Semiclassical Soliton-Type Solutions of the Hartree Equation

Transcription

1 ISSN Doklady Maemacs 7 Vol. 76 No. pp 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 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/S 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

2 776 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

3 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

4 778 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

5 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 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 (989).. L. P. Paevsk Usp. Fz. Nauk (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 (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 (979).. M. V. Karasev ad A. V. Pereskokov Izv. Akad. Nauk Ser. Ma ().. 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 (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 (). DOKLADY MATHEMATICS Vol. 76 No. 7