Semiclassical interpretation of Wei–Norman factorization for SU(1,1) and its related integral transforms

J. Ma h. Phys. 61, 082107 (2020); h ps://doi.o g/10.1063/1.5143586 61, 082107 © 2020 Au ho (s). Semiclassical in e p e a ion o Wei–No man ac o iza ion o SU(1, 1) and i s ela ed in eg al ans o ms Ci e as: J. Ma h. Phys. 61, 082107 (2020); h ps://doi.o g/10.1063/1.5143586 Submi ed: 25 Decembe 2019 . Accep ed: 25 July 2020 . Published Online: 19 Augus 2020 Julio Gue e o , and Manuel Be ondo ARTICLES YOU MAY BE INTERESTED IN Ma hema ics o he classical and he quan um Jou nal o Ma hema ical Physics 61, 082101 (2020); h ps://doi.o g/10.1063/1.5135015 Cohe en s a es o posi ion-dependen mass apped in an in ini e squa e well Jou nal o Ma hema ical Physics 61, 082102 (2020); h ps://doi.o g/10.1063/5.0015418 A quan um sys em wi h a non-He mi ian Hamil onian Jou nal o Ma hema ical Physics 61, 082106 (2020); h ps://doi.o g/10.1063/5.0011098 Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp Semiclassical in e p e a ion o Wei–No man ac o iza ion o SU(1,1)and i s ela ed in eg al ans o ms Ci e as: J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 Submi ed: 25 Decembe 2019 •Accep ed: 25 July 2020 • Published Online: 19 Augus 2020 Julio Gue e o1,2,a) and Manuel Be ondo3 AFFILIATIONS 1Depa men o Ma hema ics, Facul y o Expe imen al Sciences, Uni e si y o Jaen, Campus Las Lagunillas s/n, 23071 Jaén, Spain 2Ins i u e Ca los I o Theo e ical and Compu a ional Physics, Uni e si y o G anada, Fuen enue a s/n, 18071 G anada, Spain 3Depa men o Physics and As onomy, B igham Young Uni e si y, P o o, U ah 84602, USA a)Au ho o whom co espondence should be add essed: [email p o ec ed] ABSTRACT Wep esen anin e p e a iono he unc ionsappea ingin heWei–No man ac o iza iono hee olu ionope a o o aHamil onianbelong- ing o he SU(1,1) algeb a in e ms o he classical solu ions o he Gene alized Caldi ola–Kanai (GCK) oscilla o (wi h ime-dependen mass and equency). Choosing P2,X2, and he dila ion ope a o as a basis o he Lie algeb a, we ob ain ha , ou o he six possible o de ings o he Wei–No man ac o iza ion o he e olu ion ope a o o he GCK Hamil onian, h ee o hem can be exp essed in e ms o i s classical solu ions and he o he h ee in ol e he classical solu ions associa ed wi h a mi o Hamil onian ob ained by in e ing he mass. In addi ion, we gene alize he Wei–No man p ocedu e o compu e he ac o iza ion o o he ope a o s, such as a gene alized F esnel ans o m and he A nold ans o m (andi s gene aliza ions),ob aining alsoin hesecases asemiclassical in e p e a ion o he unc ions in he exponen so he Wei–No man ac o iza ion. The singula i ies o he unc ions appea ing in he Wei–No man ac o iza ion a e ela ed o he caus ic poin s o Mo se heo y, and he exp ession o he e olu ion ope a o a he caus ics is ob ained using a limi ing p ocedu e, whe e he Fou ie ans o m o he ini ial s a e appea s along wi h he Guoy phase. Published unde license by AIP Publishing. h ps://doi.o g/10.1063/1.5143586 I. INTRODUCTION The Wei–No man (WN) ac o iza ion me hod1,2 allows us o exp ess he e olu ion ope a o (o p opaga o ) o a sys em o i s -o de linea di e en ial (ope a o ) equa ions y′( )=A( )y( ) as a p oduc o a ini e numbe o exponen ials, in he case in which A( ) is an elemen o a ini e-dimensional Lie algeb a. I has mul iple applica ions, one o he mos impo an being he ac o iza ion o he e olu ion ope a o associa ed wi h Sch ödinge ’s equa ion, whe e A( ) is p opo ional o he quan um Hamil onian. WN ac o iza ion leads o a se o nonlinea di e en ial equa ions o he unc ions [he e usually deno ed by gi( )] appea ing in he exponen ial ope a o s. In many si ua ions, hese equa ions a e o Ricca i ype, which admi a ans o ma ion, unde sui able changes in he dependen a iable, in o second-o de linea di e en ial equa ions. One o he mos aluable examples in quan um mechanics a e hose o quad a ic Hamil onians in posi ion Qand momen um P, which expand hesu(1,1)Liealgeb a. The ime-dependen case,whose mo e gene alexp essionisknown as heGene alizedCaldi ola–Kanai (GCK) Hamil onian,3–5 has impo an applica ions in as di e se ields as ion ap physics,6pho onics la ices,7and cosmology.8Thus, a me hod o inding he solu ion in his gene al case is c ucial. The WN ac o iza ion me hod applies in his case, and we show in his pape ha , o a basis o he quad a ic Lie algeb a gi en by P2,Q2, and QP, he second-o de linea di e en ial equa ion ob ained om he Ricca i equa ion is he Eule –Lag ange (EL) equa ion associa ed wi h he classical e sion o he GCK Hamil onian in h ee o he possible o de ings o he exponen ials. Fo he o he h ee o de ings, he second-o de linea di e en ial equa ion is he Eule –Lag ange (EL) equa ion associa ed wi h a classical Hamil onian ha is a mi o e sion o he o iginal GCK Hamil onian, whe e he mass is in e ed. This p o ides a geome ic and J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-1 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp semiclassical in e p e a ion o he WN ac o iza ion o he SU(1,1) case, since all he in o ma ion p o ided by he e olu ion ope a o is ob ained om he solu ions o he EL equa ion. Simila esul s can be ound in he li e a u e9,10 (see also Re s. 11 and 12), bu in hose pape s, jus one o de ing has been conside ed. This s udy could also be gene alized o o he Lie g oups such as SU(2), which will be conside ed elsewhe e. Fo he GCK case, he e olu ion ope a o can be conside ed as a linea in eg al ans o m13 (see also Appendix B). Associa ed wi h he e olu ion ope a o , he e a e o he linea in eg al ans o ms ha ela e di e en GCK sys ems. These a e he F esnel ans o m14 and he A nold ans o m,5along wi h hei gene aliza ions. The di e ence be ween he la e wo is ha he A nold ans o m is, up o a local phase, a poin (o geome ic) ans o ma ion, i.e., he in eg al ans o m educes o a local ope a o . The p ice o be paid o his simpli ica ion is an ex a di eomo phism in ime in such a way ha he e olu ion imes a e di e en in he wo sys ems. Poin ans o ma ions a e p e e able o non- i ial in eg al ans o ms since hey p ese e poin (o geome ic) symme ies o he sys em ha in he case o he GCK oscilla o a e gi en by he Sch ödinge g oup.5We p o ide in his pape a modi ica ion o he WN me hod ha allows us o ob ain also a ac o iza ion o hese in eg al ans o ms. These exp essions seem o be new in he li e a u e. One o he p oblems o he WN ac o iza ion me hod is i s local cha ac e in ime, in he sense ha he unc ions gi( ) appea ing in he exponen ialsdi e ge o speci ic ini e alues o . Thisp oblemisanalyzed o heSU(1,1)case,andi is ela ed o he ac ha no allelemen s o he g oup can be w i en in a ac o ized way (i.e., he ac o iza ion is no on o). This p oblem is no speci ic o he WN me hod bu appea s also in o he app oaches such as Feynman’s pa h in eg al me hod o compu ing he p opaga o .15,16 The singula alues o he unc ions gi( ) a e ela ed o he caus ics appea ing in he pa h in eg al me hod and a e also ela ed o he caus ics appea ing in Mo se heo y.17 They a e also ela ed o he ocal poin s appea ing in Fou ie op ics.18 The compu a ion o he p opaga o a he caus ics in he pa h in eg al app oach in ol es he use o highe o de pe u ba ions. In ou case, we use a ine- uning analysis o he limi a he caus ic poin s o de i e i s exp ession. In addi ion, he phase jumps o he p opaga o in he pa h in eg al me hod15,16 appea ing when passing h ough a caus ic, which a e ul ima ely ela ed o he Maslo index and he Maslo co ec ion, and wi h he Guoy phase in op ics,19 a e also easily ob ained in he WN me hod. In summa y, he exis ence o singula i ies in he unc ions gi( ) can be easily handled wi hin he WN me hod and does no p e en he compu a ion o he e olu ion ope a o o all imes. The p oblem o he singula i ies o gi( ) and he phase jumps o he wa e unc ion when c ossing a singula i y seems no o be p esen in o he wo ks such as Re s. 9and 10. The eason is ha in hose pape s, he e olu ion ope a o is applied o a pa icula ini ial s a e (numbe s a e o cohe en s a e o he s anda d ha monic oscilla o ), ob aining explici exp essions o he e ol ed wa e unc ion ( his is made possible by he ac o ized o m o he e olu ion ope a o p o ided by he WN me hod) ha do no depend explici ly on gi( ) bu di ec ly on he solu ionso he ELequa ion, whichdo no possess singula i ies. Howe e , he phasejumps appea ingin ou app oach when c ossinga caus ic can s ill be ound in hei solu ions, in a o m o an a c an unc ion o he quo ien o he wo classical solu ions, which expe iences a jump o πwhen he denomina o is ze o (i.e., a caus ic poin s). In a sense, ou app oach, ocusing on he e olu ion ope a o , is mo e gene al han ha o Re s. 9and 10 whe e e y pa icula ini ial s a es a e conside ed. Howe e , we ha e o ackle he p oblem o he singula i ies (inhe en o he WN ac o iza ion), and his is hidden in Re s. 9and 10 due o he nice p ope ies o he ini ial s a es conside ed. The con en o his pape is as ollows: In Sec. II, he WN ac o iza ion is e iewed and pa icula ized o he Sch ödinge case in Sec. II A. In Sec. III, he case o he GCK oscilla o is conside ed and he e olu ion ope a o is ac o ized using wo ep esen a i e o de ings. The unc ions gi( ) a e ela ed o solu ions o he EL equa ions associa ed wi h he o iginal classical Hamil onian and a mi o e sion o i . In Sec. IV, he F esnel ans o m is discussed and gene alized, p o iding a way o compu e a ac o iza ion by modi ica ion o a WN me hod. In Sec. V, he A nold ans o m and i s gene aliza ions a e e iewed, p o iding also a ac o iza ion by a modi ied WN (MWN) me hod. In Sec. VI, some pa icula e sions o he F esnel ans o m a e eco e ed o pa icula cases o he MWN me hod. In Sec. VIII, he pa icula case o he CK oscilla o , o cons an equency and damping coe icien , is discussed in de ail. In pa icula , he p oblem o he caus ic poin s [whe e he unc ions gi( ) di e ge] is discussed, and he compu a ion o he e olu ion ope a o oge he wi h he phase jumps a hese poin s is p o ided wi hin he WN me hod. Finally, a e he Conclusion, a couple o appendices a e p o ided: Appendix A, which discusses he g oupoid p ope y o he e olu ion ope a o in he amewo k o he WN me hod, and Appendix B, con aining a e iew o he mos impo an and use ul esul s abou linea in eg al ans o ms. II. THE WEI–NORMAN FACTORIZATION METHOD Conside he i s -o de linea ini ial alue p oblem (IVP), whe e he p ime indica es de i a i e wi h espec o , y′( )=A( )y( ), y( 0)=y0, (1) wi h y( ),y0∈Vand y0being he ini ial condi ion a 0. He e, Vis a ec o space (which can be in ini e-dimensional) and A( ) is a amily o endomo phism o V(i.e., a amily o ma ices o linea ope a o s). By Pica d’s exis ence and uniqueness heo em20 (pa icula ized o sys ems o homogeneous linea equa ions), i A( ) is con inuous and ∥A( )∥is bounded on some in e al I⊂Rcon aining 0, hen he IVP (1) has a unique solu ion on I. Inmos examplesinphysics,A( )is(up oa ac o ) heHamil onianinaclassicalo quan ummechanicalsys em,andEq.(1)co esponds o Hamil on’s o Sch ödinge ’s equa ion, espec i ely. The case o he Sch ödinge equa ion will be ea ed in Sec. II A. I is con enien o w i e he solu ion y( ) as J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-2 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp y( )=U( , 0)y( 0)∀ ∈I, (2) whe e U( , 0) is known as he ma izan ,21 e olu ion ope a o , p opaga o , ans e ma ix, e c., depending on he con ex . Using he e olu ion ope a o , Eq. (1) is ans o med in o ∂U( , 0) ∂ =A( )U( , 0), U( 0, 0)=IV, (3) wi h IVbeing he iden i y au omo phism o V. Fo V ini e-dimensional, i is easily checked ha U( , 0) is an au omo phism o V o all , 0∈I, since we ha e ha (see, o ins ance, Re . 22) de U( , 0)=de U( 0, 0)exp(∫ 0 (A(s))ds)=exp(∫ 0 (A(s))ds)(4) since de U( 0, 0)=1 and A( ) is bounded on I. The in ini e-dimensional case will be discussed in Sec. II A. The amily o e olu ion ope a o s e i ies he g oupoid p ope y, U( 2, 1)U( 1, 0)=U( 2, 0), U( 1, 0)=U( 0, 1)−1(5) ∀ 0, 1, 2∈I. See Appendix A o a discussion o he g oupoid p ope y in he con ex o he Wei–No man ac o iza ion. Only in he ime-independen case A( )=A,∀ ∈I, we ind ha U( , 0)=U( − 0) and U( ) de ines a 1-pa ame e g oup, U( 1+ 2)=U( 1)U( 2), (6) wi h Aas he in ini esimal gene a o . Equa ion (3) is mo e gene al han Eq. (1) and has he ad an age ha he ini ial condi ion, U( 0, 0)=IV, is ixed. Fo simplici y o no a ion, we shall simply w i e U( )≡U( , 0) and ake 0=0 in mos o he cases, bu we should keep in mind he dependence o U( ) on he ini ial ime 0. In wha ollows, we shall assume ha A( ) can be w i en as a ini e sum, A( )=n ∑ i=1αi( )Xi∀ ∈I, (7) whe e B={Xi,i=1,...,n=dimG} o ms a basis o a Lie algeb a G, ealized as endomo phisms o V, wi h commu a ion ela ions gi en by [Xi,Xj]=n ∑ k=1CijkXk(8) and Cijk a e known as he s uc u e cons an s o he Lie algeb a Gand he basis B. Then, he WN heo em1,2 s a es ha he e olu ion ope a o U( ) can be ac o ized as U( )=eg1( )X1eg2( )X2⋅⋅⋅egn( )Xn,gi(0) =0, i=1,...,n, (9) whe e he unc ions gi( ),i=1,...,nsa is y a se o non-linea i s -o de di e en ial equa ions. This cons uc ion holds, in gene al, only in an open subin e al J⊂Icen e ed a 0=0 (local heo em). The e a e some impo an cases whe e he ac o iza ion is alid o all ∈I (global heo em), namely, o bo h sol able algeb as and he case o 2 ×2 eal ma ices.2 A. Wei–No man ac o iza ion applied o Sch ödinge ’s equa ion The quan um e olu ion equa ion analogous o Eq. (1) is gi en by he Sch ödinge equa ion [no e he p esence o he imagina y uni he e as compa ed o Eq. (1)], i∂ ∂ ∣Ψ( )⟩=H( )∣Ψ( )⟩,∣Ψ( 0)⟩=∣Ψ0⟩, (10) J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-3 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp whe e we ha e used Di acs’s ke no a ion, i.e., ∣Ψ( )⟩,∣Ψ0⟩∈DH⊂H, and His a (in ini e-dimensional, in gene al) Hilbe space. He e, H( ) is a amily o essen ially sel -adjoin ope a o s ac ing on a common dense domain DH⊂H. Unde mild condi ions o H( ), he Sch ödinge equa ion has a unique solu ion ∣Ψ( )⟩ o any ∣Ψ0⟩∈H alid o all ∈R. In oducing he e olu ion ope a o Uas ∣Ψ( )⟩=U( , 0)∣Ψ0⟩∀ ∈R, (11) his di e en ial equa ion can be ans o med in o [no e again he p esence o he imagina y uni as compa ed o Eq. (3)] i∂U( , 0) ∂ =H( )U( , 0), U( 0, 0)=IH. (12) In his in ini e-dimensional case, a gene aliza ion o Eq. (4) also applies, and using he ac ha H( ) is essen ially sel -adjoin (and he p esence o he imagina y uni ), we conclude ha U( , 0) is a amily o uni a y ope a o s, sa is ying he g oupoid p ope y (5). Only in he case H( )=H0,∀ ∈R, his amily cons i u es a 1-pa ame e g oup wi h in ini esimal gene a o H0. As ea lie , we shall simply w i e U( ) o U( , 0), bu we should keep in mind he dependence on 0i he Hamil onian is ime-dependen . Assuming ha he Hamil onian H( ) can be w i en as H( )=n ∑ i=1αi( )Xi∀ ∈R, (13) whe e, now, Xi,i=1,...,n, a e essen ially sel -adjoin ope a o s ha ing he same common dense domain DH⊂H, and closing he Lie algeb a G[wi h commu a ion ela ions (8)], he WN ac o iza ion (9) o he e olu ion ope a o U( ) can also be pe o med. The ac o iza ion will be alid in an in e al J⊂R. The same conside a ions abou he local o global cha ac e o he ac o iza ion also apply he e since his depends (excep o some pa icula cases such as ha o 2 ×2 eal ma ices) on he s uc u e cons an s o he Lie algeb a Gand no on he pa icula ep esen a ion (ei he as ma ices o ope a o s) o he Xi. III. EVOLUTION OPERATOR FOR GCK We shall now conside he speci ic case o he gene alized Caldi ola–Kanai (GCK)3,4 quan um Hamil onian, H( )=1 2m( )P2+1 2m( )ω2( )X2, (14) whe e X≡xand P≡−i∂ ∂xa e he usual dimensionless posi ion and momen um ope a o s in one dimension. Bo h he mass m( ) and he equency ω( ) a e aken o depend on ime. This Hamil onian appea s in many physical si ua ions, gene alizing he s anda d ha monic oscilla o [ω( )=ω0,m( )=m0] and he ime-dependen (pa ame ic) ha monic oscilla o [m( )=m0] appea ing, o ins ance, in ion aps.6 The gene al case appea s, o ins ance, in some cosmological models.8 Gi en ha he Hamil onian is ime-dependen , he co esponding ime e olu ion ope a o U( ) does no ake he simple ex book expo- nen ial o m. Ins ead, we shall ollow he WN ac o iza ion app oach o i s compu a ion, and o ha pu pose, we i s p oceed o ind he Lie algeb a associa ed wi h ou Hamil onian. The h ee ope a o s K=1 2 ∂2 ∂x2=−P2 2, (15) V=X2 2, (16) and S=i 2D, (17) wi h D=1 2(XP +PX) as he dila ion ope a o , o m a basis o an su(1,1) algeb a. No e ha K,V, and Da e He mi ian ope a o s, and hus, S is an i-He mi ian. The Hamil onian in Eq. (14) is ob iously an elemen o his algeb a, H( )=−K m( )+m( )ω2( )V, (18) whe e he dila ion elemen Sis no p esen . Howe e , due o he commu a ion ela ions o he Lie algeb a su(1,1), Swill appea in he Wei–No man ac o iza ion. We could ha e conside ed he mos gene al elemen o he Lie algeb a, including a e m in Sas well as in he Hamil onian (see Re s. 9and 10 o a de ailed s udy o his case in he WN app oach). Howe e , he dila ion elemen So he Lie algeb a is no ealizable in mos physical applica ions, and in any case, i can be easily emo ed by a gauge ans o ma ion.23 J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-4 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp The co esponding commu a ion ela ions de ining he su(1,1) Lie algeb a a e [K,V]=2S,[K,S]=K,[V,S]=−V. (19) In oking now he Wei–No man ansa z (see Re s. 1and 2and Sec. II), we can exp ess he ime e olu ion ope a o U( ) as a p oduc o h ee exponen ial ope a o s, U( )=eg1( )M1eg2( )M2eg3( )M3, (20) whe e BM≡{M1,M2,M3}is a gi en pe mu a ion o he basis {K,V,S}. Fo con enience, we ha e labeled he indices o he g unc ions acco ding o he o de ing o he exponen ials. Each exponen in egi( )Mi,i=1,2,3 consis s o a p oduc o an unknown ime-dependen unc ion gi( ) and he co esponding basis ope a o Mi. Since hese ope a o s do no commu e, he explici o m o he unc ions gdepends on he chosen pe mu a ion BM[al hough he exp ession o U( ) does no depend on i ]. The ad an age wi h his ac o ized o m (as opposed o he exponen ial o a sum) is ha he applica ion o each exponen ial ac o egi( )Mi o a ke is s aigh o wa d, e en o ime-dependen unc ions gi( ). No e ha each exponen ial ope a o egi( )Mico esponds o a one-pa ame e ans o m subg oup gi en in Eq. (B11) o Appendix B (see also Re . 13). This ealiza ion cons i u es a wo old ep esen a ion ( he me aplec ic ep esen a ion) o he symplec ic g oup o canonical ans o ma ions. A. O de ing B1 Fo he pa icula o de ed basis B1={V,S,K}, he exp ession o U( ) is chosen as U( )=eg1( )Veg2( )Seg3( )K,U( =0) =I. (21) Applying he gene al desc ip ion o he WN ac o iza ion me hod explained in Sec. II, we de i e he h ee coupled di e en ial equa ions sa is ied by he unc ions gi( ), −ig′ 1( )=W( ) m0g1( )2−m0ω2( ) W( ),g1(0) =0, −ig′ 2( )=2W( ) m0g1( ), g2(0) =0, −ig′ 3( )=W( ) m0eg2( ),g3(0) =0, (22) whe e W( )=m0 m( ), wi h m0=m(0). I should be s essed ha , om hese equa ions, g1( ) and g3( ) a e pu e imagina y, while g2( ) is eal. These, oge he wi h he he mi ici y o he ope a o s K,V, and D, gua an ees ha he e olu ion ope a o U( ) is uni a y. This u ns ou o be he same condi ion o he es o his pape and will no be u he discussed. The i s equa ion is an uncoupled Ricca i equa ion o g1( ), and once sol ed, he second and hen he hi d equa ions a e sol ed by quad a u es. In Sec. VIII, we p o ide he explici solu ions o hese equa ions o he usual Caldi ola–Kanai case3,4 m( )=m0e2γ and ω( )=ω0. Fo gene al ime-dependen unc ions m( ) and ω( ), we shall ins ead ans o m he Ricca i equa ion in o a second-o de linea di e en ial equa ion o gain some physical insigh . Wi h he s anda d change o unc ions,24 g1( )=im0u′2( ) W( )u2( ), (23) he equa ion o u2( ) esul s in u″ 2( )−W′( ) W( )u′2( ) + ω( )2u2( )=0, u2(0) =1, u′2(0) =0. (24) This is he Eule –Lag ange equa ion associa ed wi h he classical Hamil onian co esponding o Eq. (14). I is he mos gene al equa ion o an oscilla o wi h ime-dependen equency and mass (o damping coe icien ). See Re . 5 o he d i en case and also Re . 9in he amewo k o he WN app oach. In e ms o he unc ion u2( ), g2( ) can be sol ed as g2( )=−logu2( )2. (25) J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-5 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp Thus, he ac o s eg1( )Veg2( )Sappea ing in he ope a o U( ) ep esen a dila ion by 1 u2( ) ollowed by a mul iplica ion by he phase exp(im0u′ 2( ) 2W( )u2( )x2). Once we know g2( ), g3( ) is exp essed as g3( )=i m0∫ 0 W( ′) u2( ′)2d ′=i m0 u1( ) u2( ), (26) whe e u1( )=u2( )∫ 0 W( ′) u2( ′)2d ′,u1(0) =0, u′1(0) =1, (27) is a new, independen solu ion o Eq. (24). In e ms o he solu ions u1and u2, we ha e W( )=u′1( )u2( )−u1( )u′2( ). Thus, W( ) u ns ou o be he W onskian o he wo undamen al solu ions o Eq. (24). The ac o eg3( )Kappea ing in he ope a o U( ) ep esen s anin eg al ans o mknown as heF esnel p opaga o (o F esnel ans o m) [see Appendix B, he i s line in Eq. (B11)]. The F esnel p opaga o is a non-poin (o non-geome ic) ans o m, i.e., i is no ob ained, up o a phase, as a change o a iables in he a gumen o he unc ion (see Appendix B). In op ics, i accoun s o he p opaga ion o wa es in ee space, and in quan um mechanics, i is esponsible o he ime e olu ion o a ee non- ela i is ic pa icle. B. O de ing B2 I we choose a di e en o de ing, o ins ance, B2={K,S,V}, he e olu ion ope a o is ac o ized as U( )=eh1( )Keh2( )Seh3( )V. (28) Applying again he WN ac o iza ion me hod, he coupled di e en ial equa ions sa is ied by he unc ions hi( ) a e −ih′1( )=−m0ω2( ) W( )h1( )2+W( ) m0,h1(0) =0, −ih′2( )=2m0ω2( ) W( )h1( ), h2(0) =0, −ih′3( )=−m0ω2( ) W( )e−h2( ),h3(0) =0. (29) Using again he change o unc ions h1( )=−iW( ) ′ 2( ) m0ω2( ) 2( ), (30) he equa ion o 2( ) akes he ela ed o m ″ 2( ) + (W′( ) W( )−2ω′( ) ω( )) ′ 2( ) + ω2( ) 2( )=0, 2(0) =1, ′ 2(0) =0. (31) The ic ion coe icien in his equa ion, W′( ) W( )−2ω′( ) ω( ), can be in e p e ed as co esponding o a ime-dependen mass ˜ m( )=m2 0ω2 0 m( )ω( )2, whe e ω0=ω(0) has been in oduced o con enience. Hence, Eq. (31) is he Eule –Lag ange equa ion associa ed wi h he new classical Hamil onian ˜ H( )=1 2˜ m( )p2+1 2˜ m( )ω2( )x2. (32) No e ha ˜ H( ) is H( ) wi h he unc ions mul iplying Kand Vin e changed. In he pa icula case o cons an equency ω( )=ω0, he new ic ion coe icien has he opposi e sign. The e o e, Eq. (31) co esponds o he Eule –Lag ange equa ion o a mi o pa icle in he sense o he Ba eman dual sys em.25 In e ms o he unc ion 2( ), h2( ) can be sol ed as h2( )=log 2( )2, (33) analogous o Eq. (25), bu wi h he opposi e sign. J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-6 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp In his case, he ac o s eh2( )Seh3( )Vappea ing in he ope a o U( ) ep esen mul iplica ion by he phase exp(−iW( ) ′ 2( ) m0ω2( ) 2( )x2) ollowed by a dila ion by 2( ). Once we know h2( ), h3( ) is exp essed as h3( )=−im0∫ 0 ω2( ′) W( ′) 2( ′)2d ′=−im0ω2 0 1( ) 2( ), (34) whe e 1( )= 2( ) ω2 0∫ 0 ω( ′)2 W( ′) 2( ′)2d ′, 1(0) =0, ′ 1(0) =1, (35) is a new solu ion o Eq. (31). In e ms o he solu ions 1and 2, we ha e ha ′ 1( ) 2( )− 1( ) ′ 2( )=ω( )2 W( )is he new W onskian o he wo undamen al solu ions o Eq. (31). Ou o he o he ou emaining o de ed basis, wo o hem lead o simila esul s o he basis B1wi h he same Eule –Lag ange equa- ion (24), while he o he wo a e simila o B2wi h he same Eule –Lag ange equa ion (31) and will no be u he discussed he e [see Re . 26 o a de ailed s udy o he Wei–No man me hod o he g oup SL(2,R), whe e he same basis is conside ed and he six o de ings a e discussed o he pu pose o ob aining non-linea supe posi ion p inciples]. IV. GENERAL FRESNEL TRANSFORM In his sec ion, we conside he possibili y o ela ing a GCK sys em, wi h equency ω( ) and mass m( ) o he ee pa icle. Tha is, we a e in e es ed in a uni a y ans o ma ion F( ), called he Gene al F esnel ans o m (GFT), om he Hilbe space H o solu ions o he Sch ödinge equa ion o he GCK sys em (14) o he co esponding Hilbe space H0 o he ee pa icle H0=−1 m0K, (36) whe em0=m(0)as inSec.III.We ake henomencla u e om Re .14,whe e hey deno eagene allinea in eg al ans o m(see Appendix B) wi h he in eg al ke nel gi en by (B5) by a gene al F esnel ans o m since i con ains he F esnel p opaga o as a pa icula case. Deno ing by U( ) and U0( ) he e olu ion ope a o s o he GCK sys em and he ee pa icle, espec i ely, he ollowing diag am is commu a i e and all ope a o s appea ing in i a e uni a y: whe e G( )=F( )−1. F om he diag am, i can be immedia ely seen ha G( )=U( )U0( )−1. The co esponding di e en ial equa ion o G( ) is i∂G( ) ∂ G( )−1=H( )−G( )H0G( )−1,G(0) =I, (37) gene alizing he Sch ödinge equa ion o he e olu ion ope a o [Eq. (12)]. This equa ion can also be sol ed by using he WN ac o iza ion echnique since bo h H( ) and H0belong o he same su(1,1) algeb a. Using he same basis as in Sec. III, o he speci ic o de ing B1, he exp ession o G( ) is chosen as G( )=e 1( )Ve 2( )Se 3( )K, (38) whe e he unc ions i( ),i=1,2,3 e i y he new non-linea equa ions −i ′1( )=W( ) m0 1( )2−m0ω2( ) W( ), −i ′2( )=2W( ) m0 1( ), −i ′3( )=W( ) m0e 2( )−1 m0. (39) J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-7 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp The equa ion o 3has an ex a e m −1 m0as compa ed o Eq. (22), e lec ing he ac ha we a e now sol ing o he gene al F esnel ans o m [Eq. (37)], ins ead o he e olu ion ope a o U( ) [Eq. (12)]. As in he case o he e olu ion ope a o , we shall ans o m he p esen Ricca i equa ion in o a second-o de linea di e en ial equa ion o gain some physical insigh . Wi h he s anda d change o unc ions,24 1( )=im0u′2( ) W( )u2( ), (40) he equa ion o u2( ) is u″ 2( )−W′( ) W( )u′2( ) + ω( )2u2( )=0, u2(0) =1, u′2(0) =0, (41) equi alen o Eq. (24). As abo e, his is he Eule –Lag ange equa ion associa ed wi h he classical Hamil onian co esponding o Eq. (14). In e ms o he unc ion u2( ), g2( ) can be sol ed as 2( )=−logu2( )2. (42) Once we know 2( ), 3( ) is exp essed as 3( )=i m0∫ 0(W( ′) u2( ′)2−1)d ′=i m0(u1( ) u2( )− ), (43) whe e u1( )=u2( )∫ 0 W( ′) u2( ′)2d ′,u1(0) =0, u′1(0) =1, (44) isanewsolu iono Eq.(41).In e mso hesolu ionsu1andu2,weha eW( )=u′1( )u2( )−u1( )u′2( ), heW onskiano he wo undamen al solu ions o Eq. (41). No e he p esence o he e m −i m0in Eq. (43), as compa ed o Eq. (26), whose o igin is he abo emen ioned ex a e m appea ing in he equa ion o g3in Eq. (39). In addi ion, he gene al F esnel ans o m can be ac o ized in a di e en basis and/o o de ing. In pa icula , i can be ac o ized in he o de ing gi en by B2, wi h esul s simila o hose o Sec. III B. The uni a y ans o ma ion G( ) maps he ee pa icle in o an a bi a y GCK sys em. Using he explici o m o hese solu ions, we con i m ha G( )=U( )U0( )−1 o his pa icula case. Thegene al F esnel ans o mcan be u he gene alized oa ans o ma ion ela ingana bi a y GCKsys em o heha monic oscilla o o e en ela ing wo a bi a y GCK sys ems. Howe e , in any case, he gene al F esnel ans o m will be a non-poin ans o ma ion (see Appendix B) since i will always include an exponen ial ac o e 3( )K( he F esnel p opaga o ), which is a non- i ial in eg al ope a o in his ep esen a ion. I is desi able o build a ans o m whose ac o iza ion does no in ol e he F esnel p opaga o , being he e o e a poin ans o ma ion. This will be conside ed in Sec. V. V. ARNOLD TRANSFORM In his sec ion, we conside he possibili y o ela ing wo di e en GCK sys ems, namely, sys em 1 wi h equency ω1( ) and mass m1( ) and sys em 2 wi h equency ω2( ) and mass m2( ), h ough a poin ans o ma ion, wi h he pay-o o an addi ional di eomo phism in ime. Tha is, we a e in e es ed in a ans o ma ion A( ) [and i s in e se B( )≡A( )−1] om sys em 1 o sys em 2 such ha he ollowing diag am is commu a i e and all ope a o s appea ing in i a e uni a y: In his diag am, H(1) 1is he Hilbe space o solu ions o he Sch ödinge equa ion o sys em 1 a ime 1and H(2) 2is he Hilbe space o solu ions o he Sch ödinge equa ion o sys em 2 a ime 2.U1( 1) and U2( 2) a e he co esponding e olu ion ope a o s o sys ems 1 and 2. A he bo om line o his diag am, H0 1= 2=0 ep esen s he iden ical Hilbe spaces H(1) 1=0≡H(2) 2=0. J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-8 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp Wi h his solu ion, he exp ession o gi( ) a e easily compu ed using (51), g1( )=−ie2γ sin(ωd )(γ(ω2 0+Ω2)sin(ωd )+ωd(ω2 0−Ω2)cos(ωd )) (γ2+Ω2)sin2(ωd )+γωdsin(2ωd )+ω2 dcos2(ωd ), g2( )=−log⎛ ⎝e−2γ ((γ2+Ω2)sin2(ωd )+γωdsin(2ωd )+ω2 dcos2(ωd )) ω2 d⎞ ⎠, g3( )=0, (77) whe e he las unc ion is ze o by cons uc ion (poin ans o ma ion). In his case, he in o ma ion is encoded in he di eomo phism in ime appea ing in Eq. (45). I should be s essed ha he solu ion b( ) o he gene alized E mako equa ion ne e anishes; he e o e, he unc ions giha e no singula i ies (excep a in ini y). Thus, he A nold ans o m o he ha monic oscilla o has a global cha ac e ; see Sec. VII. See also Re . 30 o a de ailed discussion o his case. D. A nold ans o m o he ee pa icle In he case o he A nold ans o m o he ee pa icle, he exp ession o he gi( ) is g1( )=−im0ω2 0e2γ sin(ωd ) γsin(ωd )+ωdcos(ωd ), g2( )=−log(e−γ ωd(γsin(ωd ) + ωdcos(ωd ))2 , g3( )=0, (78) whe e, again, he las unc ion is ze o by cons uc ion. As in he case o he e olu ion ope a o o he F esnel ans o m, he gidi e ges o he same ini e alues k. In ac he A nold ans o m o he ee pa icle, he GFT, and he e olu ion ope a o coincide when Ψ0(x)=ψ(x,0); see Sec. VII. See also Re s. 5and 31 o a discussion o his case. IX. CONCLUSIONS In his pape , we ha e applied he Wei–No man me hod, wi h SU(1,1) algeb a, o he GCK oscilla o , ob aining a ac o iza ion o he e olu ionope a o in e mso unc ions gi( )appea ing in heexponen ial ac o s ha can beexp essed in e ms o he solu ions o he Eule – Lag ange equa ions associa ed wi h he classical e sion o he GCK Hamil onian o a mi o e sion o i (wi h in e ed mass), depending on he chosen o de ed basis o he Lie algeb a. In his way, we p o ide a semiclassical in e p e a ion o he WN me hod, building he e olu ion ope a o in e ms o classical solu ions. We also p o ide ac o iza ions, by means o a modi ied WN me hod, o o he in eg al ans o ms such as he F esnel ans o m and he A nold ans o ms and hei gene aliza ions, which ela e di e en GCK sys ems in a uni a y way. The A nold ans o m is cha ac e ized by he condi ion o being a poin ans o ma ion (i.e., a local ans o m), implying ha i s ac o iza ion in ol es only wo ac o s. The pay-o o his simplici y is he need o an ex a epa ame e iza ion in ime, implying ha he A nold ans o m maps solu ions o one sys em in o he o he bu wi h di e en , al hough ela ed, e olu ion imes. One o he p oblems wi h ac o iza ion echniques such as he WN me hod is he impossibili y o ob aining he ac o iza ion o ce ain ime alues s, whe e all he unc ions gidi e ge. These co espond o ze os o u2( ) and a e usually deno ed in he li e a u e ocal poin s o caus ics. We ha e been able o ob ain he exp ession o he e olu ion ope a o s a he caus ics using a ine- uning analysis o he limi ing p ocess when app oaches s, ob aining ha he e olu ion ope a o in ol es he Fou ie ans o m o he ini ial s a e, ollowed by a escaling and a local phase. In addi ion o his, he e is a pa i y ans o ma ion and a phase jump when c ossing he caus ic. The example o he CK oscilla o (wi h cons an equency and damping coe icien ), as he simples example o ime-dependen Hamil onian wi h SU(1,1) symme y, is ho oughly discussed. The echniques de eloped in his pape can also be applied o o he g oups, such as SU(2), wi h applica ions in he case o a spin in he p esence o a ime-dependen magne ic ield32 o in pho onic sys ems such as wa eguide a ays wi h a z-dependen e ac ion index and couplings.33 I would also be in e es ing o gene alize ou cons uc ion o o he cases whe e he WN me hod is no di ec ly applicable bu whe e we can s ill use he in e ac ion pic u e,34 apply a mean ield app oxima ion,35 o a pe u ba i e ea men . A u he in e es ing s udy would be he cons uc ion o Wigne unc ions using he WN me hod, hus making a connec ion be ween he semiclassical desc ip ion p o ided by he Wigne unc ions and he one p o ided in his pape . J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-15 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp ACKNOWLEDGMENTS J.G. hanks he Spanish Minis e io de Ciencia, Inno ación y Uni e sidades o inancial suppo (G an Nos. FIS2017-84440-C2-2- P and PGC2018-097831-B-I00). M.B. acknowledges he hospi ali y o he Uni e si y o Jaén and he Ins i u e Ca los I o Theo e ical and Compu a ional Physics (Uni e si y o G anada). APPENDIX A: GROUPOID PROPERTY OF THE EVOLUTION OPERATOR IN THE WEI–NORMAN FORMULATION In his appendix, we shall del e in o he g oupoid p ope y sa is ied by he e olu ion ope a o gi en in Eq. (5). The main di e ence o he g oupoid p ope y (5) wi h espec o he g oup p ope y (6) [apa om he ob ious dependence in wo pa ame e s ( , 0) in he case o g oupoid] is ha he composi ion o wo e olu ion ope a o s U( 1, 0) and U( 2, ′1), U( 2, ′1)U( 1, 0), equi es ′1= 1, i.e., no all elemen s o he g oupoid can be composed. In addi ion, he main di e ence o he g oupoid wi h espec o he semig oup is ha in he semig oup, he exis ence o he in e se is no gua an eed o all elemen s, no e en he exis ence o he iden i y elemen . The s uc u e o a g oupoid is mo e simila o a g oup han o a semig oup. Elemen s o a g oupoid can be uni a y, o ins ance (as in he case o he Sch ödinge equa ion), bu elemen s o a semig oup canno be uni a y, as i happens in he case o he e olu ion unde he hea equa ion o in open sys ems. Le uss udy heimplica ionso heg oupoidp ope yin heWei–No mancon ex ,i.e.,wha a e hecondi ions equi edon he unc ions giin he WN ac o iza ion in o de o he e olu ion ope a o o sa is y he g oupoid p ope y (5). To simpli y he no a ion, we shall conside he case o he GCK oscilla o s udied in Sec. III, whe e he Lie algeb a is SU(1,1). We shall also es ic o he o de ing B1o Sec. III A o conc e eness. Conside he ac o ized e olu ion ope a o gi en in Eq. (21), which, es o ing he dependence on he ini ial ime 0, can be w i en as U( , 0)=eg1( , 0)Veg2( , 0)Seg3( , 0)K,U( 0, 0)=I. (A1) In his case, he g oupoid p ope y (5) o he Wei–No man ac o ized e olu ion ope a o is w i en as U( 2, 0)=eg1( 2, 0)Veg2( 2, 0)Seg3( 2, 0)K=U( 2, 1)U( 1, 0) =eg1( 2, 1)Veg2( 2, 1)Seg3( 2, 1)Keg1( 1, 0)Veg2( 1, 0)Seg3( 1, 0)K. (A2) I can be shown ha his equa ion implies he ollowing es ic ion o he unc ions gi: g1( 2, 0)=g1( 2, 1) + eg2( 2, 1)g1( 1, 0) 1−g1( 1, 0)g3( 2, 1), g2( 2, 0)=g2( 2, 1) + g2( 1, 0)−log[1−g1( 1, 0)g3( 2, 1)]2, g3( 2, 0)=g3( 1, 0) + eg2( 1, 0)g3( 2, 1) 1−g1( 1, 0)g3( 2, 1). (A3) I should be s essed ha hese equa ions a e compa ible wi h he uni a i y condi ions men ioned in Sec. III A, namely, ha g1and g3 a e pu e imagina y, while g2is eal. In addi ion, he condi ion o he in e se in (5) implies g1( 0, 1)=−e−g2( 1, 0)g1( 1, 0) 1−e−g2( 1, 0)g1( 1, 0)g3( 1, 0), g2( 0, 1)=−g2( 1, 0)−log[1−e−g2( 1, 0)g1( 1, 0)g3( 1, 0)]2, g3( 0, 1)=−e−g2( 1, 0)g3( 1, 0) 1−e−g2( 1, 0)g1( 1, 0)g3( 1, 0). (A4) Equa ions (A3) a e o unc ional ype; he e o e, he ques ion a ises i hese unc ional equa ions a e equi alen o he WN equa ions (22) (unde he assump ion o di e en iabili y o a leas con inui y o he gi). The answe is a i ma i e, bu some mino modi ica ions should be made o Eq. (22) o make explici he dependence on 0, J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-16 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp −ig′ 1( , 0)=1 m( )g1( , 0)2−m( )ω2( ), g1( 0, 0)=0, −ig′ 2( , 0)=2 m( )g1( , 0), g2( 0, 0)=0, −ig′ 3( , 0)=1 m( )eg2( , 0),g3( 0, 0)=0, (A5) whe e he p ime indica es he de i a i e wi h espec o he i s a gumen . We es o e he use o m( ) ins ead o W( ) in he equa ions since now m0=m(0) does no make sense in his se ing. Now, we a e going a s ep o wa d, and we calcula e he ini ial eloci ies [using he ini ial condi ions in Eq. (A5)], g′ 1( 0, 0)=−im( 0)ω2( 0), g′ 2( 0, 0)=0, g′ 3( 0, 0)=i m( 0). (A6) These condi ions a e clea ly alid o any alue o 0; he e o e, we can ew i e Eqs. (A5) as g′ 1( , 0)=g′ 3( , )g1( , 0)2+g′ 1( , ), g1( 0, 0)=0, g′ 2( , 0)=2g′ 3( , )g1( , 0), g2( 0, 0)=0, g′ 3( , 0)=g′ 3( , )eg2( , 0),g3( 0, 0)=0. (A7) I now we compu e he de i a i es g′i( , 0) as g′ i( , 0)=limh→0gi( +h, 0)−gi( , 0) hand exp ess gi( +h, 0) in e ms o gj( +h, ) and gj( , 0) using Eq. (A3), we a i e o Eqs. (A7) [using also ha g′2( , )=0 om Eq. (A6)]. Fo comple eness, we p o ide he exp ession o he WN unc ions gi( , 0) in e ms o he solu ions o he classical equa ions o mo ion, gene alizing Eqs. (23), (25), and (26). Fi s o all, we should ob ain he solu ions o Eqs. (24) and (27) bu by sa is ying he same ini ial condi ions a a bi a y 0ins ead o 0=0. Deno ing by u1( , 0) and u2( , 0) hose solu ions, hey a e ela ed o u1( ) and u2( ) by a canonical ans o ma ion, (u1( , 0) u2( , 0))=1 W( 0)(u2( 0)−u1( 0) −u′2( 0)u′1( 0))(u1( ) u2( )). (A8) Now, i can be checked ha he ollowing unc ions g1( , 0)=im( )u′2( , 0) u2( , 0), g2( , 0)=−logu2( , 0)2, g3( , 0)=i m( 0) u1( , 0) u2( , 0), (A9) sa is y Eq. (A5). APPENDIX B: LINEAR INTEGRAL TRANSFORMS AND ITS RELATION TO LINEAR CANONICAL TRANSFORMATIONS IN PHASE SPACE Linea in eg al ans o ms13,14 a e ope a o s cha ac e ized by an in eg al ke nel, ˆ CΨ(x)=∫∞ −∞C(x,x′)Ψ(x′)dx′, (B1) whe e C(x,x′) is he in eg al ke nel o he ans o m ˆ C. A well-known example o his kind o ans o m is Fou ie T ans o m (FT), whe e he in eg al ke nel is CFou ie (x,x′)=1 √2πe−ix′x, wi h i s gene aliza ions o he F ac ional Fou ie T ans o m (F FT), wi h ke nel J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-17 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp CF FT(x,x′)=1 √2πisin θei(x2+x′2)cos θ−2xx′ 2 sin θ. (B2) In he p e ious equa ion, choosing θ=ω , we eco e he p opaga o desc ibing ime e olu ion in a ha monic oscilla o , Cosc(x,x′, )=√mω 2πihsin(ω )eimω  h(x2+x′2)cos(ω )−2xx′ 2 sin(ω ), (B3) whe e he dimensional cons an s (m,ω, and h) ha e been es o ed. Ano he impo an in eg al ans o m, wi h applica ions in op ics, is he F esnel T ans o m, wi h ke nel CF esnel(x,x′,z)=1 iλzeiπ λz(x−x′)2, (B4) desc ibing p opaga ion in ee space along he zdi ec ion o wa es o wa eleng h λ. A na u al gene aliza ion o hese ans o ms consis s in conside ing he mos gene al quad a ic polynomial in xand x′in he exponen o he in eg al ke nel, hus leading o wha is known as linea canonical ans o m13 o Gene alized F esnel T ans o m (GFT),14 wi h ke nel CM(x,x′)=1 √2πibeiax′2−2xx′+dx2 2b, (B5) which is he exponen ial o a quad a ic o m ela ed o he ma ix M=(a b c d), (B6) sa is ying de (M)=ad −bc =1. Thus, M∈SL(2,R)≈Sp(1,R). No e ha cdoes no appea in he exp ession o he ke nel, gi en ha c=ad−1 b om he condi ion on he de e minan . Equa ion (B5) is no well de ined in he case b=0, i.e., o lowe iangula ma ices. In his case, d=1 a≠0, and he in eg al ke nel can be w i en as13 CM(b=0)(x,x′)=1 √aeic 2ax′2δ(x−x′/a). (B7) The esul ing in eg al ans o ma ion in his case is ˆ CM(b=0)Ψ(x)=1 √aeic 2ax2Ψ(x/a) (B8) and is called geome ic o poin ans o ma ion. A poin ans o ma ion in phase space is a canonical ans o ma ion (q,p)→( Q,P) induced by a coo dina e ans o ma ion, i.e., i  Q= Q(q, ), hen P=J(q, )−1p, whe e J(q, ) is he Jacobian o he coo dina e ans o ma ion. A poin ans o ma ion does no in ol e an in eg al (and i is he e o e local), co esponding o a dila ion by 1/a ollowed by a mul iplica ion by a phase quad a ic in x. The ac o 1 √amul iplying he unc ion ensu es he uni a i y o he dila ion. No e ha he GFT ans o m is a uni a y ans o ma ion wi h he usual L2(R) scala p oduc . The ac ion on ope a o s Xand Pby his uni a y ans o ma ion is ˆ CMXˆ C† M=dX −bP ≡X′, ˆ CMPˆ C† M=−cX +aP ≡P′. (B9) Thus, (X′ P′)=M−1(X P), and he e o e, ˆ CMcons i u es a uni a y wo old ep esen a ion o he g oup SL(2,R)≈Sp(1,R), known as he me aplec ic ep esen a ion. 1. One-pa ame e ans o m subg oups Since he GFT ans o m cons i u es a uni a y ep esen a ion o he SL(2,R) g oup o linea canonical ans o ma ions in phase space, we can use well-known ac o iza ion o mulas in e ms o unipa ame ic subg oups. Le us de ine he subg oups M =(1b 0 1),Mg=(1 0 c1),Md=⎛ ⎝a0 01 a⎞ ⎠. (B10) J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-18 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp Explici ly, ˆ CM Ψ(x)=1 √2πib∫∞ −∞dx′ei 2b(x−x′)2Ψ(x′), ˆ CMgΨ(x)=eic 2x2Ψ(x), ˆ CMdΨ(x)=1 √aΨ(x/a). (B11) These ans o ma ions a e well known in he op ics li e a u e. The ans o ma ion gene a ed by M is known as he F esnel p opaga o in ee space (in quan um mechanics, i co esponds o he e olu ion ope a o o a ee Galilean pa icle), he ans o ma ion gene a ed by Mg isknown as hequad a u e phase ope a o (o Gauss–Weie s ass ope a o ), and he ans o ma ion gene a ed byMdis known as he dila ion ope a o (in quan um mechanics, i co esponds o a squeezing ope a o ). 2. In ini esimal gene a o s o in eg al ans o ms I is use ul o ind he di e en ial ope a o s ha gene a e he in eg al ans o m o each one o he unipa ame ic subg oups, in he sense ha ˆ CM( )Ψ(x)=e NΨ(x). (B12) The di e en ial ope a o Ncan be compu ed by di e en ia ion wi h espec o he pa ame e a =0 (assuming ha =0 co esponds o heiden i y ans o m).Fo eachoneo heunipa ame icsubg oups discussed inAppendixB 1, heco espondingin ini esimal gene a o s a e13 N =i 2 ∂2 ∂x2=iK, Ng=i 2x2=iV, Nd=−(x∂ ∂x+1 2)=−2S, (B13) whe e we ha e aken b= ,c= , and a=e in Eq. (B10). Thus, up o cons an s, hey coincide wi h he ope a o s K,V, and Sde ined in Eqs. (15)–(17), showing ha ope a o s such as he e olu ion ope a o , he gene al F esnel ans o m, o he A nold ans o m a e, in ac , linea in eg al ans o ms ha can be w i en as he p oduc o he unipa ame ic subg oups o in eg al ans o ms discussed in Appendix B 1. REFERENCES 1J. Wei and E. No man, “Lie algeb aic solu ion o linea di e en ial equa ions,” J. Ma h. Phys. 4, 575–581 (1963). 2J.WeiandE.No man,“Onglobal ep esen a ionso hesolu ionso linea di e en ialequa ionsasap oduc o exponen ials,”P oc.Am.Ma h.Soc.15,327–334(1964). 3P. Caldi ola, “Fo ze non conse a i e nella meccanica quan is ica,” Il Nuo o Cimen o 18, 393–400 (1941). 4E. Kanai, “On he quan iza ion o he dissipa i e sys ems,” P og. Theo . Phys. 3, 440–442 (1948). 5V. Aldaya, F. Cossío, J. Gue e o, and F. F. López-Ruiz, “The quan um A nold ans o ma ion,” J. Phys. A: Ma h. Theo . 44, 065302 (2011). 6G. Ha a i, Y. Ben-A yeh, and A. Mann, “P opaga o o he gene al ime-dependen ha monic oscilla o wi h applica ion o an ion ap,” Phys. Re . A 84, 062104 (2011). 7B. M. Rod íguez-La a, P. Aleahmad, H. M. Moya-Cessa, and D. N. Ch is odoulides, “E mako –Lewis symme y in pho onic la ices,” Op . Le . 39, 2083 (2014). 8L. Pa ke , “Quan ized ields and pa icle c ea ion in expanding uni e ses. I,” Phys. Re . 183, 1057 (1969). 9¸S. A. Büyüka¸sık and Z. Çayiç, “Exac ly sol able He mi e, Lague e, and Jacobi ype quan um pa ame ic oscilla o s,” J. Ma h. Phys. 57, 122107 (2016). 10¸S. A. Büyüka¸sık, “Squeezing and esonance in a gene alized Caldi ola–Kanai ype quan um pa ame ic oscilla o ,” J. Ma h. Phys. 59, 082104 (2018). 11A. Palma and I. Ped aza, “On he ime-dependen solu ions o he Sch ödinge equa ion,” in Topics in he Theo y o Chemical and Physical Sys ems edi ed by S. Lahma , J. Ma uani, S. Wilson, and G. Delgadp-Ba io (Sp inge , 2007), pp. 147–150. 12A. Palma, M. Villa, and L. Sando al, “On he ime-dependen solu ions o he Sch ödinge ’s equa ion. II. The one-mode ield pe u bed ha monic oscilla o ,” In . J. Quan um Chem. 112, 2441–2443 (2012). 13K. B. Wol , In eg al T ans o ms in Science and Enginee ing (Sp inge , Bos on, 1979). 14D. F. V. James and G. S. Aga wal, “The gene alized F esnel ans o m and i s applica ion o op ics,” Op . Commun. 126, 207–212 (1996). 15P. A. Ho á hy, “The Maslo co ec ion in he semiclassical Feynman in eg al,” Open Physics Cen . Eu . J. Phys. 9, 1 (2011). 16B. K. Cheng, “Exac e alua ion o he p opaga o o he damped ha monic oscilla o ,” J. Phys. A: Ma h. Gen. 17, 2475 (1984). 17J. Milno , Mo se Theo y, Annals o Ma hema ics S udies Vol. 51 (P ince on Uni e si y P ess, 1963). 18J. W. Goodman, In oduc ion o Fou ie Op ics (Robe s and Company Publishe s, 2005). 19A. E. Siegman, Lase s (Uni e si y Science Books, Cali o nia, 1986). 20E. A. Codding on and N. Le inson, Theo y o O dina y Di e en ial Equa ions (Ta a Mcg aw-Hill Publishing Co., L d., New Delhi, 1955). J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-19 Published unde license by AIP Publishing Jou nal o Ma hema ical Physics ARTICLE sci a ion.o g/jou nal/jmp 21H. F. Bake , “On he in eg a ion o linea di e en ial equa ions,” P oc. London Ma h. Soc. s1-35, 334–378 (1902). 22S. Blanes, F. Casas, J. A. O eo, and J. Ros, “The Magnus expansion and some o i s applica ions,” Phys. Rep. 470, 151–238 (2009). 23J. Gue e o and F. F. López-Ruiz, “The quan um A nold ans o ma ion and he E mako –Pinney equa ion,” Phys. Sc . 87, 038105 (2013). 24N. Kh. Ib aglmo , “G oup analysis o o dina y di e en ial equa ions and he in a iance p inciple in ma hema ical physics ( o he 150 h anni e sa y o Sophus Lie),” Russ. Ma h. Su . 47, 89 (1992). 25J. Gue e o, F. F. López-Ruiz, V. Aldaya, and F. Cossío, “Symme ies o he quan um damped ha monic oscilla o ,” J. Phys. A: Ma h. Theo . 45, 475303 (2012). 26J. F. Ca iñena, G. Ma mo, and J. Nasa e, “The non-linea supe posi ion p inciple and he Wei–No man me hod,” In . J. Mod. Phys. A 13, 3601–3627 (1998). 27S. Wang, H.-Y. Fan, and H.-C. Yuan, “F esnel ope a o , squeezed s a e and Wigne unc ion o Caldi ola–Kanai Hamil onian,” Mod. Phys. Le . A 26, 1433 (2011). 28H.-y. Fan and L.-y. Hu, “Co espondence be ween quan um-op ical ans o m and classical-op ical ans o m explo ed by de eloping Di ac’s symbolic me hod,” F on . Phys. 7, 261–310 (2012). 29S. C uz y C uz and Z. G ess, “G oup app oach o he pa axial p opaga ion o He mi e–Gaussian modes in a pa abolic medium,” Ann. Phys. 383, 257–277 (2017). 30J. Gue e o and F. F. López-Ruiz, “On he Lewis–Riesen eld (Dodono –Man’ko) in a ian me hod,” Phys. Sc . 90, 074046 (2015). 31J. Gue e o, V. Aldaya, F. F. López-Ruiz, and F. Cossío, “Un olding he quan um A nold ans o ma ion,” In . J. Geom. Me hods Mod. Phys. 09, 1260011 (2012). 32M. En íquez and S. C uz y C uz, “Exac ly sol able one-qubi d i ing ields gene a ed ia nonlinea equa ions,” Symme y 10, 567 (2018). 33B. M. Rod íguez-La a, R. El-Ganainy, and J. Gue e o, “Symme y in op ics and pho onics: A g oup heo y app oach,” Sci. Bull. 63, 244 (2018). 34R. Román-Anchey a, M. Be ondo, and J. Récamie , “Pa ame ic oscilla o in a Ke medium: E olu ion o cohe en s a es,” J. Op . Soc. Am. B 32, 1651 (2015). 35M. Be ondo and J. Récamie , “Dipole induced ansi ions in an anha monic oscilla o : A dynamical mean ield model,” Chem. Phys. Le . 503, 180 (2011). J. Ma h. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-20 Published unde license by AIP Publishing