ˜7 ˜7 Ø DÉ    Ù  Ê ˜[ ˜\ Ú



dhlhjh_lZd`_dZdb y\ey_lkylhevdhgme_\ufijb[eb`_gb_fiheghc .  fh^_ebkhhl\_lkl\mxs_c Ij_h[jZah\Zgb_ ZgZeh]bqghhkms_kl\ e_gghfmk biha\hey_lihemqblv^\mf_jgucZgZeh]mjZ\g_gbc 

 ; N P M  ; N P  M   5 M   ; N P M  ; N P M  ; N P  M

; DE M

EN P M  

 '7DE M    EN  P M = ∆  7N  P M − + ∆  7N  P M −   '7 ∆7   N  P 

(

)

(

)

FZljbpZ $M  hl\_qZxsZy kbkl_f_ mjZ\g_gbc   iylb^bZ]hgZevgZy ih jy^dZ 1 – 1   H[h[s_gb_ gZ lj_of_jguc kemqZc hq_\b^gh hgh ijb\h^bl d k_fb^bZ]hgZevghcfZljbp_ $M k^bZ]hgZevgufbwe_f_glZfb  5 M   Ihjy ^hd wlhc fZljbpu ± 1 – 1  – 1   ]^_ 1  ± qbkeh ©^bkkbiZlhjguo iehkdh kl_cª 1 – 1  

†G_h^ghjh^gu_p_ihqdb G_h^ghjh^gufb gZau\Z_f p_ihqdb ^bkkbiZlhjh\ hlebqZxsboky ^jm] hl ^jm]Z \k_fb iZjZf_ljZfb O N › O N  F9  N › F9  N  λ N ≠ λ N + 

W U  N › W U  N  N =  1 −  JZkkfhljbf [ZeZgk l_ieZ \ ^bkkbiZlhj_ 0 N  gZ M fpbde_ ' i_jbh^_ HimkdZyh^bgZdh\mx^ey\k_o^bkkbiZlhjh\iehsZ^v l_iehi_j_^Zxsboih\_joghkl_c 6bf__f

[

]

∆4N  M = F9  N O N ∆7N  M  ∆4 N+ M = Õ M T  NM − − N − T N M−  N +  

 WM

T  NM   N

WM

Ô

>7aN  M W  7aN  M W @ G W

]^_

5N  N

 7  7N  M  5N  N  N  M

5N  N

U N   U N U N 

O N ON 

IjbjZ\gb\ZyZddmfmebjm_fh_dhebq_kl\hl_ieZ '4N  M ihlhdh\hfm '4N M 

ihke_g_keh`guoij_h[jZah\ZgbcgZoh^bf

∆7N  M τM DN

ON  J N  F9  N

(

)

D =  N ∆  γ N  7N  M  O  N  N

UN  J N  5N  N 



UN  γ N = γ N + γ N +  5N  N

MjZ\g_gb_   y\ey_lky ZgZeh]hf mjZ\g_gby   b i_j_oh^bl \ g_]h ijb \ujh`^_gbbg_h^ghjh^ghklb D N D  O N O  J N  J N J N  I_ j_oh^\ hlkj_^gboaZpbdel_fi_jZlmj^bkkbiZlhjh\dgZqZevguf ^ey ^Zggh]hpbdeZ bdijbjZs_gbyfboijb\h^bldM>P\b^Z

D N M  N & N  M  D N M N & N  M  D N M N  & N  M &N M

'7N  M  D N M  N '7

EN  M 



 J N D N M N   J N 

 ÕUN  + ÄN   D N MN =    ÕM  Ihkdhevdm ^ey ]jZgbqguo ^bkkbiZlhjh\ p_ihqdb  N    N



1  [ZeZgkh\u_ mjZ\g_gby ZgZeh]bqgu   b   ihegZy kbkl_fZ ba 1]h mjZ\g_gby \ a

D N MQ  lj_o^bZ]hgZevgZy b i_j_oh^bl \ fZljbpm $ M  ijb \ujh`^_gbb g_h^ghjh^ghklb  γ N − = γ N = = γ N + =  F_lh^uj_r_gby khhl\_lkl\mxllZdh\uf^ey 

fZljbqghf \b^_ ZgZeh]bqgZ   fZljbpZ $ M



†G_klZpbhgZjgu_p_ihqdb JZkkfhljbf g_h^ghjh^gmx p_ihqdm ^bkkbiZlhjh\ ^ 0 N `  N =  1 −  

iZjZf_ljudhlhjuobaf_gyxlky\h\j_f_gb>ey M ]hpbdeZ M =  - wlhba f_g_gb_hibku\Z_lkynhjfmeZfb    τ  ON  M − ON  M − O N τ = ON  M τ = ON  M −  + ε NO M   ε NO M =    τ M  ON  M −        τ  F9  N  M − F9  N  M −   F9  N τ = F9  N  M τ = F9  N  M − τ  + ε NF M  ε NF M =   τ M  F9  N  M −         τ  λ N  M − λ N  M −   λ N τ = λ N  M τ = λ N  M −  + ε Nλ M   ε Nλ M =  τ M   λ − N M      

Ihkdhevdm gZ g_[hevrbo ijhf_`mldZo \j_f_gb W M  hlghkbl_evgh_ baf_g_gb_ iZjZf_ljh\fZeh^Ze__iheh`bfqlh

HNQ M







  HNQ M ¹ HNP M   Q P

O F O 

Baf_g_gb_\h\j_f_gbl_fi_jZlmju^bkkbiZlhjZ 0 N 

a 7N  M  W

È WØ 7N  M   '7N  M É Ù  ÊW M Ú

È WØ 7N  M   7N  M  7N  M  É Ù ÊW M Ú a G 7N  M

'7N  M

GW  WM



:ddmfmebjm_fh_l_jy_fh_ 0 N gZ M fpbde_l_ieh τM

a ∆4N  M = 6 ON  M τ F9 N  M τ G 7N  M = ∆4 N M − ⋅ ΦMN 

∫

'4N M  Ihlhdh\h_l_ieh





[

]

∆ 4 N+ M = 6 τ M T N+ M − T N− M 

]^_

T N M





6ON  M F9  N  M '7N  M )MN    HNO M  HNF M 

 WM

WM

Ô

>7aN  M W  7aN  M W @ G W 5N  N  W



 WM

T N M

(

WM

Ô

>7aN  M W  7aN  M W @ G W 5N  N  W

)

 a a  W 5N − N (Õ) = a ρN − M (Õ) + a ρN  M (Õ)  U N M 

ON  M  W  O N  M  W



Kmq_lhf −

T N+ M = T N− M =

7N + M − 7N  M 5 N MN−+ 

7N  M − 7N − M 5 N M−−N

]^_

)MN 

Φ MN − =

 7N + M − − 7N  M − ∆7N + M − ∆7N  M   M  Φ  N +  −  +    

 7N  M − − 7N − M − ∆7N  M − ∆7N − M   M  Φ N −  −  +    







U N  M  HNO M  HNO M  U N  M  HNO  M  HNO  M





 5N M N 

(

)



(

ρ N − M − ε NO−  M − εNλ−  M + ρN  M − εNO M − ε Nλ M

(

 5 N M−− N 5 N M−− N =

)

(

)

 ÒN − M − + ÒN  M −  



) 

Ih^klZ\eyygZc^_ggu_\ujZ`_gby\mjZ\g_gb_[ZeZgkZ\ 0 N

'4N  M

ihemqbf

È  W UN  M   M Ø ) N Ù '7N  M É W Ú Ê M

>

'4N M 

@

'  J N M   ( N M 7N  M   '  ( N M 7N  M   N

 N

A^_kvh[hagZq_gh

ρN  M − ON M   M  U N  M   γ N M−− =  J N   D N  M  5N M N  5 N M−− N   J N M  J N M  J N M  ( N M = ρN  M − Φ MN −    ( N M   U N  M  ) MN  ( N M  ( N M   ( N M 

W UN  M 



?kebiZjZf_lju^bkkbiZlhjZ\h\j_fy M ]hpbdeZihklhyggughkdZqdhh[jZa ghf_gyxlkyijb W W M ijbgbfZyagZq_gby Q N  M  ν = O  F9  λ lh\_ebqbgu

HNQ M h[jZsZxlky\gmebLh]^Zkh]eZkgh b \gmebh[jZsZxlkyb

\_ebqbgu

)MN  )MN 

( N M 

( N M   



DZdke_^m_lba lh]^Z )MN

bmjZ\g_gb_ i_j_oh^yl\mjZ\g_ gb_ hlebqZykvebrvbg^_dkhf© M ªmiZjZf_ljh\ ?keb ijb W W M  iZjZf_lju kdZqdhh[jZagh g_ baf_gyxlky hklZ\Zykv ih klhyggufblhbg^_dk© M ªfh`_l[ulvhims_gbkh\iZ^_gb_j_^mpbjh\Zggh]h

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'W  7  [  O  W  7  [ W  7  [  O  W



'W '[ 7  [ W 

  O '[7  [ W = DτU  ∂ [ +  ∂ [ +7  [ W      



<ijb\_^_gguonhjfmeZohkms_kl\e_ghjZaeh`_gb_\jy^uL_cehjZihkl_i_ gyf W M  b O  k ihke_^mxs_c aZibkvx \ hi_jZlhjghf \b^_ >ey '7N  M 7N  M  7N  M bf__f

'7N  M  7  [ W  W M  7  [ W

Ë Û WM ÌW M ˜ W  ˜W  Ü7  [ W   ÌÍ ÜÝ





Iheh`bf W M O D W U Lh]^Ze_\ZybijZ\ZyqZklb [m^mlhij_^_eylvky  b khhl\_lkl\_gghijb W M W U \gboIhemq_ggh_ihegh_mjZ\g_gb_ . fh^_ebbf__l\b^

(

)

        D    ∂ W + ÕU ∂ W + − D ∂ [ +  ÕU ∂ [ + + ÕU ∂ W +  ÕU ∂ W +7  [ W =       Ijb oZjZdl_jguo \j_f_gZo ijhp_kkZ l_iehi_j_ghkZ f_gvrbo beb h^gh]h ih jy^dZ k W U  g_h[oh^bfh bkihevah\Zlv mjZ\g_gb_   beb _]h . ZgZeh]   Ih f_j_ \hajZklZgby wlh]h oZjZdl_jgh]h \j_f_gb \   fh`gh hl[jZku\Zlv qe_guk\ukhdbfbkl_i_gyfb W U hklZ\eyyqe_gukgbadbfbkl_i_gyfb>eyoZ jZdl_jguo \j_f_g ihjy^dZ W !! W U  fh`gh hl[jhkblv qe_gu k W U  WU  b l^ qlh^Z_lgme_\h_ijb[eb`_gb_±mjZ\g_gb_Nmjv_IjboZjZdl_jguo\j_f_gZo ihjy^dZ W   W g_h[oh^bfhhklZ\blvqe_guk W U ijb_s_f_gvrbo± W U b WU b l^ l_ ihemqZ_f i_j\h_ \lhjh_ b ihke_^mxsb_ . ijb[eb`_gby iheghc . fh^_eb   MjZ\g_gb_ i_j\h]h ijb[eb`_gby ke_^m_l ba   ijb  P •   WP U

 ∂7 ∂ 7  D  ∂ 7   + ÕU ∂ W −D  − ÕU  =    ∂W ∂[   ∂[ 
(

)

˜7 ˜7  WU  ˜W ˜W 

D



˜7 D Ë ˜ È ˜7 Ø ˜7 Û  D    W É Ù U Ì ˜[   Í ˜W Ê ˜[  Ú ˜[  ÜÝ

D  Ë ˜ È ˜7 Ø ˜ È ˜7 Ø Û WU ÌD É  Ù    É  Ù Ü   Í ˜W Ê ˜[ Ú ˜W Ê ˜[ Ú Ý



Hp_gdZihjy^dZ\_ebqbguqe_gh\ kh^_j`Zsboq_l\_jlmxb\lhjmxijh ba\h^gu_ihdhhj^bgZl_ihdZau\Z_lqlhi_j\ucbagbogZ^\Zihjy^dZf_gv r_\lhjh]hWlhiha\hey_l\\_klb©ijb[eb`_gb_ªhl[jhkb\\ ihke_^ gbcqe_gqlh^Z_l 

( + ÕU ∂W ) ∂∂7W − D ∂ 7  =   

∂[ 



MjZ\g_gb_   ± \lhjh]h ihjy^dZ ih W  ihwlhfm gZqZevgu_ mkeh\by aZ^Zqb Dhrb^eyg_]h

7  [



M [ 

∂7 = Ψ  [  ∂W W = 



Ke_^h\Zl_evgh j_r_gb_   g_ kh\iZ^Z_l k j_r_gb_f mjZ\g_gby \h \lhjuo kdh[dZol_kj_r_gb_fmjZ\g_gbyNmjv_ H[hagZqbf

∂7 ∂ 7 −D = )  [ W ≠   ∂W ∂[ 



Lh]^Zih^klZgh\dZ \ bj_r_gb_ihke_^g_]h^Zxl

)  [ W

)  [ H[S W W U  )  [ = Ψ  [ − D

G ϕ G [





Nmgdpby 7  [ W l_i_jve_]dhgZoh^blkybaj_r_gbyg_h^ghjh^gh]hmjZ\g_gby   ]^_ ijZ\Zy qZklv ^ZgZ   ©Bklhqgbd l_ieZª )  [ W  h[mkeh\e_g gZ qZevghcg_h^ghjh^ghklvxl_fi_jZlmjubkdhjhklb__baf_g_gbyhgbf__lj_ eZdkZpbhggucoZjZdl_jbijb W W U khklZ\ey_lf_g__k\h_ci_j\hgZqZev ghc\_ebqbguZijb W =  ÕU ±f_g__ AZ^ZqZ>\mf_jgh_ . ijb[eb`_gb_>ey^bkdj_lgh]hmjZ\g_gby^\m f_jghcp_ihqdb i_j_oh^d . fh^_ebhkms_kl\ey_lkyZgZeh]bqghMjZ\ g_gb_i_j\h]hijb[eb`_gbybf__l\b^  Ë  D W  D  W ˜  ³ ˜ ˜  ˜\ U W W Ì  U [ Í









ÛÜÝ7  [ W

  

Z\lhjh]h  ËÈ   Ø   D W  W    D  W  ˜  ³ ˜ W ˜ U W U U ˜W ˜ [  ˜ \ W U W ÌÊ Ú   Í



A^_kv ³







ÛÜÝ7  [ W

 

˜[  ˜\ Hp_gdZqe_gh\k ˜[ b ˜\ \ ihdZau\Z_lqlhb\wlhf

kemqZ_fh`gh\uibkZlvmjZ\g_gb_©ijb[eb`_gbyª

  WU ˜W ÈÊ ˜˜7W  D³7 ØÚ



(817)

j_r_gb_ dhlhjh]h ZgZeh]bqgh j_r_gbx   Gme_\h_ ijb[eb`_gb_ \  \gh\v^Z_lmjZ\g_gb_Nmjv_^\mf_jgh_  AZ^ZqZ . fh^_evg_h^ghjh^ghcaZ^Zqb<kemqZ_g_h^ghjh^ghcp_ ihqdbdh]^ZdhglbgmZebaZpbyhkms_kl\ey_lkygZhkgh\_mjZ\g_gby [h e__keh`gh]hq_f lj_[mxlky^hihegbl_evgu_hlghkbl_evgh khhl \_lkl\by '  b . fh^_e_c >ey nhjfmebjh\db bo ij_^iheZ]Z_f   ^bZf_lju \k_o ^bkkbiZlhjh\ h^bgZdh\u ON O N  1    nmgdpbb F9 F9  [  b O O  [ dhlhjufbZiijhdkbfbjmxlkyiZjZf_lju F9N b O N y\eyxlky^hklZ lhqgh ]eZ^dbfb qlh \hafh`gh _keb Q N  Q N  Q N   Q F9  O  hlebqZxlky 

^jm] hl ^jm]Z mf_j_ggh   nmgdpby D D  [ O  [ F9  [  y\ey_lky g_ij_ ju\ghc I_j_ibr_f   \ \b^_   iheh`b\ ON O >ey iZjZf_ljh\ J Q  Q N   N  N   bf__f

J N 

O  O N 5N  N

O

  J N  O  N O N 

 O N  5N  N 

  O  N O N 

NhjfZebamy\lhjh_ij_^iheh`_gb_mkeh\b_f

 G  Ì [ O  <<  Ì [ G [   mdZaZggu_khhl\_lkl\byaZibku\Z_f\\b^_

O N  O  [  ÌN − → Ì [ − O ≅ Ì [ − ÌN + → Ì [ + O ≅ Ì [ +

G Ì [ O  G[

G Ì([ ) O  G[



Ih^klZgh\dZ \ ^Z_l

J N     J  [  J N     J  [ J N  J  [

J N   J N   

O G O  O  [ G [



Kmq_lhf  b ijZ\mxqZklv ihemqZ_f\\b^_

(

)

∆  γ N 7N  M → 'W ' [γ 7  [ W  '[J N

 FK O˜ [  J  [ VK O˜ [   

Hi_jZlhj '[J h[h[sZ_lhi_jZlhj '[  ^eyh^ghjh^ghcp_ihqdbbi_j_oh ^bl\g_]hijb O FRQVW  J [  

  O  ' [γ → ' [ = ' [ = FK O∂ [ −  = D ÕU  ∂ [ + ∂ [ +   

E_\ZyqZklvmjZ\g_gby ij_h[jZam_lkyZgZeh]bqgh bihke_g_kdhevdh ]jhfha^dbo\uqbke_gbcihemqZ_fmjZ\g_gb_i_j\h]hijb[eb`_gby

/ 7  [ W



  WU ˜W ËÌÍF9  [ ˜˜7W  ˜˜[ ÈÊO [ ˜˜7[ ØÚ ÛÜÝ



WU

W U  [

O   D  [



MjZ\g_gb_\lhjh]hijb[eb`_gbybf__l\b^

Î  WU  [ Ë ÛÞ È O Ø  Ï /  F  [ ̘ W ˜ [ É  ˜ [Ù  F9  [ ˜W ˜ [  O  [ ˜ [ Üß7  [ W Ú Ê 9 Í Ýà Ð

 

]^_ / ±hi_jZlhji_j\h]hijb[eb`_gbykh]eZkgh MjZ\g_gb_gme_\h]h ijb[eb`_gbyke_^m_lba ijb W U  bkh\iZ^Z_lkmjZ\g_gb_fNmjv_^ey g_h^ghjh^ghckj_^u

F9  [

∂7 ˜ ˜7 Ø   È O  [  Ê [ ˜ ˜[ Ú ∂W



AZ^ZqZ . fh^_evg_klZpbhgZjghcp_ihqdbDhglbgmZebaZpbymjZ\ g_gby^eyg_klZpbhgZjghcg_h^ghjh^ghcp_ihqdb bh^gh\j_f_gghdZd ihdZaZgh\†^eyg_ebg_cghc hkms_kl\ey_lkym`_ijh^_fhgkljbjh\Zggufb kihkh[Zfbgh[he__]jhfha^dZIhwlhfmkjZam\uibku\Z_fdhg_qguc\b^ih emq_ggh]hmjZ\g_gbyh]jZgbqb\Zykvi_j\ufijb[eb`_gb_f

 ∂ 7 ∂  O  ∂7   ∂  ∂7   ∂ OF  ∂7  D  − + τ +      −  OQ    + τU  U ∂W  ∂W ∂[  Ì  ∂[     OF ∂W  ∂W  ∂[  ∂  ∂  O  ∂7   ∂λ ∂  O  ∂7   + D    OQ   +  OQ    =    ∂[  ∂W  λ  ∂[  λ ∂W ∂[  λ  ∂[  



O  [ W  O O [ W  F F9 F [ W  D D  [ W O  [ W F [ W  W U  [ W O   [ W D  [ W Ijb W U  ba ke_^m_lgme_\h_ijb[eb

]^_ O

WU

`_gb_^eyg_h^ghjh^ghcg_klZpbhgZjghcp_ihqdb

˘7 ˜ O ˜7 Û ˜7  D Ì   È OQ Ø Ü Ê Ú ˜[ Ý [ ˜W ˜ O Í ˜[





<kemqZ_ O = O = FRQVW  λ = λ  = FRQVW ba ke_^m_l  MjZ\g_gb_ kh^_j`bl\k_[_hkgh\guoqZklguomjZ\g_gbckhhl \_lkl\mxsbojZaebqgufdhf[bgZpbyfaZ\bkbfhkl_ciZjZf_ljh\ O  F O hldh hj^bgZl b \j_f_gb DeZkkbnbdZpby wlbo qZklguo fh^_e_c \ZjbZglh\  ijb\_ ^_gZ\ lZ[e  Z khhl\_lkl\mxsb_ wlbf \ZjbZglZf d\ZabehdZevgu_ mjZ\g_gby l_iehi_j_ghkZke_^mxsb_ba ±\lZ[eBg^_dk©ªkhhl\_lkl\m_lih klhyggufagZq_gbyfiZjZf_ljh\ 

LZ[ebpZ


    

    

Fh^_ebkj_^u IG_h^ghjh^gZyg_klZpbhgZjgZykj_^Z W U

O = O  [ W O = O  [ W O = O  [ W O = O  [ W O = O

λ = λ  [ W λ = λ[ λ = λW λ = λ

λ = λ  [ W IIG_h^ghjh^gZykj_^Z τ U = τ U [

O = O [ O = O [ O = O [ O = O [ O = O

λ = λ[ λ = λ[ λ = λ λ = λ

λ = λ[

W U  [ W F = F  [ W F = F[ F = FW F = F

F = F  [ W F = F[ F = F

F = F[ F = F

F = F[

ΙΙΙ.G_klZpbhgZjgZykj_^Z τ U = τ U W

    



O = O W O = O W O = O W O = O W O = O

λ = λW

F = FW

λ = λ

F = FW

λ = λW

F = F

λ = λ

F = F

λ = λW

F = FW

D\Zabnbgblgu_mjZ\g_gbyfh^_eb ‹\Z jbZglZ ih lZ[e








MjZ\g_gb_











( + τU ∂W )∂W7 − D ∂ [ 7 − ∂ [  OQ λO ∂ [7   + τU O −∂W O∂W7 =  







( + τU ∂W ){∂W7 − D [∂ [ 7 − O −∂ [O∂ [7 ]}+



 O   + τ U (OF )− ∂ W (OF )∂ W7 + D  ∂ W OQ  ∂ [ 7 + (Oλ )− ∂ W λ∂ [O∂ W7  =  λ   

( + τU ∂W ){∂ W7 − D[∂ [ 7 − O −∂ [O∂ [7 ]}+  WU>O ˜ W O˜ W 7  D˜ [ O ˜ W O˜ [ 7 @  ( + τU ∂W ){∂ W7 − D[∂ [ 7 − O −∂ [O∂ [7 ]}+  W U ^F ˜ W F˜ W 7  D>˜ [ O˜ W O˜ [ 7  O ˜ W O˜ [ O˜ [ 7 @` 







Khhl\_lkl\m_l\ZjbZglmijb F = F



( + τU ∂W ){∂ W7 − D[∂ [ 7 − O −∂ [O∂ [7 ]}= 



Khhl\_lkl\m_l\ZjbZglmijb F = F





( + τU ∂W )∂W7 − D ∂ [ 7 − ∂ [  OQ λO ∂ [7   =  



LZ[ebpZ

( + τU ∂W ){∂ W7 − D[∂ [ 7 − λ−∂ [λ∂ [7 ]}= 



Ijh^he`_gb_lZ[e  











( + τU ∂W )(∂W7 − D ∂ [ 7 )+

 W U Ë OF ÌÍ



O ˜ W  OF ˜ W 7  D˜ W È OQ Ø ˜[ 7 Û ÜÝ Ê OÚ

( + τU ∂W )(∂W7 − D ∂ [ 7 )+

[



]

+ τ U (OF )− ∂ W (OF )∂ W7 + DO −∂ W O ∂ [ 7 = 

( + τU ∂W )(∂W7 − D ∂ [ 7 )+

O  W U ËO ˜ W O˜ W 7  D˜ W È OQ Ø ˜[ 7 Û Ê OÚ ÍÌ ÝÜ

( + τU ∂W )(∂W7 − D ∂ [ 7 )+

(



)

+ τ U O −∂ W O∂ W7 + DO −∂ W O ∂ [ 7 = 

( + τU ∂W )(∂W7 − D ∂ [ 7 )+

(

)

+ τ U F −∂ W F∂ W7 − Dλ−∂ W λ ∂ [ 7 = 

AZ^ZqZ  . fh^_ev g_ebg_cghc p_ihqdb I_j\h_ ijb[eb`_gb_ . fh^_ebg_ebg_cghcp_ihqdbfh`_l[ulvihemq_ghba ]^_g_h[oh^b

fhkh\_jrblvi_j_oh^u

ON  O 7 F9N  F7 O N  O 7  >jm]b_iZjZf_ljulZd`_ijbgbfZxl\b^

D N  D 7

O  7 O 7  τ UN → τ U 7 =  D 7 F 7

HimkdZy]jhfha^db_ghZgZeh]bqgu_m`_ijb\_^_ggufij_h[jZah\Zgby\uib ku\Z_fi_j\h_ijb[eb`_gb_g_ebg_cghc . fh^_eb



  ∂7 ∂  ∂7  ∂  O   ∂ 7   ∂ (FO )  ∂7  +  τ U 7 −  OQ    + τ U 7    − D 7  − τ U ∂7  λ   ∂[  ∂W ∂W  ∂W   FO ∂7  ∂W  



 ˜ È OØ ˜ È ˜7 Ø ˜  È O Ø È ˜7 Ø ˜7 Û OQ 7  WU  W U 7  OQ Ü ˜[ Ê O Ú ˜[ Ê ˜W Ú ˜7 Ê O Ú Ê ˜[ Ú ˜W ÜÝ



?keb\ \k_iZjZf_ljuiheh`blvihklhyggufblhihemqbf

˜7 ˜7 ˜7  WU   D  ˜W ˜W ˜[





l_]bi_j[hebq_kdh_mjZ\g_gb_l_iehijh\h^ghklbGme_\h_ijb[eb`_gb_ke_ ^mxs__ba ijb W U  

F 7

˜7 ˜W

˜ È ˜7 Ø O 7  Ê ˜[ ˜[ Ú



l_kh\iZ^Z_lkg_ebg_cgufmjZ\g_gb_fNmjv_ AZ^ZqZ  . fh^_ev p_ihqdb i_j_f_gghc ^ebgu I_j_oh^ d i_j\hfm ijb[eb`_gbx . fh^_eb p_ihq_d i_j_f_gghc ^ebgu ^ey m^ebgyxsboky b mdhjZqb\Zxsbokyp_ihq_d hkms_kl\ey_lkyZgZeh]bqghJZkkfhljbfm^ebgyx smxkyp_ihqdmgZhkgh\_mjZ\g_gby Iheh`bfkh]eZkgh 

W U-

WU

O

 - 





MjZ\g_gb_ kmq_lhf  kijZ\_^eb\h ijb ihklhygghc kdhjhklb m^eb g_gbyp_ihqdbdh]^Z +

 

+

- 

FRQVW Ijbi_j_f_gghckdhjhklbm^ebg_

gby dh]^Z ϑ = ϑ W  \hkihevam_fky mjZ\g_gb_f \ZjbZglZ  lZ[e   O → O W  O FRQVW  F FRQVW \dhlhjhfiheh`bf

τ U = τU+ϑ W =

O

ϑ + W

=

 ϑ+  O W    O W = O   ϑ + W   ϑ+  

L_i_jvmjZ\g_gb_\ZjbZglZlZ[eijbgbfZ_l\b^   +       + W ∂  ∂7 − D ∂ 7  + G τ Uϑ  ∂7 + D ∂ 7  =     +  τ Uϑ  G W  ∂W ∂W  ∂W  ∂[   ∂[    >ey kemqZy mdhjZqb\Zgby p_ihqdb \   ^hklZlhqgh iheh`blv W U-

W U-  

 

 

  W l_ i_j_clb -  W  -  W  Ijb [ukljhf m^ebg_gbb dh]^Z W U  W ±oZjZdl_jgh_\j_fyijhp_kkZl_iehijh\h^ghklb hl fh`ghi_j_clbd

  gh   G W ›   ijb[eb`_gbx W UW U-

 G τU+ϑ  ∂7  G τ +  ∂ 7  +  = D − Uϑ      G W  ∂W G W  ∂[  



Nhjfmebjh\dZ]jZgbqguomkeh\bc^ey . fh^_ebm^ebgyxs_ckyp_ihqdb lj_[m_lhl^_evgh]hZgZebaZImklvijb W  \gZqZe_ M ]h pbdeZ  d p_ihqd_ ^bkkbiZlhjh\ ^ 0 N `  N  1  ijbkh_^bgy_lky  1   c ^bkkbiZlhj 0 1 L_ieh\hc[ZeZgk\g_f 

'41   '41  '41   

[

6OF9 '71   M 

]

∆4 1+ + = 6  τ M T %+−  1 +  M − T  −1 + − 1  M     



Bf__f

O

76 O 

T %  1   M 





 7 1    M  T / W

O Ë ˜7 O Ø Û È 7/ W  É 7/ W  /  Ù Ü Ì Ê O  Í ˜[  Ú Ý

O

˜7/  ˜[



A^_kvbg^_dk© / ªh[hagZqZ_lih^\b`gmxdhhj^bgZlmdhgpZp_ihqdb__^ebgm / /W jZagbpZl_fi_jZlmj;hj]ZjlhgZb 0 1 jZaeh`_gZ\jy^L_cehjZ 

km^_j`Zgb_f^\moi_j\uoqe_gh\jZaeh`_gby>ey\_ebqbgu 7/  W ihemqZ_f iheh`b\ W M

   W U -

 7/ W = τM

τM

∫ 7/ W + τ G τ ≅7/ W + 

∂7/  + τ  ∂W Uϑ



Ih^klZgh\dZ \ ^Z_l

T /+ W = λ

GT ∂7 ∂  ∂7  = T W + τU+ϑ   + τU+ϑ  λ  GW ∂[ [ = / W ∂W  ∂[  [ = / W

Ijb ijbkh_^bg_gbb ^bkkbiZlhjZ 0 1  d ^bkkbiZlhjm 0 1  l_fi_jZlmjZ   i_j\h]hbagbo[ebadZd 76 ihwlhfmihlhdl_ieZd 0 1 hl;hj]ZjlhgZfZe 

IheZ]Zy_]hijb[eb`_gghjZ\gufgmexihemqZ_f 



 T W  W U-

G T GW

 J_rZymjZ\g_gb_ ^eykemqZ_\ W Uoh^bfkhhl\_lkl\_ggh

T / W #     W b   W UW U-

È W GW Ø T  H[SÉ    Ù T  W É W  W Ù Ê  UÚ

T - W

Ô



W U



FRQVW gZ



T  H[S  W W U  


gbqgufmkeh\b_fI]hjh^Z AZ^ZqZ ' b . fh^_eb k bklhqgbdZfb l_ieZ<jZkkfhlj_gguojZ g__ ' b . fh^_eyog_mqblu\Zebkvbklhqgbdbklhdb l_ieZjZaebqghcijb jh^u\_kvfZjZkijhkljZg_ggu_\j_Zevguoijhp_kkZo
'4Z

6O I Z W Z 



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  I   τ ZN = τ U (1 − N ) +  +  ZN τ U   N   F9 ∆7  

 1 

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\ i_j\hf kemqZ_  I Z = FRQVW a  I Z τ G τ =     I Z = I Z + I Z \h \lhjhf kemqZ_      a a <lj_lv_fkemqZ_dh]^Z I Z  W J7  W  J FRQVW  I Z J7 ]^_ 7 ±kj_^g__ agZq_gb_ l_fi_jZlmju aZ i_jbh^ W Z  < q_l\_jlhf kemqZ_ dh]^Z a I Z  W J  W 7  W 

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)

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Lh]^Z

 IZ= τZ

τZ

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∫



Imklv 'I Z I Z … <wlhfkemqZ_

IZ 

I Z  H P  H P …

  P 



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 (ε P )PD[ =   Hlkx^Z ke_^m_l qlh ijb ex[hf P ih]j_rghklv nhjfmeu   k P  g_

\lhjhfbgl_j\Ze_fZdkbfZevghfm H P khhl\_lkl\m_l P

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a

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a

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È O Ø É DW M Ù '7N  M Ê Ú

È O Ø '  7N  M  É  Ù I N  M  N Ê DF9 Ú



Ba mjZ\g_gby   ijb dhgdj_lbaZpbb   e_]dh ihemqblv ZgZeh]   ± mjZ\g_gb_kh^_j`Zs__ijbjZs_gbyl_fi_jZlmju>eyi_j_oh^Zd . fh^_eb\  iheh`bf

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a I  [ W  W G W

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a )  [ W

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W M ˜I     ˜W

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)  [ W

a )  [ W W U

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eh`blv F9

)  [ W

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)   [ W =

γ γ  ∂7  7  [ W    7  [ W + τ U   )   [ W = F9 F9  ∂W 

>eyq_l\_jlh]hkemqZyZgZeh]bnhjfme bf_xl\b^

)   [ W

˜  Ë J W 7  [  W  W U  J  W 7  [  W Û  F9 ÌÍ ˜W ÝÜ J  W )   [ W 7  [ W  F9



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)   [ W =

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l_fi_jZlmjgu_iheyklZgh\ylky[he__©]eZ^dbfbªb©f_^e_ggufbª\hafh`_g i_j_oh^ d jZaebqguf ijb[eb`_gbyf . fh^_e_c Ijb jZkkfhlj_gbb kbkl_f ^ey dhlhjuo / !! O  W !! W U  qlh oZjZdl_jgh ^ey [hevrbgkl\Z ijbdeZ^guo fh^_e_ci_j_ghkZ ohjhrbfijb[eb`_gb_f . fh^_e_chdZau\Z_lkygme_\h_± jZaebqgu_fh^bnbdZpbbmjZ\g_gbyNmjv_ Fh^_eyflZdh]hjh^Zihk\ys_gZg_h[hajbfZyebl_jZlmjZlhevdh\[30] ± ihjy^dZ  bklhqgbdh\  h^gZdh \hijhku f_lh^heh]bq_kdh]h oZjZdl_jZ jZk kfZljb\Zxlky qZklh mijhs_ggh [13,28,33–35,40,61,77,11,142,143,161,172–176] H[h[sZy hiul fZl_fZlbq_kdh]h fh^_ebjh\Zgby \ jZaebqguo kn_jZo Z\lhju [28,176]aZdexqZxlqlh\Z`g_crbfbbkihevam_fufbdeZkkZfbnmgdpbcy\ey xlkyZgZeblbq_kdb_klmi_gqZlu_ijhklu_  b h[h[s_ggu_ Hkgh\ghc fZlZi iZjZlhibjZxsbckygZnmgdpbbwlbodeZkkh\_]hijbeh`_gbydfh^_eyfijh p_kkh\ i_j_ghkZ ohjhrh baeZ]Zxlky \h fgh]bo jZ[hlZo \ qZklghklb \ [42– 45,48,50,52,61,75–79,109,172–192]Hkgh\u\ZykvgZwlbojZ[hlZoknhjfmebjm _f^Ze__k\hckl\Zklmi_gqZluobh[h[s_gguonmgdpbcbkihevam_fu_ijbi_ j_oh^Zo hl ' fh^_e_c d . fh^_eyf dhglbgmZebaZpby ' fh^_e_c  b hl .  fh^_e_cd ' fh^_eyfZe]_[jZbaZpbybeb^bkdj_lbaZpby . fh^_e_c ZlZd`_ ijbhij_^_e_gbbnmgdpbc=jbgZdjZ_\uoaZ^Zqbd\Zabnbgblguonmgdpbc
 [ ∈ ωN  χ N = χ N  [ =   [ ∉ ωN  OZjZdl_jbklbq_kdZynmgdpby F N  [ fh`_l[ulvij_^klZ\e_gZ\\b^_ F N  [

T [  [N   T [  [N 





]^_ T [ ±_^bgbqgZyklmi_gqZlZynmgdpbyHOw\bkZc^Z

 [ <  θ [ =   [ > 



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G [

G T [  G [ G [ G[

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Î [ … T  [ Ï [ !   Ð

Î [  G  [ Ï [ •   Ð

‡



Ô G   [ G [

G T  G   [ G[

Ô

 G   [ G [   ‡

G T  G[ 



(

>ey oZjZdl_jbklbq_kdbo nmgdpbc h[eZkl_c ZN“  ωN+ = {[ ∈ ([N − [N ]} ωN− = {[ ∈ [[N − [N )} fh`ghaZibkZlv

)

 [ ∈ ωN±  χN± = θ ±  [ − [N − − θ±  [ − [N =   [ ∉ ωN± 



JZkkfhljbfk\hckl\ZoZjZdl_jbklbq_kdbonmgdpbcBa ke_^m_l

χ N  [ = δ [ − [′  ω = N

[N

 [ ∈ ωN  δ [ − [′ G [′ =   [∉ ωN  [N −

∫



Ba  ke_^m_lqlh

G FN G[

G [  [N   G [  [N 



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^

1

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F N  [

1

G [  [ „  Z Ç N 

N

`

G [  [ „  :

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Î [ °   /   Ï [ ±  /   Ð

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F N  [  F M  [

:

G [  [ „  Z  G [  [ „  Z N

  G [  [ „  G [  [ „„ : ZM ZN  F M  [



ZN

M

:

 G [ „  [ „„  Z

G NM Z M

OG NM 

M

ZN



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 F N  [  Ki_pbnbq_kdbf O

k\hckl\hfoZjZdl_jbklbq_kdbonmgdpbcy\ey_lkybob^_fihl_glghklvbebjZ \_gkl\hex[hcp_ehckl_i_gbnmgdpbb_ckZfhcKmq_lhf bf__f

F N  [  F M  [

:

F N F M 

F N  [ F M  [ FQN  [

G NM Z M  F N  [  :

:

G NM F N  [ FN  [ FQN   [

F N  [ 

ZN  

 F N  [

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7  [  [ ∈ ωN  a !N {7  [ } = 7N  [ = χ N  [ 7  [ =  [ ∉ ωN  



DZd\b^ghba hi_jZlhjijh_dlbjh\Zgby !N bf_xsbcfmevlbiebdZlb\ gh_ij_^klZ\e_gb_±oZjZdl_jbklbq_kdmxnmgdpbx χ N [ iha\hey_lkhihklZ

a

\blv ]eh[Zevghc nmgdpbb 7 = 7 [  ]^_ [ ∈ Ω  ehdZevgmx nmgdpbx 7N  [ 

hlebqgmx hl gmey ebrv \ h[eZklb [ ∈ ωN  LZdZy hi_jZpby [u\Z_l g_h[oh^b fhcijbij_h[jZah\Zgbb . fh^_e_c Ijb i_j_oh^_ hl . fh^_eb d ' fh^_eb dZ`^hc ]eh[Zevghc nmgdpbb 7  [ g_h[oh^bfhkhihklZ\blvjy^qbk_e±kj_^gbol_fi_jZlmj\h[eZklyo ωN l_fi_jZlmju ^bkkbiZlhjh\ 0 N N =  1  Wlh hkms_kl\ey_lky ijbf_g_gb_f

N hi_jZlhjZ^bkdj_lbaZpbb ' ' N ^ 7  [ `

Ô ZN :



F N  [ 7  [ G [

7  [ 

 ZN Z N

7N  N =  1 

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a

a

∑ χN  [ 7  [ =∑ χN  [ 7  [ = χΩ  [ 7  [ = 7  [  N



N



Ba ihemqZ_f

' ^ 7  [ `

7  [

Ç ' N ^ 7  [ ` Ç F N  [ 7N  N

N

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 [P = 

O P = 

ijyfZy 

^ K °   K `  η = U P =   djm] kn_jZ   η = U P = 

ZP





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ZNP

Î[N  ÏU  ÐN

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Î[  P  ÏU  P   Ð



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KN

 1  \gmlj_ggbo

K N  N

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(

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I η ∈ / ω NP  b

h[eZkl_c





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jZkkfhljbf

Z NP  ]^_

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k_f_ckl\h

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P

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a = πP

ηN+

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∫

ηN

^



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‡

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Ç JQP <QP K  JQP

JK Iheh`b\ JK

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JQP

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ZQ

 P

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P
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Q 

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eZklb Z NP l_jZaebqgZ^ey P

   

^

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I [

.P [ K  I K

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Z

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ω+P



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jZah\Zgbcbgl_]jbjm_fu_ih [ bih K \ Z P  I K ° / ZP ±deZkknmgd

pbckbgl_]jbjm_fuf\ Z P fh^me_f
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I η = . −P ξ η  . P ξ η  I η

ω+P ω P +

=



= I η′  . −P ξ η  . P ξ η′ Ba ke_^m_lbkdhfh_ij_^klZ\e_gb_

. −P ξ η  . P ξ η′

ω+P

ω+P ω P +

= δ +P η − η′ 







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5 I W

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(

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(

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(

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V

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=

N

∑ τ M  \lhjhc rdZeu AgZq_gb_ M =

a 70  W N g_jZ\gh\_kghcl_fi_jZlmjukhhl\_lkl\m_ljZ\gh\_kghcl_fi_jZ lmj_ 7 N hij_^_ey_fhc©Z^bZ[ZlbaZpb_cª^bkkbiZlhjZijb W W N     bf_gm_fu_ \bjlm  Bf__l ©ijhf_`mlhqgu_ª agZq_gby 7 ° 7 N 7N 7N





Zevghc l_fi_jZlmjhc baf_gyxs_cky kh \j_f_g_f ebg_cgh gZ bgl_j\Ze_ W ° W N  Ijbhij_^_e_gbbg_jZ\gh\_kghcl_fi_jZlmjulZdbfh[jZahfhliZ^Z_lg_ h[oh^bfhklv \ IEDJ b \ g_ij_ju\ghklb rdZeu R7   I hij_^_e_gby ]^_ 9 0 ±h[t_ffZdjhlhqdb\\h^blh[t_fgmxl_ieh_fdhklv F9 dZd©i_j_\h^ghc dhwnnbpb_glª rdZeu R7  \ rdZem R8  ii  ± ke_^mxl ba hij_^_e_gbc fZdjhlhqdbb^bkkbiZlhjZbi_j\h]hgZqZeZl_jfhklZlbdb LZdbfh[jZahfmkljZg_gb_Zjl_nZdlh\IDEJbg_jZ\gh\_kghcl_fi_jZlm juhkms_kl\e_ghaZf_ghcwlboihgylbcgZihgylby;hj]ZjlhgbdbfZdjhlhqdZ ^bkkbiZlhjhij_^_e_gb_g_jZ\gh\_kghcl_fi_jZlmju VIjhba\h^kl\hwgljhibb L_jfbg ©ijhba\h^kl\hª \ aZ]heh\d_ ± ^Zgv ljZ^bpbb ^Ze__ [m^_f ]h\h jblvh[wgljhih]_g_a_Hkgh\gh_mjZ\g_gb_y^jZiZjZ^b]ful_jfh^bgZfbdb± mjZ\g_gb_ [ZeZgkZ wgljhibb M;W  \u\h^blky ba khhlghr_gby =b[[kZ Wlb we_f_glul_hjbb±Zjl_nZdlu‹‹\lZ[eV  Knhjfmebjm_fmkljZgyxsb_Zjl_nZdlunhjfmeu=b[[kZ [ b )  Khhl ghr_gb_=b[[kZ bkdexqZ_lkybay^jZiZjZ^b]fu <\_^_gb_\j_f_gb\ l_jfhklZlbdml_i_j_oh^dl_jfh^bgZfbd_ hkms_kl\ey_lky fh^_evx ojhgh ]_g_jZpbb kbevgh]h ^bkkbiZlhjZ V  b hij_^_e_gb_f l_fi_jZlmju ijb\_ ^_gguf\dhgp_ij_^u^ms_]hV  HibkZgb_ ijhp_kkh\ i_j_ghkZ \ kiehr guokj_^Zohkms_kl\ey_lkygZhkgh\_fh^_e_cp_ihq_d^bkkbiZlhjh\]e  Ke_^kl\b_fmkljZg_gbyZjl_nZdlZkhhlghr_gby=b[[kZy\ey_lkymkljZg_ gb_k\yaZggh]hkgbf^jm]h]hZjl_nZdlZ±M;W<V[uehihdZaZghqlhjZa 

ebqgu_ M;W ± mjZ\g_gby wgljhih]_g_aZ \ dhg_qghf kq_l_ \k_]^Z k\h^ylky d mjZ\g_gbyfl_iehi_j_ghkZHlghkbl_evghagZq_gbyM;W^eyZgZebaZwgljhih ]_g_aZ nhjfmebjh\db \ZjbZpbhgguo ijbgpbih\ hibkZgby i_j_dj_klguo wn n_dlh\aZf_lbfke_^mxs__   < nhjfmeZo ^ey wgljhih]_g_aZ g_l g_h[oh^bfhklb lZd dZd hgb g_ kh ^_j`Zl bgnhjfZpbb ^hihegbl_evghc ih hlghr_gbx d '  b . fh^_eyf l_i ehi_j_ghkZ  Ze__[m^ml ij_^eh`_gu l_jfh^bgZfbq_kdb_ \u\h ^u khhlghr_gbc \aZbfghklb   qlh iha\hebl bkdexqblv ba iZjZ^b]fu ©f_lh^fbdjhh[jZs_gby\j_f_gbªdZdZjl_nZdl 

eywlh]hi_j_oh^ZkfaZ^Zqm\V fh`ghgZclb ijbjZs_gby ^bkdj_lguo Zddmfmebjm_fuo  'V$'N  \   b ihlhdh\hc

 'VT 'N \ wgljhibcjZ\guo^jm]^jm]m ∆σ$' N

=

∆σT' N =

∆σN'

7

=

∫

7N −

G 4N 90 F9 ∆7  = 7N − 7N −





:gZeh]bx[110]fh`ghmemqrblvlZddZdZgZeh]hfnmgdpbbEZ]jZg`Zbf_x s_c jZaf_jghklv wg_j]bb  fh`gh \aylv \f_klh 'V  \_ebqbgm '4 7'V  kh\ iZ^Zxs_cjZaf_jghklbIjbgpbigZbf_gvr_]h^_ckl\byf_oZgbdb τ

60 =



∫ /T T τ G τ = PLQ 



τ



[m^mqb©i_j_\_^_gªgZ^bkkbiZlhjgucyaudijbgbfZ_l\b^ τN

67 =



Q

∫ 7N − ∆σN' G τ = ∆8 τN Q = PLQ 





Ihkdhevdm baf_g_gb_ \gmlj_gg_c wg_j]bb ^bkkbiZlhjZ gZ ex[hf ' i_jbh^_ ihklhyggh '8   const)ba \ul_dZ_lmkeh\b_

W N Q Wlhmkeh\b_jZ\ghkbevghlj_[h\Zgbx
PLQ W N Q Q°> ‡

W N ‡

PLQ   bebQ

W  W 1  N 



‡ lZddZdih  O  D



LZdbf h[jZahf ijbgpbi gZbf_gvr_]h ^_ckl\by wd\b\Ze_gl_g ijbgpbim fbgbfZevghklb '  i_jbh^Z Ihkdhevdm bkoh^guf imgdlhf [ueh \ujZ`_gb_ ^ey^bkdj_lghcwgljhibb ijb\uqbke_gbb dhlhjhc l_fi_jZlmjZ ^bkkb 

iZlhjZijb W ° W N hklZ\ZeZkvihklhygghcbjZ\ghc 7N − kdZqdhfm\_ebqb

\Zykv^h 7N ijb W W N lhjZkkfZljb\Z_fuckemqZckhhl\_lkl\m_lg_ebg_cgh fm ^bkkbiZlhjm ^ey dhlhjh]h mkeh\b_ Z^^blb\ghklb ojhghrdZeu beb ©ijZ\behhl[hjZª g_\uihegy_lky >ey ebg_cgh]h ^bkkbiZlhjZ kh]eZkgh  
W N Q

PLQ

W   1  N  

W N 

WU   1  N  



IhdZ`_fqlhijbgpbi=Pb]e_jZ±fZdkbfZevghckdhjhklbwgljhih]_g_ jZpbb − wd\b\Ze_gl_g ijbf_gbl_evgh d ^bkkbiZlhjm ijbgpbim gZbf_gvr_]h ^_ckl\by Kdhjhklv wgljhih]_g_jZpbb gZ N f ' i_jbh^_ \ kbem ihklhygkl\Z \_ebqbguijbjZs_gbywgljhibbghjZaebqgh]h^eyebg_cgh]hbg_ebg_cgh]h ^bkkbiZlhjh\  hij_^_ey_lky hlghr_gb_f 'V N W N Q  Kh]eZkgh ijbgpbiZ =Pb]e_jZ

'V N W N Q

PD[  W N Q

PLQ 



LZdbfh[jZahfh[ZkjZ\gb\Z_fuoijbgpbiZwd\b\Ze_glgulZddZdk\h^ylkyd ijbgpbimfbgbfZevgh]hojhgh]_g_aZ−fbgbfZevghklb N ]h ' i_jbh^Zdhlh jucmkljZgy_lZjl_nZdl±gZebqb_fgh]hqbke_gguo_ckl\bl_evgh ba   ke_^m_l qlh \ ebg_cghf kemqZ_ ijb Q  wlh agZq_gb_ Q  [ueh h[hkgh\Zgh k_fvx jZaebqgufbkihkh[Zfb 

> 1  N    @W N 

W 



<kbem  b   ]^_

67 N

'8 1  N W N 

67

'8 W

PLQ  N

 1 

∆7   ∆8 1 − N = 90 F9 7V − 7N +    '8 9 0 F9 '7    

  

A^_kv '8 1  N  bf__l kfuke ©^_n_dlZª ijb 7V ! 7  beb  ba[uldZ  7V < 7 \gmlj_gg_c wg_j]bb ^bkkbiZlhjZ ih hlghr_gbx d ;hj]Zjlhgm gZ N f ' i_jbh^_Z '8 ±fbgbfZevgh\hafh`gh_baf_g_gb__]h\gmlj_gg_cwg_j ]bb LZdbf h[jZahf khhlghr_gby   bf_xl kfuke l_jfh^bgZfbq_kdh_ 

^_ckl\b_gZdZ`^hf N frZ]_ N  1 h^bgZdh\hbjZ\ghk\h_fmfbgbfZev ghfmagZq_gbx < g_ebg_cghf kemqZ_  Q ‡ 
6 7C N = ∆8 1C− N τ N  ∞ = 6 7C = ∆ 8 C W = PLQ 



]^_bg^_dkhf©gª\\_jomh[hagZq_gg_ebg_cguckemqZcZ

∆8 1C− N = 90 F9 [7V − 7N + ∆7 ] ∆8 C = ∆8  



<h[hbokemqZyo±ebg_cghfbg_ebg_cghffbgbfZevgu_agZq_gbyl_jfh^bgZ fbq_kdh]h^_ckl\bykh\iZ^ZxlZhlebqby\e_\uoqZklyo b kdZ au\Zxlky \ hlebqbb ojhgh]_g_jZpbc g_ebg_cgh]h b ebg_cgh]h ^bkkbiZlhjh\ Ijb wlhf ' i_jbh^u g_ebg_cguo ^bkkbiZlhjh\ f_gvr_ q_f m ebg_cguo  W N ‡  W N  qlhk\b^_l_evkl\m_lh[hevr_cbgl_gkb\ghklbijhl_dZgbyg_eb g_cguoijhp_kkh\i_j_ghkZ Ex[hiulgZgZgZr\a]ey^ke_^mxsZyZgZeh]by<we_f_glZjghcd\Zglh \hcl_hjbbba\_klgZnhjfmeZ

(Q  (P

KQQP

'(QP 



hibku\ZxsZy baemq_gb_ d\ZglZ wg_j]bb ijb i_j_oh^_ we_dljhgZ k mjh\gy gZ  ]^_ [he__gbadbcmjh\_gv?kebiheh`b\ Q QP '7 QP '7 QP ±i_jbh^baem qZ_fhc\hegui_j_ibkZlv \\b^_

'(QP '7 QP

K

PLQ 6N 



]^_ 6N ±©d\Zglh\h_ª^_ckl\b_lh\b^gZZgZeh]byf_`^m b \dh lhjhcfbgbfZevghfm©d\Zglh\hfmª^_ckl\bx 6N K khhl\_lkl\b_fbgbfZev gh_l_jfh^bgZfbq_kdh_^_ckl\b_ 67  DZdba\_klghnmgdpby^_ckl\by^himkdZ_lij_^klZ\e_gby±nZdlhjbaZpbb gZwg_j]bxb\j_fybgZbfimevkbdhhj^bgZlm?kebqZklhl_ QQP fukhihklZ \beb fbgbfZevgh_ oZjZdl_jgh_ \j_fy '7 QP  i_jbh^ dhe_[Zgbc  lh ^ebg_ \hegu O  _kl_kl\_ggh khihklZ\blv fbgbfZevgh_ jZkklhygb_ '[O  Lh]^Z ba nhjfmeuE>_;jhceyke_^m_l

'[O 'S

K

PLQ 6N 



l_nhjfmeu b _klvke_^kl\byfbgbfmfZd\Zglh\h]h^_ckl\by Khhlghr_gby \aZbfghklb jZkkfhljbf gZ ijbf_j_ kh\f_klgh ijhl_dZx sbo ijhp_kkh\ l_iehijh\h^ghklb b ^bnnmabb [113] Dhgklblmlb\gu_ mjZ\g_ gbybf_xl\b^

,T

/TT ;T  /TP ;P  , P = /PT ; T + /PP ; P   

]^_ , ; ±ihlhdbbl_jfh^bgZfbq_kdb_kbeukhhl\_lkl\_ggh /LN ±dbg_lbq_ kdb_dhwnnbpb_gluMfgh`bfi_j\h_bamjZ\g_gbc gZ ;T Z\lhjh_± gZ ;P bkeh`bfbo

, T ;T  , P ; P

/TT ;T   /PP ;P   /TP  /PT ;T ;P 



A^_kve_\ZyqZklv_klvkdhjhklvwgljhih]_g_jZpbb[113]dhlhjZy\kbemijbg pbiZ=Pb]e_jZ^he`gZ[ulvfZdkbfZevgZKe_^h\Zl_evgh

/TT ;T   /PP ;P   /TP  /PT ;T ;P

PD[ 



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/TT = / PD[  /PP = / PD[  / + /PT PD[ = /TP PD[ + /PT PD[  PP  TP TT  Ihke_^g__bakhhlghr_gbc kijZ\_^eb\h_keb

/TP PD[ = /PT PD[ 



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U [  K 

0   [  K U [ 9

0   [ U [  K 9

0   [  K  9

Wnn_dlb\gucihlhdqZklbpd k_q_gbx [  hij_^_ey_lky jZagbp_c ihlhdh\ qZk lbphlk_q_gbc  [  K b  [  K 

T1

  - 1   - 1      

>

@



]^_bg^_dku©±ªb©ªkhhl\_lkl\mxlk_q_gbyf  [  K b  [  K Z -±kj_^ gb_ kdhjhklb ©iju`dh\ª qZklbp hij_^_ey_fu_ l_fi_jZlmjZfb \ khhl\_lkl \mxsbok_q_gbyo

ε =

P ϑ 

 = N7  - 

È N Ø Ê P Ú



7

-  7

->7  [ @ 

]^_ N ±ihklhyggZy;hevpfZgZ 7 ±l_fi_jZlmjZHlihlhdZqZklbp T 1 i_j_c ^_fdihlhdmfZkku TU 

T 1 P 9

  - U  -U      JZaeZ]Zynmgdpbb\jy^uL_cehjZihkl_i_gyf K bh]jZgbqb\Zykvqe_gZfbeb g_cgufbih K ihemqbf TU

>

@

 ∂ Ò Ò ∂ϑ ∂7   Tρ = − Kϑ  [  +    ∂[ ϑ ∂7 ∂[  Ba ke_^m_lqlh

 ˜- - ˜7

˜ Ë È N Ø OQ ˜7 Ì Ê P Ú Í



Û   OQ 7 Ü  Ý

  7





HdhgqZl_evghgZoh^bf

TU

˜U U ˜7 Ø  '7  '7 È  Ê ˜[ 7 ˜[ Ú

  K- 7  



Ihemq_ghmjZ\g_gb_^eyihlhdZfZkkuh^ghbamjZ\g_gbc 
1   1 7  ϑ → ϑ7 = ϑ7 7 = FRQVW 7 U  U [ ]^_ H7

H

TX =

H7 1 7  9

N7  ±kj_^gyywg_j]byghkbl_e_cwg_j]bbIhemqZ_f

 [(ϑ7 ÒX )− − (ϑ7 ÒX )+ ] =  −  Kϑ7  ∂ ÒX + ÒX ∂ϑ7 ∂7        ∂[ ϑ7 ∂7 ∂[  

?keb \   kh\_jrblv ij_h[jZah\Zgb_ ZgZeh]bqgh_   b \hkihevah \Zlvkykhhlghr_gb_f

8 Æ 1 = F9 7  ÒX = 7 7 = 9 9

lhihemqbf

TX = − Ì7

∂7  O 7 ∂[

 K- 7 F9   7



Qlh[umq_klv\ebygb_gZihlhdl_ieZh^gh\j_f_gghb^ms_]hijhp_kkZ^bnnm abb d TX  g_h[oh^bfh ^h[Z\blv khklZ\eyxsmx h[mkeh\e_ggmx fZkkhi_j_gh khf

Ihegucihlhdl_ieZ[m^_l

ε T X ' =  Tρ  P



T XΣ = T X + T X ' 



Ba  ± ke_^m_l

∂7 ∂ρ  − '7 7  ∂[ ∂[ U'  7 H  '7  '7 7   O 7  7 P 7 T XΣ = −λ Σ 7

O 6 7

Nhjfmehc  ^Z_lky \lhjh_ dhgklblmlb\gh_ mjZ\g_gb_ hij_^_eyxs__ iehlghklvihlhdZl_ieZh[mkeh\e_ggh]hkh\f_klghijhl_dZxsbfbijhp_kkZfb l_iehijh\h^ghklb b ^bnnmabb ?keb khihklZ\blv ihemq_ggu_ mjZ\g_gby   b   k bo ©dZghgbq_kdbfbª ij_^klZ\e_gbyfb   b \hkihevah \Zlvky\ujZ`_gbyfb^eyl_jfh^bgZfbq_kdbokbe[211]

;T lhg_f_^e_gghihemqbf

 ˜7  ;P 7 ˜[

N7 ˜U  UP ˜[



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=E:<: NMG>:F?GL:EVGUCOJHGH:JL?N:DL >bjZd kdZaZe dZdlh qlh ijh]j_kk \ l_hj_ lbq_kdhcnbabd_qZklh[u\Z_lk\yaZgkij_h^he_ gb_fij_^jZkkm^dh\Wlh\iheghcf_j_hlghkblky dgZklhys_fmkemqZx BIjb]h`bgBKl_g]_jk
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= W + =   gZoh^beZkv \ Imklv fZl_jbZevgZy lhqdZ \ fhf_gl \j_f_gb W lhqd_ [  hkb Ox \^hevdhlhjhcb^\b]ZeZkv^Zevr_\ijZ\hlh_klv\klh

jhgm [ !  < fhf_glu \j_f_gb t 1( + ) = τ0  t (2+ ) = 2τ0  t 3( + ) = 3τ0  t (4+ ) = 4τ0 

>

@

lh_klvijb W   °  W   lhqdZhdZau\ZeZkvgZjZkklhygbyohlgZqZeZ [



hkb 2[  jZ\guo [  [  [  [  = /  lh _klv ijhreZ imlv [W + = [ = /  ?keb wlh ^\b`_gb_ ©h[jZlblvª lh ^\b]Zykv \ ©h[jZlghfª \j_f_gb

t ( −) ∈ [0,− t (4+ )] ijb t1( −) = −τ0 fZl_jbZevgZylhqdZ^he`gZbf_lvdhhj^bgZlm

[ = [ ijb t (2−) = −2τ0 –dhhj^bgZlm [ [ ijb t 3( −) = −3τ0 –dhhj^bgZlm [ = [  b ijb t (4−) = −4τ0  – [ = [  =   DZaZehkv [u ©h[jZlgh_ª ^\b`_gb_ ihkljh_gh lhqdZ \ha\jZsZ_lky hl [ /  d [   ^\b]Zykv \ ©h[jZlghfª \j_ f_gb t ( − ) ∈ [0,−4τ0 ]  Ijh\_jbf h^gZdh dZd k\yaZgu ©ijyfh_ª b ©h[jZlgh_ª \j_fy^eydZ`^hclhqdbfZjrjmlZ [

t

(+)

= t (0+ )

= 0  Z t ( −)

=

− t (4+ )

 [

= −4τ0  >ey [

/ bh[jZlgh >ey x = x0 = 0

[ t1( + ) = τ0  Z t ( −) = −3τ0  b

l^K\yavijyfh]hbh[jZlgh]h\j_fzg^eyex[h]h [ = [L L =  

t i( + ) + t i( −) = t (4+ ) = 4τ0 



Qlh `_ lh]^Z hagZqZ_l mkeh\b_ ©h[jZs_ggh]hª ^\b`_gby W  W " Ba 

ke_^m_l qlh W L + = W L −  lhevdh ijb L Ijb^jm]boL ›  wlhckbff_ljbb (t iheh`blv W L +

i = 2. 

=

  [ = [  t (2+ ) = 2τ0  t (2−) = −2τ0 

=| −t |) \h\k_g_lBh[jZlgh_keb\

− W L− lhihemqbfqlh W L +

= W +  = τ  = − W L − lh_klv

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( )

( )

V0 t (0−) → −V4 t (4+ ) h[jZs_gb_kdhjhklb kgZ]ey^ghcdZjlbghc©ijhp_kkZ

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W  W  [L W   [L  W  9L  → − 9L   9L W → − 9L −W  A^_kv [L b 9L –dhhj^bgZlubkdhjhklbqZklbpHkgh\gh_mjZ\g_gb_f_oZgbdb II aZdhgGvxlhgZ fh`ghaZibkZlv\\b^_

PL

G  [L GW

)L L

 1 



beb\wd\b\Ze_glghf\b^_

G 9L )L   GW >_ckl\bl_evgh ijb W  W  9L → − 9L  h[_ wlb nhjfu bg\ZjbZglgu Ijb wlhf\h[hbomjZ\g_gbyoijZ\ZyqZklvkbeZ )L hklZ_lkybg\ZjbZglghc PL

)L  [L  W = )L  [L −W 



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P[   + 8  [ = (   9 –kdhjhklv U(x – ihl_gpbZevgZyZ ?–ihegZy

]^_ m–fZkkZ [ G [ G W wg_j]byfZl_jbZevghclhqdbMjZ\g_gb_ \[]ij_^klZ\e_gh\\b^_

G[ GW

  (  8  [  P



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G[  = ± 9  + 8  [ 8  [ = GW P

[

∫ )  [ G [ 



[

9 –gZqZevgZykdhjhklvqZklbpuBa b \ul_dZ_lqlhbf_xlky^\Z 

mjZ\g_gby– bmjZ\g_gb_khhl\_lkl\mxs__agZdm©–ªi_j_^dhjg_f

G[     (  8  [  GW P Ba ke_^m_lqlhijb W  W mjZ\g_gb_ \©h[jZlghf\j_f_gbªih   [  W  a\hey_l hij_^_eblv ©w\hexpbx gZaZ^ª lh _klv j_r_gb_ \b^Z [  − −  Ijb W = W  =   [  [ Ba ke_^m_l G [ = 9 − = − 9 +  9 + =   −   G W W = 

 (( − 8  [ )  P



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[ = [   9  = 9 



Hij_^_ebf djbl_jbc dhjj_dlghklb HI: g_\uiheg_gb_ dhlhjh]h [m^_l k\b^_l_evkl\h\Zlv h g_dhjj_dlghklb wlh]h ihgylby Kh]eZkgh hij_^_e_gbx HI:h[jZs_gb_\j_f_gb W  W \mjZ\g_gbb^\b`_gbykh\f_klghkbaf_g_ gb_fagZdZm 9  9 → −9 ^he`ghiha\heblvihemqblv©h[jZlgmxªljZ_d lhjbx

[  − = [−W  Imklv

[  =   b ijb ©ijyfhfª ^\b`_gbb

 W = W  + >  9 >   qZklbpZ ^\b`_lky \ijZ\h \ klhjhgm o > 0 Ijb h[  −

+

jZlghf ^\b`_gbb  W = W = −W <  9 <   qZklbpZ ^\b`_lky \e_\h \ klhjhgm \k_ [hevrbo _ [ _  [ <   ?keb hij_^_e_gb_ HI: dhjj_dlgh lh ^he`gh\uihegylvkykhhlghr_gb_

[ − (− W ) = − [ + (W ) [  + W >   W >  



Fh`gh\\_klblZd`_mkeh\b_

[ − (− W ) <   W >  



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G [ GW



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 const

( D

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D W [  + W = [W  + = 9 W +     >ey©h[jZs_gbyª\ gZ^hiheh`blv W  W  9 → −9 Wlh^Z_l D W [  − = [W  − = [−W = −9 −W +  = [  + W >   



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W

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∫

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∫

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[  − = [−W = −9 −W + , −W = 9 W + , W = [  + W >   




  K  8  [    ijb

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 b ke_^m_lg_dhjj_dlghklvHI:
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 −



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[  + = [W = 9 [

 

 N    ω =   P 

= [−W = (− 9 )

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 (   9 =   P 

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G[  N   9 W = = G W  P 

 

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⋅ [ 



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x (42)

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x5(3)

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