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k}P ^&YerYRT|a`b\ λ g`d\qY4`uszfqZ[oPaefqcbUfTPa|6P`dZiURSVZ[jh`dPaRNoBY L IPaRTS`d\qY4`uszfqZ[oP&ejhZrY4U&`d\qY
ZrRq\qU&
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ZrjPae[jhUP`uszfqZ[oP&eUcdSVYLc_UamW4Pa|RqZi`bnTSVY ε `d\TP`op\NPacpP&o8`dYcdZiLYLj`b\TY
jh`dPaRNSqPacpS"SVY^zZ[P`bZrU&R"U&?`d\qYlSVZrW"YRTjbZrU&RqerYLjd&j TnToB`bnTPa`bZrU&RTjUaD`b\TYQfTPacpPaW"Y`bYcpjUaD`d\qYQW"YLSVZrnqW
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jdoP&eiY l gtqnq`
kxYW4P s*`b\TZiRq{Ua PjP
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`b\T$Y TnTo8`dnTP`dZiURTjmZ[jQl jbW"P&eie ε 1 V`b\qYLR`b\qY~W"Ujh`QYB¯DYLoB`bZr^&Y~ZrR6`bYcpP&oB`bZrU&RUaI`b\qY~k}P ^&Y-jxkmZ`d\
`b\TYcpPaRTSqU&WW"YLSVZrnqW kmZreieDUVoonqc}km\qYR l ∼ λ z`b\NP`mZrjL6`b\qYk}P ^&YerYRT|a`b\Zrj}oBUW"fTPacpPatqerYm`dU
`b\TY_oBU&cdcdYe[P`bZrU&ReiYLRq|a`d\0,v QRTS]`d\qZrj}ZrR6`bYLcdPo8`bZrU&RkmZreie:t?Y~U&tTjbYcd^PatqerYkm\qYR]`b\qY~fTcbUfTPa|6P`bZrU&R
SVZ[ju`pPaRToY Z[jerP&cb|Y
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RTSVYLYLSWLUVSVYLcbR`bYLop\TRqLU&erU&∼|s]λεZ[jP&tqerY_`bUfqcdUVSVnToBYTt?YcpjlU&^&YLcbs*6nTPaerZ`us ε 1 v QUkxY^&YLc
YRT|&ZrRqYYLcdjxPaZrW P`}nTjbZiRq|^&YLcbs"erU&Rq|Tt?Ycp.j -VP&oB`bnTP&eiers`b\qYLsPaerk}P szjnTjbYlTt?Ycpjxkm\qUjbYlerYRT|a`b\Nj
)
"
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c
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P`km\qZ[op\`b\qYlZiRNnqYRToYQUaDcpPaRTSVUW"RqYLjdj
P&RTS
Z[j
t?YLoBUW"YLjUaU&cpSVYLc v}\TP`Zrjkm\zs`b\qY"P&jbszWfq`bUa`dZro_<cdP&W"YkxU&cd{
`b\TY~cbYLeiYL^ P&R6`}URqY&qP&1RTS`b\qY~URqY`b\TPa`QkmZiere0t?Y_P&SqSVcdYLjdjbYLS
ZrR`b\qZ[jQop\TεP&fV`bYL1cLv L ∼ λε
ª > ª > HgE 'FgG C gJ gE ?G2KMA 6E :H qCLK¡F?A> }\qcdU&nT|&\qUnV`_`b\TZrj"op\TP&fV`bYLckxY
jb\TPaereNoU&RTjbZrSqYc%`d\qYVop\Tcb6SVZiRT|&Yc%Y-6nTP`dZiUR0v2
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L
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jbU&ernV`bZrU&RTj}U&2`d\qY<U&cdW
iut + uxx = 0.
km\qYLcbY k ∈ R Zrj`d\qYkxP ^YRznqWtNYLcLvy}\qY*fq\TPjhYU&l`d\qZrj
WURqUVop\qcdU&W4P`dZro]kxP ^YoLPaR$tNY
kmcdZ`b`bYRP&j k(x − kt) qjhU~`d\TP`x`b\TYfq\NP&jbYl^&YLeiUVoZ`us"Ua:`d\qZ[jk}P ^&YPafqf?YLP&cdj`bUtNYYL6nTPaeN`bU k v
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kmZi`b\PaRZiRTZ`dZrP&eoBU&RNSVZ`dZiURPa` t = 0 u(t = 0, x) = u (x) Ikm\qYcdY u ∈ L v jhUeinq`bZrU&R
fqcdUVoBYLSqnqcbY<Uc0`d\qZrj2fqcdU&tqerYW Zrj2tzs£TU&nqcdZrYc:`dcdP&RTju<UcbW*v RTY%Tcpju`gf?Ycb<U&cdW4j0PZicdYLoB`I£TU&nqcdZiYLc
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,F
0
0
2
û(0, k) =
1
2π
Z
∞
u(0, x)e−ikx dx.
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RTP&cbs
SVZi¯DYcdYR6`bZ[Pae0Y-6nTP`dZiURTj.
−∞
2
}\qYQjbU&ernV`bZrU&R"P`P&Rzs~`bZrW"Y t oLPaR"tNYmUtV`dP&ZiRTYLSt6sPafqfTeiszZrRq|`b\qYm
Rz^&YLcdjbYm£qUnqcbZrYc%2cdP&RTjh<U&cdW
<
u£ ût = −ik 2 û =⇒ û(t, k) = û(0, k)e−ik t .
Vop\qYLW4P`bZ[oP&eiers&VkxY\TP ^&Y u(t, x) =
DFT
−→
u(0, x)
↓
u(t, x)
IFT
←−
Z
∞
û(t, k)eikx dk.
−∞
û(0, k)
↓
"!$#%'&#()&*%+*-,.()&()#/
û(t, k)
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`b\TYW"U6ju`~RTPa`bnqcpPae%PafTfqcbU6P&op\jbZrRToBY"PaRzsk}P ^&YoPaRtNY
SqYLjdoBcdZit?YLSP&j`b\qY
jbnqf?YcdfNU6jhZi`bZrU&RUa
jbnTop\YLeiYLW"YR6`dP&cbs~k}P ^&Y`bcpPaZrRTj0tzs£TU&nqcdZiYLcI`bcpPaRTjh<U&cdW*v 0Y` û(x) tNYx`d\qYmPaW"fqerZ`dnTSVYQP` x ∈ R
UaPW"U&RqUVop\qcdU&W4P`dZro_kxP ^Y u(t, x) = exp(−ik t)û(x) `dcdP ^YerZiRq|ZiR`b\qYU&RqYSVZrW"YRTjbZrU&RTP&e
W"YLSVZrnqWXSVYLjdoBcdZit?YLS]ZrR£IZr|Tv &v%}\qYW"YLSqZinqW!Zrj}\TU&W"U&|YRqYLU&nTjxU&nq`djbZrSVY`d\qY~jhe[Pat P&RTS
`b\TY~kxP ^Y u UtNYLszjx`d\qYVop\qcdSqZiRq|Yc}YL6nTPa`bZrU&R iu + u = 0 v mooBUcdSqZiRq|eis û jbPa[0,`bZ[juL]TY-j
/,-21 -
-
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4
: 9D-2=
-
4
4
7
2
t
2
ûxx + k û = 0
xx
-
?><- A +
E
8
? jbU`d\TP`QZi`Q\TPjx`b\qY<UcbW
<U&c
û(x) = eikx + Rε e−ikx
PaRNS
<U&c x ≥ L.
û(x) = T e
}\qYoU&W"fqerYBz®^P&einqY-S cpPaRNSVU&W ^P&cbZ[PatqerYLj PaRTS PacdY`b\qYcdY TY-o8`dZiUR$P&RTS `bcpPaRTjbW"Zrjh
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k}P ^&YRznqWtNYLc k P&RTS
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jh`dP`dZiURTPacds&2Ycd|&UVSVZ[oaPaRTSLYcdUaW"YLPaRfqcdUVoBYLjdj V v}\qYSVZrW"YRTjbZiURqerYLjdj~fTPacpPaW"Y`bYc ε > 0
op\TP&cdPo8`bYLcbZrY-jx`b\qYPaW"fqerZ`dnTSVYUa%`b\TYcpPaRTSVUWf?Ua`dYR6`bZ[Pae®vl}\qYjboLPae[PacmTYLerS û jbPa`bZ[juNYLjLq<U&c
x≤0
ε ikx
ε
x ∈ [0, L]
ε
B
ûxx + (k 2 − εV (x))û = 0.
}\qYoBU&R6`dZiRznqZi`us*Ua P&RTS Pa`
PaRTS
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`b\TY`ukU"f?U&ZrR`mt?U&nqRNûSqPacdsoBûURTSVZi`bZrxU&RT=j' 0 x = L
Pa` x = 0, ikû − û = 0 P` x = L.
ikû + û = 2ik
x
;
x
ei(kx−k
2
x
t)
T ε ei(kx−k
Rε e−i(kx+k
2
iut + uxx = εV (x)u
t)
t)
-
0
L
2
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x
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`b\NP`%\TP&j2TRTZ`dYmWUW"YR6`djIUaDPaereVU&cpSVYcpjIPaRTSZrjIcpPafqZ[SVers~W"ZVZrRq|TvIwY}W4P s~`d\qZiRT{~<U&cZrRTju`pPaRToY
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I V LGa ba
K
K&OQP F
ε
loc
F
RTSUJ FWV HX8Y 0Y)J ZJ [\XL ]
1
1
ln |T ε |2 (k, L) = − ε .
L
Lloc
_§i¤ &¥ p¦ p¤¤ ¦ &¤ ]¦ ¤
L→∞
F
V XS
α(k) 2
1
=
ε + O(ε3 ),
ε
Lloc
2k 2
MHbKcXd
ε]
α(k) :=
Z
∞
duE[V (0)V (u)] cos(2ku).
0
}\TYjh`bnTSqsUaQ`d\qY*YBVfNURqYR6`dZrP&et?Y\TP ^zZrU&cUaQ`d\qY]`bcpPaRTjbW"Z`b`bZr^6Zi`us |T | oP&R't?Y
B
¨
SVZr^zZrSVY-SZrR6`bU`ukUju`dYfTjLv£2Zrcpju`I`b\qY}erUVoPaerZrLP`dZiURerYRq|&`b\Zrjjb\qUkmR_`dUt?Y}YL6nTPae6`dU`d\qYxZiRz^&YLcdjbY
UaT`b\qY 0s6PafqnqRTU^Yzf?U&RTYR6`PjbjbUVoBZ[P`dYLS`bU`b\TYcpPaRTSqU&W UjdoBZreie[P`dU&c v +(k −εV (x))v = 0 v
zY-oBU&RNS~`b\TYYzfNPaRTjbZiUR_UaT`b\qY 0s6PafqnTRqU^YVfNURqYR6`UaT`d\qYxcdP&RTSVU&W U6jboZiererPa`bUc2Z[joBU&W"fqnq`bYLS0v
wYjh\NPaere:Tcpju`m`dcdP&RTju<UcbW`b\qYtNUnqRTSqP&cbs
^P&einqYfqcbUtqerYW
xZrR6`bUP&R*ZiRqZi`bZ[Pae2^PaernqY
fqcdU&tqerYW*V`d\TP`QZ[jmW"U&cdY`bcpP&oB`dP&tqeiY~jbZiRNoBY`b\TY_cbYLeiYL^PaR6`QnNPaR6`bZi`bZrYLjmkmZrere0`b\TYR*tNYP&SqP&fV`bY-S`dU
`b\TY~RTP`dnqcdP&eDTei`bcpP`bZrU&R*U&2`d\qY~cdP&RTSVUW fqcdUzoYLjdj V v
RNjhZ[SVY`b\TY~fNYLch`dnqcbt?YLS*jberP&t*kYYBVfTP&RTS
ZiR`d\qY<U&cdW
û
HXXd
ε 2
2
xx
B
;
km\qYLcbY P&RTS PacdYcbY-jhf?YLoB`bZr^&Yers*tTP&op{zk}PacpS <|U&ZrRq|`dU`d\qYeiY ` PaRNS<U&cdk}PacpS <|U&ZrRq|
`dU
`b\TY~cbZr|&\6A` W"UVSVBYLjmSqYBTRqY-St6s
û(k, x) = A(k, x)e−ikx + B(k, x)eikx ,
ikû − ûx ikx
e ,
2ik
}\qYfqcdUVoBYLjdj (A, B) Z[jQjbU&ernV`bZrU&RUa
A=
d
dx
A
B
= P (k, x)
A
B
,
B=
ikû + ûx −ikx
e
.
2ik
P (k, x) =
iε
V (x)
2k
B
}\qYt?U&nqRTSTPacdsoBURTSVZi`bZrU&RTj xcdYLP&S]ZrR`dYcdW"jQUa A P&RTS
B
;
1
−e−2ikx
e2ikx
−1
.
w Y
ZiR6`dcbUVSVnToY"`b\qYfqcbUfTPa|6P`dU&c Y IZMv Y&v`b\qY4<nTRTSqPaW"YLR`pPaeW"Pa`bcdZjhUeinq`bZrU&RUa}`b\qYeiZrRqYLP&c
jbszjh`bYLWUa2SqZ¯DYcdYR6`bZ[PaeDY-nNP`bZrU&RNj.
v£qcdU&W jbszW"WY`bcdZiY-jZrR ?v Zi`
Z[jmPafqfTP&cbYLR6`x`d\TP`QZi (a, b) Z[jQP"jhUeinVY`dZiUR]=U&P Y xYkm(0)Zi`b\=`b\qIY~ZrRqZi`bZ[PaegoU&RTSqZ`dZiURTj.
A(k, L) = 0, B(k, 0) = 1.
x
;
`b\TYR
d
ZrjmP&RqUa`d\qYcQjbU&ernV`dZiUR]erZiRTYLPacders
ZiRNSVYf?YRTSqYR6`QUa (a, b) v}\znTjQkY~oP&R]kmcdZi`bY
(b , a )
a(k, 0) = 1, b(k, 0) = 0,
∗
∗
B
Y (k, x) =
a(k, x)
b(k, x)
b∗ (k, x)
a∗ (k, x)
.
}\qYW"UVSVYLj A P&RTS B W4P s
t?Y~YBVfqcdYLjdjbYLS
ZrR`bYLcbW4j}UaI`b\TY~fqcbUfTPa|6P`dU&c £qcdU&W `b\TYZ[SVYR6`bZi`us
SVY-SVnToBY`d\TP`
A(k, x)
B(k, x)
~PafTfqeiZrYLS<U&c
= Y (k, x)
x = L
A(k, 0)
B(k, 0)
.
PaRTS`d\qYt?U&nqRNSqPacdsoU&RTSqZ`dZiURTj kxYoP&R
B
zZrRToBY`b\qYW4P`bcdZi P \TPjm`dcdPoBY_LYcdUTq`d\qYSqYB`bYLcbW"ZrRTPaR6`Ua%`b\TYW4P`dcbZi Y ZrjoBURTju`pPaR6`L?Z®v Yv
qjhU`d\TP`QkxY|&YB`}`d\qY~YRqYLcb|soBURTjbYcd^ Pa`bZrU&R]cbYLerPa`bZrU&R
|a(k, x)| − |b(k, x)| = 1
B
Rε (k, L) = −(b∗ /a)(k, L),
2
T ε (k, L) = (1/a)(k, L).
2
|Rε (k, L)|2 + |T ε (k, L)|2 = 1.
8
? }\qY`bcpPaRTjbW"Z`b`bZr^zZ`us |T | Z[jxYL6nTPaeN`bU 1/|a| (k, L) vIwYZiR6`dcbUVSVnToY v = ae + be v
`
jdP`bZ[jhTYLjx`d\qY~YL6nTPa`bZrU&R
kmZi`b\
`d\qYZiRqZi`bZ[PaeDoBURTSVZi`bZrU&R v(0) =v 1 +v (k(0) =− εV−ik)vvI
=gk0,YSVYRTUa`bYtzs γ (k) `b\qY 0s6PafqnqRTU^
YBVf?U&RqYLR6`:`b\TPa`g|&U^YcdRTjN`b\qY%Yzf?U&RTYR6`bZ[Paea|cbUk}`d\U&z`b\qY6nTP&R`dZ`us r(k, L) := |v| +|v | /k ε 2
2
ikx
−ikx
2
xx
ε
x
2
γ ε (k) = lim
x
2
2
1
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(k
)|
E
|T
(k,
)|
|T
(k
,
)|
E T ε ( 2 )2 =
dkdk 0 .
ε
4π 2 0
ε2
ε2
0
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(|T (k, L/ε )|, |T (k , L/ε )|)
ε
2
ε
0
2
0
]
8
P&RTS
|T ε (k, L/ε2 )|
? P&cbYP&jbszWfq`bUa`dZroLPaereisnqRToBUcbcdYe[P`dYLS:VjbU`b\TPa`
|T ε (k 0 , L/ε2 )|
2
Z ∞
1
ε L 2 ε→0
2
2
E T ( 2 ) −→
|f (k) |T̄ (k, L)| dk ,
ε
2π 0
km\qZ[op\]fTcbU^YLj`b\qY_oU&Rz^&YLcb|YRToYlU& T (L/ε ) ZiR L v
0Y`]nTjP&jdjbnqW"Y`b\TPa`]`d\qYZrRToU&W"ZiRT| k}P ^&YZ[jRTP&cbcdUkmtTP&RTS:}`b\NP`]Z[j`bU jdP s `b\TPa``b\TY
jbfNY-o8`bcpPaeDoBUR6`bYR6` f Zrj}oBURToBYLR6`bcpP`bY-S4PacdU&nqRNS"`b\qYoP&cbcdZrYck}P ^&YLRznqWt?Yc k PaRTS\TP&jRTPacdcdUk
tTP&RTSVkmZ[Sz`b\ ¡jbW4PaereiYLc2`d\TPaR atTnV`%erP&cb|Yc2`b\TP&R ε BvI}\qYLR T (L) SVY-oP sVjIYBVfNURqYR6`dZrP&eierskmZ`d\
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ε
2
2
0
2
(
α(k0 )
1
L1
ln T (L) ' −
L
8k02
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R `d\qY`bZrWYSqU&W4PaZrR'`d\qY*eiUVoP&eiZrLPa`bZrU&R
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`bZrU&RUa`b\qY"W"UVSVnqerZ%Ua`d\qY`bcpPaRTjbW"ZrjdjhZrU&RoBUzYB
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PaRTS T (k , L/ε ) PacdYoU&cdcbYLerPa`bY-S
U&RTeis
Zi |k − k | ≤ ε v
T (k, L/ε )
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`b\TYcdYLjbnqei`~UaxPSVY`bYcdW"ZiRTZrjh`bZ[o"oBU&Rz^U&ernV`bZrU&RU&`d\qY4ZiRTZ`dZrP&e%fqnqe[jhYvxY\TZiRTS`b\TZrj<cdU&R6`fTPacb`
YW"YLcb|YLjI`b\TY hoUVSqP }km\qZrop\"Z[j%ZrRToBU\qYcdYR6`LtqnV`W4P soU&R6`dP&ZiRNj%W"Ujh`Ua?`b\qYm`dcdP&RTjbWZi`h`dYLSYRq
Ycd|&sv cdPa`b\qYLcmoBU&Rz^zZrRToBZrRq|"\qYnTcbZ[ju`dZroYzfTerP&RTP`dZiURoLPaR*tNY<UnqRTSZrR© = 0NVYLo8`dZiUR v}\qY
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cdYLjbnqe`pj%oP&Rt?Yx<U&nqRTSZrR*© = ¬ zv gYB`nTj%PjbjbnqW"Yx`b\TPa``b\qY}<cdYL6nqYLRToBsoU&R6`bYLR6`U&T`d\qYmZiRTZ`dZrP&e
fqnqe[jbYZ[jloBURToBYLR6`bcpP`bY-SPacdU&nqRNS`b\qYoPacdcdZiYLcQkxP ^YRznqWtNYLc k vl
URqYPaRNPaers6LYLjQ`b\qYYRqYLcb|s
oBUR6`bYR6`Ua`d\qY<cbUR`fTP&ch`Ua`d\qYkxP ^YZrR`b\TY<cpPaW"YLkUcb{ZiR6`dcbUVSVnToYLS\qYcdYP&tNU^Y&N`b\qYLR
U&RTY_TRTSTjl`d\TP`Zi`SVY-oP sVjQYVfNURqYR6`bZ[PaererskmZi`b\`b\qY"jbZiLYU&%`b\qY"jhe[PatkmZ`d\P]op\TPacpP&oB`bYcdZ[ju`dZro
erYRq|&`b\'km\qZ[op\'Z[j_`d\qY ¦ §i¤ eiUVoP&eiZrLPa`bZrU&RerYRq|&`b\ 2k /α(k ) v QUkxY^&YLcLP&jjh\TUkmR\qYcdY
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ª > ¬ > E qAIG2F VE 'F?A2KMJ]F -J&K VC-FTEz> }\qY~cdP&RTSVUW\TPacdW"U&RqZ[oUjdoBZreie[P`dU&c 2I V bP
ba
0
ε
2
Q4
ε
2
0
7+
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2
0
,+
,:
+
,+
D
!
,
0
K
,+
B
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,- 84
:
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0
E
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© ¬ = ª ¬6^zZrtqcpP`bZrU&RNj
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ε>0
ytt + (1 + εη(t))y = 0
! !
!
!
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G := lim
1
ln r(t),
t
r(t) =
p
|y(t)|2 + |yt (t)|2 .
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W"U&W"YR6`mUaI`b\TY~WUVSVnTerPa`bZrU&R
t→∞
<
8 Gp := lim
1
ln E [r(t)p ] ,
pt
km\qYLcbY E ju`pPaRTSTj_<Uc_`b\TY]Yzf?YLoB`dPa`bZrU&R'kmZi`b\ cbY-jhf?YLoB``bU`b\TY]SVZ[ju`dcbZrtqnV`dZiUR U&m`b\qYfqcdUzoYLjdj
v"
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<Uc~Y^&YLcbs v4nq`
η
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`b\qYRqU&pRTeiZrRqYLP&c
f?UkYLcQ<nqRToB`bZrU&R |.| PaRNS*`b\qYerZiRqY-Pacju`pP`bZ[jh`bZ[oPaeP ^&YLcdP&|&ZrRq| E [.]8v}\qYcpPaRTSqU&WXW"Pa`bcdZ
fqcdUVSVnTo8`pj`b\qYLU&cds'PafqfqerZrYLj`bU`b\qY*fqcdU&tTeiYLW 8v £TU&c"ZrRTju`pPaRToY]erYB`4nTj"PjbjbnqW"Y`d\TP``b\TY
cpPaRTSVUW fqcdUVoBYLjdj η Zrj
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^PaernqYLj}UR
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^PaernqYLj`pPa{&YLRtzs :Z`~Z[jfTcbU^YLSZrROmYuv%© #g0}\qYLU&cdYW a`d\TP``b\qYLcbY"YzZ[jh`djPaRPaRTP&eis6`bZ[o
<nqRToB`bZrU&R g(p) jbnTopm\]`b\TPa'` t→∞
p
p
C ;
,&
^
;
1
ln E [|r(t)|p ]
t
1
lim ln r(t)
t→∞ t
ln r(t) − tg 0 (0)
√
t
lim
t→∞
^;
8e =
g(p),
=
g 0 (0)
dist.
−→
P&eiW"Ujh`QjhnTcbYLeis ,
N (0, g 00 (0)).
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ZrjQWURqUa`dU&RqYZiRTocbY-P&jbZiRT|Tv}\TZrjlfqcbU^YLj}ZrRfTPacb`bZ[oBnTerP&cm`b\TPa`
ZrjQRqU&RV
p 7→ g(p)/p
cpPaRTSVUW*v%
R*oLP&jbYUaRqU&RVfqZrYLoYkmZ[jhY~oBURTjh`dPaR6`mfqcdUVoBYLjdjbYLj η q^PacdZiUnTj}^&YLcdjbGZrU&RT=jxgUaI(0)`b\TY_Pat?U^&Y
`b\TYU&cdYW YBVZ[ju`km\TZrop\szZrYe[S`b\qYjbP&WY
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YBVfqcdYLjdjbZiURU& Z[j^&YcdsZrR6`bcdZroLP`dY&?Y^&YLR*<U&c^&YLcbsjhZrW"fqeiYcdP&RTSVUW!fqcdUVoBY-jbjbYLj η vl
R`b\qY
<U&ererUkmZiRq|jbnqtTjbg(p)
nqtTjbYLo8`dZiURTj"kxY*jb\TP&eieQSVYLcbZr^&YoeiU6jhY-S'<UcbW YVfqcbY-jbjbZrU&RTj<U&c"`d\qY 0s6PafqnTRqU^
YBVf?U&RqYLR6`dj G PaRTS G ZrR`d\qY<cpPaW"YkxU&cd{4km\qYcdY`b\qY~RTU&Z[jhYerY^&YLe:Zrj}erUk~v
wYPjbjbnqW"Y<cbUWRqUk U&R]`b\NP`}`b\qYfqcdUVoBY-jbj = f (m(t)) km\qYLcbY f ZrjmP"jbW"U6U&`b\PaRTS
t?U&nqRTSqYLS"<nqRNo8`bZrU&RPaRTS m Z[jP&R
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^fqcdUVoBY-jbjkmZ`d\ZrRVTRqZi`bY-jhZrW4Pae?|&YRTYcpP`bUc
U&RPW"P&RqZi<U&e[S M kmZi`b\ZrR6^P&cbZ[PaR6`fTcbUtTPatqZrerZ`us π(dm) vwY4Pae[jbUPjbjbnqW"Y`d\TP` f (m) \TP&Qj
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!
!
!
&
p
Z
f (m)π(dm) = 0.
M
8
? .
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(r(t), ψ(t))
PaerYR6`}`bU y(t) = r(t) cos(ψ(t)) y (t) = r(t) sin(ψ(t))
UI V LQKbJXL
F
Xd
K
V
[
V OL X
2I V XL )L
9D
;
t
B
r(t)
kmZi`b\
ψt (t)
= r0 exp
PaRNS
Z
t
0
q(ψ(s), m(s))ds ,
= h(ψ(t), m(t)),
q(ψ, m) = q0 (ψ) + εq1 (ψ, m)
h(ψ, m) = h0 (ψ) + εh1 (ψ, m)
q1 (ψ, m) = −f (m) sin(ψ) cos(ψ),
q0 (ψ) = 0,
£qcdU&W ?v
ZrjxPP&cb{U^fqcdUVoBY-jbjU&R4`d\qYjbfTP&oY
km\qYLcbY S SVYRqU&`bY-j%`b\TY
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Bo ZrcpoBnqW<YcdYRNoBYlU&(ψ,:`b\qm)YnqRqZi`moBZrcdoeiYkmZ`d\]ZrRVTRqZi`bY-jhZrW4Pae?|&YLRqYcpPS`dU&×c ML = Q + h(ψ,
m)
kmZi`b\ZrRz^PacdZrP&R`WY-P&jbnqcdY p̄(ψ, m)dψπ(dm) km\qYLcbY p̄ oLPaRt?YU&tV`pPaZrRqYLSPjQ`d\qY"jhUeinV`dZiURUa
v moLoBUcdSVZrRq|`bU"`d\qY`b\qYLU&cdYW U& cpPanqYLe©«ª *V:`d\qY~erU&Rq|&M`dZiW"YtNYL\TP ^zZiUc}Ua oPaR
&
L p̄ = 0
t?Y~YBVfqcdYLjdjhY-S
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h1 (ψ, m) = −f (m) cos2 (ψ).
h0 (ψ) = −1,
1
1
∂
∂ψ
∗
B
G=
Z
q(ψ, m)p̄(ψ, m)dψπ(dm).
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M
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M
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tzs ZrRTjb{6s©«¬ #z2nqRTSVYLc PaRNS lcbRTU&e[SY`lPae®vx© qgnqRTSVYLc v
wY_jh\TP&eie0PjbjbnqW"Y<cbUWRqUk U&R`b\NP`
P&RTSkYerUzU&{
<UcQPaRYBVfTP&RTjhZrU&RUa kmZi`b\
cdYLjbfNY-o8`x`bU ε 1 v%}\qYjh`bcpP`bYL|&s<U&ererUkQjxεoeiU6jhYL1eis`d\qYU&RqYSVY^YerU&f?YLSZiROmYBuvm© VvwGYTcpjh`
SVZr^zZrSVY`d\qY~|&YLRqYcpP`dU&c L ZiR6`dU`d\qY_jbnqW L = L + εL kmZi`b\
S 1 ×M
1/2
$
#
_
0 0
L0 = Q + h0 (ψ)
∂
,
∂ψ
1
L1 = h1 (ψ, m)
∂
.
∂ψ
ljxjb\qUkmR4ZiR© VN`d\qYfqcbUtTPatTZierZ`us"SVYLRTjhZi`us
oP&R
tNYYBVfTP&RTSVY-S4P&j
mk \qYLcbY p̄ p̄ TP&RTS p jdP`dZrjh<s L p̄ = 0 Pap̄RTS L p̄ + L p̄ = 0 Lp̄ =p̄ p̄++Lεp̄p̄ +=ε0 rp̄vrvivx+£qU&...c
U&RNoBY_`d\qYYVfTPaRTjbZrU&RU& p̄ Zrj{zRqUkmR0DZi`oP&Rt?YnTjbYLSZiR }`dU|&Zr^&Y_`b\qYYBVfTP&RTjhZrU&RU& G
P`mUcdSVYLc 2 kmZ`d\cdYLjbf?YLo8`}`dU ε 0
B
G=
1
Z
S 1 ×M
0
2
∗
0 0
∗
0 1
∗
1 0
% ;
∗
0 2
1
∗
1 1
2
2
q0 p̄0 + ε(q1 p̄0 + q0 p̄1 ) + ε2 (q1 p̄1 + q0 p̄2 ) (ψ, m)dψπ(dm) + O(ε3 ).
.
z ZrRToBY h ZrjoBURTju`pPaR6` = −1 p̄ Z[jm`b\TYSVYLRTjhZi`us*Ua`b\TYnqRqZi<U&cdWSVZ[ju`dcbZrtqnV`dZiURU&R S × M v £Tnqch`d\qYc p̄ jdP`dZrjhTY-j L p̄ = −L p̄ = ∂ (h p̄ ) PaRTS Z[joU&RTjbYL6nqYLR6`bers
p̄ ≡ (2π)
|&Zr^&YLR]tzs
0
0
1
0
−1
∗
0 1
1
p̄1 (ψ, m) = −
1
2π
Z
∗
1 0
ψ
1 0
∞
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a0
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c
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1
0
0.5
0
0
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0
5
x
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1
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