Pll-2001-61

OqYJIY)H6aaTap MBny3bIHHH CMBHHHIlKHH

HblOTOHOBCKA5I MTEPAUMOHHA5I CXEMA C BAPMAUMOHHbIM ltlgtYHKUMOHAJIOM

_ lllBMHrEPA JUIs PElllEHMSI 3AnAqM PACCESIHMJl o c

~ -~ ~ HanpaBJIeHO B laquoJournal of Computational Methods FERMI AS_0 in Sciences and Engineeringraquo~

i-o AUG 2 1 2001u_n=-r=w i__0o

_r=w LlaRARY-2001

1 BBelJeBHe

3anaIH paCCefIHHfI B fInepHoA ltpH3HKe CBfI3aHbI C BbIIHCJIeHHeM aMnJIHTynbI pacshy

CefIHHfI ltPa30BbIX cnBHrOB H napaMeTpOB CMernHBaHHfI BXOnfImHX B onpeneneHHe

aCHMnTOTHIeCKOA BOJIHOBOA ltPYHKIJHH ypaBHeHHfI IllpenHHrepa C KOpoTKoneAcTBYshy

IOmHM nOTeHIJHaJIOM V )lJIfI perneHHfI 3anaIH paCCefIHHfI HCnOJIb3YIOTCfI Pa3JIHIshy

HbIe MeTonbI OnHHM H3 pacnpOCTpaHeHHbIX nonxonOB fIBJIfIeTCfI BapHaIJHOHHbIA

MeTOn IllBHHrepa B aTOM nonxone HCnOJIb3yeTciI CJIenylOmee BbIpa)KeHHe nJIfI Bashy

pHaIJHOHHoro ltPYHKIJHOHaJIa

1 (kIVI1J~+raquo)(1J~)IVlk)f(kk) = (1)

211 (1]~)IV - VGoVI1]~+raquo) KOTopoe n03BOJIHeT BbIIHCJIHTb aMnJIHTynbI H ltPa3bI pacceHHHH C 60JIbrneA TOIHo-

CTblO IeM B 60PHOBCKOM npH6JIH)KeHHH [12] 3necb 11])- np06HbIe ltPYHKIJHH Goshy

cBo6onHaH ltPYHKIJHH rpHHa B pa60Tax [3 4] 6bIJIO nOKa3aHO ITO MO)KHO nOCTashy

TOIHO altpltpeKTHBHO HaxonHTb perneHHH 3anaIH paCCefIHHfI C nOMOmblO Pa3JIHIHbIX

HTepaIJHOHHbIX cxeM nOCTpoeHHbIX Ha OCHOBe BapHaIJHOHHoro ltPYHKIJHOHaJIa IllBHHshy

repa (1) HCnOJIb3YH cenapa6eJIbHoe npH6JIH)KeHHe nJIfI KOpoTKoneAcTBYlOmero no-

V(N) = E[j=l VI1Ji)dW)(1]iV (2)

di = (1]iI V I1]j)middot

TeM He MeHee np06JIeMbI nOCTpoeHHH YCToAIHBbIX HTepaIJHOHHbIX cxeM n03BOJIHshy

IOmHX nOJIYIHTb perneHHH C 3anaHHoA TOIHOCTblO fIBJIfIIOTCfI aKTYaJIbHbIMH H B

HaCTOfImee BpeMfI

B pa60Te [5] 6bIJIa nOCTpoeHa YCToAIHBaH HTepaIJHOHHaH cxeMa perneHHfI 3anaIH

pacceHHHH nJIfI ypaBHeHHH IllpenHHrepa Ha OCHOBe HenpepbIBHoro aHaJIOra MeTona

HblOToHa (HAMH) C nOMOmblO nonOJIHHTeJIbHoro BapHaIJHOHHoro ltPYHKIJHOHaJIa

XIOJIbTeHa B naHHoA paOOTe 9TOT MeTOn npHMeHeH nJIH perneHHfI 3anaIH pacceshy

fIHHH C 3anaHHoA TOIHOCTblO B paMKaX HHTerpaJIbHoro ypaBHeHHH HCnOJIb3yeTcH

BapHaIJHOHHbIA ltPYHKIJHOHaJI IllBHHrepa (1)

CTPYKTypa PaOOTbI CJIenylOmaH Bo BTOPOM pa3neJIe npenCTaBJIeHbI 06mHe nOJIOshy

)KeHHH Heo6xonHMbIe nJIfI nocTpoeHHH YCToAIHBbIX HTepaIJHOHHbIX cxeM perneHHfI

1

2

CneKTpaJIbHbIX 3allaq Ha OCHOBe HAMH llOKa3aHa B03MOLKHOCTb BbIBOlla Pa3JIHqshy

HbIX H3BeCTHbIX HTepaUHOHHbIX CXeM llJIH CneKTpaJIbHbIX 3allaq Ha ellHHOA OCHOBe

HAMH

B TpeTbeM pa3lleJIe C nOMOmbIO BapHaUHoHHOrO ltPYHKUHOHaJIa IlIBHHrepa 3allaqa

paCCeHHHH CltP0PMYJIHpOBaHa KaK 3allaqa Ha c06cTBeHHbIe 3HaqeHHH OTHOCHTeJIbHO

napbI HeH3BeCTHbIX ltpa30BOro CllBHra H BOJIHOBOA ltPYHKUHH llOCTpOeHa YCToAqHshy

BaH HTepaUHOHHaH cxeMa Ha OCHOBe HAMH Hero MOllHltpHKaUHA B OKpeCTHOCTH

peIIIeHHH

B qeTBepTOM pa3lleJIe 3ltpltpeKTHBHOCTb npellJIOLKeHHOA HTepaUHOHHoA cxeMbI nposhy

lleMOHCTpHpOBaHa Ha qHCJIeHHbIX peIIIeHHHX 3allaqH ynpyroro pacceHHHH C nOTeHshy

UHaJIOM Mop3e H co CltpepHqeCKOA HMOA

HeIIpepbIBHbIA aHaJIOr MeTOl1a HblOTOHa

PaCCMOTpHM cxeMY peIIIeHHH CneKTpaJIbHbIX 3allaq Ha OCHOBe HenpepbIBHoro aHashy

JIora MeTOlla HbIOToHa KOTopaH 3aKJIIOQaeTCH B 3aMeHe HCXOllHOA HeJIHHeAHOA CTa-

UHOHapHoA 3allaQH

fP(aAy) = 0 (3)

OTHOCHTeJIbHO HeH3BeCTHoA z = (A y) E Rn x Y Y ~ B npH ltpHKcHpoBaHHoM

Ha60pe ltpH3HQeCKHX BeKTop-napaMeTpOB a E Rm 3BOJIIOUHOHHOA 3allaQeA KOIIIH [6]

dz(t)fP(az(t))--u- = -fP(az(t)) (4)

z(O) = zoo (5)

3lleCb t (0 t lt (0) - HenpepbIBHbIA napaMeTp OT KOTOPOro 3aBHCHT TpaeKTOshy

pHH z(t) fP - npOH3BOllHaH (l)peIIIe Zo - 3JIeMeHT B OKpeCTHOCTH HCKOMOro peIIIeHHH

z = (Ay) ypaBHeHHH (3) ~OKa3aTeJIbCTBO CX0llHMOCTH HenpepbIBHoA TpaeKshy

TOpHH z(t) npH t -+ 00 K peIIIeHHIO z npH YCJIOBHH HenpepbIBHocTH fP H fP H

cymecTBoBaHHH OrpaHHQeHHOrO onepaTopa (fP)-1 B oKpecTHocTH z OCHOBaHO Ha

cymecTBoBaHHH HHTerpaJIa 3allaQH KOIIIH (4)-(5)

(6)

2

I(HcKpeTHaH armpOKCHMaUHH rro rrapaMeTPY t 3anaIH (4)-(5) Ha OCHOBe MeTona

9itJIepa CBOnHT ee K pellIeHHIO rrOCJIenOBaTeJIbHOCTH JIHHeitHblx 3anaI

P(a Zk)~Zk = -P(a Zk) (7)

Zk+l = Zk + Tk~Zk

npHIeM CneUHaJIbHbIM BbI6opOM rrapaMeTpa Tk MO)KHO onTHMH3HpoBaTb CKOPOCTb

H YCToitIHBOCTb CXOnHMOCTH Zk ~ z [7 8] I(JIH KJIaCCHIeCKoit CneKTpaJIbHoit

3auaIH OTHOCHTeJIbHO rrapbI Z = A II E Rn x Y HemmeitHoe ypaBHeHHe (3)

MO)KHO rrpenCTaBHTb B BHne

(H(a) - Al)P ) P(a A 11) = = o (8)

( F(aAW)

3neCb H(a) - onepaTOp B rHJIb6epTOBoM npOCTpaHCTBe a F(a A II) - nonOJIHHTeJIbshy

Hblit cPYHKUHOHaJI HanpHMep

a) (II II) - 1 = 0 YCJIOBHe HOPMHPOBKH (9)

6) (II (H (a) - AI) II) = 0 YCJIOBHe opTOrOHaJIbHOCTH

I(JIH pellIeHHH CrreKTpaJIbHbIX 3anaq (8) npHMeHHMa HTepaUHOHHaH cxeMa (7) I(BYXshy

KOMpOHeHTHaH CTpYKTypa cPYHKUHH P H B03MO)KHOCTb H3MeHeHHH BHna cPYHKUHshy

OHaJIa F B HTepaUHHX n03BOJIHIOT nOJIYIHTb llIHPOKHit Ha60p HTepaUHOHHbIX rrposhy

ueCCOB C perYJIHpyeMbIMH cBoitcTBaMH B TOM IHCJIe H H3BeCTHble MeTonbI 06paTHbIX

HTepaUHit 06paTHble HTepaUHH C p)JIeeBCKHM cnBHrOM H npyrHe

IhepaUHoHHaH cxeMa (7) nJIH ypaBHeHHH (8) rrpH cPHKcHpoBaHHoM 3HaIeHHH

BeKTop-napaMeTpa a npencTaBJIHeT Ha Ka)KnOM llIare l-nepaliHit CHCTeMY OTHOCHshy

TeJIbHO HTepaUHoHHoit rronpaBKH ~Zk = ~Ak ~Wk

(H - Ak)~Pk = -(H - )kI)Wk + ~)kWk (10)

FH)kl Wk)~)k + F~()k IIk)~IIk = -F()k Ilk)

B 3aBHCHMOCTH OT crroc06a pellIeHHH Hoit CHCTeMbI H BbI60pa cPOPMbI cPYHKshy

UHOHaJIa F MO)KHO nOJIYIHTb pa3Hble H3BeCTHble HTepaUHOHHble cxeMbI pellIeHHH

CneKTpaJIbHbIX 3anaq IlpencTaBJIHH ~ II k B BHne

(11)

3

rrOJIy-qaeM CJIe1lYIOmee BbIpaJKeHHe 1lJIlI ~Ak

~Ak= 1+(Wkl Wk) 2(Wkl Uk)

npH Tk = 1 rrOJIy-qaeM CJIe1lYIOmee BblpaJKeHHe LIJI1I HOBbIX rrpH6JIHJKeHHtt

Wk+l = ~Ak(H - AkI)-IWk (12)

A A + H(W W) k+l = k 2(W (H-gtI) IW)

OTCIO1la BH1lHO -qTO rrOJIy-qeHa H3BeCTHall cxeMa o6paTHblx HTepaUHlt

npH HCrrOJIb30BaHHH ltPYHKUHOHaJIa F B ltpopMe (96) rrOJIy-qaeM CJIe1lYIOmyIO CH-

CTeMY OTHOCHTeJIbHO HTepaUHOHHbIX rrorrpaBOK

HCrrOJIb3Yll rrepBoe ypaBHeHHe 3Tott CHCTeMbI rrOJIy-qaeM H3 BTOPOIO ypaBHeHHlI

ECJIH orrepaTOp H CaMOCOrrpllJKeHHbItt TO

(13)

(14)

B COOTHOlIIeHHe (12) 1laeT

HJIH

Wk+l = ~Ak(H - AkI)-lWk (15)

_ (WHW) Ak+l - (11)11)) bull

4

rrOJIy-qaeM CJIe1lYIOmee BbIpaJKeHHe 1lJIlI ~Ak

~Ak= 1+(Wkl Wk) 2(Wkl Uk)

npH Tk = 1 rrOJIy-qaeM CJIe1lYIOmee BblpaJKeHHe LIJI1I HOBbIX rrpH6JIHJKeHHtt

Wk+l = ~Ak(H - AkI)-IWk (12)

A A + H(W W) k+l = k 2(W (H-gtI) IW)

OTCIO1la BH1lHO -qTO rrOJIy-qeHa H3BeCTHall cxeMa o6paTHblx HTepaUHlt

npH HCrrOJIb30BaHHH ltPYHKUHOHaJIa F B ltpopMe (96) rrOJIy-qaeM CJIe1lYIOmyIO CH-

CTeMY OTHOCHTeJIbHO HTepaUHOHHbIX rrorrpaBOK

HCrrOJIb3Yll rrepBoe ypaBHeHHe 3Tott CHCTeMbI rrOJIy-qaeM H3 BTOPOIO ypaBHeHHlI

ECJIH orrepaTOp H CaMOCOrrpllJKeHHbItt TO

(13)

(14)

B COOTHOlIIeHHe (12) 1laeT

HJIH

Wk+l = ~Ak(H - AkI)-lWk (15)

_ (WHW) Ak+l - (11)11)) bull

4

8Ta ltp0pMYJIa BMeCTe C Bblpa)KeHHeM (12) nJIlI npH6JIH)KeHHlI AkH npHBonHT K

H3BeCTHoA cxeMe o6paTHblx HTepanHA C p3JIeeBCKHM CnBHroM

IIoMHMo HenpepbIBHoro aHaJIOra MeTona HbIOToHa MO)KHO paCCMOTpeTb Henpeshy

pbIBHbIA aHaJIOr MOnHltPHnHpOBaHHoro MeTona HbIOTOHa OH npenCTaBJIlIeTClI 3Boshy

JIIOnHOHHbIM npOneCCOM

ltp(az(t))d~~t) = -ltp(az(t)) (16)

z(O) = zo (17)

rne z-HeKOTopbIA ltpHKCHpOBaHHbIA 3JIeMeHT H3 OKpeCTHOCTH HCKOMOro pemeHHlI z

8TOT nonxon naeT HTepanHOHHbIe cxeMbI THna (7) B KOTOPbIX onepaTOp ltp(a z(t))

Tpe6yeTClI o6paTHTb JIHmb OnHH pas B CneKTpaJIbHbIX 3anaqax Korna HeH3BeCTHoe

z COCTOHT H3 nBYX KOMnOHeHT A H II MO)KHO ltpHKcHpoBaTb KaK o6e TaK H OnHY

H3 3THX KOMnOHeHT B 3aBHCHMOCTH OT TOro HaCKOJIbKO xopomo H3BeCTHO COOTBeTshy

CTBYIOrnee npH6JIH)KeHHe K HCKOMOMY pemeHHIO HanpHMep npH ltpHKCHpOBaHHOM

3HaqeHHH Ak = x H3 CHCTeMbI (10) CltpYHKnHOHaJIOM F B ltpopMe (9a) nOJIyqaeM

(18)

(19)

PemeHHe ypaBHeHHlI (18) MO)KHO npenCTaBHTb B BHne

(20)

llPH 3TOM Vk paBHO

(21)

a Wk paBHO

(22)

IIoncTaBJIlIlI BblpalKeHHlI (21) H (22) B (20) nOJIyqaeM

5

3

TIPH T1 = 1 HMeeM

- 1W1+1 = (gt1+1 - gt)(H - gt1I)- WI (23)gt _ x + l+(IlI Ill)

1+1 - 2(1lI (H_~l-lll1)

ITO HBJIHeTCH cxeMoA 06paTHbIx HTepaUHA C ltpHKCHpOBaHHbIM CnBHroM 2hy CXeMY

ueJIeco06pa3HO HCIIOJIb30BaTb B COIeTaHHH C 0PToroHaJIH3aUHeA HaAneHHOrO IIpHshy

6JIH)KeHHH llI1+1 KO BCeM Y)Ke HaAneHHbIM C06CTBeHHbIM 3JIeMeHTaM Wn rne m

HOMep C06CTBeHHOrO 3JIeMeHTa IIpH YCJIOBHH ITO 3TOT Ha60p IIpenCTaBJIHeT OpTOshy

roHaJIbHYIO CHCTeMY C HeKOTOpbIM BeCOM

CPOPMYJIHpOBKa 3aAaqH paCCeHBHH KaK 3aAaqH Ba

co6cTBeBBhIe 3BaqeBHH Ba OCBOBe BapHaDHoBBoro

clgtYBKDHoBana IIIBHBrepa H YCTOitqHBaH

HTepauHOBBaH cxeMa

1 PaccMoTpHM 3anaIY pacceHHHH nJIH panHaJIbHoro ypaBHeHHH IIIpenHHrepa Ha

IIOJIYOCH 0 lt r lt 00 C KOpoTKoneAcTBYIOlUHM IIOTeHnHaJIOM V(r) IIpH 3anaHHOM HMshy

IIyJIbCe k

~ + ~pound _ l(l + 1) + k 2 ) llI(r) = 2V(r)II(r) (24)( dr2 r dr r2

KOTopaH 3aKJIIOIaeTCH B HaXO)KneHHH peryJIHPHbIX peIIIeHHA II (r) C aCHMIITOTMIeshy

CKHMH YCJIOBHHMH

II(r)rl r-+O (25)

T() ASin(kr - rrl2 +61) I r ) r -+ 00 (26)

r

H BbIIHCJIeHHH ltpa3bI 61 3necb 1 Op6HTaJIbHbIA MOMeHT TIOJIO)KHB A = (k cos 6)-1

H paccMaTpHBaH perYJIHpHble peIIIeHHH II(r) C aCHMIITOTHKOA

T( ) sin(kr - rrl2) +coskr - rrl2)tg61 I r IV kr r -+ 00 (27)

3aMeTHM ITO ORH npenCTaBHMbI B BHne [1 2]

6

rlle

ht(r) = -YI(7) + ijl(r) rlt = minrr rgt = maxrr (29)

jl( r) cltpepHlIeCKHe ltPYHKIJHH BecceJIlI Y( r) - ltPYHKIJHlI HeitMaHa

ACHMrrToTHKa perneHHlI Wr) rrpH r -+ 00 HMeeT BHll

00 Wr) -+ jlkr) - 2k 10 j[kr)V(r)W(r)r2drhtkr) (30)

0603HalIHM

V[-1(61) = 2[00 j[kr)Vr)W(r)r2dr (31)middotil

C rrOMOlIJblO 3TOrO Bblpa)KeHHlI MOJKHO rroJlytJHTb CJleJlYIOlIJee TO)KJJeCTBO

00

lJi(r) + 2k1 i(krlt)ht(krraquoV(r)lJi(r)r2dr = itkr)v(t5t21OO i(kr)V(r)lJi(r)r2dr (32)

HJIH

00 00

lJi(r) - 2k 1 i(krlt)y(krraquoV(r)lJi(r)r2dr =gt(6)2i(kr) 1 i(kr)V(r)lJi(r)r2dr (33)

3llecb

(34)

orrpelleJIlIeT 3HalIeHHe HCKOMoro ltpa30BOro CllBHra 6[ = 61 k) rrpH ltpHKcHpoBaHHoM

3HalIeHHH HMrryJIbCa k

BBelleM HHTerpaJIbHble orrepaTopbI

A(rr)W(r) = W(r) - 2kJo j[(krdYl(krraquoV(r)W(r)r2dr (35)

B(r r)Wr) = 2jl( kr) Jooo jlkr) V(r)W(r)r 12 dr

Torlla (33) rrpHMeT BHJl

(A(r r) - gtB(r r))Wr) = 0 (36)

rlle gt = -k cot 61 CrreKTpaJIbHblit rrapaMeTp ilorrOJIHHM ypaBHeHHe (36) YCJIOBHeM

opTOrOHaJIbHOCTH

Fgt Ill) = Wr )V(r )r2 (Ar r) - gtB(r r )W(r)) = O (37)

vb YCJIOBHlI OPTOroHaJIbHOCTH CJIe)lyeT lITO

gt _ (W(r)V(r)r2 Ar r)W(r)) (38)

- (Wr)V(r)r2 B(r r)W(r))

7

9Ta ltp0pMYJIa aHaJIOrH-qHa p3JIeeBCKOMY -qaCTHOMY B 3aIla-qe IlHCKpeTHOrO CrreKTpa

OrrepaTOpa gtHeprHH OIlHaKO HMeeT JIHllIb CBOftCTBO cTauHOHapHOCTH a He 3KCTpeshy

MaJIbHOCTH B OTJIH-qHe OT CTaHIlapTHoft rrOCTaHOBKH 3aIla-qH pacceHHHH ypaBHeHHH

(36) (37) rr03BOJIHIOT CltP0PMYJIHpOBaTb CrreKTpaJIbHYIO 3aIla-qy OTHOCHTeJIbHO rrapbI

z = x W rrpH-qeM coocTBeHHblft 3JIeMeHT WorrpeIleJIeH C TO-qHOCTblO IlO KOHCTaH-

TbI

~(z) = ~~~r~~~W_ AB)W) =O (39)

9BOJIIOuHoHHoe ypaBHeHHe (4) IlJIH 3Toft 3aIla-qH rrpHoopeTaeT ltPOpMy

(A shydw

xB)- shydt

dXBwshy =

dt -(A shy xB)W (40)

ECJIH orrepaTOp V(A shy xB)W caMocorrpHLKeHHblft H BblrrOJIHHeTCH YCJIOBHe

TO HMeeT MeCTO COOTHOIIIeHHe

(Ir2 V(A - gtB) ~~) + (Ir2 V ((A - gtB) ~~ - BI ~~) ) = -(Ir2 V(A - gtB)I) (41)

HCrrOJIb3YH (40) 113 (41) HMeeM

dW)(

Wr2 V(A - XB)di = o (42)

TorIla HCrrOJIb3YH rrpeIlCTaBJIeHHe

dw oW oW dX (43)di = fit + ox dt

H3 (40) HMeeM

~~ = -W (44)

(A - xB)~~ = Bw

llo)JcTaHoBKa (43) (44) B COOTHOllIeHHe (42) IlaeT

dX (WVr2 (A - xB)W) -= (45)dt (WVr2 BW)

HJIH B HHTerpaJIbHOM BHIle

00 dgt = 10 Il(r)V(r)r2dr - 2k 1000 loco I(r)V(r)i (krdYI (krraquoV(r)I(r)r2 r2drdr _ gt (46)

dt 2 (It il(kr)V(r)I(r)r2dr l )2

8

Cl)YHKnHOHaJI rrpaBoA qaCTH (46) COBrralJaeT C BapHanHOHHhIM ltPYHKnHOHaJIOM IllBHHshy

repa (1)

ilPH lJHCKpeTH3anHH BhIpa)KeHHe (33) rrpHHHMaeT BHlJ

N N

i(ri) - 2k L GijV(rj)rjtji(rj) = X(8)2il(kri) Lil(krj)V(rj)rjtji(rj) i = rn (47) j=l j=l

ilaJIee MO)KHO 3arrHcaTb

(A shy XB)q(r) = 0 (48)

rlJe

Ahj = Jij shy 2kGij V(rj)rJej (49)

Bij = 2jl(kri)jl(krj)V(rj)rjej (50)

B 3TOM cJIyqae rrOJIyqaeM CJIenyIOIUHA HTepanHoHHhIA rrponecc

(X - XnB)Un = Bln _ (qnVr2 Aqn) (51)JLn - (qnVr2 Bqn) - An

ln+l = ln +Tn(Vn +UnJLn)

Xn+1 = Xn +TnJLn

n = 012 Ao lo 3alJaHhI H lJJIlI BhI60pa HTepanHOHHoro mara Tn HCrrOJIb3Yshy

eM crroco6 OCHOBaHHbIA Ha MHHHMH3anHH HeBH3KH [8J llPH qHCJIeHHOM pemeHHH

Ha caMOM lJeJIe rrJIOXO 06YCJIOBJIeHHOA (041 107 ~ cond ~ 030 1010) CHCTeMhI

(51) HCrrOJIb30BaHhI rrporpaMMhI HOBOro rraKeTa JINRLINPACK [12]

2 OTlJeJIbHO paCCMOTpUM 3anaqy paCCelIHHlI lJJIlI OlJHOMepHoro ypaBHeHHlI Illpeshy

lJHHrepa Ha BceA OCH (-0000) C KOpOTKOlJeAcTBYIOIUHM rrOTeHnHaJIOM V(x) rrpH

3anaHHOM HMrrYJIbCe k

C= +k) W(X) = 2V(x)W(X) (52)

9

rlle

ht(r) = -YI(7) + ijl(r) rlt = minrr rgt = maxrr (29)

jl( r) cltpepHlIeCKHe ltPYHKIJHH BecceJIlI Y( r) - ltPYHKIJHlI HeitMaHa

ACHMrrToTHKa perneHHlI Wr) rrpH r -+ 00 HMeeT BHll

00 Wr) -+ jlkr) - 2k 10 j[kr)V(r)W(r)r2drhtkr) (30)

0603HalIHM

V[-1(61) = 2[00 j[kr)Vr)W(r)r2dr (31)middotil

C rrOMOlIJblO 3TOrO Bblpa)KeHHlI MOJKHO rroJlytJHTb CJleJlYIOlIJee TO)KJJeCTBO

00

lJi(r) + 2k1 i(krlt)ht(krraquoV(r)lJi(r)r2dr = itkr)v(t5t21OO i(kr)V(r)lJi(r)r2dr (32)

HJIH

00 00

lJi(r) - 2k 1 i(krlt)y(krraquoV(r)lJi(r)r2dr =gt(6)2i(kr) 1 i(kr)V(r)lJi(r)r2dr (33)

3llecb

(34)

orrpelleJIlIeT 3HalIeHHe HCKOMoro ltpa30BOro CllBHra 6[ = 61 k) rrpH ltpHKcHpoBaHHoM

3HalIeHHH HMrryJIbCa k

BBelleM HHTerpaJIbHble orrepaTopbI

A(rr)W(r) = W(r) - 2kJo j[(krdYl(krraquoV(r)W(r)r2dr (35)

B(r r)Wr) = 2jl( kr) Jooo jlkr) V(r)W(r)r 12 dr

Torlla (33) rrpHMeT BHJl

(A(r r) - gtB(r r))Wr) = 0 (36)

rlle gt = -k cot 61 CrreKTpaJIbHblit rrapaMeTp ilorrOJIHHM ypaBHeHHe (36) YCJIOBHeM

opTOrOHaJIbHOCTH

Fgt Ill) = Wr )V(r )r2 (Ar r) - gtB(r r )W(r)) = O (37)

vb YCJIOBHlI OPTOroHaJIbHOCTH CJIe)lyeT lITO

gt _ (W(r)V(r)r2 Ar r)W(r)) (38)

- (Wr)V(r)r2 B(r r)W(r))

7

9Ta ltp0pMYJIa aHaJIOrH-qHa p3JIeeBCKOMY -qaCTHOMY B 3aIla-qe IlHCKpeTHOrO CrreKTpa

OrrepaTOpa gtHeprHH OIlHaKO HMeeT JIHllIb CBOftCTBO cTauHOHapHOCTH a He 3KCTpeshy

MaJIbHOCTH B OTJIH-qHe OT CTaHIlapTHoft rrOCTaHOBKH 3aIla-qH pacceHHHH ypaBHeHHH

(36) (37) rr03BOJIHIOT CltP0PMYJIHpOBaTb CrreKTpaJIbHYIO 3aIla-qy OTHOCHTeJIbHO rrapbI

z = x W rrpH-qeM coocTBeHHblft 3JIeMeHT WorrpeIleJIeH C TO-qHOCTblO IlO KOHCTaH-

TbI

~(z) = ~~~r~~~W_ AB)W) =O (39)

9BOJIIOuHoHHoe ypaBHeHHe (4) IlJIH 3Toft 3aIla-qH rrpHoopeTaeT ltPOpMy

(A shydw

xB)- shydt

dXBwshy =

dt -(A shy xB)W (40)

ECJIH orrepaTOp V(A shy xB)W caMocorrpHLKeHHblft H BblrrOJIHHeTCH YCJIOBHe

TO HMeeT MeCTO COOTHOIIIeHHe

(Ir2 V(A - gtB) ~~) + (Ir2 V ((A - gtB) ~~ - BI ~~) ) = -(Ir2 V(A - gtB)I) (41)

HCrrOJIb3YH (40) 113 (41) HMeeM

dW)(

Wr2 V(A - XB)di = o (42)

TorIla HCrrOJIb3YH rrpeIlCTaBJIeHHe

dw oW oW dX (43)di = fit + ox dt

H3 (40) HMeeM

~~ = -W (44)

(A - xB)~~ = Bw

llo)JcTaHoBKa (43) (44) B COOTHOllIeHHe (42) IlaeT

dX (WVr2 (A - xB)W) -= (45)dt (WVr2 BW)

HJIH B HHTerpaJIbHOM BHIle

00 dgt = 10 Il(r)V(r)r2dr - 2k 1000 loco I(r)V(r)i (krdYI (krraquoV(r)I(r)r2 r2drdr _ gt (46)

dt 2 (It il(kr)V(r)I(r)r2dr l )2

8

Cl)YHKnHOHaJI rrpaBoA qaCTH (46) COBrralJaeT C BapHanHOHHhIM ltPYHKnHOHaJIOM IllBHHshy

repa (1)

ilPH lJHCKpeTH3anHH BhIpa)KeHHe (33) rrpHHHMaeT BHlJ

N N

i(ri) - 2k L GijV(rj)rjtji(rj) = X(8)2il(kri) Lil(krj)V(rj)rjtji(rj) i = rn (47) j=l j=l

ilaJIee MO)KHO 3arrHcaTb

(A shy XB)q(r) = 0 (48)

rlJe

Ahj = Jij shy 2kGij V(rj)rJej (49)

Bij = 2jl(kri)jl(krj)V(rj)rjej (50)

B 3TOM cJIyqae rrOJIyqaeM CJIenyIOIUHA HTepanHoHHhIA rrponecc

(X - XnB)Un = Bln _ (qnVr2 Aqn) (51)JLn - (qnVr2 Bqn) - An

ln+l = ln +Tn(Vn +UnJLn)

Xn+1 = Xn +TnJLn

n = 012 Ao lo 3alJaHhI H lJJIlI BhI60pa HTepanHOHHoro mara Tn HCrrOJIb3Yshy

eM crroco6 OCHOBaHHbIA Ha MHHHMH3anHH HeBH3KH [8J llPH qHCJIeHHOM pemeHHH

Ha caMOM lJeJIe rrJIOXO 06YCJIOBJIeHHOA (041 107 ~ cond ~ 030 1010) CHCTeMhI

(51) HCrrOJIb30BaHhI rrporpaMMhI HOBOro rraKeTa JINRLINPACK [12]

2 OTlJeJIbHO paCCMOTpUM 3anaqy paCCelIHHlI lJJIlI OlJHOMepHoro ypaBHeHHlI Illpeshy

lJHHrepa Ha BceA OCH (-0000) C KOpOTKOlJeAcTBYIOIUHM rrOTeHnHaJIOM V(x) rrpH

3anaHHOM HMrrYJIbCe k

C= +k) W(X) = 2V(x)W(X) (52)

9

9Ta ltp0pMYJIa aHaJIOrH-qHa p3JIeeBCKOMY -qaCTHOMY B 3aIla-qe IlHCKpeTHOrO CrreKTpa

OrrepaTOpa gtHeprHH OIlHaKO HMeeT JIHllIb CBOftCTBO cTauHOHapHOCTH a He 3KCTpeshy

MaJIbHOCTH B OTJIH-qHe OT CTaHIlapTHoft rrOCTaHOBKH 3aIla-qH pacceHHHH ypaBHeHHH

(36) (37) rr03BOJIHIOT CltP0PMYJIHpOBaTb CrreKTpaJIbHYIO 3aIla-qy OTHOCHTeJIbHO rrapbI

z = x W rrpH-qeM coocTBeHHblft 3JIeMeHT WorrpeIleJIeH C TO-qHOCTblO IlO KOHCTaH-

TbI

~(z) = ~~~r~~~W_ AB)W) =O (39)

9BOJIIOuHoHHoe ypaBHeHHe (4) IlJIH 3Toft 3aIla-qH rrpHoopeTaeT ltPOpMy

(A shydw

xB)- shydt

dXBwshy =

dt -(A shy xB)W (40)

ECJIH orrepaTOp V(A shy xB)W caMocorrpHLKeHHblft H BblrrOJIHHeTCH YCJIOBHe

TO HMeeT MeCTO COOTHOIIIeHHe

(Ir2 V(A - gtB) ~~) + (Ir2 V ((A - gtB) ~~ - BI ~~) ) = -(Ir2 V(A - gtB)I) (41)

HCrrOJIb3YH (40) 113 (41) HMeeM

dW)(

Wr2 V(A - XB)di = o (42)

TorIla HCrrOJIb3YH rrpeIlCTaBJIeHHe

dw oW oW dX (43)di = fit + ox dt

H3 (40) HMeeM

~~ = -W (44)

(A - xB)~~ = Bw

llo)JcTaHoBKa (43) (44) B COOTHOllIeHHe (42) IlaeT

dX (WVr2 (A - xB)W) -= (45)dt (WVr2 BW)

HJIH B HHTerpaJIbHOM BHIle

00 dgt = 10 Il(r)V(r)r2dr - 2k 1000 loco I(r)V(r)i (krdYI (krraquoV(r)I(r)r2 r2drdr _ gt (46)

dt 2 (It il(kr)V(r)I(r)r2dr l )2

8

Cl)YHKnHOHaJI rrpaBoA qaCTH (46) COBrralJaeT C BapHanHOHHhIM ltPYHKnHOHaJIOM IllBHHshy

repa (1)

ilPH lJHCKpeTH3anHH BhIpa)KeHHe (33) rrpHHHMaeT BHlJ

N N

i(ri) - 2k L GijV(rj)rjtji(rj) = X(8)2il(kri) Lil(krj)V(rj)rjtji(rj) i = rn (47) j=l j=l

ilaJIee MO)KHO 3arrHcaTb

(A shy XB)q(r) = 0 (48)

rlJe

Ahj = Jij shy 2kGij V(rj)rJej (49)

Bij = 2jl(kri)jl(krj)V(rj)rjej (50)

B 3TOM cJIyqae rrOJIyqaeM CJIenyIOIUHA HTepanHoHHhIA rrponecc

(X - XnB)Un = Bln _ (qnVr2 Aqn) (51)JLn - (qnVr2 Bqn) - An

ln+l = ln +Tn(Vn +UnJLn)

Xn+1 = Xn +TnJLn

n = 012 Ao lo 3alJaHhI H lJJIlI BhI60pa HTepanHOHHoro mara Tn HCrrOJIb3Yshy

eM crroco6 OCHOBaHHbIA Ha MHHHMH3anHH HeBH3KH [8J llPH qHCJIeHHOM pemeHHH

Ha caMOM lJeJIe rrJIOXO 06YCJIOBJIeHHOA (041 107 ~ cond ~ 030 1010) CHCTeMhI

(51) HCrrOJIb30BaHhI rrporpaMMhI HOBOro rraKeTa JINRLINPACK [12]

2 OTlJeJIbHO paCCMOTpUM 3anaqy paCCelIHHlI lJJIlI OlJHOMepHoro ypaBHeHHlI Illpeshy

lJHHrepa Ha BceA OCH (-0000) C KOpOTKOlJeAcTBYIOIUHM rrOTeHnHaJIOM V(x) rrpH

3anaHHOM HMrrYJIbCe k

C= +k) W(X) = 2V(x)W(X) (52)

9

Cl)YHKnHOHaJI rrpaBoA qaCTH (46) COBrralJaeT C BapHanHOHHhIM ltPYHKnHOHaJIOM IllBHHshy

repa (1)

ilPH lJHCKpeTH3anHH BhIpa)KeHHe (33) rrpHHHMaeT BHlJ

N N

i(ri) - 2k L GijV(rj)rjtji(rj) = X(8)2il(kri) Lil(krj)V(rj)rjtji(rj) i = rn (47) j=l j=l

ilaJIee MO)KHO 3arrHcaTb

(A shy XB)q(r) = 0 (48)

rlJe

Ahj = Jij shy 2kGij V(rj)rJej (49)

Bij = 2jl(kri)jl(krj)V(rj)rjej (50)

B 3TOM cJIyqae rrOJIyqaeM CJIenyIOIUHA HTepanHoHHhIA rrponecc

(X - XnB)Un = Bln _ (qnVr2 Aqn) (51)JLn - (qnVr2 Bqn) - An

ln+l = ln +Tn(Vn +UnJLn)

Xn+1 = Xn +TnJLn

n = 012 Ao lo 3alJaHhI H lJJIlI BhI60pa HTepanHOHHoro mara Tn HCrrOJIb3Yshy

eM crroco6 OCHOBaHHbIA Ha MHHHMH3anHH HeBH3KH [8J llPH qHCJIeHHOM pemeHHH

Ha caMOM lJeJIe rrJIOXO 06YCJIOBJIeHHOA (041 107 ~ cond ~ 030 1010) CHCTeMhI

(51) HCrrOJIb30BaHhI rrporpaMMhI HOBOro rraKeTa JINRLINPACK [12]

2 OTlJeJIbHO paCCMOTpUM 3anaqy paCCelIHHlI lJJIlI OlJHOMepHoro ypaBHeHHlI Illpeshy

lJHHrepa Ha BceA OCH (-0000) C KOpOTKOlJeAcTBYIOIUHM rrOTeHnHaJIOM V(x) rrpH

3anaHHOM HMrrYJIbCe k

C= +k) W(X) = 2V(x)W(X) (52)

9

4

KOTopaH 3aKJIlOqaeTCH B HaXO)l(IIemrH perYJIHpHbIX pemeHHtt W(r) C aCI1MnTOTHWshy

CKHMH YCJIOBHHMH

w(x) ~ 0 X -t -00 (53 )

W(X) ~ Asin(kx +0) X -t 00 (54)

H BbIqHCJIeHHH ltpa3bI O B 3TOM CJIyqae CJIO)l(HO HaIIHcaTb pemeHHe B HHTerpaJIbshy

HOM BHIIe C nOMOIUblO nepBOrO rpaHHqHOrO YCJIOBHH (53) PaccMoTpHM CJIerWlOUlee

rpaHHqHOe YCJIOBHe BMeCTO (53)

w(x) -t 0 X -t a a lt 0 (55)

3aMeTHM ITO pemeHHH npeIICTaBHMO B BHIIe

w(x) sin(k(x - a)) - -k2(-k) (XI sin(k(xlt a)) cos(kxraquoV(x)w(x)(Lr (J6)COS a Ja

3IIeCb xlt = minxx xgt = maxxx ACHMnTOTHKH pemeHHH W(x) HMelOT BHII

Ilrx) -+ 0 x -t a (57) 00

Ilr(x) -+ sin(k(x - a)) - kco(ka) cos(kx) 1 sin(k(x - a))V(x)Ilr(x)dx x -t 00 (58)

0603HaqHM 00

v(o) = - kco(ka) 1 sin(k(x - a))V(x)w(x)dx (59)

TorIIa MO)l(HO HanHcaTb CJIellYlOIUee TO)l(IIeCTBO

00

w(x) + kc(ka) 1 sin(k(xlt - a))cos(kxraquoV(x)W(x)dr

oo

= A(0) __2____ sin(k(x a)) ra sin(k(x - a))V(r)W(x)dzJ (60)k cos(ka) Ja

-1 A(0) = -ljv(0) = -co-s-(k-a-)t-a-n~(0--)-+-s-in-(k-a-) (61)

onpeIIeJIHeT 3HaqeHHe HCKOMoro ltpa30BOrO CIIBHra 0 = o(k) OTClOIIa A = cl~~)

IJaJIee MbI MO)l(eM HCnOJIb30BaTb npeIIJIO)l(eHHbItt BbIme MeTOII

qHCJIeHHhle npHMephl H o6cY)KneHHe

IJJIH aHaJIH3a TOqHOCTH BbIqHCJIHTeJIbHott cxeMbI YII06HO paCCMalpHBalh IIpHMeshy

pbI C H3BeCTHbIMH aHaJIHTHqeCKHMH pemCIUfHMH

10

41 lloTeHUHan Mop3e

PaCCMOTpRM ypaBHeHRe (52) C rrOTeHURaJIOM Mop3e [10]

V(x) = M D (e- 20(x-xa) _ 2e-a (x-xaraquo) (62)

AHaJIRTRqecKoe pell1eHRe [I (x) COOTBeTCTBYlOIUee HerrpepbIBHoMY crreKTpy k gt 0

RMeeT BRn [l1J

e-O5

w(x) = ~ (evC u F(-d + 05 - isl- 2is e) - e-1Wel$ F(-d + 05 + is 1 + 2is e)) (63)

rne F - BblpO)KneHHaH rRrrepreOMeTpRqeCKaH ltPYHKURH

~ = 2de-o (x-xlt) d = v2M15 s =~ w = argf(I+2is)+argf(-d+05-is) (64) Q Q

R f - raMMa-ltPYHKURH

Oc06eHHocTblO paccMaTpHBaeMoit 3anaqH HBJIJIeTCJI TO qTO He3aBHCHMaJI rrepeshy

MeHHaJI x H3MeHJIeTCJI Ha Bceit OCH -00 lt x lt 00 rrpHqeM

V(x) ---+ +00 x -t -00 V(x) ---+ 0 x -t 00 (65)

a ltPYHKUHJI [I (x) HMeeT aCRMrrTOTHKH BHna

[I(x) ---+ 0 x -t -00 [I(x) ---+ sin(kx + 8) x -t 00 (66)

3neCb 8 - HCKOMaJI ltpa3a paCCeJIHHH - HMeeT BHn

8 = -kxa - s In(2d) -- w (67)

IIo3TOMY HH)Ke rrpenCTaBJIeHbI pe3YJIbTaTbI nBYX paCqeTOB C rpaHHqHbIMH YCJIoshy

BHJIMH (25) H (53) rr03BOJIJIlOIUHe oueHHTb BKJIan rrOTeHUHaJIa V (x) Ha OTpe3Ke

x E [-a 0] B pell1eHRe

B qHCJIeHHOM rrpHMepe 3HaqeHHJI rrapaMeTpOB

M = 8876 D = 0104 Q = 067 Xo = 209 (68)

COOTBeTCTBYlOT 3HaqeHHJIM rrapaMeTpOB B pa60Te [5] 8TOT rrOTeHUHaJI H306pa)KeH

Ha pHC 1 TOqHOCTb qHCJIeHHbIX pe3YJIbTaTOB 3aBHCHT OT rrapaMeTpOB Pa3HOCTHOit

ceTKH 1h = Xmin = Xo Xj = Xo + jh j = 1 N Xn = x max B Ta6JI 1 H

11

30

V(x)

20

10

x o

i i

o 6 9

PHC 1BH)1 nOTeHlIHaJI8 Mop3e V(x) C

n8p8MeTp8MH M=8876 D=0104

a=067 xa=209

It(x) 10

05

00

-05

-10 -t--o----------------- x -5 0 5 10 15 20 25 30 35

PHC 2BH)1 ltPYHKUHH Ij(x) npH k=020

CnJIOIllH8H JIHHHH p8CQeT no ltp0pMYJIe (63)

nYHKTHpH8H JIHHHH QHCJIeHHblit pe3YJIbT8T

Ta6JIHU8 1 ([)a3bl pacceHHHH -0 x E (0 35)

k h =01 hj2 hj4 h-tO -0[5J

020 12599077 12618001 12622719 12624285 1262287

014 09000893 09015871 09019605 09020844 09019789

010 06495453 06506877 06509724 06510669 06509868

008 05217001 05226381 05228720 05229496 05228852

lE- 4 6568483E shy 4 6580794E shy 4 6583863E shy 4 6584882E 4 658407E shy 4

Ta6JIHU8 2 ([)a3bl pacceHHHH -0 x E (-535)

k h =01 h-tO -OAH

020 12557224 12576763 12581631 12583246 12583252

014 08968501 08983957 08987808 08989086 08989090

010 06471009 06482795 06485731 06486705 06486709

008 05197008 05206686 05209097 05209897 05209899

1E-4 6542430E shy 4 6555129E shy 4 6558293E shy 4 6559343E shy 4 6559345E shy 4

12

30

V(x)

20

10

x o

i i

o 6 9

PHC 1BH)1 nOTeHlIHaJI8 Mop3e V(x) C

n8p8MeTp8MH M=8876 D=0104

a=067 xa=209

It(x) 10

05

00

-05

-10 -t--o----------------- x -5 0 5 10 15 20 25 30 35

PHC 2BH)1 ltPYHKUHH Ij(x) npH k=020

CnJIOIllH8H JIHHHH p8CQeT no ltp0pMYJIe (63)

nYHKTHpH8H JIHHHH QHCJIeHHblit pe3YJIbT8T

Ta6JIHU8 1 ([)a3bl pacceHHHH -0 x E (0 35)

k h =01 hj2 hj4 h-tO -0[5J

020 12599077 12618001 12622719 12624285 1262287

014 09000893 09015871 09019605 09020844 09019789

010 06495453 06506877 06509724 06510669 06509868

008 05217001 05226381 05228720 05229496 05228852

lE- 4 6568483E shy 4 6580794E shy 4 6583863E shy 4 6584882E 4 658407E shy 4

Ta6JIHU8 2 ([)a3bl pacceHHHH -0 x E (-535)

k h =01 h-tO -OAH

020 12557224 12576763 12581631 12583246 12583252

014 08968501 08983957 08987808 08989086 08989090

010 06471009 06482795 06485731 06486705 06486709

008 05197008 05206686 05209097 05209897 05209899

1E-4 6542430E shy 4 6555129E shy 4 6558293E shy 4 6559343E shy 4 6559345E shy 4

12

2 npellCTaBJIeHbI qHCJIeHHble pe3YJIbTaTbI BeJIHqHH -8 llJIH BbIqHCJIeHHH KOTOPbIX

HCnOJIb30BaJIHCb ltP0PMYJIbI (28) H (56) COOTBeTCTBeHHO BKJIall HHTepBaJIa x E

[-50] B 8 COCTaBJIHeT BeJIHqHHY nOpHllKa 10-2 bull

CXOllHMOCTb CeTOqHbIX pelIIeHHit 8h COOTBeTcTByeT TeOpeTHqeCKoit O(h2) TaK

KaK BeJIHqHHa

(69)

B Ta6JI 2 3KCTpanOJIHpOBaHHble 3HaqeHHH BeJIHqHHbI -8 cpaBHHBaIOTCH C aHashy

JIHTHqeCKHMH 3HaqeHHHMH -8AH BbIqHCJIeHHbIMH no ltp0pMYJIe (67) AHaJIH3 peshy

3YJIbTaTOB nOKa3bIBaeT ~iTO 3KCTpanOJIHIIHH npH h -+ 0 06ecneqHBaeT TOqHOCTb

BblqHCJIeHHH ltpa3bI ~ 10-5 + 10-6 bull Ha pHC 2 npHBelleHbI TOqHoe H qHCJIeHHoe

pelIIeHHH l1(x) npH k = 02 OHH HaXOllHTClI B xopOllieM COrJIaCHHbull

42 llOTeHlJHan cqepHQeCKoA 1IMbI

PaCCMOTpHM ypaBHeHHe (24) npH 1 = 0 C nOTeHIIHaJIOM cltpepHqeCKoit lIMbI

V(r) = - Vo lt 0 r lt ro (70) Orgt roo

AHaJIHTHqecKoe pelIIeHHe l1(r) npH k -+ 0 Ha HHrepBaJIe r E (0 ro) HMeeT BHll

l1o(r) ---t 1 sin(v2Vor) (71) cos(y2Voro) y2Vor

BeJIHqHHa ao llJIHHa pacceHHHH - onpelleJIHeTCH qepe3 80 no ltp0pMYJIe

-lim(k cot(80raquo= _ ((tan(y2Voro) _ 1) ro)-1 (72)k-+O ~ro)

B 3TOM npHMepe HaM llOCTaTOqHO BbIqHCJIHTb ao Ha HHTepBaJIe (0 s r s ro)

0603HaqHM bo = v2Voro H B pacqeTax HCnOJIb3yeM ro = 1 B Ta6JI 3 H 4 npell-

CTaBJIeHbI QRCJIeHHhIe pe3YJIbTaTbI BeJIHqRHbl ao nOJIyqeHHoit npH HCnOJIb30BaHHH

ltP0PMYJIbl (28) CXOllHMOCTb CeTOqHbIX pelIIeHHit 8h COOTBeTCTBYeT TeOpeTHqeCKoit

O(h2) TaK KaK BeJIHqRHa

(73)

13

li(r) li(r) 00 00

-02 -02

-04 -04

-06 -06

-08 -08

r-10 r -10 00 02 04 06 08 10 00 02 04 06 08 10

a 6

PHC 3 BH)J ltPYHKDHH W(r) npH 60 = 1f H a) k=1E-2 6) k=1E-5 CnJIOlllHall JIHHHH pacleT no

ltp0pMYJIe (71) nYHKTHpHaH JIHHHlI - lHCJIeHHhIA pe3YJIbTaT

Ta6JIHDa 3 1JHCJIeHHhIe pe3YJIbTaThI npH ao r E (01) k=1E-5

bo h =002 h-+O

021f -63966702 -63967619 -63967848 -63967924 -63967925

041f -06899402 -06900324 -06900554 -06900630 -06900631

061f 03799557 03798608 03798371 03798292 03798292

081f 07758865 07757809 07757545 07757457 07757457

101f 10002012 10000502 10000125 10000000 10000000

Ta6JIHDa 4 QHCJIeHHhIe pe3YJIbTaThI npH ao r E (01) k=lE-2

AHbo h = 002 h2 h4 h-+O a0

0211 -63968484 -63969401 -63969630 -63969706 -63967925

0411 -06900041 -06900963 -06901193 -06901269 -06900631

0611 03799134 03798186 03797949 03797870 03798292

081f 07758527 07757471 07757207 07757119 07757457

101f 10001730 10000220 09999843 09999717 10000000

14

li(r) li(r) 00 00

-02 -02

-04 -04

-06 -06

-08 -08

r-10 r -10 00 02 04 06 08 10 00 02 04 06 08 10

a 6

PHC 3 BH)J ltPYHKDHH W(r) npH 60 = 1f H a) k=1E-2 6) k=1E-5 CnJIOlllHall JIHHHH pacleT no

ltp0pMYJIe (71) nYHKTHpHaH JIHHHlI - lHCJIeHHhIA pe3YJIbTaT

Ta6JIHDa 3 1JHCJIeHHhIe pe3YJIbTaThI npH ao r E (01) k=1E-5

bo h =002 h-+O

021f -63966702 -63967619 -63967848 -63967924 -63967925

041f -06899402 -06900324 -06900554 -06900630 -06900631

061f 03799557 03798608 03798371 03798292 03798292

081f 07758865 07757809 07757545 07757457 07757457

101f 10002012 10000502 10000125 10000000 10000000

Ta6JIHDa 4 QHCJIeHHhIe pe3YJIbTaThI npH ao r E (01) k=lE-2

AHbo h = 002 h2 h4 h-+O a0

0211 -63968484 -63969401 -63969630 -63969706 -63967925

0411 -06900041 -06900963 -06901193 -06901269 -06900631

0611 03799134 03798186 03797949 03797870 03798292

081f 07758527 07757471 07757207 07757119 07757457

101f 10001730 10000220 09999843 09999717 10000000

14

B Ta6JI 3 H 4 3KCTpanOJIHpOBaHHbie 3HaIeHHlI BeJIHIHHbl ao cpaBHHBaIOTClI C

aHaJIHTHIeCKHMH 3HaIeHHlIMH a~H BblIHCJIeHHblMH no ltp0pMYJIe (72) AHaJIH3 peshy

3YJIbTaTOB nOKa3bIBaeT ITO 3KCTpanOJIlIUHlI npH h -J 0 06ecneIHBaeT TOIHOCTb

BbIIHCJIeHHlI ao ~ 10-6 7 10-7 npH k = IE - 5 H ~ 10-3 npH k = IE - 2 Ha

pHC 3 llaHO cpaBHeHHe TOtHoro H tHCJIeHHOrO pellIeHHH IJI (x) npH b = 7f OHHo

HaXOUlITClI B xopOllIeM COrJIaCHH

5 3aKJlI01JeBHe

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H3BeCTHbIX ltpa30BOrO CUBHra H BOJIHOBOft ltPYHKUHH C nOMOWbIO BapHaUHoHHoro

ltPYHKUHOHaJIa IlIBHHrepa 9ltpltpeKTHBHOCTb npeUJIO)KeHHoft HTepaUHoHHoft cxeMbI

npoUeMOHCTpHpOBaHa Ha TOIHO pewaeMbIX npHMepax ynpyroro paCCelIHHlI C noshy

TeHUHaJIOM Mop3e H co cltpepHIeCKoft lIMoft I1peUJIO)KeHHblft noUxoU UonYCKaeT

nplIMOe 0606weHHe Ha MHoroMepHble H MHoroKaHaJIbHble 3aUaIH paCCelIHHlI npH

nOUXOUlIweM BbI60pe annpOKCHMaUHH peweHHft HanpHMep C nOMOWbIO cenapa6eJIbshy

HblX nOTeHUHaJIOB annpOKCHMaUHH BeftTMeHa a TaK)Ke B BHUe np06HbIX ltPYHKUHft

C 3aUaHHbIMH BapHaUHOHHbIMH napaMeTpaMH B 06JIaCTH UeftcTBHlI nOTeHUHaJIa BHe

3TOft 06JIaCTH 3aUaIOTClI aCHMnTOTHIeCKHe COCT01lHHlI C HeH3BecTHoft aMnJIHTyUoft

paccelIHHlI napaMeTpbI KOTOPOft HaXOUlITClI C nOMOWbIO npeUJIO)KeHHoft HTepaIlHshy

OHHoft cxeMbI

JIHTepaTypa

[1] MeccHa A KBaHTOBalI MexaHHKa M HaYKa 1978 T 1

[2] MeccHa A KBaHTOBalI MexaHHKa M HaYKa 1979 T 2

[3] 3y6apeB A JI 9lJA5I 1978 T 9 BbIn 2 C 453

15

[4] 3y6apeB A 1 BapHaUHoHHblfi rrpHHUHrr lllBI1Hrepa B KBaHTOBofi 1fXaHUKe

11 3HeproH3J(aT~ 1981

[5] BUHHUKHfi C II nY3bIHUU H B CmpHoB 1O Cj15IJ(cplIan (m3HKct 1990

T 52 BbIrr 4(10) C 1176

[6] )KaHnaB T nY3bIHHH n BI I KBMuMltIgt 1992 T 32 N 1 C 3

[7] )KaHrIaB T nY3bIHUH H BI I )KBMHMltIgt 1992 T 32 N 6 C 846

[8] EpMaKoB B B KaJlHTKHH H Hj IKBMHMltIgt T 21 C 491 1981

[9] ltIgtmorre 3 3auaQH ITO KBaHTOBofi MexaHHKe M MHP 1974 T 1

[10 Alhasid Y et al IIAnn Phys 1983 V 148 P 346

[II EMeJlbHHeHKo r A M Up nperrpHHT OH5I11 Pll-2000-287 1ly6na 2000

PYKonHCh nocrynHJIa B H3llaTeJIhCKHH OTlleJI 6 anpeIDI2001 rOlla

16

Qynyyn6aarap 0 n3blHHH MB BHHHuKHH CM Pll-2001-61 HbIOrOHOBCKasJ HrepanHOHHasJ CXeMa C BapHanHOHHbIM cpyHKUHOHanOM Il1BHHrepa JllliI perneHIDI 3auaqH paCCeSIHIDI

)InSI perneHIDI 3auaqH paCCeSIHIDI c 3auaHHOH TOQHOCTblO nocTpoeHa YCTOHQHshyBasJ HTepanHOHHasJ cxeMa Ha OCHOBe HenpepblBHoro aHanora MeTO)la HhIOTOHa 3ashynaQa paCCeSIHIDI cpopMynHpyeTCSI KaK 3auaQa Ha c06CTBeHHble 3HaQeHIDI OTHOCHshyTenbHO naphI HeH3BeCTHhlX cpaJOBoro C)lBHra H BOJIHOBOH CPyHKUHH C nOMOHblO BapHauHOHHoro CPyHKUHOHana Il1BHHrepa 3cpcpeKTHBHOCTh npellJIOgtKeHHOH HTeshyPaQHOHHOH cxeMhl npO)leMOHCTpHpOBaHa Ha TOQHO pernaeMhIX npHMepax 3auaQH ynpyroro paCCeSIHIDI C nOTeHUHanOM Mop3e H co ccpepHQeCKOH SIMOH

Pa60Ta HbmOJIHeHa B JIa60paTopHH HHcpopMaUHOHHhIX TeXHOJIOIHH H JIa6opashyTOpHH TeOpeTHQeCKOH CPH3HKH HM HHEoroJII060Ba OIUlH npH nOlIllepgtKKe PltIgtltIgtI1 rpaHThI N 00-01-00617 N 00-02-16337 N 00-02-81023 Bel 2000 a

npenpHHT 06benHHeHHoro HHCTH-ryra j1nepHblX HccnenOBaHHH Jly6Ha 2001

nepeBO)l aBTOPOB

Chuluunbaatar 0 Puzynin 1 V Vinitsky Sl Pll-2001-61 A Newtonian Iteration Scheme with the Schwinger Variational Functional for Solving a Scattering Problem

A Newtonian iteration scheme has been constructed for solving a scattering problem with using the Schwinger variational fucntional The scattering problem is formulated as an eigenvalue problem with respect to a pair of the unknowns a phase shift and a wavefunction An efficiency of the proposed iteration sheme and its accuracy are demonstrated on exact solvable examples of elastic scattering problem with Morze and spherical potentials

The investigation has been performed at the Laboratory of Information Technologies and at the Bogoliubov Laboratory of Theoretical Physics lINR and supported by the RFBR Grants No 00-01-00617 No 00-02-16337 No 00-02-81023 Bel 2000 a

Preprint of the Joint Institute for Nuclear Research Dubna 2001

[4] 3y6apeB A 1 BapHaUHoHHblfi rrpHHUHrr lllBI1Hrepa B KBaHTOBofi 1fXaHUKe

11 3HeproH3J(aT~ 1981

[5] BUHHUKHfi C II nY3bIHUU H B CmpHoB 1O Cj15IJ(cplIan (m3HKct 1990

T 52 BbIrr 4(10) C 1176

[6] )KaHnaB T nY3bIHHH n BI I KBMuMltIgt 1992 T 32 N 1 C 3

[7] )KaHrIaB T nY3bIHUH H BI I )KBMHMltIgt 1992 T 32 N 6 C 846

[8] EpMaKoB B B KaJlHTKHH H Hj IKBMHMltIgt T 21 C 491 1981

[9] ltIgtmorre 3 3auaQH ITO KBaHTOBofi MexaHHKe M MHP 1974 T 1

[10 Alhasid Y et al IIAnn Phys 1983 V 148 P 346

[II EMeJlbHHeHKo r A M Up nperrpHHT OH5I11 Pll-2000-287 1ly6na 2000

PYKonHCh nocrynHJIa B H3llaTeJIhCKHH OTlleJI 6 anpeIDI2001 rOlla

16

Qynyyn6aarap 0 n3blHHH MB BHHHuKHH CM Pll-2001-61 HbIOrOHOBCKasJ HrepanHOHHasJ CXeMa C BapHanHOHHbIM cpyHKUHOHanOM Il1BHHrepa JllliI perneHIDI 3auaqH paCCeSIHIDI

)InSI perneHIDI 3auaqH paCCeSIHIDI c 3auaHHOH TOQHOCTblO nocTpoeHa YCTOHQHshyBasJ HTepanHOHHasJ cxeMa Ha OCHOBe HenpepblBHoro aHanora MeTO)la HhIOTOHa 3ashynaQa paCCeSIHIDI cpopMynHpyeTCSI KaK 3auaQa Ha c06CTBeHHble 3HaQeHIDI OTHOCHshyTenbHO naphI HeH3BeCTHhlX cpaJOBoro C)lBHra H BOJIHOBOH CPyHKUHH C nOMOHblO BapHauHOHHoro CPyHKUHOHana Il1BHHrepa 3cpcpeKTHBHOCTh npellJIOgtKeHHOH HTeshyPaQHOHHOH cxeMhl npO)leMOHCTpHpOBaHa Ha TOQHO pernaeMhIX npHMepax 3auaQH ynpyroro paCCeSIHIDI C nOTeHUHanOM Mop3e H co ccpepHQeCKOH SIMOH

Pa60Ta HbmOJIHeHa B JIa60paTopHH HHcpopMaUHOHHhIX TeXHOJIOIHH H JIa6opashyTOpHH TeOpeTHQeCKOH CPH3HKH HM HHEoroJII060Ba OIUlH npH nOlIllepgtKKe PltIgtltIgtI1 rpaHThI N 00-01-00617 N 00-02-16337 N 00-02-81023 Bel 2000 a

npenpHHT 06benHHeHHoro HHCTH-ryra j1nepHblX HccnenOBaHHH Jly6Ha 2001

nepeBO)l aBTOPOB

Chuluunbaatar 0 Puzynin 1 V Vinitsky Sl Pll-2001-61 A Newtonian Iteration Scheme with the Schwinger Variational Functional for Solving a Scattering Problem

A Newtonian iteration scheme has been constructed for solving a scattering problem with using the Schwinger variational fucntional The scattering problem is formulated as an eigenvalue problem with respect to a pair of the unknowns a phase shift and a wavefunction An efficiency of the proposed iteration sheme and its accuracy are demonstrated on exact solvable examples of elastic scattering problem with Morze and spherical potentials

The investigation has been performed at the Laboratory of Information Technologies and at the Bogoliubov Laboratory of Theoretical Physics lINR and supported by the RFBR Grants No 00-01-00617 No 00-02-16337 No 00-02-81023 Bel 2000 a

Preprint of the Joint Institute for Nuclear Research Dubna 2001

Qynyyn6aarap 0 n3blHHH MB BHHHuKHH CM Pll-2001-61 HbIOrOHOBCKasJ HrepanHOHHasJ CXeMa C BapHanHOHHbIM cpyHKUHOHanOM Il1BHHrepa JllliI perneHIDI 3auaqH paCCeSIHIDI

)InSI perneHIDI 3auaqH paCCeSIHIDI c 3auaHHOH TOQHOCTblO nocTpoeHa YCTOHQHshyBasJ HTepanHOHHasJ cxeMa Ha OCHOBe HenpepblBHoro aHanora MeTO)la HhIOTOHa 3ashynaQa paCCeSIHIDI cpopMynHpyeTCSI KaK 3auaQa Ha c06CTBeHHble 3HaQeHIDI OTHOCHshyTenbHO naphI HeH3BeCTHhlX cpaJOBoro C)lBHra H BOJIHOBOH CPyHKUHH C nOMOHblO BapHauHOHHoro CPyHKUHOHana Il1BHHrepa 3cpcpeKTHBHOCTh npellJIOgtKeHHOH HTeshyPaQHOHHOH cxeMhl npO)leMOHCTpHpOBaHa Ha TOQHO pernaeMhIX npHMepax 3auaQH ynpyroro paCCeSIHIDI C nOTeHUHanOM Mop3e H co ccpepHQeCKOH SIMOH

Pa60Ta HbmOJIHeHa B JIa60paTopHH HHcpopMaUHOHHhIX TeXHOJIOIHH H JIa6opashyTOpHH TeOpeTHQeCKOH CPH3HKH HM HHEoroJII060Ba OIUlH npH nOlIllepgtKKe PltIgtltIgtI1 rpaHThI N 00-01-00617 N 00-02-16337 N 00-02-81023 Bel 2000 a

npenpHHT 06benHHeHHoro HHCTH-ryra j1nepHblX HccnenOBaHHH Jly6Ha 2001

nepeBO)l aBTOPOB

Chuluunbaatar 0 Puzynin 1 V Vinitsky Sl Pll-2001-61 A Newtonian Iteration Scheme with the Schwinger Variational Functional for Solving a Scattering Problem

A Newtonian iteration scheme has been constructed for solving a scattering problem with using the Schwinger variational fucntional The scattering problem is formulated as an eigenvalue problem with respect to a pair of the unknowns a phase shift and a wavefunction An efficiency of the proposed iteration sheme and its accuracy are demonstrated on exact solvable examples of elastic scattering problem with Morze and spherical potentials

The investigation has been performed at the Laboratory of Information Technologies and at the Bogoliubov Laboratory of Theoretical Physics lINR and supported by the RFBR Grants No 00-01-00617 No 00-02-16337 No 00-02-81023 Bel 2000 a

Preprint of the Joint Institute for Nuclear Research Dubna 2001