ISSN 64 564 Doklady Maemacs 7 Vol. 76 No. pp. 775 779. Pleades Publsg Ld. 7. Orgal Russa Te V.V. Belov M.F. Kodra eva E.I. Smrova 7 publsed Doklady Akadem Nauk 7 Vol. 46 No. pp. 77 8. MATHEMATICAL PHYSICS Semclasscal Solo-Type Soluos of e Harree Equao V. V. Belov a M. F. Kodra eva b ad E. I. Smrova a Preseed by Academca V.P. Maslov Ocober 5 6 Receved May 8 7 DOI:.34/S645647534 INTRODUCTION Cosder e model Harree equao (self-cosse feld equao) a eeral feld U() w a raslao-vara self-aco poeal V( y) y : Ĥ κ ----Ψ Ĥ κ Ψ Ψ ( ) Ĥ κ [ Ψ] Ĥ κ Ψ( y ) V( y) dy pˆ Ĥ ------ U( ) pˆ. m () () Here κ are real parameer ad ( ] s a small real parameer. Ts uary-olear equao plays a fudameal role quaum eory ad olear opcs [] parcular e eory of a Bose Ese codesae [] ad e descrpo of collecve ecaos molecular cas ad DNA molecules [3 4]. Equaos of ype () ave bee eesvely suded. We oe oly [5 6] were umerous refereces ca be foud. A class of uary-olear equaos coag equaos of form () bu wou sgulares er coeffces was suded [7 9]. Te egevalue problem for operaor () w a sgular self-aco poeal was aalyzed [] by applyg a sgular verso of e WKB meod w e elp of referece equaos. a Moscow Sae Isue of Elecrocs ad Maemacs (Teccal Uversy) Bol so Treksvyael sk per. 3/ Moscow 98 Russa e-mal: belov@mem.edu.ru kaerasm@bk.ru b Deparme of Maemacs ad Sascs Memoral Uversy of Newfoudlad S. Jo s AC 5S7 Caada e-mal: mkodra@ma.mu.ca Gve Eq. () w smoo poeals U() ad V(τ) e goal of s work s o cosruc localzed formal asympoc soluos as ad esabls e codos uder wc ese soluos ca be reaed as semclasscal solo soluos (semclasscal solos). Frs e ype of localzed asympoc soluos of eres s descrbed e lear case.e. we κ (). To do s we cosder e evoluo of a compressed coere sae e semclasscal appromao. Specfcally we se Ψ Ψ ( ) N ep -- ( p ) ep -- ( B (. (3) Here (p ) s a arbrary fed po of e pase space B s a arbrary symmerc mar w a posve magary par (ImB > ) N (π /4 (deimb) /4 ad ( ) s e er produc. I s well kow a a major po e semclasscal approac s a e cosruco of semclasscal asympocs for a quaum problem s reduced o e cosruco of soluos o e equao of moo of e correspodg classcal sysem ad o e sudy of s geomerc ad opologcal properes. For lear quaum mecacs (w κ ) e correspodg classcal sysem s a a pror Hamloa sysem w e Hamloa fuco H H (p ): ṗ U( ) ẋ p. (4) Te geomerc objecs geerag a semclasscal aswer s Caucy problem () (3) wc s smple from e po of vew of e geeral eory [ ] are e [ Λ r ]-zero-dmesoal Lagraga mafold Λ w a Maslov comple germ r were Λ s a po o e pase rajecory of sysem (4) p P( p ) p cl () X( p ) cl () wc sars a e po (p ) a. 775
776 BELOV e al. Te geeral formulas of e WKB Maslov comple meod gve e followg leadg erm of e asympocs (mod 3/ ) of problem () (3) (a κ ): Ψ( N ep -- ( S cl () ( p cl () cl () ) ep -- ( cl () BC () ( cl () ( de C ( /. (5) Here S cl () s e classcal aco S cl cl τ ( ) ------------- U( cl ( τ) ) dτ (6) ad e marces B() ad C() (defg e comple germ r {(w r) w BC wz r ()z}) are e mar soluos o e Caucy problem for e varaoal sysem (learzao of sysem (4) e egborood of ) Λ Ḃ U'' ( cl () ) B Ċ C B( ) B C( ) (( δ j (7) Asympoc soluo (5) (Gaussa wave packe) s localzed as e egborood of Λ o e classcal rajecory: lm supp Ψ cl () ad lm supp Ψ p cl () were Ψ s e Fourer rasform of Ψ. Suc solo-ype soluos cao be erpreed as semclasscal solos because ese wave packes dsperse as (a mes ~ ---------- were < δ < ). Ecep for e case δ we U() s e poeal of a quadrac oscllaor e dspersos of coordaes ad momea calculaed from e saes Ψ( ) (5) crease as o more slowly a a lear fuco of. I urs ou a e focusg effec due o e egral oleary a κ leads (a leas for cove selfaco poeals) o e esece of localzed Gaussa-packe asympocs of Eq. () a are smlar form o (5) for wc e dspersos of coordaes ad momea are bouded fucos of me [ ). Suc asympoc soluos are aurally erpreed as Gaussa semclasscal solos. Obvously e geeralzao of e above germ cosrucos o olear quaum sysems s a orval ask. Specfcally eve e saeme of e correspodece problem o classcal resuls as s raer problemac because s uclear ( coras o e lear case) wa we sould mea by e classcal. equaos of moo correspodg o quaum equao () w a olear Hamloa Ĥ κ (). Te followg seco gves a aswer o s problem based o e covara approac developed [4 5] wc s a eeso of e well-kow Erefes approac [3] o e dervao of classcal equaos of moo e appromao as for e (lear) Scrödger equao. Specfcally le Ψ( be a eac soluo (or a -appromae soluo ) o olear equao (). By e classcal pase rajecory p of a quaum parcle e sae Ψ Ψ( we mea a vecor fuco z Ψ ( (P( X( ) a depeds smooly o ad as compoes a are e meas of e coordae ad momeum operaors e sae Ψ: P( Ψ Ψ ad X( Ψ Ψ were s e er produc ( ). p. EQUATION OF CLASSICAL MECHANICS FOR THE HARTREE EQUATION IN THE CLASS OF SEMICLASSICALLY LOCALIZED STATES Le Z( (P( X( ) be a smoo oeparameer famly of pase rajecores (w parameer. Defe a class K of fucos depedg o ad [ ) wc s called e class of fucos semclasscally localzed o e rajecory Z( as : K ΦΦ φ ------------------------- X( ) ep -- ( S ( ) ( P ( ) X( ) (8) were S( ) s a smoo real fuco of ad S( ) ad φ(ξ ) s e Scwarz space S( ) w respec o ξ. Lemma. For fucos Φ from e class K e ceered momes α ( of order α α sasfy e esmaes α ( Φ Φ{ ˆ z} α Φ α ( ------------------------------ O ( α / ) Φ α α ( α were { ˆ z } α s a operaor w e Weyl symbol ( z) α z z Z( ( p; ) X( ad p p P(. Teorem. Te classcal sysem (mod 3/ ) assocaed w e olear operaor ad e class K Ĥ κ DOKLADY MATHEMATICS Vol. 76 No. 7
SEMICLASSICAL SOLITON-TYPE SOLUTIONS OF THE HARTREE EQUATION 777 (wc s called e Hamlo Erefes sysem) as e form ṗ U( ) κ τ V( ) --Sp( U'' ( ) κv'' ( y) )σ y (9) ẋ p JM κ ( ) M κ ( )J ( p ). Here s a real block symmerc mar σ pp σ p σp σ p σ σ σ p σ M κ ( ) σ pp σ pp U'' ( ) κv ττ '' ( ) U'' ( ) U( ) ----------------- j () ad J s a sadard symplecc mar. Proof skec. Sce e evoluo operaor of Eq. () s uary for soluos Ψ( of s equao we ave e Erefes eorem for e quaum averages  ( Ψ Â Ψ were  s a self-adjo operaor ( ): d ----  ( d () Assume a ere ess a asympoc (mod N ) N soluo Ψ( K o Eq. (). Usg relao () for operaors from e Heseberg Weyl uversal evelopg algebra w geeraors Î ˆ k ˆ k X k ( ad ˆ p k pˆ k P k ( were k ad Î s e dey operaor we epad e operaor Ĥ κ a Taylor seres powers of ˆ α ad ˆ p α p (were (α α p ) α ) ake o accou e esmaes from Lemma ad eglec e ceered quaum momes of order α ( α 3) o oba sysem (9). Remark. Te dyamc varables sysem (9) amely (σ pp ) km Ψ ˆ p k ˆ p m Ψ (σ ) km Ψ ˆ k ˆ m Ψ ) ad (σ p ) km -- Ψ ( ˆ k ˆ p m ˆ p m ˆ k )Ψ for k m wc are of order -- [ Ĥ Â] κ-- y Ψ* ( y )[ V( y) Â]Ψ y d ( ). ake o accou e fluece of quadrac quaum flucuaos of coordaes ad momea abou er lmg values X( ) ad P( ). Tus e classcal equaos deped regularly o e small parameer of e semclasscal appromao e quaum problem. Ts fac s of key mporace for e cosruco of localzed asympocs of e Harree equao e class of fucos K (8).. COHERENT STATES OF THE HARTREE EQUATION For problem () (3) le p (κ deoe e projeco oo of e soluo (P( X( ( ) o e Caucy problem o e erval [ T] T > for Hamlo Erefes equao (9) w al daa duced by Ψ ( (3): X Ψ Ψ P Ψ Ψ p ad Ψ ( were e blocks of s mar obvously ave e form σ -- σ pp -- [B B ] ad σ p 4 4 B *B -- [B B ]. Here e symbols * ad o e mar 4 * B deoe e Herme cojugae comple cojugae ad raspose respecvely. By aalogy w lear eory (see [7] Iroduco) e Hamlo sysem s called a varaoal sysem w a self-aco: Ḃ JM κ ( X () ) B () Ċ C were X () s e leadg erm of e epaso X( X () X () O( ) ad e mar JM κ () s defed (). Deoe by B B() ad C C() e mar soluo o s sysem w e al daa B B ad C. Teorem. Le e poeals U() ad V(τ) be fucos of e class C (3) ( ). Te for [ T] e asympoc soluo (mod 3/ ) o problem () (3) e ( ) orm of e rg-ad sde s localzed e egborood of e po Λ (κ ad as e form Ψ ν ( N ν ep -- ( S κ ( ( P ( ) X( ----- X( BC ep ( () ( X( ---------------------. de C () (3) Here e real pase S κ ( (e aalogue of e aco (6 s Λ DOKLADY MATHEMATICS Vol. 76 No. 7
778 BELOV e al. S κ ( Ẋ ----- ( τ U( X τ ( ) ) dτ κv( ) κ -- Sp ( V ττ '' ( )σ ( τ dτ. (4) To prove e eorem we epad e equao coeffces Taylor seres e egborood of Λ (κ apply Teorem ake o accou a e las equao sysem (9) s e well-kow La equao e verse scaerg meod w respec o e mar J ad e apply Lemma below. Lemma. Le A() be e Caucy mar of sysem (). Te e soluo o e Caucy problem for e sysem JM κ (X ( M κ (X (J ( s gve by e formula Ψ ( A () Ψ A (). 3. SEMICLASSICAL SOLITONS OF THE HARTREE EQUATION WITH NO EXTERNAL FIELD (5) Assumg a U() () we sae e problem for Eq. () w more geeral Caucy daa a ose (3). Namely le Ψ be e Fock saes of a muldmesoal oscllaor: Ψ Ψ ν ( ) N ν Ψ ( )φ ν ------------- ν (6) were Ψ ( s defed (3) ad φ ν s a muldmesoal Herme polyomal of mul-de ν wc s represeed erms of e geeralzed creao operaors φ ν ------------- Λˆ ν () were Λˆ ν () [ Λˆ ()] ν [ Λˆ ()] ν ad Λˆ j () ------ ------- j (Imw j ) (see e.g. []). Here w j (j ) are e colum vecors of B. I e case U() e sysem of equaos for (9) as cosa coeffces. Terefore e Hamlo Erefes sysem as well as e varaoal sysem self-aco () w e al daa duced by al fuco (6) ca easly be egraed. Te al codos for sysem (9) ave e form X P p σ σ pp D ν (( ν j )δ j ) σ p --D 4 ν (7) Assume a sysem () s sable;.e. e self-aco poeal sasfes e codo ω j (8) Deoe by (j ) e egevalues of e mar κv ττ '' (). Lemma 3. Le codo (8) be sasfed ad le e frequeces be o resoa;.e. ω l ω m lm l m. (9) Te e soluo o Caucy problem (9) (7) s gve by e formulas were Γ A lm ad B lm are cosa real marces ad a d lm k l m s l ad e l (l m ) are real vecors from wose eplc form ca easly be derved usg formulas (5) (for σ ) w e Caucy mar -- [ B 4 D ν B B *D ν B ] -- [ B 4 D ν D ν B ]. κv ττ '' ( ) >. ( a X ( ) p a ) ---------------------------- ( d lm cos( ( ω l ω m lm l m k lm s (( ) ω l ω m ( e l cos( ( ω l s l s( ( ω l ) l P ( ) Ẋ( a κ τ V( ) σ ( Γ ( cos( ( ω l ω m A lm A lm l m s (( ω l ω m B lm ) A ll cos( ω l ) l B ll s( ω l ) cos[ R] R / s[ R] Rs[ R] cos[ R] R κv ττ '' ( ) DOKLADY MATHEMATICS Vol. 76 No. 7
SEMICLASSICAL SOLITON-TYPE SOLUTIONS OF THE HARTREE EQUATION 779 ad e frs par of equaos sysem (9) (for X( P( ). Lemma 3 ad Teorem mply e followg resul. Teorem 3. Le codos (8) ad (9) be sasfed ad le V(τ) C 3 ( ). Te e asympoc semclasscal solo soluo o problem () (6) as e form Ψ ν ( N ν ep -- ( S κ ( ( P ( ) X( ep ----- ( X( ĊC () ( X( --------------Λˆ ν () () C () S κ ( Ẋ ----- τ ( ) dτ κv( ) κ -- Sp ( V ττ '' ( )σ ( τ ) dτ. Here e fucos X( ad σ () are defed Lemma 3; C() sasfes e sysem Ċ κv''()c C() Ċ () B Λˆ ν ν ν () [ Λˆ () ] [ Λˆ () ] Λˆ j () ------ (( z* j () pˆ ) ( ż* j () ĊC () z* j () ( X( ; ad z j () j are e colums of e mar C(). CONCLUSIONS I olear quaum mecacs w e model Hamloa Ĥ κ gve by () we ave derved eplc formulas () for Gaussa wave packes a do o dsperse e semclasscal appromao a leas e case of raslao-vara oresoa cove poeals. Te cosruco of suc wave packes e case of a arbrary eeral elecromagec feld requres a addoal sudy. I e case of omogeeous felds e formulas for odspersve solos wll be gve elsewere. ACKNOWLEDGMENTS Ts work was suppored by e Russa Foudao for Basc Researc projec os. 5--968 ad INI 5--. M.F. Kodra eva ackowledges e suppor of e Naural Sceces ad Egeerg Researc Coucl of Caada. REFERENCES. Y. La ad H. A. Haus Pys. Rev. A 4 854 866 (989).. L. P. Paevsk Usp. Fz. Nauk 68 64 653 (988). 3. A. S. Davydov Solos Molecular Sysems (Naukova Dumka Kev 984; Kluwer Dordrec 99). 4. V. D. Lako ad N. S. Falko Compuers ad Supercompuers Bology (Izevsk Moscow) [ Russa]. 5. E. H. Leb ad B. Smo Commu. Ma. Pys. 53 85 (977). 6. J. M. Cadam ad R. T. Glassey J. Ma. Pys. 6 (975). 7. V. P. Maslov Comple Markov Cas ad Feyma Coual Iegral (Nauka Moscow 975) [ Russa]. 8. V. P. Maslov Sovrem. Probl. Ma. 53 (978). 9. M. V. Karasev ad V. P. Maslov Sovrem. Probl. Ma. 3 45 (979).. M. V. Karasev ad A. V. Pereskokov Izv. Akad. Nauk Ser. Ma. 65 33 7 ().. V. P. Maslov Operaor Meods (Nauka Moscow 973) [ Russa].. V. P. Maslov Te Comple WKB Meod for Nolear Equaos (Nauka Moscow 977; Brkäuser Basel 994). 3. P. Erefes Zscr. P. 45 455 457 (97). 4. V. V. Belov ad A. Yu. Trfoov A. Pys. (New York) 46 () 3 8 (996). 5. V. V. Belov A. Yu. Trfoov ad A. V. Sapovalov Teor. Ma. Fz. 3 46 (). DOKLADY MATHEMATICS Vol. 76 No. 7