duheri kshamata

tathakathit duheri-vihir kshamata hii quantum yantriki, quantum kshetra siddhant aanee itaratra vividh bhautik ghatana kinwa ganiteey gunadharmanchya shodhasathi lakshaniy swarasya asalelya anek chaturthansh sambhavyatepaiki eka aahe kaaran tee anek prakarananmadhye atee-saralikaran naan karata spasht gananela anumati dete.
asha prakare yuklidikrut kshetra siddhantat chhadm-shastriy sanrachana mhanoon instantonchi sankalpana spasht karnyasathi 'samamiteey duheri-vihir kshamata' anek varshe eka model mhanoon kaam karate. sopya quantum yantrik sandarbhat hii kshamata fenman paath integralschya mulyankanasathi eka model mhanoon kaam karate. kinwa urja eigenvalue spashtapane midvanyachya uddeshane vividh paddhatindware schroedinger samikaranache samadhan.
dusarikade, "ulat samamiteey duheri-vihir kshamata" hii, kshay daranchi ganana karnyasathi aanee asymptotic vistaranchya mothya kramachya vartanachya shodhasathi schroedinger samikaranat eka kshullak kshamata mhanoon kaam karate.
quartic potentialcha tisra prakar mhanje 'vikshipt sadhe harmonic ausilator' kinwa 'shuddha anharmonic ausilator' jyamadhye purnapane swatantra urja spectrum aahe.
sambhavya chaturthansh sambhavyatecha chautha prakar mhanje varr dilelya pahilya donpaiki ekacha 'asamamit aakar'.
duheri vihir aanee itar chaturthansh kshamatevar vividh paddhatinni upchar kele jaau shakatat-mukhya paddhati mhanje (aaa) vikshobh paddhat (buub. dingle aanee etch.jeey.doubleu. mueller-cursten) jyasathi seema atty ladane aavashyak aahe, (buub) doubleu. keey. bii. paddhat aanee (qa) paath integral paddhat..
serve prakarananvar echchya pustakat tapshilvar upchar kele jaatat.jeey.doubleu. mueller-cursten. mathieu functionschya asymptotic vistaranche mothe kram vartan aanee tyanchya eigenvalues (jyanna vaishishtyapurn sankhya dekhil mhanatat) heey aarchya pudhil peparmadhye ghetale gele aahe.buub. dingle aanee etch.jeey.doubleu. mueller.
samamiteey duheri vihir.
sahityatil mukhya swarasya (kshetra siddhantashi sambandhit karanansathi) samamiteey duheri-vihirinvar (sambhavya) aanee tethe quantum yantrik ground statevar kendrit aahe. sambhavyatechya madhyavarti kubadyatoon bogda kadhane samavisht asalyane, yaa sambhavyatesathi schroedinger samikaranachya aayjan energychi ganana kshullak nahee. ground statechya babatit instanton aanee anty-instanton mhanoon olakhalya jaanarya seudoclassical sanrachanandware madhyasthi kelee jaate. spasht swarupat hii hyperbolic functions aahet. chhadmashastriy sanrachana mhanoon heey naisargikaritya ardhashastriy babinmadhye disatat-(mothya pramanat vegale kelelya) instanton-anty-instanton jodyanchi berries hii patal vayoo andaje mhanoon olakhali jaate. shevati midavaleli ground steet aayjan energy hii eka abhivyakti aahe jyat instantonchya euclidean kruticha ghatank asato. hii eka abhivyakti aahe jyat ghatak sootra _ 1 aahe aanee mhanoon tyache varnan (shastriyadrushtya) nonperterbative effect mhanoon kele jaate.
samamiteey duheri-well swayam-parasparsanvadasaha scalar feeled siddhantchya path avibhajya siddhantatil instanton sanrachanechi sthirata instantanbaddalchya lahan dolananche samikaran vaparoon tapasali jaate. heey samikaran poshle-teller samikaran aahe asey aadhalate (i.eae. naun-negative igenvalues, poshle-teller potentials schroedinger samikaranasarkhe itar kramankache vibhedak samikaran. eigenvalueschi naun-negativity hii instantonchya sthirateche soochak aahe.
varr sangitalyapramane, instanton heey euclidean velechya anant reshevar paribhashit kelele chhadm kann sanrachana aahe jeey sambhavyatechya done vihirinmadhye sanvad sadhate aanee pranalichya jaminichya sthitisathi jababdar asate. uchch, i sathee sambandhit jababdar sanrachana.eae. uttejit, avastha hii euclidean velechya vartulavar paribhashit keleli niyatkalik kshanika asatat jee spasht swarupat jacobian lambvartulakar karyanchya (trikonamitiy karyanche samanyikaran) sandarbhat vyakt kelee jaatat. yaa prakarananmadhye paath integralchya mulyankanat tyanusar lambvartulakar integralscha samavesh asato. yaa niyatkalik instantansbaddalchya chhotya chadhautaranche samikaran heey lamey samikaran aahe jyachi uttare lamey functions aahet. asthiratechya prakarananmadhye (ulat duheri-vihirichya kshamatesathi) yaa samikaranat nakaratmak eigenvalues aahet, jee yaa asthirateche soochak aahet, i.eae. kshay.
dingle aanee mularchya vikshobh paddhaticha vapar (moolatah mathieu samikaranala lagoo kele jaate, i.eae. cosine potentials schroedinger samikaran) sathee quartic potentialsathi schroedinger samikaranachya parameter samamiticha vapar karane aavashyak aahe. eka sambhavyatechya done kimananpaiki ekachya aaspas vistarato. yavyatirikt yaa paddhatisathi ativyaptichya kshetranmadhye upayanchya vividh shakhanchi julani aavashyak aahe. seema atincha vapar kelyane shevati (niyatkalik kshamatechya babatit) ashabdik parinam hoto.
khaleel swarupat samamiteey duheri-vihir sambhavyatesathi schroedinger samikaranatil mapdandanchya drushtine
sootra _ 3 sathee eigenvalues aadhaltat (mueller-curstenche pustak, sootra (18.175b), pii pahaa. 425)
spashtapane hii eigenvalues asymptotically (sootra _ 6) potentialchya harmonic bhagatoon apekshit kelyapramane ksheen hotat. parinamachya ashant bhagachya atty paryayane summ kinwa visham aahet heey sootra 7 aanee sootra 8 madhye pahaa (mathioo functions, lamey functions, prolet spheroidal wave functions, oblet spheroidal wave functions aanee itaransathi sambandhit parinamanpramane).
kshetra siddhant sandarbhat vareel samamiteey duheri-vihir kshamata anekada lihili jaate (sootra _ 9 heey eka scalar kshetra aahe) 
aanee instanton heey neuton-sarkhya samikaranache samadhan sootra _ 11 aahe.
(sootra _ 13 haah euclidean vel aahe), i.eae.
sootra _ 16 baddalchya _ 15 yaa lahan chadhautaranche samikaran
poshle-teller samikaran aahe (poshle-teller kshamata pahaa)
sobat
serve aaygan moolye sootra _ 19 sakaratmak kinwa shoonya asalyane, instanton sanrachana sthir aahe aanee tyat konatahi kshay nahee.
sootra _ 20 chya adhik samanya prakaranat shastriy upay mhanje niyatkalik instanton 
jethe sootra _ 22 heey niyatkalik jacobian lambvartul karya sootra _ 23 chey lambvartulakar mapank aahe. yaa samanya prakaranat lahan chadhautar samikaran lamey samikaran aahe. maryada sootra _ 24 madhye samadhan sootra _ 25 vacuum instanton samadhan banate,
ulat duheri vihir kshamata.
yaa sambhavyatesathi schroedinger samikaranachi eigenvalue prapt karnyasathi, ativyapi kshetrantil upayayojananchya julanisah aanee seema atty ladnyabarobarach (duheri-vihirinpeksha bhinn) vikshobh siddhant punha vaparla jaau shakato. tathapi, yaa prakaranat, sambhavyatechya madhyavarti gartet vistar hoto. tyamule vareel parinamanpeksha parinam vegale aahet.
khaleel swarupat ulate duheri-vihirichya sambhavyatesathi schroedinger samikaranatil mapdandanchya drushtine
sootra _ 28 sathee eigenvalues aadhaltat (mueller-curstenche pustak, sootra (18.86), pii pahaa. 503)
yaa abhivyakticha kalpanik bhag seechya parinamashi sahmat aahe.emm. bender aanee tii.tii. woo (tyanche sootra (3.36) pahaa aanee sootra _ 30 aanee tyanchya sanketan sootra _ 31 madhye seet qaraa). haah parinam vikshobh siddhantachya mothya kramachya vartanachi charcha aanee tapasat mahattvachi bhoomika bajavato.
shuddha anharmonic ausilator.
khaleel swarupat shuddha anharmonic ausilatorsathi schroedinger samikaranatil mapdandanchya drushtine
sootra _ 33 sathee eigenvalues aadhaltat
adhik sandnya sahajpane mojalya jaau shakatat. itar serve prakarananpramanech, 35 aanee 36 yaa sutranmadhye vistarache gunank paryayi summ kinwa visham aahet heey pahaa. quartic potentialsathichya vibhedak samikaranachya upayancha haah eka mahattvacha pailu aahe.
sarvasadharan tippanya.
duheri vihir aanee ulat duheri vihirinsathi vareel parinam paath integral paddhatidvare dekhil prapt kele jaau shakatat (tethe niyatkalik instantance, see. eff. dware. instantance), aanee doubleu. keey. bii. paddhat, jaree lambvartul samakalan aanee gamma karyachya karyachya andajasah, yaa sarvanmule ganana adhik katheen hote. parinamanpaiki 'q'-'q', sootra _ 8-sootra _ 8 madheel badalanmadhil vikshobhkari bhagacha samamiti gunadharm keval schroedinger samikaranatun vyutpattimadhye prapt keela jaau shakato joou parinam midvanyacha changla aanee yogya marg aahe. yaa nishkarshala mathioo samikaran aanee lamey samikaran yasarkhya itar itar kramankachya vibhedak samikarananchya tapasanyadvare samarthan dile jaate, jeey tyanchya eigenvalue samikarananmadhye samaan gunadharm darshavtat. shivay yaa pratyek prakaranat (duheri-vihir, ulat duheri-vihir, cosine kshamata) shastriy sanrachanebaddalachya lahan chadhautaranche samikaran lamey samikaran aahe.

