
svayam-samyuktha oopparettar

ganithashaasthrathil, ananthamaaya-dimensional complexes vekttar space "vii" yill oru svayam-adjayint oopparettar aantharika uthppanna soothravaakyam _ 1 (thulyamaayi, parimitha-dimensional kesil oru hermissian oopparettar) oru leanier mapp "eiii" ("vii" ll ninnu thanne) aanu, athu athintethaaya adjaayintaanu. v" oru nischitha oorthoonoormal adisthaanamulla parimitha-alavaanengil, ithu "a" yude matricks oru hermissian matricks ayirikkumenna vyavasthaykku thulyamaanu, i.iii., athinte konjugettu transpose "a" nuu thulyamaanu. parimitha-dimensional spectral sidhaantham anusarichu, "v" nuu oru oorthoonoormal adisthaanamundu, eee adisthaanavumaayi bandhapetta "a" yude matricks yadhaartha samkhyakalil entrikal ulla oru diagonal matricks aanu. ekapaksheeyamaaya alavilulla hilbert edangalil oopparettarmaarkku eee aashayathinte saamaanyavalkkaranangal prayogikkunnathinekkur eee lekhanam kaikaaryam cheiyunnu.
functional analysis, kwaandam mechanics ennivayil selph-adjayint oopparettarmaare upayogikkunnu. kwaandam mechanicsil avayude praadhaanyam kwaandam mekkaaniksinte diarac-voonn neuman formulationilaanu, athil sthaanam, momentam, angular momentam, spin thudangiya bhauthika nireekshanangale hilbert speysile svayam-adjayint oopparettarmaar prathinidheekarikkunnu. hamiltonian oopparettar formula _ 2 aanu prathyeka praadhaanyam
ithu oru nireekshikkaavunna oru yadhaartha pottensial feeldile "m" pindamulla oru kanikayude motham oorjjavumaayi poruthappedunnu. paridhiyillaattha oopparettarmaarude oru pradhana vibhagamaanu deferential oopparettarmaar.
ananthamaaya alavilulla hilbert speysukalile svayam-adjayint oopparettarmaarude ghadana adisthaanaparamaayi parimitha alavilulla kesinode saamyamullathaan. athaayath, yadhaartha moolyamulla gunana oopparettarmaarkku ekathaanamaayi thulyamaanengil maathram oopparettarmaar svayam-adjayint aanu. anuyojyamaaya parishkkaranangaliloode, eee phalam ananthamaaya alavilulla edangalile paridhiyillaattha opparettarmaarilekku vyaapippikkaan kazhiyum. ellaayidathum nirvachikkapetta oru svayam-adjayint oopparettar anivaryamaayum parimithamayathinaal, paridhiyillaattha kesil domaine prashnatthil oraal kooduthal shradhikkendathundu. ithu kooduthal vishadamaayi chuvade vishadeekarikkunnu.
nirvachanangal.
formula _ 4 paridhiyillaathathaayirikkade (i.iii. parimithamaaya domaine formula oopparettar) parimithamaaya domaine formula 5 ulla formula 6 parimithamaaya alavilullappol eee vyavastha yaanthrikamaayi nilanilkkunnu, kaaranam oru parimithamaaya alavilulla oro leanier oopparettarinum formula 7 aanu.
randaamathe" vaadathil aantharika uthppanna soothravaakyam _ 8 konjugettu-leanier ayirikkatte. sankeernamaaya hilbert edangalkku maathrame ithu baadhakamaakoo. nirvachanam anusarichu, adjayint oopparettar formula _ 9 pravartthikkunnathu 11 enna moolakangalude formula adangiya subpace formula _ 10-laanu, athinu oru formula _ 12 undu, athaayath oro formulakkum formula _ 13 _ 14 krameekarana formula _ 15 leanier oopparettar formula _ 16 nirvachikkunnu.
oru (ekapaksheeyamaaya) oopparettar formulayude graph formula _ 4 aanu settu formula _ 18 oru oopparettar formula _ 19 formula _ 21 aanenkil formula _ 4 vipuleekarikkumennu parayappedunnu.
saandramaayi nirvachikkapetta oopparettar formula _ 4 neey cemetric ennu vilikkunnu
thaazhe kaanichirikkunnathupole ellah soothravaakyangalkkum, soothravaakyam 27 aanenkil maathram soothravaakyam 4 samamithiyaan.
29 enna soothravaakyam vyakthamaayi paranjaal, formula 30 umm formula 31 umm oro svayam-adjayint oopparettarum samamithiyaanengil paridhiyillaattha saandramaayi nirvachikkapetta oopparettar formula 4 neey svayam-adjayint ennu vilikkunnu. neremarichu, oru cemetric oopparettar formula _ 4, athinu formula _ 30 svayam-adjayint aanu. bhauthikashaasthrathil, hermissian enna padam samamitiyum svayam-adjayint oopparettarmaareyum orupole soochippikkunnu. iva randum thammilulla sookshumamaaya vyathyasam pothuve avaganikkappedunnu.
oro formulakkum (aavashyamillaatha-parimithappedutthiya) oopparettar formulakku 36 nuu "ellaayidathum parimithappedutthiya" vipareethamaanengil oru upasett formula _ 34 neey resolvent settu (allengil regular settu) ennu vilikkunnu. 37 enna pooraka soothravaakyathe spectrum ennu vilikkunnu. parimithamaaya alavukalil, formula _ 38 eejan moolyangal maathramaayi adangiyirikkunnu.
parimithamaaya svayam-adjayint oopparettarmaar.
oru bounded oopparettar "a" svayam-adjayint aanenkil 
ellah formulakalkkum 'h' ll '40', '11' formulakalkkum. a" ennathu samamitiyum soothravaakyavum aanenkil, hellinger-toplits sidhaantham anusarichu "a" ennathu nirbandhamaayum parimithappedutthiyirikku.
oro bounded leanier oopparettar "t": oru hilbert space "h"-ll "h" "" h "" formula 43-ll ezhuthaam, avide "a": "h" "" h "" umm "b": "h" "" h "" umm bounded selph-adjayint oopparettarmaaraanu.
parimithamaaya svayam-adjayint oopparettarmaarude savisheshathakal.
h" oru hilbert space ayirikkatte, formula _ 44 formula _ 45 ll nirvachichirikkunna oru bounded selph-adjayint leanier opparettaraayirikkatte. 
cemetric oopparettarmaar.
"kurippu cemetric oopparettarmaare mukalil nirvachichirikkunnu."
"a" "ennathu samamithiyaan" "a" "a".
oru paridhiyillaattha, saandramaayi nirvachikkapetta oopparettar formula _ 4, formula _ 68 aanenkil maathram, iff-part adjayint oopparettarude nirvachanathil ninnu nerittu pinthudarukayaanu. formula _ 4 samamithiyaanennu karuthuka, ulppeduthal formula _ 70 pinthudarunnu
kauchi-bunyakovsky-schwarz asamathvam oro soothravaakyathinum
samathvam kaaranam samathwa soothravaakyam _ 73 nilanilkkunnu
oro soothravaakyathinum _ 75 enna soothravaakyathinte saandrathayum aantharika uthppannathinte apachayavum.
ellaayidathum nirvachikkapetta oru samamiti oopparettar parimithavum svayam-adjayintum aanennu hellinger-toplits sidhaantham parayunnu.
"a" ennathu samamithiyaan ′ ′ "x" ′ "′" "x" "′ ′ r".
nirvachanathil ninnu nerittu pinthudarunna bhagam maathrame ulloo (mukalil kaanuka). iff-part theliyikkaan, aantharika uthppanna soothravaakyam _ 77 "aadya" vaadathil aanti-leanier randaamathethil leaniorumaanennu saamanyatha nashtappedaathe anumaanikkuka. (vipareeta saahacharyathil, pakaram nammal formula _ 78 upayogichaanu pravartthikkunnathu). formula _ 4-nte samamiti dhruveekarana identityil ninnu pinthudarunnu
oro soothravaakyathinum ithu baadhakamaan _ 81
||("a" − "λ") "x"|| ≤ "d" ("λ")  ω||"x"||.
oru svayam-adjayint oopparettarude spectrum yadhaarthamaanennathinte thelivaayi eee property upayogikkunnu.
formula _ 82 formula _ 83 umm formula _ 84 umm nirvachikkuka, moolya formula _ 85 formula _ 86 muthal shariyaayi nirvachikkappettittundu, samamiti kaaranam formula _ 87 aanu. pinne, oro soothravaakyathinum _ 88 num oro soothravaakyathinum _ 89
evide soothravaakyam _ 91
vaasthavathil, kauchi-schwarz asamathvam kondu soothravaakyam _ 92 aakatte,
formula 94 aanenkil formula 95, formula 96 ennivaye "bounded daun" ennu vilikkunnu.
oru lalithamaaya udaaharanam.
mukalil soochippichathupole, spectral sidhaantham selph-adjayint oopparettarmaarkku maathrame baadhakamaakoo, pothuve cemetric oopparettarmaarkku allah. ennirunnaalum, eee ghattathil namukk eejanvektarukalude oorthoonoormal adisthaanamulla oru cemetric oopparettarude lalithamaaya udaaharanam nalkaam. (eee oopparettar yadhaarthathil "adisthaanaparamaayi svayam-samyukthamaanu.") thaazhepparayunna oopparettar" "a" "nuu oru compact invezhs undennu kaanaan kazhiyum, athaayath anubandha deferential samavaakyam" "ap" "=" "g" "chila integral (athinaal compact) oopparettar" "g" "aanu pariharikkunnath". compact cemetric oopparettar "jii" nuu poornnamaaya aijanvektarukalude kanakkakkavunna oru kudumbamundu. a" ennathinum ithu thanne parayaam.
sankeernamaaya hilbert space "l" 2 [0,1] umm deferential oopparettarum pariganikkuka.
athirthi vyavasthakal thripthippeduthunna [0,1]-ll sankeernamaaya moolyamulla ananthamaaya deferentiable functionukalaaya "f" ulkkollunna formula _ 98-noppam 
appol aantharika uthppannathinte bhagangal thammilulla samyojanam "a" samamithiyaanennu kaanikkunnu. bhagangalaal rund thavana samyojanam nadatthaanum formula _ 100-nulla thannirikkunna athirthi vyavasthakal bhagangalaal samyojanathile athirthi nibandhanakal aprathyakshamaakumenn urappaakkunnuvenn parisodhikkaanum vaayanakkaarane kshanikkunnu.
eiii" yude eejan functionukal sinasoyidukalaanu.
yadhaartha eejan moolyangal "n" 2π2-ll; saine functionukalude ariyappedunna oorthoganaalitti cemetric enna savisheshathayude anantharafalamaayi pinthudarunnu.
eee oopparettarude saamaanyavalkkaranangal njangal chuvade pariganikkunnu.
svayam-adjayint oopparettarmaarude spectrum.
formula _ 4 oru paridhiyillaattha samamiti opparettaraayirikkatte. formula _ 104 aanenkil maathram formula _ 103 svayam-adjayint aanu
anivaryamaaya svayam-samyojanam.
oru cemetric oopparettar "a" ellaayppozhum closs cheyyaavunnathaanu; athaayath, "a" yude graph closs cheiyunnathu oru oopparettarude graph aanu. a" yude closser svayam-adjayint aanenkil oru cemetric oopparettar "a" adisthaanaparamaayi svayam-adjayint aanennu parayappedunnu. thulyamaayi, "advitheeya" svayam-adjayint ekstenshan undengil "a" adisthaanaparamaayi svayam-adjayint aanu. praayogikamaayi paranjaal, oru svayam-adjayint oopparettar undaayirikkunnathu oru svayam-adjayint oopparettar ullathupole thanne nallathaanu, kaaranam svayam-adjayint oopparettare labhikkunnathin maathram closher edukkendathundu.
udaaharanam "f" ("x") → "x"· "f" ("x").
sankeernamaaya hilbert space "l" 2 (r), oru functione "x" kondu gunikkunna oopparettar enniva pariganikkuka
a" yude domaine ellah "l" 2 functionukaludeyum formula _ 106-nte space aanu, athinu formula _ 107 square-integrable aanu. appol "a" ennathu svayam-samyukthamaanu. maruvashathu, "a" nuu eejan functionukalonnumilla. (kooduthal krithyamaayi paranjaal, "a" nuu "normalise cheyyaavunna" eejanvektarukalonnumilla, athaayath, yadhaarthathil "a" nirvachichirikkunna hilbert speysilulla egenvectorukal.)
nammal pinneedu kaanum pole, selph-adjayint oopparettarmaarkku valare pradhanappetta spectral propertykal undu; avar vaasthavathil pothuvaaya alaveduppu edangalile gunana oopparettarmaaraanu.
cemetric vezhsas selph-adjayint oopparettarmaar.
mukalil charcha cheythathupole, oru cemetric oopparettarum oru selph-adjayint (allengil adisthaanaparamaayi selph-adjaayint) oopparettarum thammilulla vyathyasam sookshumamaanengilum, spectral sidhaanthathile sidhaantham selph-adjaayint aanennathinaal ithu pradhaanamaan. ivide nammal vyathyaasathinte chila vyakthamaaya udaaharanagal charcha cheiyunnu; pothuvaaya sidhaanthathinaayi cemetric oopparettarmaarude vipuleekaranangalekkurichu chuvadeyulla vibhaagam kaanuka.
domainukalekkurichulla oru kuripp.
oro svayam-adjayint oopparettarum samamithiyaan. neremarichu, formula _ 108 svayam-adjayint aaya oro cemetric oopparettarum. 109 enna soothravaakyam 110 enna soothravaakyathekkaal karsanamaayi valuthaakkunna samamiti oopparettarmaarkku svayam-adjayint aakaan kazhiyilla.
athirthi vyavasthakal.
hilbert space oru bounded domainile functionukalude oru space aanenkil, eee vyathyaasangal kwaandam physixile parichithamaaya oru prashnavumaayi bandhappettirikkunnu oraalkk oru oopparettare nirvachikkaan kazhiyilla-momentam allengil hamiltonian oopparettar polullava-oru bounded domainil "boundari vyavasthakal" vyakthamaakkathe. ganithashaasthraparamaayi, athirthi vyavasthakal thiranjedukkunnathu oopparettarkku anuyojyamaaya oru domaine thiranjedukkunnathinu thulyamaanu. udaaharanathinu, hilbert space formula _ 111 (edavelayile [0,1] square-integrable functionukalude space) pariganikkuka. plaankinte sthiraangam 1nuu thulyamaayi krameekarichukondu saadhaarana soothravaakyam upayogichu namukk eee sthalathe oru "momentam" oopparettar "eiii" nirvachikkaam
athirthi vyavasthakal thiranjedukkunnathinu thulyamaaya "a" ennathinu nammal ippol oru domaine vyakthamaakkanam. naam thiranjedukkukayaanenkil
appol "a" ennathu samamithiyalla (kaaranam bhagangal thammilulla samyojanathile athirthi padangal aprathyakshamaakilla).
naam thiranjedukkukayaanenkil 
thudarnnu bhagangal upayogichu samyojanam upayogichu, "a" samamithiyaanennu eluppathil parisodhikkaan kazhiyum. eee oopparettar adisthaanaparamaayi svayam-adjayint allah, ennirunnaalum, adisthaanaparamaayi "a" yude domainil njangal valareyadhikam athirthi vyavasthakal vyakthamaakkiyathinaal, ithu adjayintinte domainine valare valuthaakkunnu. (eee udaaharanam chuvadeyulla "udaaharanagal" vibhaagathilum charcha cheyyappedunnu.)
prathyekichum, "a" ennathinu mukalil domaine thiranjedukkumbol, "a" yude closher formula _ 115-nte domaine ithaanu
athesamayam "a" yude adjayint formula _ 9-nte domaine ithaanu
athaayath, klosarinte domaine "a" nte domaininte athe athirthi vyavasthakalundu, ithu kurachu karsanamaaya minusamarnna anumaanamaan. athesamayam, "a" ll "valareyadhikam" athirthi vyavasthakal ullathinaal, formula _ 9 nuu "valare kurachu" undu (yadhaarthathil, eee saahacharyathil onnumilla). bhagangal upayogichu samyojanam upayogichu formula _ 121-nuu formula _ 120 kanakkaakkukayaanenkil, edavelayude iruvashathum formula _ 122 aprathyakshamaakumennathil, bhagangal upayogichu samyojanathile athirthi nibandhanakal radhaakkaan formula _ 123-leeii athirthi vyavasthakalonnum aavasyamilla. athinaal, mathiyaya minusamarnna function formula _ 123 formula _ 126 ulla formula _ 9nte domainilaanu.
klosarinte domain adjayintinte domain yojikkathathinaal, "a" adisthaanaparamaayi svayam-adjayint allah. ellaathinumupari, oru pothuvaaya phalam parayunnath formula 127-nte adjayintinte domaine "a"-yude adjayintinte domaine thulyamaanu ennaanu. athinaal, eee saahacharyathil, formula 127-nte adjayintinte domaine formula 127-nte domaininekkaal valuthaanu, ithu formula 127 svayam-adjayint allennu kaanikkunnu, nirvachanam anusarichu "a" adisthaanaparamaayi svayam-adjayint allah ennaanu ardhamaakkunnathu.
a" enna domainil nammal valareyadhikam athirthi vyavasthakal earppeduthi ennathaanu munbathe udaaharanathile prashnam. periodic boundari vyavasthakal upayogikkuka ennathaanu domaininte mikacha thiranjeduppu
eee domainil, "a" adisthaanaparamaayi svayam-samyukthamaanu.
eee saahacharyathil, spectral sidhaanthathinaayulla domaine prashnangalude prathyaaghaathangal namukk manassilakkaan kazhiyum. domaininte aadya choys (athirthi vyavasthakalonnumillaathe) upayogikkukayaanenkil, formula 133-nte ellah function formulakalum igenvectorukalaanu, aigan moolya formula 134 aanu, athinaal spectrum muzhuvan sankeernna thalamaanu. nammal domaininte randaamathe choys upayogikkukayaanenkil (direclet athirthi vyavasthakalode), "a" nuu egenvectorukal illa. domaininte moonnaamathe choys (periodic boundari vyavasthakalode) upayogikkukayaanenkil, "a" ennathinu "a" enna function formula _ 135 ennathinu oru oorthoonoormal adisthaanam namukk kandethaan kazhiyum. athinaal, eee saahacharyathil "a" svayam-adjayint aaya oru domaine kandethunnathu oru vittuveezhchayaan domaine cheruthayirikkanam, angane "a" samamithiyayirikkum, pakshe formula _ 136 aayi valuthayirikkanam.
eaka sheshiyulla shroadinger oopparettarmaar.
samamitiyum (adisthaanaparamaayi) svayam-adjayint oopparettarmaarum thammilulla vyathyaasathinte kooduthal sookshumamaaya udaaharanam kwaandam mechanicsile shroadinger oopparettarmaaril ninnaan varunnath. pottensial energy ekamaanenkil-prathyekichu pottensial paridhiyillaathathaanenkil-anubandha shroadinger oopparettar adisthaanaparamaayi svayam-adjayint aayirikkunnathil paraajayappettekkam. oru alavil, udaaharanathinu, oopparettar
ithu adisthaanaparamaayi sugamavum athivegam shayikkunnathumaaya pravartthanangalude idathil svayam yojikkunnilla. eee saahacharyathil, avashya svayam-samyojanathinte parajayam adisthaana classical sisttathile oru patholaji prathifalippikkunnu 138 pottensial enna soothravaakyamulla oru classical kanika parimithamaaya samayathinullil ananthathayilekku rakshappedunnu. eee oopparettarkku oru "advitheeya" svayam-adjayint illa, pakshe "ananthathayile athirthi vyavasthakal" vyakthamaakkikkond labhicha svayam-adjayint ekstenshanukal ithu amgeekarikkunnu. (formula _ 2 oru yadhaartha oopparettaraayathinaal, ithu sankeernamaaya samyojanathode sanjarikkunnu. athinaal, aparyapthatha soochikakal yaanthrikamaayi thulyamaanu, ithu svayam-adjayint ekstenshan undakunnathinulla vyavasthayaanu. thaazhe koduthirikkunna cemetric oopparettarmaarude vipuleekaranangalekkurichu charcha kaanuka.)
eee saahacharyathil, sugamavum athivegam shayikkunnathumaaya functionukalude sthalathu nammal thudakkathil formula _ 2 nirvachikkukayaanenkil, adjayint "athe" oopparettar aayirikkum (i.iii., athe soothravaakyam nalkiyirikkunnu) ennaal saadhyamaaya ettavum valia domainil, athaayath
appol formula _ 142 oru cemetric oopparettarallennu kaanikkan kazhiyum, ithu formula _ 2 adisthaanaparamaayi svayam-adjayint allennu theerchayaayum soochippikkunnu. vaasthavathil, formula _ 142 nuu shudhamaaya saangalppika eejan moolyangalulla egenvectorukal undu, ithu oru cemetric oopparettarkku asaadyamaanu. formula _ 144-leeii rund padangal thammilulla radhaakkal kaaranam eee vichitramaaya sambhavam saadhyamaan formula _ 142-nte domainil function formula _ 122 undu, athinu formula _ 148 allengil formula _ 149 enniva formula _ 150-ll vevvere illa, ennaal formula _ 142-ll sambhavikkunna avayude samyojanam formula _ 150-laanu. ithu formula _ 145-neey nonsimetrik aakaan anuvadikkunnu, formula _ 154-umm formula _ 155-umm samamiti oopparettarmaaraanenkilum. 138 enna repelling pottensial formula 157 enna parimithamaaya pottensial formula upayogichu maattisthaapikkukayaanel itharathilulla radhaakkal sambhavikkilla.
shroadinger oopparettarmaar svayam-adjayint allengil adisthaanaparamaayi svayam-adjayint aakanulla vyavasthakal vividha paatapusthakangalil kaanam, udaaharanathinu, referenceukalil pattikappedutthiyirikku dreasin, shubin, haal, reed, saimon enniva.
spectral sidhaantham.
bhauthikashaasthra saahithyathil, oru svayam-adjayint oopparettarinu eejanvektarukalude oorthoonoormal aditharayundennu spectral sidhaantham palappozhum prasthaavikkappedunnu. ennirunnaalum, bhauthikashaasthranjarkku "thudarchayaaya spectrum" enna prathibhaasathekkurichu nannayi ariyaam; athinaal, avar oru "oorthoonoormal adisthaanathekkurichu" samsaarikkumbol avar classic ardhathil oru oorthoonoormal adisthaanam "allengil" athinte chila thudarchayaaya analog "ennaanu ardhamaakkunnathu. udaaharanathinu, momentam oopparettar formula _ 158-nte kaaryatthil, bhauthikashaasthranjar parayum, egenvectorukal function formula _ 159 aanennu, avaa hilbert space formula _ 150-ll vyakthamalla. (bhauthikashaasthranjar parayum, egenvectorukal "normalisable allathava" aanennu.") bhauthikashaasthranjar thudarnnu eee" "egenvectorukal" "oru thudarchayaaya ardhathil oorthoonoormal aanennu parayum, avide saadhaarana chronecker deltta formula _ 161 oru diarac deltta function formula _ 162 upayogichu maattisthaapikkunnu".
eee prasthavanakal ganithashaasthranjarkku aswasthathayundaakkunnathi thonnumenkilum, oru pothu soothravaakyam _ 163 functione "superposition" aayi prakadippikkaan anuvadikkunna forrier parivartthanam upayogichu avaye karsanamaakkaam (i.iii.eee functionukal formula _ 163-ll illengilum, forrier transform momentam oopparettare "diagonalise" cheiyunnu; athaayath, ithu formula _ 166 kondu gunanathinte oopparettaraayi parivartthanam cheiyunnu, ivide formula _ 166 formulayude variablaanu.
oru oopparettar gunana oopparettarinu ekathaanamaayi thulyamaanennu kaanichu athine "diagonalise" cheyyaanulla saadhyatha pole spectral sidhaanthathe pothuve prakadippikkaan kazhiyum. spectral sidhaanthathinte mattu pathippukal samanamaaya reethiyil oru svayam-adjayint oopparettarkku yadhaarthathil chodyam cheyyappedunna hilbert spacil illaatha "egenvectorukal" undayirikkaamenna ashayam pidichedukkaan uddeshichullathaanu.
spectral sidhaanthathinte prasthavana.
hilbert edangalil "a", "b", "h", "k" enniva bhaagikamaayi nirvachichirikkunna oopparettarmaar "u": "h" "k" enna ekeekritha parivartthanam undengil maathrame ekathaanamaayi thulyamaakoo.
oru gunana oopparettare inipparayunna reethiyil nirvachikkunnu ("x", σ, μ) oru ganyamaaya additteev alaveduppu sthalavum "x" ll "f" oru yadhaartha moolyamulla alakkaavunna function aakatte. oru oopparettar formula _ 169
mukalil valathuvashath "l" 2 ulla si yude sthalamaanu domaine, athine gunana oopparettar ennu vilikkunnu.
spectral sidhaanthathinte oru pathipp inipparayunna reethiyil prasthaavikkam.
spectral sidhaanthathinte mattu pathippukal mukalil linkucheythirikkunna spectral sidhaantha lekhanathil kaanam.
unitory (athinaal parimithappedutthiya) oopparettarmaarude spectral sidhaanthathilekku kurachukondu paridhiyillaattha svayam-adjayint oopparettarmaarude spectral sidhaantham theliyikkaan kazhiyum. eee kuravu adutha vibhaagathil nirvachichirikkunna svayam-adjayint oopparettarmaarkkaayi "kayley transform" upayogikkunnu. t ennathu f kondu gunikkukayaanenkil, t yude spectrum f-nte avashya paridhi maathramaanennu nammal sradhichekkaam.
functional calculus.
spectral sidhaanthathinte oru pradhana prayogam oru "functional calculus" nirvachikkuka ennathaanu."athaayath, formula 171 riyal lainile oru function formula 172 oru selph-adjayint oopparettarumaanenkil, oopparettar formula 173 nirvachikkaan njangal aagrahikkunnu. formula 172 nuu aijanvekttar formula 175 nte yadhaartha oorthoonoormal adisthaanam undengil, aijanvekttar formula 175 umm aijanvelvu formula 179 umm ulla oopparettaraanu formula 173. formula 172 nuu thudarchayaaya spectrum ulla kesilekk eee ashayam vyaapippikkuka ennathaanu functional kaalkkulasinte lakshyam.
kwaandam physixil prathyeka praadhaanyamulla formula 172 hamiltonian oopparettar formula 2 umm formula 183 oru exponensialumaanu. eee saahacharyathil, pravarthanaparamaaya calculus oopparettare nirvachikkaan namme anuvadikkanam.
kwaandam mechanicsile samaya-parinamam nirvachikkunna oopparettar.
spectral sidhaantham urappunalkunnathupole-formula _ 122 kondu gunanathinte oopparettaraayi "t" yude praathinidhyam nalkumbol-functional calculus swabhaavam kaanikkan eluppamaanu "h" ennathu r-ll oru parimithamaaya yadhaartha moolyamulla boral functionaanengil, "h" ("t") ennathu composition formula _ 186 upayogichu gunanathinte oopparettaraanu.
vyakthithvathinte pariharam.
thaazhepparayunna notteshan avatharippikkunnath pathivaanu.
formula 188 ennathu interwell formula 189-nte characteristic function (indikkettar function) aanenkil, projection oopparettarmaarude kudumbamaaya e "t" (λ) neey "t"-yude aidantittiyude resolution ennu vilikkunnu. koodaathe, "tii" yude samagramaya praathinidhyam theliyikkaan kazhiyum
durbalamaaya oopparettar topology upayogichu mukalilulla oopparettar integralinte nirvachanam oru scaler moolyamulla steeltjas integralaayi kuraykkaam. ennirunnaalum, kooduthal aadhunika chikilsakalil, eee praathinidhyam saadhaaranayaayi ozhivaakkunnu, kaaranam mikka saankethika prashnangalum functional calculus vazhi kaikaaryam cheyyaan kazhiyum.
bhauthikashaasthra saahithyathile roopeekaranam.
bhauthikashaasthrathil, prathyekichu kwaandam mechanicsil, spectral sidhaantham mukalil paranjathupole spectral sidhaanthavum boral functional calculus inipparayunna reethiyil diarac notteshan upayogichu samyojippikkunna reethiyilaan prakadippikkunnathu
h" svayam-adjayint aanenkil "f" oru boral functionaanengil,
koode
h" yude muzhuvan spectrathinum mukaliloode integral kadannupokunnu. h" ennathu aijanvektarukal si "e" diagonalise cheythathaanennu notteshan soochippikkunnu. atharamoru notteshan poornnamaayum aupachaarikamaanu. dairaakkinte notteshanum munbathe vibhaagavum thammilulla saamyam kaanaan kazhiyum. aidantittiyude resolution (chilappol projection moolyamulla alavukal ennu vilikkunnu) rank-1 projection formula 193-maayi aupachaarikamaayi saamyamullathaan. diarac notteshanil, (projective) alavukal poornnamaayum aupachaarikamaaya vasthukkalaaya eejan vaalyookalum eejanstettukalum vazhiyaan vivarikkunnath. oraal pratheekshikkunnathupole, ithu swathuathinte prameyathilekkulla kadannupokaline athijeevikkunnilla. randaamathe formulationil, alakkunnathinu munbu sisttam formula 194 ll thayyarakkiyittundel, formula 194 nte spectral alavu upayogichaanu alavukal vivarikkunnath. pakaramaayi, oraalkk eejanstettukalude ashayam samrakshikkanum athu aupachaarikamaakkunnathinukam karsanamaakkaanum thaalpparyamundengil, oraalkk samsthaana idam anuyojyamaaya rigged hilbert space upayogichu maattisthaapikkaam.
sidhaanthathe aikyathinte pariharam ennu paraamarshikkukayaanenkil
cases formula _ 197 ennathu oru hermissian "echu", oru scue-hermissian (scue-hermissian matricks kaanuka) oopparettar formula _ 198 ennivayude aakethukayaan, oraal biorthoganal basis settine nirvachikkunnu.
spectral sidhaantham ingane ezhuthuka
(atharam oopparettarmaar scattering sidhaanthathil prathyakshappedunna sandarbhathinaayi feshback-phano partitioning reethi kaanuka).
cemetric oopparettarmaarude vipuleekaranangal.
thaazhepparayunna chodyam niravadhi sandarbhangalil uyarnnuvarunnu hilbert space "echu" leeii oru oopparettar "eiii" samamithiyaanengil, athinu eppozhaanu svayam-adjayint ekstenshanukal ullathu? savisheshamaaya oru svayam-adjayint ekstenshanulla oru oopparettare adisthaanaparamaayi svayam-adjayint ennu parayappedunnu; thulyamaayi, oru oopparettar athinte closher ("a" yude graph closher cheiyunna oopparettar) svayam-adjayint aanenkil athu adisthaanaparamaayi svayam-adjayint aanu. pothuve, oru cemetric oopparettarinu niravadhi svayam-adjayint ekstenshanukal undakam allengil onnumilla. athinaal, athinte svayam-adjayint ekstenshanukalude oru vargeekaranam njangal aagrahikkunnu.
avashya svayam-samyojanathinulla aadya adisthaana maanadandam inipparayunnavayaanu
oopparettarmaarude formula 201, formula 202 ennivaykku nissaramaaya kernelukal undengil maathrame "a" adisthaanaparamaayi svayam-adjayint aakoo. athaayath, formula _ 9 nuu aigan value formula _ 204 allengil formula _ 205 ulla oru igenwector undengil maathram "a" svayam-adjayint aakunnathil paraajayappedunnu.
oru svayam-adjayint oopparettarude kayley parivartthanavum aparyapthatha soochikakalum eee prashnam kaanunnathinulla mattoru maarggamaan. (adacha oopparettarmaarumaayi edapazhakunnathu palappozhum saankethika soukaryapradamaanu. ellah cemetric oopparettarmaarum closs cheyyaavunnavaraanennu ariyappedunnathinaal, cemetric kesil, clossedness aavasyakatha thadassangalonnum srishtikkunnilla.)
ivide, "rann", "dome" enniva yathaakramam chithratheyum (mattoru vidhathil paranjaal, shreni) domaine soochippikkunnu. athinte domainil w ("a") isometric aanu. koodaathe, 1 −w ("a") nte paridhi "h" ll saandramaanu.
neremarichu, eathenkilum bhaagikamaayi nirvachicha oopparettar "yuu" athinte domainil isometric aayirikkukayum (athu adaykkanamennilla) 1 − "yuu" saandramaayirikkukayum cheythaal, oru (advitheeya) oopparettar ezz ("yuu") undu.
angane
oopparettar s ("u") saandramaayi nirvachikkappettathum samamithiyumaan.
mappingukal dablyu, ezz enniva parasparam vipareethamaanu.
mapping dablyu-yeei kayley transform ennu vilikkunnu. ithu eathenkilum samamiti saandramaayi nirvachikkapetta oopparettarumaayi bhaagikamaayi nirvachikkapetta isometry bandhappeduthunnu. mappingukal w, s enniva monodon aanennu shradhikkuka ithinartham "b" ennathu saandramaayi nirvachikkapetta cemetric oopparettar "a" vipuleekarikkunna oru cemetric oopparettaraanenkil, appol w ("b"), athupole thanne s-num, w ("a") vikasippikkunnu.
ithu udanadi namukk "a" nuu svayam-samyuktha vipuleekaranathinu aavasyamaayathum mathiyayathumaaya oru vyavastha nalkunnu, inipparayunna reethiyil
hilbert space "echu" ll bhaagikamaayi nirvachichirikkunna isometric oopparettar "vii" nuu dome ("vii") nte norm klosaril savisheshamaaya isometric extensionundu. adacha domainullah bhaagikamaayi nirvachikkapetta isometric oopparettare paarshyal isometry ennu vilikkunnu.
oru bhagika isometry "vii" nalkumbol, "vii" yude kuravu soochikakal domaininteyum shreniyudeyum oorthoogoonal poorakangalude alavayi nirvachikkappedunnu
oru oopparettarude cemetric ekstanshanukalum athinte kayley transforminte isometric ekstanshanukalum thammil oru bijection undennu nammal kaanunnu. anubandha isometric ekstenshan yoonittariyaanengil maathram cemetric ekstenshan svayam-adjayint aanu.
rund aparyapthatha soochikakalum poojyamaanengil maathram oru cemetric oopparettarinu savisheshamaaya oru svayam-adjayint ekstenshan undu. atharamoru oopparettar adisthaanaparamaayi svayam-samyukthanaanennu parayappedunnu. adisthaanaparamaayi svayam-adjayint allatha cemetric oopparettarmaarkku ippozhum oru canonical svayam-adjayint ekstenshan undayirikkaam. noonn-negatteevu" cemetric oopparettarmaarude (allengil kooduthal pothuve, thaazhe parimithappedutthiyirikku oopparettarmaar) kaaryavum ithaanu. eee oopparettarmaarkku ellaayppozhum canonical aayi nirvachikkapetta friedriches ekstenshan undu, eee oopparettarmaarkku namukk oru canonical functional calculus nirvachikkaan kazhiyum. vishakalanathil sambhavikkunna pala oopparettarmaarum thaazhe parimithappedutthiyirikku (laplasian oopparettarude negatteevu polullava), athinaal eee oopparettarmaarude avashya adjayintnas prashnam nirnaayakamalla.
kwaandam mechanicsile svayam-adjayint ekstenshanukal.
kwaandam mechanicsil, nireekshanangal svayam-adjayint oopparettarmaarumaayi poruthappedunnu. oru-paaraameettar unitory grouppukalekkurichulla stoon sidhaantham anusarichu, svayam-adjayint oopparettarmaar krithyamaayi time evalution oopparettarmaarude unitory grouppukalude ananthamaaya janarettarmaaraanu. ennirunnaalum, hamiltonian samamithiyulla deferential oopparettarmaar ulppedunna oru samaya-parinaama samavaakyamaayi niravadhi bhauthika prasnangal roopappeduthiyittundu. atharam sandarbhangalil, onnukil hamiltonian adisthaanaparamaayi svayam-adjayint aanu, eee saahacharyathil bhauthika prashnathinu savisheshamaaya parihaarangalund allengil ananthathayile vividha tharatthilulla athirthi vyavasthakalkko vyavasthakalkko anusrithamaayi hamiltoniyante svayam-adjayint ekstenshanukal kandethanulla oru sramam.
udaaharanam. thudakkathil minusamarnna kompaakt support functionukalil nirvachichirikkunna _ 209 enna pottensial formula oru dimensional shroadinger oopparettar adisthaanaparamaayi svayam-adjayint aanu (athaayath, oru svayam-adjayint closher undu) pakshe athinullathalla. 55umm 86umm pejukalilulla 'dreasin aandu shubin' allengil haalile 9.10-aamm vakupp kaanuka.
210 enna soothravaakyathinte avashya svayam-samyojanathinte paraajayathinu oru kanikayude classical dynamicsil pottensial formula 211 ulla oru prathiroopamunda classical kanika parimithamaaya samayathinullil ananthathayilekku rakshappedunnu.
udaaharanam. oru kanika pakuthi variyil neengunnathin svayam-adjayint momentam oopparettar "p" illa. ennirunnaalum, oru pakuthi variyile oru "swatantra" kanikayude hamiltonian soothravaakyam _ 212 nuu vividha tharatthilulla athirthi vyavasthakalkku anusrithamaayi niravadhi svayam-adjayint extensionukalundu. bhouthikamaayi, eee athirthi vyavasthakal uthbhavasthaanathulla kanikayude prathifalanangalumaayi bandhappettirikkunnu (reed saiman, vol.2 kaanuka).
voonn nyoomaante soothravaakyangal.
a" ennathu samamithiyil saandramaayi nirvachichirikkunnuvennu karuthuka. appol "a" yude eathenkilum samamiti vipuleekaranam "a" * nte niyanthranamaan. vaasthavathil, dome ("a" *) enna nirvachanam prayogikkunnathiloode "a" "b", "b" samamiti "b" "a" * "nalkunnu.
akheesar, glassman paraamarsangalil ivaye voonn nyoomaante soothravaakyangal ennu vilikkunnu.
udaaharanagal.
adisthaanaparamaayi svayam-adjayint allatha oru cemetric oopparettar.
nammal aadyam hilbert space formula _ 213 umm deferential oopparettarum pariganikkunnu.
[0,1] ll thudarchayaayi deferentiable cheyyaavunna sankeernna moolyamulla functionukalude sthalathu nirvachichirikkunnu, athirthi vyavasthakal thripthippeduthunnu
bhagangal kondu samyojippikkunnathiloode kaanikkan kazhiyunnathupole "d" oru cemetric oopparettaraanu. n" +, "n" − (thaazhe nirvachichirikkunnathu) ennee edangal samavaakyathinte vitharana pariharangal yathaakramam nalkunnu.
avaa "l" 2 [0,1] ll undu. eee parihara edangal oronnum yathaakramam "x" "" e "" −x "," x "" "e" "x" ennee functionukalaal srishtikkappedunna 1-dimensional aanennu oraalkk kaanikkan kazhiyum. ithu kaanikkunnath "dii" adisthaanaparamaayi svayam-adjayint allah, marichu svayam-adjayint extensionukalundu ennaanu. eee svayam-adjayint ekstenshanukal unitory maappingukalude "n" + → "n"-nte space upayogichu paaraameettar cheiyunnu, ithu eee saahacharyathil yoonit sarkkil t aanu.
eee saahacharyathil, formula _ 217-nte domaininte nirvachanathile athirthi vyavasthakalude "thettaaya" thiranjeduppu moolamaanu avashya svayam-adjayintanukalude parajayam. formula _ 217 oru phastt oordar oopparettaraayathinaal, formula _ 217 samamithiyaanennu urappakkaan oru athirthi vyavastha maathrame aavashyamulloo. mukalil nalkiyirikkunna athirthi vyavasthakal otta athirthi vyavastha upayogichu maattisthaapikkukayaanel
appol "d" ippozhum samamithiyayirikkum, ippol vaasthavathil, adisthaanaparamaayi svayam-adjayint aayirikkum. athirthi vyavasthakalude eee maattam "dii" yude oru prathyeka svayam-samyuktha vipuleekaranam nalkunnu. mattu adisthaanaparamaayi svayam-adjayint ekstenshanukal varunnath phoom formula _ 221-nte athirthi vyavasthakal adichelppikkunnathil ninnaan.
eee lalithamaaya udaaharanam oru ooppan settu "m" ll cemetric deferential oopparettarmaarude "p" yude svayam-adjayint ekstenshanukalekkurillu oru pothuvaaya vastutha vyakthamaakkunnu. eejan value speysukalkkidayilulla ekathaana bhoopadangal avaa nirnnayikkunnu. 
ivide "p" dist ennathu "p"-nte vitharana vipuleekaranamaanu.
sthiramaaya-gunaka oopparettarmaar.
sthiramaaya gunakangalulla deferential oopparettarmaarude udaaharanam nammal aduthathaayi nalkunnu. anuvadikkuka.
yadhaartha" gunakangalulla r "n" ll oru bahupadamaayirikkuka, avide α oru (parimitha) koottam multy-induxukalkku mukalilayirikkum. angane
oppam
naam notteshanum upayogikkunnu
appol oopparettar "p" (d) ennathu r "n"-ll compact sapporttinte ananthamaaya deferentiable functionukalude space nirvachichirikkunnu.
ithu adisthaanaparamaayi "l" 2 (r "n") ll svayam-samyukthamaanu.
kooduthal pothuve, compact sapporttinte ananthamaaya deferentiable complexes-valued functionukalil pravartthikkunna leanier deferential oopparettarmaare pariganikkuka. m" ennathu r "n" nte oru thuranna upavibhaagamaanengil"
ivide "a" α ennathu (sthiramaayirikkanamennil) ananthamaaya deferentiable functionukalaanu. "p" oru leanier oopparettaraanu.
p" nuu anusrithamaayi mattoru deferential oopparettar undu, "p" yude aupachaarika adjayint."
spectral multiplisity sidhaantham.
oru svayam-adjayint oopparettarude gunana praathinidhyam, valare upayogapradamaanengilum, oru canonical praathinidyamalla. svayam-adjayint oopparettarmaaraaya "a", "b" enniva eppol unitaayi thulyamaanennu nirnnayikkaan eee praathinidhyatthil ninnu oru maanadandam verthirichedukkunnathu eluppamallennu ithu soochippikkunnu. nammal ippol charcha cheiyunna ettavum mikacha dhaanya praathinidhyatthil spectral multiplisity ulppedunnu. eee phalangalude vrithathe "spectral multiplisityude hahn-hellinger sidhaantham" ennu vilikkunnu.
ekeekrithamaaya bahuswarata.
nammal aadyam "ekeekrithamaaya bahuswarata" nirvachikkunnu
nirvachanam. oru svayam-adjayint oopparettar "a" nuu ekeekritha gunanam "n" undu, avide "n" ennathu 1 ≤ "n" ≤ ω aanu, "a" ennathu "f" (λ) = λ oonn function upayogichu gunanathinte oopparettar m "f" nuu ekathaanamaayi thulyamaanengil maathram
ivide h "n" ennathu "n" enna alavilulla oru hilbert space aanu. m "f" yude domainil r-ll vekttar moolyamulla functionukal si adangiyirikkunnu, athaayath
negatteevu allatha, ennaavunna additteev alavukal, ν, avaa disjoyint boral settukalil pinthunaykkunnundel maathram paraspara ekavachanamaanu.
eee praathinidhyam inipparayunna ardhathil savisheshamaanu ore "a" yude atharam rund praathinidyangalkku, avaykku ore alavukolukal 0 enna ardhathil thulyamaanu.
nerittulla integralukal.
hilbert speysukalude nerittulla integralukalude bhasha upayogichu spectral multiplisity sidhaantham parishkarikkaam
spectral sidhaanthathinte gunana-pravartthaka pathippil ninnu vyathyasthamayi, direct-integral pathipp savisheshamaan, athaayath alavu thulyatha klaas (allengil thathulyamaayi athinte alavu 0) advitheeyamaayi nirnnayikkappedukayum alakkaavunna function formula _ 233 mikkavarum ellaayidathum nirnnayikkappedukayum cheiyunnu. function formula _ 234 ennathu oopparettarude spectral multiplisity functionaanu.
svayam-adjayint oopparettarmaarude vargeekarana phalam namukk ippol prasthaavikkam rund svayam-adjayint oopparettarmaar unitaayi thulyaraanu, (1) avarude spectra settukalaayi sammathichaal, (2) avarude direct-integral praathinidyangalil drishyamaakunna alavukalkku poojyatthinte alavu ore settukalum (3) direct integralile alavumaayi bandhappett avarude spectral multiplisity functionukal mikkavarum ellaayidathum yojikkunnu.
udaaharanam laplasian ghadana.
r "n" leeii laapleshyanaan oopparettar
mukalil soochippichathupole, laplasian forrier parivarthanatthaal konaakrithiyilaanu. oru oopparettar enna nilayil ithu noonn-negatteevu aayathinal laplasian-δ-nte "negatteevu" pariganikkunnath yadhaarthathil kooduthal swaabhaavikamaanu; (deerghavrithaakrithiyilulla oopparettar kaanuka).
shudhamaaya pooyint spectrum.
h" nuu "a" ennathinu igenvectorukal adangiya oorthoonoormal adisthaanam {"" ei "}" i "ωi undengil maathrame" h "-ll oru svayam-adjayint oopparettar" a "-nuu shudhamaaya pooyint spectrum undaakoo.
udaaharanam. harmonic oosilettarinte hamiltonian oru quadratic pottensial "vii" undu, athaayath
eee hamiltonian shudhamaaya pooyint spectrum undu; kwaandam mechanicsile boundu stettu hamiltonianmark ithu saadhaaranamaanu. munbathe oru udaaharanathil choondikkaanichathupole, oru paridhiyillaattha cemetric oopparettarinu hilbert space basis roopappeduthunna egenvectorukal undennathinu mathiyaya vyavastha athinu compact invezhs undu ennathaanu.

