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Semiclassical Localization In Phase Space.

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'' 2(,,:7 (*  < *   @  T+- @   T+-6 1    '  1') :,, (  1 , /1 @  @  * )    '  )  9 8e0(:1 ( (*E1') ,:, @1 '6  C *  /1   5* 1/  :E7 (* ,   / (  ') ,,: * 1   (*;= (@1 @  (* ,:   Y / ; @  :' ,: ' ) @,, (*Q;  )  ; @,F'H7  e 1, <  T+-6   T+-Z  ; 1 .+ 2? 1,: (* (* 1H 7 ': )  &0  '10@e= (*) , @'':) , ,: &  < *:)+* ;, $ 7 '  := (* L90 @ 90  e' S 4< e,  a ;= (@1 '  (*'6E 7J;  )  e=  0,:(8 ;,:   (* ;  0) '   *B'(EE 7J;=  8* )  (*( 7     ,/ e   F  ? >*;,:) &? 1  0,'? 1 ?(* ;  0) '    '@!* S' 7Ib  e=1,:'   9  < F ; @ (:) 0, (*Q  ;  )  ; @,H' 7  e 1,:' '* <  ':=  ;, e  * $ 4  O  8 <  Z   S8   <   "  ; : ) 8 ; , < *   <      @( 'B(*  1''1 e ()+@   <      ? ) D(*T H' 7  e 1,F /  '@ @,/1( e ( E1G ;*'  1 *B,/ e    L 0 2(0 E )+* @ ) , e' S 4@ e, ' ( *D,:/1 e  !) ,'') @,  e' S 4< e, '  ;* 7 ') @,#(  '?(* 'D  , @1 '?   (*@ < '  ,:8, 7E(*B) , '') ,F e' S 4@ e,B) 1   G' ; 1   /D(   '! E ;;,) @(:1 <  < ,,I ') 0'' ) 1T  ;, >Y;  <  '5F@ E ,,: ;)6 8 9=  2( @, ;  @(1 5@  (* ) 1   H' ;=   /( (@)  < *)+* ' (* '< B ) ,,: % 6 C B D  ( 0 ) 1  4  *' " ( 7 , ', < )  e   0)    F 0 (* E1  F(* J7 D  /  K ,:E:(* 1   @ :=  ; 1 .+ 2  '0,: < *:)+* '* < ' * < (**:/1*   / 7E,E:) '61= ;' 0  8 9=  2(,= ;= (@1 ' '  '  :) , @'':) , ,:E:  `   (*  E,&(1  % e D  ,:,8 ;(:A ) ;' 0 I : 9  9 ,  ;= (@1 1 '1 ) 1T  ; ) B  : 1,:   E,    e= *? /1  0 ) (:1 '!  # e= (*?H' ;= ) ( ,8) 10 9 /B 0 ) (:1 9) 10 9 /B(* 90 e    :/1 , 4@,0 '  % e  ,: <   C * * *7 D  _/  K ,:E:!(* 1   '($ 7 ' (* @ O     .  O  ,: <6 &!   41, "!     #%$    < *  !     T+-(' %   D+-L O)  ( '?*BL90  /7 '* ,, @  /7 O@  &  B  ' (*!) @ 1 :) , ` 0.4 ,,:#E '0 !1 !   ? ) !(*!*:/1*   /7Qe  * $4 1 ?>*;= ) (@(:1 4@,0 ' C  ;' 0  89=  2(,F;  @(1 ' I ' 1 ,87 ;   1 (* ;  )  ;,'S7Ibe=1,   0 (* T  ;=1 .+ 2B) , ''  ;= (@1 ' ) @ +e  )+* @ () (  D  e,7      / (*@ *(7 )+*  /1 (*1') ,,@(:1 ' ! *:/1*:, 7 1') ,,@(: /D 0 ) 1   <  ,:,   < $717I )   @  J + @    2(  ' )  1   ( '  ;* ' 'G;)  <  ?>*;  ) (*@ )+*  /1B (*  :'  ,@( ( E)  1 ) @,  ( ' 1  @1      =: 1  L90:  '?(*@@ ;  @(1  ) (' %;,   <  4@' B  Y  T+-     '5  :, 4@,0  1'E (*   E*1E1/1  10'!@F  /1  ?1 ?: < *    '! '  *ES' 7  e=,8   I(* ' ) @,,  4 10 : : 2( / (,F ;= (@1   )  e  4I: <  ' Y L90  2(8 D @1  (* ) @ 1 :) , ( (@ ' 1  (  < *1' /1  (( /  0 ) 1 :'  C *B   ,@(:1 )    ) (: / 4 10  : 2( / (,2 ;  @( '  A ;' 0  8 9=  2(,9 ;= (@1 ' :' * C * 1  $ - /1  4 < *)+*   ':=  ;,  R 4 '1 E  'F* @ 1 !)   B 1  @)+* )  1 :) ,@( (@ ' 1  (   4 10   2 /1 ,F ;  @(1 "  < &(*     "  ?   M < C * ) @ 1 :) ,!( (@ ' 1  ( ')  ,' e= )+* () (  D  e,7 (*  ) 1 1 *  1:''1 ,:/1 e  6(* ) , '':) ,5 e' S 4< e, ' #(*( 7  E(* @,/1( e ( 01 G ;*:'E'   *  '  e  0: 0, (* 1    0 ( 0:'  1@  7I: /1  B7 J P (* B) 1 0/2@(:1 < &(* 4 10   2 /1 , ;  @( '/1: 41 '5 >@) (:, 7* 1   ;  ' H 4  / 0I(1E1G ;*' '#@ *,/ e  ;' *0 B  8 9=  2(, ;= (@1 '  "  ? >  (0   '1EB ;;,) @(:1 '  (* ) 1 ) ( ;(' ? 41 , ;  '   : L90  2(0 )+* 1'  1    ,:,0'.( (( (*  60' C * '. 1 ?' * L90  20  /1 ) :E 7 * 1   < *)+* )  e  ;   41 0' / (**7 D  _/ K ,E: (* 1   @  - /11  4 ' *      /18 4 '6 )+*  )   D (  (* '  :) ,'') @, e  * $ 4 1  ,: 1'.,,= :/1  0 ) 1 ' # H' 7 '  < *1' ) , '') ,F,:E: '? /1I )  ' 7 '* @?(*? 7 e  ) 1   L90 B :'(  e0I(  * *:/1*   / 7 ,:E:  : :)+9 / (*=  e  * $ 4  @6(* ) , '') , ` .0 4 ,,: ;  e  e:,&E 7  '&E 71 -  /  1 M  e,:,  *'BE  ' (*@ M ; 1'&( G' ; )  ,:     0 41, O    * <  @ ' ' < *    %    '?*B'   :/1  0 ) 1 '  C *'' (* '.(  /1 '. :/11 10'6  '0,: @  /1  (, @(0  eI+:  '    L 0 2(0 )+*1' ' ) &   '  ;  B :) (:1 0   <  ,:,      @(* <   ) 1  :(:1 '  (* 1,:, <  / < B'* < ? >T  ;, '   :') 0''  0 * 5 L90 '(:1 '  , @ (%  L90  2(0  /I :) &E 71 C *' ) 1   ;.;,:) @1 :'!* ()    0,  < *:)+* ' e @' 1 E* e' H 4@@(:1 D(* @!(* (: E( 41,0I(  ;  ((1 '  480 :  2( /1 (,! ;  @(1 ''I)  @ < &(* * ) , '') ,   <   ' /   (* '  1 (* (* 1S 7 480 :  2( /1 (,  ;  @( ' )     :  &('( (@)   ;= :I :) 1G e:(' @?* ) , '':) , S' 7 '(   C *'D:' < *:)+* ) @ e  ? >*;  ''   E'D*M (*D @ 10' 1 0,  < *)+* ) 1   ) ' *Y  L90  2(0   )+*  :) ,#H' ;= )  0 < &(* (* ) , @'':) , ;  :I ) 1H e:' 9  E:'!@(* *  5@E1'!( =  ;I('(Q  ;   4?  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') 8 e= e=  1  < *)+* ) 1 2+: Q  ' V   < *  (*) ,'') , '( 0) 0  '  ;;  @ ?: (*?,:E: V%X N "  /18 4?B'*1 ! ? 4  < *  : D /1  : 2(' 9'* <  / * < *  1 0, '     ;I(  C *    ( < e )+8(*:'# ;; 2)+* 2) 1T  ;   < :* (* ; ? 4I:10'1  2' (*  ,&(*10/1* * 90 e  @?S' 7 '(  '  < *:)+* (*:') @ e= < ;;,: ') 1 '  ( e:, 7 , /  :' ;, * 7 ':) , E   / *' (Y  e  ) ,   F @(0 b  V '  ) 1 '( 2 ' Y  ; 1  * ,: & V%X N ' ;0  8, 7    ,  1 (*Y  ;,* 7I') , ;=1: 2  4I: <   * 'M  e= 0'.(   0 *  C *' :' @( T  ; : 7  ) 1 LI3 N  16*  ; L < *)+*  ;  .('  1 *E/1  (,! ? 4  < )+*  (@) ( (*' )+*  ;I( ?  ) 1 2+: ' '1E  <   '  16*  ;JB C *  41 (,:., ;:) (0 6@ * ' E) ,'') ,9,E: * @   /1 '  $* :') 0''1 ':% ( ?L :' (* 5 L 0 2(0 E )+* @ )  , L90  2(&( ' e= ) 1EB) ,:1'E) , '':) , L90 @ 9: ' : O1 * '.+@ '  */*8, 7 1') ,,@(H 7     /E*  6 1/1,: <  4 ,: /* (  '? 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EB  :/1* e=1 * I D   :'  , 4@,0  2' <  * $46+) :,87 ''0E  1*  10   #'  1 *B  /1  )(7) 1  :1 ' 1 $< :,, e   ` /1  /1 '0e  & 1, 5;* @' 'H; ) @ C *:' ' (*  @'1 < *,7 '0)+* '(@( '  ) ,,: ` /1 (@ /1 @ F C *?7 *4be   '0  *1 0/1*,:7   * : (* 1 H7 ' <  ,, ? 41 ,: ;  ' (*   : 0'  L90  2(0   )+*  :) ':'(*YL 0 2( D @(:1 51    2 /1 e,   A 6 H' 7 ' E' #'   / U6 '+JL 0 J 4@,0 ;* '  0 ) ( '  C * (* 1H 7  (* ' '.+@ '?' , '') 1T  ;, ( :, 7 ? 41 ,: ;   =,:*10/* (*  ? >I'. '1E   @(E 2(' I6 7J 2 6 77G1N   a0 @H' ( ? 41 , ;  /  (,!(* 1H 7  '0)+* '(@( ' =,1 /D*,:  '  * ( 41 , ;  9?#*B(* 1H 7 #4 10 :  2 /1 ,  ;  ((1 '  ;,:? > ;* 'B 0 ) 1 ' :   1K G 0,N  ;;  @  /: * L90  2(8 D @1 B)   :1   < 5(S 7(? ;;:, 7(* '  ; I)  0 5(?  ,,: ;)  1H e: < ?' (*  < @   1 )   ) 1 '  ) 1=  ;,? >%4@,:0 T ;* @'? 0 ) 1 '  7  ) 1= 2ML  ;,? > 4@,0 ;* ' < E'.(0* 7 '1E , ) , L90 '.(:1 'B  , @ ( 1') ,:, @1H 7 : 9 /1 ,' < :(* ) 1=  0 ) (:1 '  G' ;  )  ,:8, 7 (* @=  ;: / :  0)  e,7 (* : @/1 @H 7M;  .  " Y  ;  ' 2  *    0 ( U6 7J N 1 * ) ( !@  ;' 0  : 9  2 ,  ;= (@1  1') ,:, @1H 7 : 9 /1 ,' < :* ) =  ;,:? >%4@,:0 = ;* ' 0 ) ( F C *:'   ' *: 9  0) (  @,E1'! 8, 79(:)? > (  ' '  )  ''(S 71 1   D( ,,:0' (@ (* ;  e,  6''0E (*@         ,+-  < :*  ; @ (, : 9  2 ,F ;= (@1   S+- :''  *   ) =  ;,:? > 4@,:0 < :*    :'T  % N * <  * $ 4    S+-8 "  ,+-1D+-.    < *  " T+-       ,+-   :'* 'H7  e 1, @    <  1 E/1  (@,;' 0 I :9  9 ,  ;  @(1 < B?>*;  ) ?: ,   / 1    ': :,  5 1 0,  Je0 ' ) (* 'S7  e=, < ,, D, /1  e R  ; 18, 7 1E @,  Je0 0'  'EI@(*  0 ) 1 1 ;*''H; )  &6:'6A; :1 H7D @6) , @ * < (D? 4@,0 @ :  ) 1T;, > @ /10E 2'   S+-1T+-  C *B  (*  ,E1'  ,:7 ) > ( ' ' ;   4 :  '  <  < $ 7 (  > (   'EI*  0 ) (:1 () 1T;, > @ /10E 2' < *)+* <  )  0'  1 10 ) '  ;,: '.#E 7J;=  ` /1  /1 '(@( ' < :(*D) 1=  ;,? >b;* @'  0 ) ( @  2: D ' ' 9(* C * ':= ) *  2 '(@( ' < *)+* <  < :,, '0. 7   +@,  7I ) ( 2I 27I )  *( 7   )  )  9 (@  1 b  ; 1: 9?:'?' 0 E) : 2 '0. 7 (*  1  A  C * (  ) 1*  2 '.+@ * R ' e=  0'  (* ,:: ((0  1 T  ; @ (:) 0, E 7J;=  '+( ' < :* DE1 B1 , 'W' ;  ) :' :, 7    ( / ON     /   ; ;  .( ' 9'  /    G0P *1 ?GN  0 G1J  "  < ,,  Re,7D(*:'6@ 1') ,:, @1 H7  0 ) (:1 * L90   (@) ;* '  0 ) (:1 < *)+* :' ) 1 )  2( @(  1  ; 1: 9 : A;*' 'G; )  - >*;,:) &(,:7 < :b  ) 1*  2'.+@ :' /18 4 e,7 A2C 1O IF   , .  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O 1i &O .#}*)!   | ~ #)   ~ &' . < = # V'  2  2 O V#% ? 2 #) Ÿ%2 & | % .*!%;  % 1>35143/6 >  398;:!< Q  = # ? = Q   = # S?xq ' µ F@G0H~I :!< s  t  Q u =" ?žK!:¦Y!< F <‡: Q    ?Ef+ 1 35143/6 >398;:!< ?  ^?Üq A©C Q = = Q = 7 w  ‰ f 7 w   5 f— u µ  …  ‚  t † s  u ƒt 2Q ="   ¤ ? s   t   Q u = " ³  ¤ ?   ¤ { | † †5ˆ ‰† ‰†    ‚   t „ƒ…   … ‚ ƒ t „ƒ…  ‚ „ƒ…   t ' Œ „ƒ…   ' … ‚ „ƒ… ¿t '  Œ „ƒ…  ' µ u µ ' u    …  ‚  … Œ  ˆ    ' … '  Œ „ƒ…    ' … ' Œ „ƒ…   ' t † y y ¡  g   ¥ … ' ¤ { | { |  }O  …  ‚ Qt  ' † '    u µ  † †‰ˆ ‰† 5†   ‚   t „ƒ…   … ‚ ƒ t „ƒ…  ‚ „ƒ…   t ' Œ „ƒ…   ' … ‚ „ƒ… ¿ t '  Œ „ƒ…  '  y {¦| …  ‚  … Œ  ˆ    ' … '  Œ „ƒ…   ' … ' Œ „ƒ…    ' t   '   Š Œ … Œ  ˆ    ' …  '  Œ „ƒ…   ' … ' Œ „ƒ…    ' ‰† = )  V+ 3$ ~ PD*#  z // u V µ ¤ ) Ù  ) 0 .  |  .! # Q ¨"1! ## d  2  #    ' = # ' ; 7  = 7 # |  ~ #  ¡ s ƒ t  = s   t   ¥ Q  =" ?  y  ‚   t ! † … ‚  t „ƒ…   †5ˆ  ‚ „ƒ… ¿  t  ' Œ „ƒ…   ‰†‰ˆ  ‚ !¿ t  ' ! † ' ˆ ‚  t  ' Œ „ƒ…   5†  {¦| …  ‚  …   … ' Œ „ƒ…    …   '  ' t   '  Š Œ …   … ' Œ „ƒ…    …   ' ! 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