IAOe?IIAEUeIA AEAAeAI?ss IAOE OE?A?IE

?INOEOOO IAOAIAOEEE

AeAEAOeUeEEE Ieaen?e TH??eiae/

OAeE 513.88

AeEOA?AIOe?AEUeI?

OA

INAAAeIAeEOA?AIOe?AEUeI?

IIA?AOI?E

IA

IANE?I*AIIIAEI??IEO

IIIAIAEAeAO

01.01.01 – iaoaiaoe/iee aiae?c

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

aeieoi?a o?ceei-iaoaiaoe/ieo iaoe

Ee?a 1999

Aeena?oaoe??th ? ?oeiien.

?iaioo aeeiiaii a ?inoeooo? iaoaiaoeee IAI Oe?a?ie

Iaoeiaee eiinoeueoaio:

aeaaeai?e IAI Oe?a?ie,

aeieoi? o?ceei-iaoaiaoe/ieo iaoe

AA?ACAINUeEEE TH??e Iaea?iae/,

?inoeooo? iaoaiaoeee IAI Oe?a?ie,

caa?aeoaa/ a?aeae?eo

Io?oe?ei? iiiiaioe:

/eai-ei?aniiiaeaio IAI Oe?a?ie, aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

IAO?EIA Aeieo?i sseiae/, ?inoeooo? iaoaiaoeee IAI Oe?a?ie,

caa?aeoaa/ a?aeae?eo

/eai-ei?aniiiaeaio IAI Oe?a?ie, aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

ssAe?AIEI Ieoaeei Eineiiae/, Ee?anueeee iaoe?iiaeueiee oi?aa?ne-

oao ?i. Oa?ana Oaa/aiea, i?ioani?

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

NIIEssIIA Ieaa Aai?a?eiae/, Iineianueeee aea?aeaaiee oi?aa?ne-

oao ?i. I.A.Eiiiiiniaa, ?in?y, i?ioani?

I?ia?aeia onoaiiaa: O?ceei-oaoi?/iee ?inoeooo iocueeeo oaiia?aoo? ?i.

A.?.A??e?ia IAI Oe?a?ie

Caoeno a?aeaoaeaoueny 1 /a?aiy 1999 ?ieo i 15 aiaeei?

ia can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 26.206.01

i?e ?inoeooo? iaoaiaoeee IAI Oe?a?ie ca aae?anith:

252601 Ee?a 4, INI, aoe. Oa?auaie?anueea, 3

C aeena?oaoe??th iiaeia iciaeiieoenue a a?ae?ioaoe?

?inoeoooo iaoaiaoeee IAI Oe?a?ie

Aaoi?aoa?ao ?ic?neaii 27 ea?oiy 1999 ?ieo.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee

aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

i?ioani?
Ia?aaa?c?a N.A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeueiinoue oaie

Oai??y aeeoa?aioe?aeueiex oa inaaaeiaeeoa?aioe?aeueiex iia?aoi??a, ui
ae?thoue o i?inoi?ax ooieoe?e iane?i/aiii? e?eueeino? ci?iiex, ? iaeiei
?ciaeaaaeeea?oex ?icae?e?a iane?i/aiiiaei??iiai aiae?co. C iaeiiai aieo,
neeueiith iioeaaoe??th ?icaeoeo oe??? aaeoc? ? canoinoaaiiy o
iaoaiaoe/i?eoa oai?aoe/i?e o?ceoe?. C ae?oaiai aieo, oeae ?icaeoie
aeiaaa? canoinoaaiiy ?aeae oa iaoiae?a aaiiao???, oiiieia??, oai???
i??e, oai??? iia?aoi??a, ui aea? yne?aaee i?eeeaae ?ioa?iieyoe?? ??ciex
aaeocae iaoaiaoeee. Oae? iia?aoi?e ?icaeyaeaeenue o ?iaioax aaaaoueix
aaoi??a (N.Aeueaaaa??i, .I.Aa?acainueeee, O.I.Aa?ac?i, I.A?oee,
?.I.Aaeueoaiae, E.A?inn, TH.A.Eiiae?aoue?a, I.?ueieia?, I.A.Niieyiia,
I.TH.O?aii?eia oa

iaoe?ia oice/ia eioa?aoo?a). A?eueoa /anoeia ?nioth/eo ?ia?o i?enay/aia

aeiaaeeo e?i?eiiai oaciaiai i?inoi?o. Aeiaaeie iae?i?eiex oaciaex

i?inoi??a a?ae?a?a? ia iaio aaaeeeao ?ieue o iaoaiaoe/iiio aiae?c? oa

iaoaiaoe/i?e o?ceoe?

(iai?eeeaae, o eaaioia?e oai??? iiey oa noaoenoe/i?e o?ceoe?). Ia?ax?ae

aei oeueiai aeiaaeeo i?ecaiaeeoue aei aaaaoueix o?oaeiiu?a o ca’yceo ?c
neeaaeiith

aaiiao?e/iith no?oeoo?ith iane?i/aiiiaei??iex iiiaiaeae?a, ye?

i?e?iaeiuei c’yaeythoueny o canoinoaaiiyx, oa a?aenooi?no? caaaeueii?

oai??? oaeex iiiaiaeae?a. Oiio aeine?aeaeaiiy eiaeiiai i?eeeaaeo oaeiai

oeio aeiaaa? ?icaeoeo iniaeeai? oaxi?ee.

Oe?eaaei oa aaaeeeaei eeanii aeeoa?aioe?aeueiex iia?aoi??a ia
iane?i/aiiiaei??iex iiiaiaeaeax ? iia?aoi?e Ae???xea, anioe?eiaai? c
a?aan?anueeeie i??aie ia iane?i/aiiex aeiaooeax eiiiaeoiex ?eiaiiaex
iiiaiaeae?a. Oae? iia?aoi?e c’yaeythoueny o ca’yceo c a?ao/anoeie
iiaeaeyie noaoenoe/ii? iaxai?ee. ??ci? aniaeoe aeine?aeaeaiiy oeex
iia?aoi??a oa a?aeiia?aeiex i?aa?oi ?icaeyaeaeenue aaaaoueia aaoi?aie
(Ae.No?oe, A.Caaa?eeinueeee, N.Aeueaaaa??i, TH.A.Eiiae?aoue?a,
I.?ueieia? oa ?io?).

Caoaaaeeii, ui o aeiaaeeo e?i?eiiai ni?iiaiai i?inoi?o
noixanoe/ia aeeiai?ea, ui a?aeiia?aea? a?aan?anuee?e i???, iiaea aooe
iiaoaeiaaia ??cieie niiniaaie (iaoiae oi?i Ae???xe?, aaciina?aaeiy
iia?aoi?ia eiino?oeoe?y ia?e?anueei? i?aa?oie ?c caaeaiei aaia?aoi?ii oa
i?aex?ae, ui aeei?enoiao? noixanoe/i? aeeoa?aioe?aeuei? ??aiyiiy). O
aeiaaeeo eiiiaeoiex ni?iiaeo i?inoi??aia?o? aeaa i?aexiaee aeia?a
?icaeiai?. Cie?aia, aaciina?aaeiy eiino?oeoe?y i?aa?oie, ui aeei?enoiao?
yaiee aeaeyae aaia?aoi?a, aea?oueny o ?iaioax Ae.No?oea ?
A.Caaa?eeinueeiai. I?ioa i?yia iiaoaeiaa a?aeiia?aeiiai i?ioeano ca
aeiiiiiaith oai??e noixanoe/iex aeeoa?aioe?aeueieo ??aiyiue (“aeaoaa?iaa
aeeiai?ea”), ui aea? e?auee eiio?ieue eiai aeanoeainoae, cono??/a?oueny
c na?eicieie o?oaeiiuaie oa aeiaaa? ?icaeoeo a?aeiia?aeii? aiae?oe/ii?
oaxi?ee. O iaei?ino?oiio aeiaaeeo iane?i/aiiiaei??iiai oi?o oaeee
i?aex?ae aoa ?aae?ciaaiee ?.Oiee? oa Ae.No?oeii. Ia?ax?ae aei aeiaaeeo
iao?ea?aeueiex eiiiaeoiex iiiaiaeae?a i?ecaiaeeoue aei aaaaoueix
aeiaeaoeiaex o?oaeiiu?a. Oiio, iacaaaeath/e ia aaeeeo e?euee?noue ?iaio
o oeueiio iai?yieo, oeae aeiaaeie caeeoeany iaaeine?aeaeaiiei.
Cacia/eii, ui a?aenooi?noue no?oeoo?e aeaaeeiai a?eue?a?oiaa aai
aaiaxiaa iiiaiaeaeo ?iaeoue iaiiaeeeaei aeei?enoaiiy ?nioth/i? oai???
noixanoe/iex aeeoa?aioe?aeueiex ??aiyiue (NAe?) ia iane?i/aiiiaei??iex
iiiaiaeaeax. Oiio ia?oei e?ieii o iai?yieo ?aae?caoe?? i?ia?aie
“noixanoe/iiai

eaaiooaaiiy” ia i?iaeaeo-iiiaiaeaeax ? ?icaeoie iaiax?aeii?

aeeoa?aioe?aeueii-aaiiao?e/ii?

oaxi?ee oa a?aeiia?aeii? oai??? NAe?.

Aaaeeeaei iaoiaeii aeine?aeaeaiiy aeeoa?aioe?aeueiex iia?aoi??a ?
oae caaia neiaieueia /eneaiiy. Oea icia/a?, ui aeeoa?aioe?aeueiee
iia?aoi? ?icaeyaea?oueny ye ooieoe?y a?ae “aeaiaioa?iex” iia?aoi??a
iiiaeaiiy oa aeeoa?aioe?thaaiiy. ?icaeoie oeueiai /eneaiiy aaaea aei
oai??? inaaaeiaeeoa?aioe?aeueiex iia?aoi??a (IAeI). Oaeei /eiii, eean
IAeI ?, c iaeiiai aieo, i?e?iaei?i ?icoe?aiiyi eeano aeeoa?aioe?aeueiex
iia?aoi??a, oa c ?ioiai aieo, iniaenoi a?ae?a?a? aaaeeeao ?ieue o
canoinoaaiiyx. Aeiaaeie ne?i/aiiiaei??iex oaciaex i?inoi??a aea/aany
aaaaoueia aaoi?aie, cie?aia o eeane/iex ?iaioax A.I.Ianeiaa oa
E.Xuei?iaiaea?a. Iiooaeiei caniaii aea/aiiy aneiioioe/ii? iiaaae?iee
IAeI ? iaoiae eaiii?/iiai iia?aoi?a Ianeiaa, yeee aea? iiaeeea?noue
anoaiiaeoe ca’ycie aneiioioe/iex ?ica’yce?a inaaaeiaeeoa?aioe?aeueiex
??aiyiue c aaiiao???th eaa?aiaeiaex iiiaiaeae?a, ui iienothoueny
?ica’yceaie a?aeiia?aeiex aai?eueoiiiaex nenoai. Iiaeaeueoee ?ica’ycie
oeueiai iaoiaeo aea? iiaeeea?noue iiaoaeiae aneiioioe/iex ?ica’yce?a
inaaaeiaeeoa?aioe?aeueiex ??aiyiue O?aae?iaa?a, ye? a naith /a?ao
aeei?enoiaothoueny o eiino?oeoe?? IAeI c iae?i?eieie

oaciaeie i?inoi?aie.

?aae?caoe?y oaei? i?ia?aie o iane?i/aiiiaei??iiio aeiaaeeo ?
aeoaea aaaeeeaith oa o?oaeiith caaea/ath. Aeaye? eiie?aoi? eeane
iane?i/aiiiaei??iex IAeI ?icaeyaeaeenue o ?iaioax O.I.Aa?aciia,
I.Aioeea, I.A.Niieyiiaa, I.TH.X?aiiieiaa, aea ?icaeiaia o oeex ?iaioax
oaxi?ea nooo?ai aeei?enoiao? e?i?eio no?oeoo?o oaciaiai i?inoi?o. A?eueo
oiai, ni?iae iiaoaeoaaoe iane?i/aiiiaei??iee eaiii?/iee iia?aoi? Ianeiaa
cono??/athoueny c na?eicieie o?oaeiiuaie, ui iia’ycai? c aaiiao???th
iane?i/aiiiaei??iex eaa?aiaeiaex oa neiieaeoe/iex iiiaiaeae?a. Oaeei
/eiii, iaa?oue iiaoaeiaa eiie?aoiex i?eeeaae?a iane?i/aiiiaei??iex
iiiaiaeae?a oa aeaaa? neiaie?a, aeey yeex oey i?ia?aia ? ?aae?noe/iith,
? aeoaea aaaeeeaei caaaeaiiyi.

Oaeei /eiii, aaeeeee ?ioa?an aei oai??? aeeoa?aioe?aeueiex oa
inaaaeiaeeoa?aioe?aeueiex iia?aoi??a c iae?i?eieie iane?i/aiiiaei??ieie
oaciaeie i?inoi?aie, oa ?? ca’ycie ?c /eneaiieie no/anieie o?ce/ieie
oai??yie, oaeeie ye oai??y noixanoe/iiai eaaiooaaiiy eaaioiaex oa
eeane/iex a?ao/anoex iiaeaeae, oai??y aneiioioe/iiai eaaiooaaiiy,
?iaeoue oaio aeena?oaoe?? aeoaea aeooaeueiith.

Ca’ycie c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie.

?iaioo aeeiiaii ca?aeii c iaoeiaeie oaiaie oa ieaiaie Iinoeoooo
iaoaiaoeee IAI Oe?a?ie. (? aea?aeaaii? ?a?no?aoe?? – 0198U001995).

Iaoa ?iaioe.

Iaoith ?iaioe ? aea/aiiy aeayeex eeania aeeoa?aioe?aeueiex oa
inaaaeiaeeoa?aioe?aeueiex iia?aoi?ia ia ianeii/aiiiaeii?iex iiiaiaeaeax,
a naia iia?aoi??a Ae???xea a?anianueeex ii? ia ianeii/aiiex aeiaooeax
eiiiaeoiex iiiaiaeaeia oa inaaaeiaeeoa?aioe?aeueieo iia?aoi??a c
neiaieaie ia a?eueaa?oiaeo neiieaeoe/ieo i?inoi?ao ? iiiaiaeaeao,
cie?aia, iiiaiaeaeax ?ica‘yce?a iane?i/aiiiaei??ieo aaiieueoiiiaex
nenoai.

Iaoiaeeea aeineiaeaeaiiy.

O ?iaioi aeei?enoiaothoueny iaoiaee ooieoeiiiaeueiiai aiaeico,
oai?i?

noixanoe/iex aeeoa?aioe?aeueiex ?iaiyiue oa aeeoa?aioe?aeueii?
aaiiao?i?.

Iaoeiaa iiaecia iaea?aeaiex ?acoeueoaoia.

Iniiaieie ?acoeueoaoaie, ui aecia/athoue iaoeiao iiaecio ?iaioe
oa aeiinyoueny ia caoeno, ? ianooii? ?acoeueoaoe:

1. Iiaoaeiaaii eiiai?i?nii i?aaenoaaeaiiy i?aa?oi, ui ii?iaeaeaii
iia?aoi?aie

Ae???xea oe?ieiai eeano a?aan?anueeex ii? ia ianeii/aiiex aeiaooeax

eiiiaeoiex iiiaiaeaeia, oa aeiaaaeaii ?o oaea?ia?noue.

Ca aeiiiiiaith oeeo i?aaenoaaeaiiue aeiaaaeaii nooo?ao
naiini?yaeaiinoue

oaeex iia?aoi?ia Ae???xea o ?aci

ianeii/aiiiai ?aiao aca?iiaeie oa io?eiaii iien aeno?aiaeueiex

a?aan?anueeex ii? ia ianeii/aiiex aeiaooeax

eiiiaeoiex iiiaiaeaeia o oa?iiiax a?aiaee/iinoi aiaeiiaiaeii?
noixanoe/ii?

aeeiaiiee.

2. Aeiaaaeaii inioaaiiy oa ?aeeiinoue neeueieo

?ica’yce?a ianeii/aiiex nenoai noixanoe/iex

aeeoa?aioe?aeueiex ?iaiyiue ia eiiiaeoiex iiiaiaeaeax. Aeiaaaeaii

eaaciiiaa?iaioiinoue ?iciiaeieia ?ica’yce?a oaeex nenoai ia a?oiao E?.

3. ?ic?iaeaii eiino?oeoeith IAeI c neiaieaie ia aieueaa?oiaiio
oaciaiioi?inoi?i, ui aeithoue o L_2 ii aeaaeeie ii?i ia aieueaa?oiaiio
i?inoi?i.

Aeiaaaeaii iniiaii aneiioioe/ii oi?ioee neiaieueiiai /eneaiiy oaeex

iia?aoi?ia, cie?aia oi?ioee eiiooaoe?? oa eiiiiceoe??.

4. Iiaoaeiaaii aneiioioe/ii ?ica’ycee aiaeiiaiaeiex
inaaaeiaeeoa?aioe?aeueiex?iaiyiue O?aaeiiaa?a.

5. Iiaoaeiaaii aeaaa?o IAeI c neiaieaie ia i?eionoeiex neiieaeoe/iex
iiiaiaeaeax, cie?aia, iiiaiaeaeax ?ica’yce?a iane?i/aiiiaei??ieo

aaiieueoiiiaex nenoai.

6. Iia?aoi? Ae???xea aaonniai? ii?e ia aieueaa?oiaiio i?inoi?i
?aaeiciaaiiye aeaiaio aeaaa?e IAeI; aeiaaaeaii aiaeiiaiaeii aneiioioe/ii
oi?ioee

neiaieueiiai /eneaiiy.

7. ?icaeiooi iiai aaiiao?e/ii iaoiaee ?ica’ycoaaiiy aeayeex eeane/iex

aaiieueoiiiaex nenoai, ui ocaaaeueiththoue noaio Einoaioa-Aaeea?a.

Oai?aoe/ia cia/aiiy.

?acoeueoaoe oa iaoiaee, ui ?ic?iaeaii o aeena?oaoei?, aeoeaii
aeei?enoiaothoueny

o iiaeaeueoex aeineiaeaeaiiyx a iaeanoi oai??? aeeoa?aioe?aeueiex oa
inaaaeiaeeoa?aioe?aeueiex iia?aoi?ia ia ianeii/aiiiaeii?iex iiiaiaeaeax
oa ?x canoinoaaiue. Aiie iiaeooue aooe ei?enieie o iaoeiaie ?iaioi
Iinoeoooo iaoaiaoeee IAI Oe?a?ie, Ee?anueeiai aea?aeaaiiai oi?aa?neoaoo,
O?ceei-oaoi?/iiai ?inoeoooo iecueeeo

oaiia?aoo?,

Iineianueeiai iinoeoooo aeaeo?iiiee oa iaoaiaoeee, oi?aa?neoao?a i?no

Aiiia (Iiia//eia), Oi?aiea (Aaeeeia?eoaiiy), Iaini (O?aioeiy).

Iniaenoee aianie caeiaoaa/a.

Iniiaii ?acoeueoaoe aeena?oaoei? io?eiaii aaoi?ii naiinoieii. Aeeaae
eiaeiiai

ic niiaaaoi?ia /ioei aeaeieaii a aeena?oaoe?eiie ?iaioi.

Ai?iaaoeiy ?acoeueoaoia.

Iaoa?iaee aeena?oaoei? aeiiiaiaeaeenue oa iaaiai?thaaeenue ia 5-io
?a?iiaenueeiio

Neiiicioii ?c noixanoe/iiai aiaeico (Aiii, Iiia//eia, 1994),
Iiaeia?iaeiie

eiioa?aioei? “Iaoiaee noixanoe/iiai aiaeico o noixanoe/iie iaxaiioei,

oiiainax oa aiieiai?” (Aieaoaeueae, Iiia//eia, 1994), Iiaeia?iaeiie
eiioa?aioei? ?c noixanoe/iiai aiaeico oa eiai canoinoaaiue (Ia?naeue,
O?aioeiy, 1995), Iiaeia?iaeiiio naiano?i “Ooieoeiiiaeueia iioaa?oaaiiy”
(Ia?eae, O?aioeiy, 1997), Iiaeia?iaeiie eiioa?aioei? “Noixanoe/ii Aeii
98” (Ithixai, Iiia//eia, 1998), Iiaeia?iaeiie eiioa?aioei? c
iaoaiaoe/ii? oiceee (Einaaii, Ii?ooaaeiy, 1998), Iiaeia?iaeiie
eiioa?aioei? c iioaioeiaeueiiai aiaeico (Ooiin, 1998), naiiia?ax
Iinoeoooo iaoaiaoeee IAI Oe?a?ie, O?ceei-oaoi?/iiai ?inoeoooo iecueeeo
oaiia?aoo? oa iaoaiaoe/iex oaeoeueoaoia

oiiaa?neoaoia Aiiia, Aixoia, Aieaoaeueaea, Iiooiiaaia, Oi?aiea oa Xaeea.

Ioaeieaoei?.

Ca iaoa?iaeaie aeena?oaoei? iioaeieiaaii 16 iaoeiaex ?iaio.

Ia’?i oa no?oeoo?a.

Aeena?oaoe?y aeeeaaeaia ia 266 noi??ieao oa neeaaea?oueny c anooio,

3 ?icae?e?a, aeniiae?a oa nieneo oeeoiaaii? e?oa?aoo?e, ui i?noeoue 118

aeaea?ae.

INIIAIEE CIINO ?IAIOE

?icae?e 1 i?enay/aiee aea/aiith iia?aoi??a Ae???oea, ui iia’ycai? c
a?aan?anueeeie i??aie ia (iane?i/aieo) i?iaeaeo-iiiaiaeaeao. O
I?ae?icae?e? 1.1 ie iaaiai?th?ii iniiai? aaiiao?e/i? no?oeoo?e oa
?icaeaa?ii noioanoe/ia /eneaiiy ia (iane?i/aiieo) i?iaeaeo-iiiaiaeaeao.
O Ioieo? 1.1.1 ie aaiaeeii iniiai? ia’?eoe ia iicia/aiiy.

Iaoae M — eiiiaeoiee ??iai?a iiiaiaeae. Ie iicia/a?ii /a?ac d_Xu

iio?aeio ooieoe?? u ocaeiaae aaeoi?iiai iiey X. A?aeiia?aeiee a?aae??io,
aecia/aiee ca aeiiiiiaith ??iaiiai? no?oeoo?e (cdot ,cdot ),
iicia/a?oueny /a?ac nabla u. A?aeiia?aeia a?aenoaiue ia M iicia/a?oueny
/a?ac rho. Ie ?icaeyaea?ii oe?eo a?aoeo bf Z^d,,dge 1, oa aecia/a?ii
i?ino?? bf Mequiv M^bf Z^d (yeee aoaeaii iaceaaoe i?iaeaeo-iiiaiaeaeii).

O Ioieo? 1.1.2 ie aoaeo?ii aeioe/ia ?icoa?oaaiiy Tbf M oa aecia/a?ii ca
eiai aeiiiiiaith aeeoa?aioe?aeuei? oa iao?e/i? no?oeoo?e ia bf M.
Caoaaaeeii, ui i?ino?? bf M ia? no?oeoo?o aaiaoiaiai iiiaiaeaeo
(iiaeaeeth yeiai ? i?ino?? iaiaaeaieo iine?aeiaiinoae ?c ??aiii??iith
ii?iith). Oey ii?ia ia ? aeeoa?aioe?eiaiith, ui ia aea?

iiaeeeaino? aeei?enoiaoaaoe oeth no?oeoo?o iiiaiaeaea aeey iiaoaeiae
noioanoe/iiai aiae?co. C iaoith iiaeieaiiy oe??? ia?aoeiaee, ie aaiaeeii
aiaeia ??iaiiai? no?oeoo?e ia bf M. Ia aa?enoe/iiio ??ai? aeioe/ia
?icoa?oaaiiy Tbf M ? bf Z^d-nooiaiai TM. I?e?iaeii ?icaeyiooe aeayea
a?eueaa?oiaa i?ae?icoa?oaaiiy oeueiai i?inoi?o. Cao?eno?ii aaaiao
iine?aeiai?noue pin l_1 aeiaeaoieo /enae oa aecia/eii a?eueaa?o?a
i?ino??

bf T_p,x=left Xin T_xbf M:sum_kin bf Z^dp_kleft|

X_kright| ^2=sf E,

aea iicia/a? aeai?no?noue cal K oa cal K^prime.

?icaeyiaii aeiaaeie, eiee aeai?no?noue cal K oa cal K^prime ?
ii?iaeaeaiith neaey?iei aeiaooeii a?eueaa?oiaiai i?inoi?o cal K _0,

cal Ksubset cal K_0subset cal K^prime .

I?iiiceoe?y 1.4.

I?aa?oia bf T^xi ,eta(t) caaeiaieueiy? ianooiio ioe?ieo:

left| bf T^xi ,eta (t)v(x)right| _cal K^prime le

e^tc,T^xi (t)left| vright| _cal K^prime (x),

aea vin C(bf Mrightarrow cal

K^prime ) oa c oaea, ui ccdot (h,h)_cal Kge

2(b(x)h,h)_cal K+Tr_cal H(B(x)hcdot ,B(x)hcdot )_cal K

aeey aeia?eueieo xin bf M oa hin cal K.

Caoaaaeaiiy.

Iaoae ?nio? c_0 oaea, ui

ccdot (h,h)_cal K_0ge 2(b(x)h,h)_cal K_0+Tr_cal H

(B(x)hcdot ,B(x)hcdot )_cal K_0

aeey aeia?eueieo xin bf M oa hin cal K. Oiae? ie ia?ii ioe?ieo

sf Eleft| eta

(t)right| _cal K_0le e^c_0tsf Eleft| eta _0(0)right|

_cal K_0^2.

Aiane?aeie oeueiai bf T^xi ,eta (t) ae?? o i?inoi?? C(bf Mrightarrow cal
K_0), oa aeey vin C(bf Mrightarrow cal K_0) ie ia?ii

left| bf T^xi ,eta (t)v(x)right| _cal K_0le

e^tc_0,T^xi (t)left| vright| _cal K_0(x).

I?iiiceoe?y 1.5.

Iaoae ain C^2(bf M_prightarrow Tbf M_p),

Ain C^3(bf M%_prightarrow HS(cal H,Tbf M_p)),

oa, aeey aeia?eueiiai yin cal K, b(cdot )yin C^1(bf M_prightarrow cal
K),

B(cdot )yin C^1(bf M%_prightarrow HS(cal H,cal K))

??aiii??ii uiaei y. Oiae? a?aeia?aaeaiiy

bf M_pbf ni xlongmapsto (xi _x(t),eta _x(t))in bf M_ptimescal K

? aeeoa?aioe?eiaiei o na?aaeiuei-eaaae?aoe/iiio nain?.

Iane?aeie.

I?e oiiaao iiia?aaeiuei? i?iiiceoe??, i?aa?oia

bf T^xi ,eta (t)

caeeoa? ?iaa??aioiei i?ino?? C^1(bf M_prightarrowcal K^prime ).
Aiaeia?/ii iiaeia aeiaanoe, ui aiia caeeoa? ?iaa??aioieie an? i?inoi?e
C^s(bf M_prightarrow cal K^prime ),,sle k, yeui eiao?oe??ioe a,b,B
(a?aeiia?aeii A) ? k (a?aeiia?aeii k+1) ?ac?a aeeoa?aioe?eiaieie.

O I?ae?icae?e? 1.3 ie ?icaeyaea?ii aeiaaeie, eiee M=G ia? no?oeoo?o
a?oie E?.

Ie aaiaeeii nenoaio e?ai-?iaa??aioiex NAe? o oi?i? No?aoiiiae/a ia G:

dxi _k(t)=L_xi _k(t)[a_k(xi (t))dt+circ dw_k(t)],;kin bf Z^d,

labeliqi2

aea w_k — iacaeaaei? a?ia??anuee? i?ioeane o cal G (a?aeiia?aei?e
aeaaa?? E?), a_k — caaeai? a?aeia?aaeaiiy G^bf Z ^drightarrow cal G,
oa L_g iicia/a? e?aee cnoa

ia aeaiaio gin G. Ie aea/a?ii oeth nenoaio ca aeiiiiiaith iaoiae?a,
?icaeyiooeo aeua. Ie i?eionea?ii, ui:

left( iright) ;

sup_kin bf Z^dsup_xin G^bf Z^d

left| a_k(x)right| _cal GF. labelif011

Iaoae H ? ooieoe??th (neiaieii) ia cal H_-^2. Aecia/eii
inaaaeiaeeoa?ai-oe?aeueiee iia?aoi? (IAeI) H(x,ihbar D_mu ) a L_2(cal
H_-,mu ) ca aeiiiiiaith oi?ioee

H(x,ihbar D_mu )varphi (x)=F_pto x^-1F_yto p[H(fracx+y2,hbar

p)varphi (y)], labelif12

hbar in (0,1]. Acaaae? oey oi?ioea ia? o?eueee aa?enoe/iee nain. Aeae?
ie aeiaaaeaii, ui oeae ae?ac ? ei?aeoiei aeey eiie?aoieo eean?a neiaie?a
ia eiie?aoieo iaeanoyo aecia/aiiy.

O Ioieo? 2.2.2 ie ?icaeaa?ii neiaieueia /eneaiiy aeey IAeI c neiaieaie,
ye? ? ia?aoai?aiiyie Oo?’? iaiaaeaieo eiiieaeniicia/ieo i??. Ie
aeei?enoiao?ii

iiiyooy ?ioaa?aeo ooieoe??, yea i?eeia? cia/aiiy o aaiaoiaiio i?inoi??,
a?aeiinii i??e iaiaaeaii? aa??aoe??. Nii/aoeo ie aea?ii ?ioa aecia/aiiy
IAeI. Aeey Hin cal F_infty equiv cal F_infty (cal H_-^2,bf C^1)
(i?ino?? ooieoe?e, ye? ? ia?aoai?aiiyie Oo?’? eiiieaeniicia/ieo
iaiaaeaieo aeaaeeeo i?? ia cal H_+^2),

H(x,p)=int e^?(+)dmu _H(x^prime,p^prime ),

ie aecia/a?ii

widehatH=int W_fracx^prime 2U_hbar p^prime

W_fracx^prime 2dmu _H(x^prime ,p^prime ), labelin1

aea W_yphi :=e^phi (x).

Oaa?aeaeaiiy 2.1.

Oi?ioea (refin1) aecia/a? iia?aoi? widehatH a L_2(cal H_-,mu ) oa

widehatHf(x)=F_pto x^-1F_yto p[H(fracx+y2,hbar p)f(y)].

labelin3

Oaa?aeaeaiiy 2.2.

Iaoae H,Gin cal F_infty , H(x,p)=int e^?+?dtheta _H(x^prime ,p^prime ).

Oiae?:

widehatHwidehatG=widehatN_1,

widehatGwidehatH=widehatN_2, aea N_1,N_2in cal F_infty

oa

N_1(x,p)=int e^?+?G(x+hbar p^prime

/2,p-hbar x^prime /2)dtheta _H(x^prime ,p^prime ), labelif22

N_2(x,p)=int e^?+?G(x-hbar p^prime

/2,p+hbar x^prime /2)dtheta _H(x^prime ,p^prime ). labelif23

Iane?aeie 2.1. Oi?ioea eiiooaoe??. Iaxae G,Hin cal F_infty .

Oiae?

frac ihbar [widehatH,widehatG]=widehatH,G+O(hbar ^2),

labelif24

aea cdot ,cdot iicia/a? noaiaea?oio aeoaeeo Ioanniia ia cal H^2,

yea aecia/a?oueny ca aeiiiiiaith ae?aco

left H,Gright =leftlangle fracpartial Hpartial x,frac

partial Gpartial prightrangle -leftlangle fracpartial G

partial x,fracpartial Hpartial prightrangle.

Caeeoie ia? oi?io hbar ^2widehatR(hbar ) c R(hbar

)in cal F_infty ,;left| R(hbar )right| _cal F_lambda

? iaiaaeaiith ??aiii??ii a?aeiinii hbar aeey aoaeue-yeiai lambda in bf
R

_+.

Ie aea?ii oaeiae ?ioo ?ioa?i?aoaoe?th oi?ioee eiiiiceoe??. Aeey oeueiai
iai iio??ai?

IAeI ?ioiai oeio. sseui Hin cal F_infty , iia?aoi? H(x,ihbar partial
/partial x) iiaea aooe aecia/aiee ca aeiiiiiaith oi?ioee:

H(x,ihbar partial /partial x)varphi (x)=int^sim e^-frac ihbar

H(x,p)varphi (y)dpdy labelif25

(I.A.Niieyiia, I.TH.X?aii?eia). Neiaie int^sim iicia/a? ii?iae?ciaaiee

?ioaa?ae, aecia/aiee ca aeiiiiiaith oi?ioee Ia?naaaey. Ie iiaeaii oaia?

ia?aienaoe oi?ioee (refif22), (refif23) aeey neiaie?a N_1,N_2 ianooiiei
/eiii:

N_1(x,p)=H(x+fracihbar 2partial /partial p,p-fracihbar 2

partial /partial x)G(x,p), labelif27

N_2(x,p)=H(x-fracihbar 2partial /partial p,p+fracihbar 2

part?al /partial x)G(x,p). labelif28

O ne?i/aiiiaei??iiio aeiaaeeo oi?ioee eiiiiceoe?? iaaaaeai? o oaeiio
aeaeyae?

a ?iaioax I.A.Ea?anueiaa oa A.I.Ianeiaa.

O Ioieo? 2.2.3 ?icaeaa?oueny neiaieueia /eneaiiy iia?aoi??a c neiaieaie
a?eueo caaaeueiiai eeano cal E (yeee ii?iaeaeaiee neiaieaie, ui
?icaeyaeaeenue aeua, oa neiaieaie iie?iii?aeueiiai ?inoo). Oae? neiaiee
ii?iaeaeothoue, acaaae? eaaeo/e, iaiaiaaeai? IAeI. Eiaeia ooieoe?y H
eeano cal E ? ia?aoai?aiiyi Oo?’? ?iciiae?eo xi _H (yeee, acaaae?
eaaeo/e, ia ? i??ith). Ie aaiaeeii i?e?iaeio iaeanoue aecia/aiiy bf D oa
aeiaiaeeii ianooiia:

Oaa?aeaeaiiy 2.3.

I?e Hin cal E iia?aoi? widehatH (aecia/aiee ca aeiiiiiaith oi?ioee
(refif12)) ? ei?aeoii-aecia/aiei ia iaeano? bf D oa caeeoa? ??
?iaa??aioiith.

Aeey fin bf D ie ia?ii:

widehatHf=int W_fracx^prime 2U_hbar p^prime W_frac%

x^prime 2fdxi _H(x^prime ,p^prime ). labelif18

Oaa?aeaeaiiy 2.4.

Iaxae H,Gin cal E. Oiae?:

widehatHwidehatG=widehatN%

_1,;widehatGwidehatH=widehatN_2, aea N_1,N_2in cal E oa

iathoue i?noea oi?ioee (refif22), (refif23) (c theta _Hequiv xi _H).

sse iane?aeie ie io?eio?ii ocaaaeueiaiiy oi?ioee eiiooaoe?? (refif24)

ia oeae aeiaaeie.

O I?ae?icae?e? 2.3 ?icaeyaeathoueny IAeI c neiaieaie, ye? inoeeeththoue
i?e hbar to 0. O aeaeyae? oaeex iia?aoi??a io?eiothoueny ?ica’ycee
inaaaeiaeeoa?aioe?aeueiex ??aiyiue O?aae?iaa?a. O Ioieo? 2.3.1
aeiaiaeeoueny ei?aeoi?noue aecia/aiiy IAeI c inoeeethth/eie neiaieaie oa
aea/athoueny ?o aeanoeaino?.

Oaa?aeaeaiiy 2.5.

Iaxae G(x,p)=e^frac ihbar P(x,p), aea Pin cal P^2(cal H_-^2,bf R^1)
(i?ino?? ae?eniex iaia?a?aiex iie?iii?a ae?oaiai ii?yaeeo ia cal
H_-^2),

P(x,p)=++2.

I?eionoeii, ui

Cequiv det(I-K^2+K_1K_2)ne 0.

Oiae? widehatG ? iaiaaeaiei o L_2(cal H_-,mu ) iia?aoi?ii, ii?ia yeiai
aei??aith? C^-1. A?i caeeoa? ?iaa??aioiith iaeanoue bf D.

Oaa?aeaeaiiy 2.6.

Iaxae G caaeiaieueiy? oiiae iiia?aaeiuei? I?iiiceoe?? oa Hin cal E.
Oiae? aeey neiaiea N_1 eiiiiceoe?? widehatHwidehatG ia? i?noea oi?ioea
(refif27).

?icaeyiaii oaia? iia?aoi? widehatE_hbar c neiaieii

E_hbar (x,p)=e^frac ihbar S(x,p),;Sin cal F_infty . labelif47

I/aaeaeii, ui E_hbar in cal F_infty, oae ui widehatE_hbar ?
iaiaaeaiei iia?aoi?ii o L_2(cal H_-,mu ). Ianooiiee ?acoeueoao
eiio?ieth? eiai caeaaei?noue a?ae hbar .

Oaa?aeaeaiiy 2.7.

Iaxae left| Sright| _cal F aeinoaoiuei iaea. Oiae? a?aeia?aaeaiiy

(0,1]ni hbar mapsto widehatE_hbar in cal L(L_2(cal H_-,mu ))

labelif48

(aea cal L(H) iicia/a? i?ino?? e?i?eiex iaiaaeaiex iia?aoi??a a

cal H) ? iaiaaeaiei.

Ioieo 2.3.2 i?enay/aiee iiaoaeia? aneiioioe/iex ?ica’yce?a
inaaaeiaeeoa?aioe?aeueiex ??aiyiue O?aae?iaa?a. ?icaeyiaii ??aiyiiy

lbrack frac partial partial t-frac ihbar widehatH]T_t=O(hbar

), labelif62

T_0=?d,

aea H=H_t oa i?eeia? ae?eni? cia/aiiy. Aoaeaii ooeaoe ?ica’ycie o
aeaeyae? IAeI c

inoeeeth/ei neiaieii U_t. Aiane?aeie oi?ioee eiiiiceoe?? oea ??aiyiiy
caiaeeoueny

aei ??aiyiiy a?aeiinii U_t:

lbrack frac partial partial t-frac ihbar

H(x+?frac partial partial p,p-

?frac partial partial x)]U_t(x,p)=O(hbar),

U_0(x,p)=1.

Iaoiae ?ica’ycaiiy ??aiyiue oaeiai oeio (yeee ? ocaaaeueiaiiyi
eeane/iiai AEA-iaoiaeo) ?icaeaa?oueny o I?ae?icae?e? 2.4. Ca?ac ie
aaiaeeii eaa?aiae?a iiiaiaeae

Lambda _t o cal H_-^4, anioe?eiaaiee c aai?eueoii?aiii H(x+q,p-y), oa
aeine?aeaeo?ii eiai aeanoeaino?. Ie aaiaeeii a?aeiia?aeia ??aiyiiy
ia?aiino oa aeiaiaeeii ianooii? ?acoeueoaoe.

Oai?aia 2.4.

??aiyiiy (refif62) ia? ?ica’ycie T_t=T_t(H), yeee i?e aeinoaoiuei
iaeiio t

ia? aeaeyae T_t=hatU_t, aea

U_t(x,p)=e^?S_t(x,p)/hbar f_t(x,p), labelif048

aea S_t ? ii?iaeaeoth/ith ooieoe??th iiiaiaeaeo Lambda _t, oa f_t ?
?ica’yceii a?aeiia?aeiiai ??aiyiiy ia?aiino.

Oai?aia 2.5.

??aiyiiy (refif62) ia? aeiaaeueiee ?ica’ycie T_tequiv T_t(H) ianooiiiai
aeaeyaeo:

T_t=hatU_t_n-1,t_n…….hatU_t_1,t_2, labelig11

aea 0=t_1<......mu (dz)+frac hbar 2int e^mu (dz)+O(hbar ^2).

Caeeoie ioe?ith?oueny o oiiieia?? cal S.

Oeae ?acoeueoao oa eiai ocaaaeueiaiiy ia aeiaaeie i??e mu, yea
caeaaeeoue a?ae hbar, aeei?enoiao?oueny o aeiaaaeaii? ianooiii?
oai?aie.

Oai?aia 2.7. Iaxae S(x)=frac 12+W(x), aea W,phi in cal F(H_-,%

bf R^1),;Bin cal L(H_-,cal H_+). Iaxae

Hin cal D^2 (:=%

cal P^2+cal F_infty , ae?eniicia/i?). Oiae?:

beginarrayc

e^-frac ihbar S(x)H(x,ihbar partial /partial x)e^frac ihbar

S(x)phi (x)=

=H(x,S^prime (x))phi (x)+

+hbar left[ ?Tr(fracpartial ^2Hpartial p^2(x,S^prime

(x))S^prime prime (x))phi (x)+fracpartial Hpartial p%

(x,S^prime (x))phi ^prime (x)right] +

+O(hbar ^2),

endarray

aea caeeoie ioe?ith?oueny o ii?i? cal F(H_-,bf R^1).

Iaoa ianooiia iaoa (bf Ioieo 2.4.3) — aeaoe eiino?oeoe?th aneiioioe/iex

?ica’yce?a caaea/? Eio? oeio

beginarrayc

left( frac partial partial t-frac ihbar H(x,ihbar partial

/partial x)right) u(x,t)=Oleft( hbar ^2right) ,

u(x,0)=e^frac ihbar S_0(x)phi _0(x),

endarray

labelie1

aea S_0in cal D^2(cal H_-,bf R^1),;phi _0in cal F%

_infty cal (H_-,bf R^1), i?eiaeii? i?e aeinoaoiuei iaeiio t.

A?aeiia?aeii aei nxaie iaoiaeo AEA, ie aaiaeeii ??aiyiiy
Aai?eueoiia-sseia?

dotS_t(x)=H(x,S_t^prime (x)) labelie5

oa ??aiyiiy ia?aiino

dotphi_t(x)=-Tr(fracpartial ^2Hpartial p^2(x,S_t^prime

(x))S_t^prime prime (x))phi _t(x)+?fracpartial Hpartial p%

(x,S_t^prime (x))phi _t^prime (x). labelie6

oa aeine?aeaeo?ii aeanoeaino? ?x ?ica’yce?a. I?ney oeueiai ie aeiaiaeeii
ianooiiee ?acoeueoao.

Oai?aia 2.9. ?ica’ycie u_t caaea/? Eio? (refie1) ?nio? i?e t
aeinoaoiuei iaeiio ? ia? aeaeyae

u_t(x)=e^frac ihbar S_t(x)phi _t(x), labelif119

aea S_t oa phi _t ? ?ica’yceaie a?aeiia?aeii ??aiyiue Aai?eueoiia-sseia?
oa ia?aiino.

O I?ae?icae?e? 2.5 ie aeei?enoiao?ii io?eiai? ?acoeueoaoe aeey iiaoaeiae
aneiioioe/iiai eaaiooaaiiy eeano iane?i/aiiiaei??iex neiieaeoe/iex
iiiaiaeae?a (ye? ie iaceaa?ii cal F — iiiaiaeaeaie). Iiaeaeeth aeey oeex

iiiaiaeae?a ? i?ino?? cal H^2, a a?aeia?aaeaiiy neeaeee ii?iaeaeothoueny
aai?eueoii?aiaie eeano cal D^2. Ie aeei?enoiao?ii caaaeueio
eiino?oeoe?th aneiioioe/iiai eaaiooaaiiy: IAeI c neiaieaie ia cal F —
iiiaiaeae? aoaeothoueny ca aeiiiiiaith aneiioioe/iex ?ica’yce?a ??aiyiue
O?aae?iaa?a, ye? a?aeiia?aeathoue

aai?eueoii?aiai a?aeia?aaeaiue neeaeee. O Ioieo? 2.5.1 ie aecia/a?ii
a?oio i?eionoeiex eaiii?/iex ia?aoai?aiue a?eueaa?oiaiai oaciaiai
i?inoi?o oa aoaeo?ii ?? i?iaeoeaia i?aaenoaaeaiiy.

Iaxae cal K ? ae?eniei naia?aaaeueiei a?eueaa?oiaei i?inoi?ii, oa

cal K_+subset cal Ksubset K_-

eiai a?eueaa?oi-oi?aeo?anueea iniauaiiy. ?icaeyiaii i?ino?? Diff_cal
F(cal K) aeeoaiii?o?ci?a

psi :cal %

Krightarrow K oaeex, ui psi (x)=x+Bx+l(x), aea B:cal K%

_-rightarrow cal K_+ — aeayeee iaiaaeaiee e?i?eiee iia?aoi? oa

aeaiaio lin cal F_infty (cal K_-,cal K_bf C). Ie

aeiaiaeeii ianooiiee ?acoeueoao.

Oaa?aeaeaiiy 2.9. Diff_cal F(cal K) ? i?aea?oiith a?oie aeeoaiii?o?ci?a
i?inoi?o cal K.

Ie aecia/a?ii iiiaeeio Symp_cal F(cal H^2) ye neiieaeoe/io i?aea?oio
a?oie Diff_cal F(cal H^2), oiaoi i?ino?? neiieaeoiii?o?ci?a gamma
i?inoi?o cal H^2 oaeex, ui gamma in Diff_cal F(cal H^2).

Iicia/eii /a?ac ;gamma _H(t,tau ),;t,tau in [0,T]; ia?aoai?aiiy i?inoi?o
cal H^2, ii?iaeaeaia cnoaaie ocaeiaae ?ioaa?aeueiex e?eaex
aai?eueoiiiai? nenoaie, ui anioe?eiaaia c aai?eueoii?aiii H. Ie iiaeaii
ioa?aeoa?ecoaaoe i?ino??

Symp_cal F(cal H^2) ianooiiei /eiii.

Oaa?aeaeaiiy 2.10.

beginenumerate

item Aeey aoaeue-yeex t,tau in [0,T];;;gamma _H(t,tau )in Symp_cal F(%

cal H^2).

item Aeey aoaeue-yeiai gamma in Symp_cal F(cal H^2) ?nio? aai?eueoii?ai
H(t)in cal D^2 oaeee, ui gamma =gamma _H(0,1).

endenumerate

Iaxae oaia? gamma in Symp_cal F(cal H^2),;gamma =gamma _H(0,1)

aeey aeayeiai H(t)in cal D^2. Cao?eno?ii i??o mu oa eioeeee alpha , ui
caaeiaieueiythoue oiiae I?ae?icae?eo 2.2, oa aaaaeaii iia?aoi?e

T_gamma =T_t(H)_lceil t=1 labelipnl1

o i?inoi?? L_2(cal H_-,mu ), aea T_t(H) ? ?ica’yceii ??aiyiiy
O?aae?iaa?a left( refif62right) .

Oai?aia 2.10.A?aeia?aaeaiiy gamma longmapsto T_gamma labelipnl2 ?
i?iaeoeaiei aneiioioe/iei i?aaenoaaeaiiyi a?oie Symp_cal F(cal H^2).

Ioieo 2.5.2 i?enay/aiee aecia/aiith oa aeine?aeaeaiith i?eionoeiex

neiieaeoe/iex iiiaiaeae?a. Ie ?icaeyaea?ii iiiaiaeae cal M c iiaeaeeth

cal K_- oa aoeanii cal U=(U_xi ,phi _xi)_xi in bf A ,

oa aea?ii ianooiia

Aecia/aiiy 2.3-2.4.

Aoean cal U iaceaa?oueny cal F -aoeanii, yeui eiai a?aeia?aaeaiiy
neeaeee iaeaaeaoue Diff_cal F(cal K). Aeaa cal F — aoeane iaceaathoueny
aea?aaeaioieie, yeui ?x ia’?aeiaiiy ? cal F — aoeanii. Iiiaeeia an?x
aea?aaeaioiex cal F — aoean?a ia cal M iaceaa?oueny cal F-no?oeoo?ith.
Iiiaiaeae cal M c o?eniaaiith cal F -no?oeoo?ith iaceaa?oueny cal F —
iiiaiaeaeii. cal F-no?oeoo?a iiiaiaeaeo cal E iaceaa?oueny
neiieaeoe/iith cal F-no?oeoo?ith, yeui a?aeiia?aei? a?aeia?aaeaiiy

neeaeee iaeaaeaoue Symp_cal F(cal H^2). O oeueiio aeiaaeeo cal E
iaceaa?oueny neiieaeoe/iei cal F-iiiaiaeaeii.

Aeae? ie ?icaeyaea?ii i?eeeaaee cal F- iiiaiaeae?a, ye? c’yaeythoueny ye
iiiaiaeaee ?ica’yce?a iane?i/aiiiaei??iex aai?eueoiiiaex nenoai.

Ioieo 2.5.3 i?enay/aiee iiaoaeia? ?iaa??aioiex IAeI c neiaieaie,
aecia/aieie ia neiieaeoe/iiio cal F-iiiaiaeae? Xi . Ie aeiaiaeeii
ianooiiee ?acoeueoao.

Oai?aia 2.11.

Aeey aeia?eueii? i??e mu oa eioeeeeo alpha ia cal H_-, ye?
caaeiaieueiythoue oiiae I?ae?icae?eo 2.2, ?nio? aeiooie cal S(Xi )
e?i?eiex i?inoi??a iaae Xi (aecia/aiee mod;hbar ), oa a?aeia?aaeaiiy
Hlongmapsto widehatH ioanniiiai? aeaaa?e

cal E(Xi ,bf C^1) a aeaaa?o e?i?eiex iia?aoi??a, ye? ae?thoue o
i?inoi?? ia?aoei?a oeueiai aeiooea, oae?, ui ia? i?noea eiiooaoe?eia
oi?ioea

frac ihbar [hatPhi,widehatG]=widehatPhi ,G+O(hbar ).

?acoeueoaoe ?icae?eo 2 iioae?eiaai? o ?iaioao [4, 5, 6, 7, 10, 11, 14].
Caoaaaeeii, ui ?iaioe [4, 5, 6, 7, 11] ? naiino?eieie, a ?iaioe [10,
14]. — ni?eueieie c i?ioani?ii N.Aeueaaaa??i. I?e oeueiio iinoaiiaea
caaea/ oa iaaiai?aiiy ca’yce?a io?eiaieo ?acoeueoao?a c ?ioeie
iaeanoyie iaeaaeeoue iaii ni?aaaoi?ai. Aeiaaaeaiiy iniiaieo
?acoeueoao?a i?iaaaeai? aaoi?ii naiino?eii.

O ?icae?e? 3 ie aea/a?ii eiie?aoiee i?eeeaae ?icaeyiooi? aeua
eiino?oeoe??. Ie ?icaeyaea?ii IAeI a L_2 ii aaonnia?e i???. O oeueiio
eiie?aoiiio aeiaaeeo ie aeine?aeaeo?ii ?x ca’ycie ?c iia?aoi?aie
Ae???xea. sseui ie ia?ii cal H_+- eaac??iaa??aioio i??o mu ia cal H_-,
ie iiaeaii ?icaeyiooe eeane/io oi?io Ae???xea cal E_mu , anioe?eiaaio c
oe??th i??ith, oa a?aeiia?aeiee aaia?aoi?

widetildeH_mu o L_2(cal H_-,mu ) (yeee iaceaa?oueny iia?aoi?ii
Ae???xea). Oeae iia?aoi? ia iiaea aooe aecia/aiee ca aeiiiiiaith
eiino?oeoe??, ?icaeyiooi? aeua. Iacaaaeath/e ia oea, ie iieaco?ii, ui
widetildeH_mu (i?eiaeii? o aeiaaeeo aaonniai? i??e mu ) iiaea aooe
aeeth/aiee a aeaaa?o IAeI.

O I?ae?icae?e? 3.1 ie ?icaeyaea?ii eiino?oeoe?? I?ae?icae?e?a 2.2 oa 2.3
o aeiaaeeo,

eiee mu ? aaonniaith i??ith, oa aoaeo?ii ?? iaaaeeeo iiaeeo?eaoe?th. Ie

?icaeyaea?ii aaonniao i??o eta c ei??aeyoe?eiei iia?aoi?ii hbar B^-1,

aea B ? iaiaaeaiei aeiaeaoiei iai?ioiei neiao?e/iei iia?aoi?ii a cal
H, yeee caeeoa? ?iaa??aioiei i?ino?? cal H_+. O Oaa?aeaeaii? 3.1 ie
i?aenoiiao?ii ?acoeueoaoe iiia?aaei?x ?icae?e?a, aaeaioiaai? ia oeae
aeiaaeie. Aeae? ie ?icaeyaea?ii oaeiae aeiaaeie, eiee B ? iaiaiaaeaiei
(aeiaeaoiei neiao?e/iei) iia?aoi?ii. Ie aoaeo?ii a?aeiia?aeii iniauaiee
i?ino?? oa iaaiai?th?ii ocaaaeueiaiiy ?acoeueoao?a iiia?aaei?x ?icae?e?a
ia oeae aeiaaeie. O I?ae?icae?e? 3.2 ie ?icaeyaea?ii iia?aoi? Ae???xea
widetildeH_eta (Ioieo 3.2.1) oa eiai cao?aiiy widetildeH_eta ,V
(Ioieo 3.2.2) iaiaaeaiei cieco iioaioe?aeii Vin cal E(cal H_-,bf R^1).
Aa??noe/iee aai?eueoii?ai H_eta ,V, ui a?aeiia?aea? widetildeH_eta
,V, ia? ianooiiee aeaeyae:

H_eta ,V(x,p)=H_eta (x,p)+V(x)

=-frac 12(+)+V(x).

Iaxae g_V(t) ? cnoaii ocaeiaae ?ioaa?aeueiex e?eaex a?aeiia?aeii?
aai?eueoiiiai?

nenoaie. Ie aeiaiaeeii, ui H_eta ,V,Gin cal E, g_V^*Gin cal E,

gamma ^*H_eta ,V=H_eta ,V+h_eta ,V, aea h_eta ,V=h_eta ,V(gamma )in cal
E. Ianooiiee ?acoeueoao iieaco?, ui iia?aoi? widetildeH_eta ,V iiaea
aooe aeeth/aiee o aeaaa?o IAeI.

Oai?aia 3.2.

Ia a?aeiia?aei?e iaeano? aecia/aiiy, ie ia?ii:

1) I?e Gin cal E

frac ihbar [widetildeH_eta ,V,widehatG]=widehatH_eta

,V,G+O(hbar ^2). labelif65

2) I?e Gin cal D^2

(widetildeH_eta ,V+widehath_eta ,V)T_t(G)=T_t(G)widetilde%

H_eta ,V+O(hbar ), labelif67

3) I?e Gin cal E

widehat(g_V(t)^*G)exp(?twidetildeH_eta ,V)=exp(?twidetilde

H_eta ,V)widehatG+O(hbar ^2), labelif66

aea exp(?twidetildeH_eta ,V) — iaeiiia?aiao?e/ia oi?oa?ia a?oia, ui
a?aeiia?aea? widetildeH_eta ,V.

O Ioieo? 3.2.3 ie ei?ioei iaaiai?th?ii aeiaaeie, eiee B ? iaiaiaaeaiei
neiao?e/iei aeiaeaoiei iia?aoi?ii a cal H, oa aeiaiaeeii aiaeia oai?aie
3.2.

sse i?eeeaae aeei?enoaiiy ?icaeyiooi? aeua oai???, ie ?icaeyaea?ii
aeiaaeie

?aeyoea?nonueeiai aiciiiiai iiey. O Oai?ai? 3.3 ie i?aenoiiao?ii
?acoeueoaoe iiia?aaeiueiai ?icae?eo o oe?e neooaoe??. Oey oai?aia
iieaco?, ui iia?aoi? widetildeH_eta ,V_alpha ,varepsilon iiaea aooe
aeeth/aiee o aeaaa?o IAeI c neiaieaie eeano cal E((W_2^2-s(bf
R^n,q_s^-1))^2,bf C^1), aea W_2^2-s(bf R^n,q_s^-1) — i?ino?? Niaie?aa,
aecia/aiee ca aeiiiiiaith aeayei? aaaiai? iine?aeiaiino? q.

sse aoei iieacaii aeua, iiaoaeiaa aneiioioe/ieo ?ica’yce?a
inaaaeiaeeoa?aioe?aeueieo

??aiyiiue aeiaaa? ai?iiy ?ica’ycoaaoe eeane/i? ??aiyiy c neiaieaie
a?aeiia?aeieo IAeI o ?ie? ooieoe?e Aai?eueoiia. O I?ae?icae?e? 3.3 ie
iaaiaeeii aeae?eueea iaoiae?a aeine?aeaeaiiy oaeeo ??aiyiiue.

Aeia?a a?aeiiei ? iaoiae aeine?aeaeaiiy aai?eueoiiiaeo nenoai ia
i?inoi?ao c e?i?eieie

aeoaeeaie Ioanniia, ui caiaeeoue iiaoaeiao ?ica’yce?a aei caaea/?
oaeoi?ecaoe?? o a?aeiia?aei?e a?oi? E? (noaia Einoaioa-Aaeea?a), aai,
oi/i?oa, ?? aaiiao?e/iiai aa??aioo, ?ic?iaeaiiai ?aeiaiii oa
Naiaiiaei-Oyiue-Oaiuenueeei. O Ioieo? 3.3.1 ie i?iiiio?ii iiae?aiee
i?aeo?ae aeey aeiaaeeo iae?i?eieo aeoaeie. Iiaoaeiaa ?ioaa?aeueieo
e?eaeo caiaeeoueny aei caaea/? oaeoi?ecaoe?? a neiieaeoe/iiio a?oii?ae?
cal E, yeee a?ae?a?a? ?ieue oaciaiai i?inoi?o iaae ioanniiiaei
iiiaiaeaeii cal N. Oai??y oaeeo ia’?eo?a aoea ?ic?iaeaia Ea?anueiaei,
Ianeiaei oa Aaeiooaiii. O ?iaioao Ea?anueiaa oa Ianeiaa a?oiiaa
no?oeoo?a, a?aenooiy o aeiaaeeo iae?i?eieo aeoaeie Ioanniia,
cai?ith?oueny eieaeueiith no?oeoo?ith aeaaeeiai a?oii?aea cal E. A
ieie? cal xi in N i?ino?? cal E iiaea aooe i?aaenoaaeaiee ye ei-aeioe/ia
?icoa?oaaiiy T^*G c eaiii?/iith neieaeoe/iith no?oeoo?ith, aea G ?
ieieii ioey a bf R^n, ? ianooiieie no?oeoo?aie: aeiaooeii
xstackrelxi*y, ui caeaaeeoue a?ae xiin N, iia?aoi?ii ?iaa?n?? x_xi^-1,
xstackrelxi*x_xi^-1=e, oa »aeniiiaioe?aeueiei a?aeia?aaeaiiyi»
exp_xi:T_xi^*cal Nrightarrow G.

Iaoae cal N ? aeaaeeei ae?eniei iiiaiaeaeii (??iaiiaei aai
a?eueaa?oiaei) c aeoaeeith Ioanniia

left f,gright (cal xi )=,

aea aai?eueoii?a iia?aoi?

Psi :T^*cal Nlongrightarrow Tcal N

acaaae? iae?i?eii caeaaeeoue a?ae cal xi in N. I?eionoeii, ui iiiaiaeae
cal N iiaea aooe i?aaenoaaeaiee ye aeiaooie cal %N=N_+times cal N_-
aeaio ioanniiiaeo i?aeiiiaiaeae?a. Oiae? aai?eueoii?a iia?aoi? ia?
aei/io oi?io

Psi =

begintabularll

Psi _+ & Psi _+-

Psi _-+ & Psi _-%

endtabular

labeligeom2

aea Psi _+ oa Psi _- caeaaeaoue eeoa a?ae a?aeiia?aeieo cal %xi _+ oa xi
__ a oi/oe? xi in cal N oa aecia/athoue aai?eueoiiia? iia?aoi?e Psi
_+:T^*cal N_+longrightarrow T% cal N_+ ? Psi _-:T^*cal N_-longrightarrow
Tcal N_-. Cai?iaaaeeii ia cal N iiao aeoaeeo Ioanniia left^.,^.right _0,
aecia/aio aai?eueoiiiaei iia?aoi?ii

Psi _0=

begintabularlc

Psi _+ & 0

multicolumn1c0 & multicolumn1l-Psi _-

endtabular

?icaeyiaii oaciaee i?ino?? cal E iaae cal N. A?i ?iceeaaea?oueny ia
aeiaooie cal E=E_+times cal E_- oaciaeo i?inoi??a iaae cal N%_+ oa cal
N_-. Neiieaeoe/ia no?oeoo?a ia cal E_pm ? caoaeaiiyi neiieaeoe/ii?
no?oeoo?e ia cal E. Oaciaee i?ino?? %cal E_0, ui a?aeiia?aea? aeoaeoe?
left cdot ,cdot right _0, oaeiae ia? oi?io cal E_0=cal E_+times cal E_-,
? neiieaeoe/ia oi?ia ia oeueiio i?inoi?? aei??aith? oi?iai ia
iiiaeieeao.

A ieie? xi in cal N ioioiaeieii cal E=T^*G ?c %Gtimes cal N ca
aeiiiiiaith a?aeia?aaeaiiy l. Oiae? cal E_+=G_+times

cal N_+ oa cal E__=G__times cal N__ (G=G_+times

G__).

Ie io?eio?ii ianooiiee ?acoeueoao.

Oaa?aeaeaiiy 3.5.

Iioaa?aeueia e?eaa nenoaie dotxi %=Psi _0dH,;Hin I(cal N), ?c ii/aoeii a
xi _0 ia? aeaeyae xi (t)=V_x(t)^-1(xi _0)equiv V_y(t)(xi _0) labelgeom6

aea xin G_+ oa yin G_- ? ?ica’yceii caaea/? oaeoi?ecaoe??

exp _xi _0tdH(xi _0)=x(t)stackrelxi _0*y(t). labelgeom7

O Ioieo? 3.3.2 ie ia?aiineii ia iane?i/aiiiaei??iee aeiaaeie a?oiiaee
iaoiae ?ioaa?oaaiiy iane?i/aiiiai eaioethaa Oiaee, yeee ocaaaeueith?
i?aeo?ae Ieueoaiaoeueeiai, Ia?aeiiiaa oa Aoaeiaia, Aaeeaoa. Eaioetha
Oiaee ?aae?co?oueny ye aai?eueoiiiaa nenoaia

ia i?a?o? iane?i/aiiiaei??ii? a?oie E?, ? a?aeiia?aei? ?ioaa?aeuei?
e?ea? io?eiothoueny o oa?i?iao eii?e?aeiaii? ae?? ?? i?aea?oie (Oai?aia
3.4).

?acoeueoaoe ?icae?eo 3 iioae?eiaai? o ?iaioao citeDP,D5,D6,D4,AD2.

?iaioa citeDP ? ni?eueiith c A.A.Iiaeeiec?iei. I?e oeueiio iinoaiiaea
caaea/? oa ?ic?iaea iaoiaeo ?? ?ica’ycaiiy iaeaaeeoue aaoi?o.
Aeiaaaeaiiy aeayeeo aeiiii?aeieo

?acoeueoao?a i?iaaaeai? ni?aaaoi?aie ni?eueii.

AENIIAEE

Aeena?oaoeiy i?enay/aia aeoaea iiioey?iie iaeanoi no/aniiai
iaoaiaoe/iiai aiaeico — oai?i? aeeoa?aioe?aeueiex oa
inaaaeiaeeoa?aioe?aeueiex iia?aoi?ia ia ianeii/aiii-aeii?iex
iiiaiaeaeax. Oaei iia?aoi?e i?e?iaeii c’yaeythoueny o aaaaoueix iaeanoyx
iaoaiaoeee oa aiaeia?athoue aaaeeeao ?ieue o ?iciiiaiioiex caaea/ax
iaoaiaoe/ii? oa oai?aoe/ii? oiceee, oaeex ye noaoenoe/ia iaxaiiea oa
eaaioiaa oai?iy iiey. Ia aiaeiiio aiae aeiaaeeo ianeii/aiiiaeii?iex
eiiieiex oaciaex i?inoi?ia, ?icaeoie oei?? oai?i? ia iiiaiaeaeax
cono?i/a? aaaaoi ia?aoeiae, iia’ycaiex ic neeaaeiith aaiiao?i?th oaeex
iiiaiaeaeia oa aiaenooiinoth oaei? ?x oai?i?, yea ixiieth? ani aaaeeeai
i?eeeaaee.

Oaeei /eiii, aea/aiiy eiaeiiai i?eeeaaeo aeiaaa? ?icaeoeo niaoeiaeueii?
oaxiiee.

O aeenna?oaoe?? io?eiai? ianooii? iia? ?acoeueoaoe:

1) Iiaoaeiaaii eiiai?i?nii i?aaenoaaeaiiy i?aa?oi, ui ii?iaeaeaii
iia?aoi?aie Ae???xea a?aan?anueeex ii? ia ianeii/aiiex aeiaooeax
eiiiaeoiex iiiaiaeaeia. Ca aeiiiiiaith oeeo i?aaenoaaeaiiue aeiaaaeaii
nooo?ao naiini?yaeaiiinoue oaeex iia?aoi?ia Ae???xea o ?aci
ianeii/aiiiai ?aiao aca?iiaeie oa io?eiaii iien aeno?aiaeueiex
a?aan?anueeex ii? ia ianeii/aiiex aeiaooeax eiiiaeoiex iiiaiaeaeia o
oa?iiiax a?aiaee/iinoi aiaeiiaiaeii? noixanoe/ii? aeeiaiiee.

2) Aeiaaaeaii inioaaiiy oa ?aeeiinoue neeueieo ?ica’yce?a ianeii/aiiex
nenoai noixanoe/iex aeeoa?aioe?aeueiex ?iaiyiue ia eiiiaeoiex
iiiaiaeaeax. Aeiaaaeaii eaaciiiaa?iaioiinoue ?iciiaeieia ?ica’yce?a
oaeex nenoai ia a?oiao E?.

3) ?ic?iaeaii eiino?oeoeith IAeI c neiaieaie ia aieueaa?oiaiio oaciaiio

i?inoi?i, ui aeithoue a L_2 ca aeaaeeith ii?ith ia aieueaa?oiaiio
i?inoi?i.

Aeiaaaeaii iniiaii aneiioioe/ii oi?ioee neiaieueiiai /eneaiiy oaeex

iia?aoi?ia, cie?aia, oi?ioee eiiooaoe??, eiiiiceoe?? oa ae?? ia
oaeaeeiinoeeethth/o

aeniiiaioo. Iiaoaeiaaii aneiioioe/ii ?ica’ycee aiaeiiaiaeiex
inaaaeiaeeoa?ai-

oe?aeueiex ?iaiyiue O?aaeiiaa?a.

4) Iiaoaeiaaii aeaaa?o IAeI c neiaieaie ia i?eionoeiex neiieaeoe/iex

iiiaiaeaeax, cie?aia, iiiaiaeaeax ?ica’yce?a iane?i/aiiiaei??ieo

aaiieueoiiiaex nenoai.

5) Iia?aoi? Ae???xea aaonniai? ii?e ia aieueaa?oiaiio i?inoi?i
?aaeiciaaii

ye aeaiaio aeaaa?e IAeI; aeiaaaeaii aiaeiiaiaeii aneiioioe/ii oi?ioee

neiaieueiiai /eneaiiy.

?acoeueoaoe aeena?oaoei? ? iaeieie c ia?oex ?aaoey?iex ?acoeueoaoia o
oeueiio iai?yieo. Iaea? oa iaoiaee, ui ?icaeiaii o aeena?oaoei?, aoee
aeei?enoaii o aaaaoueix iioex ?iaioax (aeea. iai?eeeaae, iineeaiiy
[8,97] aeena?oaoe??). Ana oea aeicaiey? c?iaeoe aeniiaie, ui
aeena?oaoeiy ? iiaei aaaeeeaei e?ieii o oai?i? ianeii/aiiiaeii?iex
aeeoa?aioe?aeueiex oa inaaaeiaeeoa?aioe?aeueiex iia?aoi?ia, a ??
?acoeueoaoe iathoue /eneaiii ca’ycee c aaaaoueia iioeie ?acoeueoaoaie o
oeie aaeoci i iiaeooue aooe aeei?enoaii o aaaaoueix aeineiaeaeaiiyx, ui
caeieniththoueny a Oe?a?ii oa ca ?? iaaeaie.

Iniiai? iieiaeaiiy aeena?oaoe?? iioae?eiaai? o ianooiieo ?iaioao:

1. Aeaeaoeeee A.TH., Iiaeeiecei A.A. A?oiiiaie iiaeoiae e
eioaa?e?iaaieth aaneiia/iie oeaii/ee Oiaeu // Oe?. iao. aeo?i. — 1988. –
40, N 4. — N. 445-447.

2. Daletskii A. Adler scheme analogue for non-linear Poisson brackets //
Non-linear and turbulent processes in physics, ed. V.G.Bar’yakhtar,
N.S.Erokhin, V.E.Zakharov, A.G.Sitenko, V.M.Chernousenko. — Kiev: Nauk.
Dumka, 1988. — C. 48-51.

3. Aeaeaoeeee A.TH. Caaea/a oaeoi?ecaoeee a neiieaeoe/aneii a?oiiieaea

e aaieeueoiiiau nenoaiu n iaeeiaeiuie neiaeaie Ioanniia // Aeiee. AI
NNN?. — 1989. — 40, N 2. — N. 389-393.

4. Aeaeaoeeee A.TH. I eaiiie/aneii iia?aoi?a Ianeiaa ia eaa?aiaeaauo
iiiaiia?aceyo n aeeueaa?oiaie iiaeaeueth // Iao. caiaoee. — 1990. — 48,
N 6. — N. — 51-60.

5. Aeaeaoeeee A.TH. Aaneiia/iiia?iua o?aaiaiey O?aaeeiaa?a e
i?aaenoaaeaiea a?oiiu neiieaeoe/aneeo i?aia?aciaaiee aeeueaa?oiaa
oaciaiai i?ino?ainoaa // Ooieoeeii. aiaeec e aai i?ee. — 1992. — 26, N
1. — N. — 74-75.

6. Daletskii A. Representations of canonical commutation relations
defined by Gaussian measure and Gaussian cocycle // Random Operators and
Stochastic Equations. — 1994. — No.1. — C. 87-95.

7. Daletskii A. Quasi-classical approximations for a one class of
infinite dimensional pseudodifferential equations // Sel. Math. Sov. —
1994. — 13, No.2. — C. 114-125.

8. Albeverio S., Daletskii A., Kondratiev Yu. A stochastic differential
equation approach to lattice spin models with values in compact Lie
groups // Infinite Dimensional Harmonic Analysis, ed. H.Hezer, T.
Hirai, T»ubingen, 1996. — C. 1-16.

9. Albeverio S., Daletskii A., Kondratiev Yu. A stochastic
differential equation approach to some lattice models on compact Lie
groups // Random Operators and Stochastic Equations. — 1996. — 4, No.3.
— C. 227-237.

10. Albeverio S., Daletskii A. Asymptotic quantization for solution
manifolds of some infinite dimensional Hamiltonian systems // J.
Geometry and Physics. — 1996. — 19. — C. 31-46.

11. Daletskii A. Asymptotic expansions for a class of infinite
dimensional pseudodifferential operators // Methods of Func. Analysis
and Topology. — 1997. — 3, No.1. — C. 51-62.

12. Albeverio S., Daletskii A., Kondratiev Yu. Infinite systems of
stochastic differential equations and some lattice models on compact
Riemannian manifolds // Ukr. Math. J. — 1997. — 49. — C. 326-337.

13. Albeverio S., Daletskii A., Kondratiev Yu. Some examples of
Dirichlet operators associated with the actions of infinite dimensional
Lie groups // Methods of Func. Analysis and Topology. — 1997. — No.3. —
C. 1-15.

14. Albeverio S., Daletskii A. Algebras of pseudodifferential operators
in L_2 given by smooth measures on Hilbert spaces // Mathematische
Nachrichten. — 1998. — 192. — C. 5-22.

15. Albeverio S., Daletskii A., Kondratiev Yu. Stochastic evolution on
product manifolds // Generalized Functions, Operator Theory and
Dynamical Systems, ed. I. Antoniou and G. Lumer, Pitman Research Notes,
Math. Series. — 1998. — 399 — C. 1-35.

16. Albeverio S., Daletskii A., Kondratiev Yu. Stochastic analysis on
(infinite-dimensional) product manifolds // Stochastic dynamics, ed.
H.Crauel and M. Gundlach. — Springer, New York, 1999. — C. 1-25.

AIIOAOe??

Aeaeaoeueeee I.TH. Aeeoa?aioe?aeuei? oa inaaaeiaeeoa?aioe?aeuei?
iia?aoi?e ia

iane?i/aiiiaei??ieo iiiaiaeaeao. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy aeieoi?a o?ceei-iaoaiaoe/ieo
iaoe ca niaoe?aeuei?noth 01.01.01 — iaoaiaoe/iee aiae?c. — ?inoeooo
iaoaiaoeee IAI Oe?a?ie, Ee?a, 1999.

Aeena?oaoe?th i?enay/aii ?icaeoeo oai??? aeeoa?aioe?aeueieo oa
inaaaeiaeeoa?aioe?aeueieo iia?aoi??a ia iane?i/aiiiaei??ieo iiiaiaeaeao.
Aieiaia oaaaa i?eae?ey?oueny ?icaeoeo eiia??i?niiai i?aeoiaeo aei
ae?ioe/ieo aeeoa?aioe?aeueieo iia?aoi??a

ia iane?i/aiieo i?iaeaeo—iiiaiaeaeao oa aneiioioe/iei iaoiaeai
aeine?aeaeaiiy

inaaaeiaeeoa?aioe?aeueieo iia?aoi??a c neiaieaie ia a?eueaa?oiaiio
i?inoi?? (aai iiiaiaeae?). Cie?aia, iiaoaeiaai? oaea?ia? i?aa?oie, ui
anioe?eiaai? c iia?aoi?aie Ae???oea a?aan?anueeeo i?? ia
i?iaeaeo—iiiaiaeaeao. Iiaoaeiaaii neiaieueia /eneaiiy aeey aaaeeeaiai
eeano inaaaeiaeeoa?aioe?aeueieo iia?aoi??a c a?eueaa?oiaei oaciaei
i?inoi?ii oa eaac?eeane/i? ?ica’ycee a?aeiia?aeieo ??aiyiue O?aae?iaa?a.
Iiaoaeiaaii ooieoi? aneiioioe/iiai eaaiooaaiiy eeano
iane?i/aiiiaei??ieo neiieaeoe/ieo iiiaiaeae?a.

Eeth/ia? neiaa: aeeoa?aioe?eiaia i??a, iia?aoi? Ae???oea, neiaieueia
/eneaiiy, oaciaee i?ino??, iane?i/aiiiaei??iee iiiaiaeae.

Aeaeaoeeee A.TH. Aeeooa?aioeeaeueiua e inaaaeiaeeooa?aioeeaeueiua
iia?aoi?u ia

aaneiia/iiia?iuo iiiaiia?aceyo. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie aeieoi?a oeceei-iaoaiaoe/aneeo

iaoe ii niaoeeaeueiinoe 01.01.01 — iaoaiaoe/aneee aiaeec. – Einoeooo
iaoaiaoeee IAI Oe?aeiu, Eeaa, 1999.

Aeenna?oaoeey iinayuaia ?acaeoeth oai?ee aeeooa?aioeeaeueiuo e
inaaaeiaeeo-oa?aioeeaeueiuo iia?aoi?ia ia aaneiia/iiia?iuo
iiiaiia?aceyo. Iniiaiia aieiaiea oaeaeaii ?acaeoeth aa?iyoiinoiiai
iiaeoiaea e yeeeioe/aneei iia?aoi?ai ia aaneiia/iuo i?iaeaeo-
iiiaiia?aceyo e aneiioioe/aneei iaoiaeai enneaaeiaaiey
inaaaeiaeeooa?aioeeaeueiuo iia?aoi?ia n neiaieaie ia aeeueaa?oiaii
i?ino?ainoaa (eee iiiaiia?acee). A /anoiinoe, iino?iaiu oaeea?iau
iieoa?oiiu, annioeee?iaaiiua n iia?aoi?aie Aee?eoea aeaanianeeo ia? ia
i?iaeaeo-iiiaiia?aceyo. Iino?iaii neiaieueiia en/eneaiea aeey aaaeiiai
eeanna

inaaaaeiaeeooa?aioeeaeueiuo iia?aoi?ia n aeeueaa?oiaui oaciaui
i?ino?ainoaii e

eaaceeeanne/aneea ?aoaiey niioaaonoaothueo o?aaiaiee O?aaeeiaa?a.
Iino?iai ooieoi? aneiioioe/aneiai eaaioiaaiey eeanna aaneiia/iiia?iuo
neiieaeoe/aneeo

iiiaiia?acee.

Eeth/aaua neiaa: aeeooa?aioee?oaiay ia?a, iia?aoi? Aee?eoea, neiaieueiia
en/eneaiea, oaciaia i?ino?ainoai, aaneiia/iiia?iia iiiaiia?acea.

Daletskii A.Yu. Differential and pseudodifferential operators on
infinite

dimensional manifolds. — Manuscript.

Thesis for a doctor degree by speciality 01.01.01- mathematical
analysis.

Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv,

1999.

The thesis is devoted to the development of the theory of differential
and pseudodifferential operators on infinite dimensional manifolds. Such
operators appear in different parts of mathematics and mathematical
physics. The main attention is paid to the probabilistic approach of
study of second order elliptic operators on product-manifolds (infinite
products of compact Riemannian manifolds) and to asymptotic methods of
investigation of

pseudodifferential operators with Hilbert (and Hilbert manifold) phase
spaces. The thesis consists of introduction, 3 chapters and references.

The introduction is devoted to the description of the field of
investigations, a review of existing results in this field and
description of the main results of the thesis.

In the first chapter Dirichlet operators associated with differentiable
measures on product-manifolds are considered. The stochastic dynamics
generated by these operators is constructed via stochastic differential
equations approach. That is, the Dirichlet operator is realized as a
generator of a Markov process given as the solution to an infinite
system of

stochastic differential equations on the initial Riemannian manifold. An
existence and uniqueness theorem for such systems is proved and
properties of solutions are studied. In particular, quasi-invariance of
distributions of the solutions (in the case where the initial manifold
has a Lie group structure) is proved. These results are applied to the
investigation of Glauber dynamics of some lattice models of statistical
mechanics. In particular, the existence of Feller semigroups associated
with Gibbs measures on product-manifolds is proved and their explicit
construction is given. The essential self-adjointness of the generator
on the space of cylinder functions (even in the case of interactions of
infinite range) is

proved. The extremal Gibbs states are characterized in terms of
ergodicity of the dynamics. Dirichlet forms on loop and diffeomorphism
groups are defined and their closability is proved.

The second chapter is devoted to the theory of infinite dimensional
pseudodifferential operators. An algebra of such operators with symbols
on Hilbert phase spaces is constructed. These operators are constructed
as functions of elementary operators of multiplication and
differentiation symmetrized w.r.t. a smooth measure on Hilbert space.
The main asymptotic formulae of the symbolic calculus (formulae for
symbols of composition commutator of two pseudodifferential operator,
formula of action on a

quickly oscillating exponent) are proved. Quasi-classical solutions of
the corresponding pseudodifferential Schr»odinger equations in the form
of pseudodifferential operators with oscillatory symbols are obtained.
For this, the infinite dimensional version of WKB method was developed.
These solutions gave the possibility to construct an asymptotic
projective representation of the group of symplectomorphisms of the
phase space. With

the aid of this representation the symbolic calculus for
pseudodifferential operators with symbols on certain infinite
dimensional symplectic manifolds is constructed. As an example, spaces
of solutions to infinite dimensional Hamiltonian systems are considered
and their asymptotic quantization is constructed.

The third section is devoted to the study of a special case of the
construction above. Namely, pseudodifferential operators in the space of
square integrable functions w.r.t. a Gaussian measure on a Hilbert space
are considered. In this special case, the corresponding Dirichlet
operator and its potential perturbations are considered as elements of
the algebra of

pseudodifferential operators. At the end, two geometric methods of
integration of infinite dimensional classical Hamiltonian equations are
proposed (which is required by the quasi-classical approach to the
investigation pseudodifferential equations). In particular, integral
curves of Hamiltonian equations associated with certain non-linear
Poisson brackets are obtained as solutions of a factorization problem in
a symplectic groupoid.

Main results of the thesis have been published in 16 scientific
publications and reported at a number of international scientific
conferences.

Key words: differentiable measure, Dirichlet operator, Feller semigroup,
pseudodifferential operator, symbolic calculus, phase space, infinite
dimensional manifold.

Похожие записи