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

J. Math. Phys. 61, 082107 (2020); https://doi.org/10.1063/1.5143586 61, 082107 © 2020 Author(s). Semiclassical interpretation of Wei–Norman factorization for SU(1, 1) and its related integral transforms Cite as: J. Math. Phys. 61, 082107 (2020); https://doi.org/10.1063/1.5143586 Submitted: 25 December 2019 . Accepted: 25 July 2020 . Published Online: 19 August 2020 Julio Guerrero , and Manuel Berrondo ARTICLES YOU MAY BE INTERESTED IN Mathematics of the classical and the quantum Journal of Mathematical Physics 61, 082101 (2020); https://doi.org/10.1063/1.5135015 Coherent states of position-dependent mass trapped in an infinite square well Journal of Mathematical Physics 61, 082102 (2020); https://doi.org/10.1063/5.0015418 A quantum system with a non-Hermitian Hamiltonian Journal of Mathematical Physics 61, 082106 (2020); https://doi.org/10.1063/5.0011098 Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp Semiclassical interpretation of Wei–Norman factorization for SU(1,1)and its related integral transforms Cite as: J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 Submitted: 25 December 2019 •Accepted: 25 July 2020 • Published Online: 19 August 2020 Julio Guerrero1,2,a) and Manuel Berrondo3 AFFILIATIONS 1Department of Mathematics, Faculty of Experimental Sciences, University of Jaen, Campus Las Lagunillas s/n, 23071 Jaén, Spain 2Institute Carlos I of Theoretical and Computational Physics, University of Granada, Fuentenueva s/n, 18071 Granada, Spain 3Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602, USA a)Author to whom correspondence should be addressed: [email protected] ABSTRACT WepresentaninterpretationofthefunctionsappearingintheWei–NormanfactorizationoftheevolutionoperatorforaHamiltonianbelonging to the SU(1,1) algebra in terms of the classical solutions of the Generalized Caldirola–Kanai (GCK) oscillator (with time-dependent mass and frequency). Choosing P2,X2, and the dilation operator as a basis for the Lie algebra, we obtain that, out of the six possible orderings for the Wei–Norman factorization of the evolution operator for the GCK Hamiltonian, three of them can be expressed in terms of its classical solutions and the other three involve the classical solutions associated with a mirror Hamiltonian obtained by inverting the mass. In addition, we generalize the Wei–Norman procedure to compute the factorization of other operators, such as a generalized Fresnel transform and the Arnoldtransform (andits generalizations),obtaining alsoin thesecases asemiclassical interpretationfor thefunctions inthe exponentsof the Wei–Norman factorization. The singularities of the functions appearing in the Wei–Norman factorization are related to the caustic points of Morse theory, and the expression of the evolution operator at the caustics is obtained using a limiting procedure, where the Fourier transform of the initial state appears along with the Guoy phase. Published under license by AIP Publishing. https://doi.org/10.1063/1.5143586 I. INTRODUCTION The Wei–Norman (WN) factorization method1,2 allows us to express the evolution operator (or propagator) for a system of first-order linear differential (operator) equations y′(t)=A(t)y(t) as a product of a finite number of exponentials, in the case in which A(t) is an element of a finite-dimensional Lie algebra. It has multiple applications, one of the most important being the factorization of the evolution operator associated with Schrödinger’s equation, where A(t) is proportional to the quantum Hamiltonian. WN factorization leads to a set of nonlinear differential equations for the functions [here usually denoted by gi(t)] appearing in the exponential operators. In many situations, these equations are of Riccati type, which admit a transformation, under suitable changes in the dependent variable, into second-order linear differential equations. One of the most valuable examples in quantum mechanics are those of quadratic Hamiltonians in position Qand momentum P, which expandthesu(1,1)Liealgebra. Thetime-dependentcase,whose more generalexpressionisknown astheGeneralizedCaldirola–Kanai (GCK) Hamiltonian,3–5 has important applications in as diverse fields as ion trap physics,6photonics lattices,7and cosmology.8Thus, a method for finding the solution in this general case is crucial. The WN factorization method applies in this case, and we show in this paper that, for a basis of the quadratic Lie algebra given by P2,Q2, and QP, the second-order linear differential equation obtained from the Riccati equation is the Euler–Lagrange (EL) equation associated with the classical version of the GCK Hamiltonian in three of the possible orderings of the exponentials. For the other three orderings, the second-order linear differential equation is the Euler–Lagrange (EL) equation associated with a classical Hamiltonian that is a mirror version of the original GCK Hamiltonian, where the mass is inverted. This provides a geometric and J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-1 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp semiclassical interpretation of the WN factorization for the SU(1,1) case, since all the information provided by the evolution operator is obtained from the solutions of the EL equation. Similar results can be found in the literature9,10 (see also Refs. 11 and 12), but in those papers, just one ordering has been considered. This study could also be generalized to other Lie groups such as SU(2), which will be considered elsewhere. For the GCK case, the evolution operator can be considered as a linear integral transform13 (see also Appendix B). Associated with the evolution operator, there are other linear integral transforms that relate different GCK systems. These are the Fresnel transform14 and the Arnold transform,5along with their generalizations. The difference between the latter two is that the Arnold transform is, up to a local phase, a point (or geometric) transformation, i.e., the integral transform reduces to a local operator. The price to be paid for this simplification is an extra diffeomorphism in time in such a way that the evolution times are different in the two systems. Point transformations are preferable to non-trivial integral transforms since they preserve point (or geometric) symmetries of the system that in the case of the GCK oscillator are given by the Schrödinger group.5We provide in this paper a modification of the WN method that allows us to obtain also a factorization of these integral transforms. These expressions seem to be new in the literature. One of the problems of the WN factorization method is its local character in time, in the sense that the functions gi(t) appearing in the exponentialsdivergeforspecificfinite values of t. ThisproblemisanalyzedfortheSU(1,1)case,anditisrelatedtothefactthatnotallelements of the group can be written in a factorized way (i.e., the factorization is not onto). This problem is not specific of the WN method but appears also in other approaches such as Feynman’s path integral method for computing the propagator.15,16 The singular values of the functions gi(t) are related to the caustics appearing in the path integral method and are also related to the caustics appearing in Morse theory.17 They are also related to the focal points appearing in Fourier optics.18 The computation of the propagator at the caustics in the path integral approach involves the use of higher order perturbations. In our case, we use a fine-tuning analysis of the limit at the caustic points to derive its expression. In addition, the phase jumps of the propagator in the path integral method15,16 appearing when passing through a caustic, which are ultimately related to the Maslov index and the Maslov correction, and with the Guoy phase in optics,19 are also easily obtained in the WN method. In summary, the existence of singularities in the functions gi(t) can be easily handled within the WN method and does not prevent the computation of the evolution operator for all times. The problem of the singularities of gi(t) and the phase jumps of the wave function when crossing a singularity seems not to be present in other works such as Refs. 9and 10. The reason is that in those papers, the evolution operator is applied to a particular initial state (number state or coherent state of the standard harmonic oscillator), obtaining explicit expressions for the evolved wave function (this is made possible by the factorized form of the evolution operator provided by the WN method) that do not depend explicitly on gi(t) but directly on the solutionsof the ELequation, whichdo notpossess singularities. However, the phasejumps appearingin ourapproach when crossinga caustic can still be found in their solutions, in a form of an arctan function of the quotient of the two classical solutions, which experiences a jump of πwhen the denominator is zero (i.e., at caustic points). In a sense, our approach, focusing on the evolution operator, is more general than that of Refs. 9and 10 where very particular initial states are considered. However, we have to tackle the problem of the singularities (inherent to the WN factorization), and this is hidden in Refs. 9and 10 due to the nice properties of the initial states considered. The content of this paper is as follows: In Sec. II, the WN factorization is reviewed and particularized to the Schrödinger case in Sec. II A. In Sec. III, the case of the GCK oscillator is considered and the evolution operator is factorized using two representative orderings. The functions gi(t) are related to solutions of the EL equations associated with the original classical Hamiltonian and a mirror version of it. In Sec. IV, the Fresnel transform is discussed and generalized, providing a way to compute a factorization by modification of a WN method. In Sec. V, the Arnold transform and its generalizations are reviewed, providing also a factorization by a modified WN (MWN) method. In Sec. VI, some particular versions of the Fresnel transform are recovered for particular cases of the MWN method. In Sec. VIII, the particular case of the CK oscillator, for constant frequency and damping coefficient, is discussed in detail. In particular, the problem of the caustic points [where the functions gi(t) diverge] is discussed, and the computation of the evolution operator together with the phase jumps at these points is provided within the WN method. Finally, after the Conclusion, a couple of appendices are provided: Appendix A, which discusses the groupoid property of the evolution operator in the framework of the WN method, and Appendix B, containing a review of the most important and useful results about linear integral transforms. II. THE WEI–NORMAN FACTORIZATION METHOD Consider the first-order linear initial value problem (IVP), where the prime indicates derivative with respect to t, y′(t)=A(t)y(t), y(t0)=y0, (1) with y(t),y0∈Vand y0being the initial condition at t0. Here, Vis a vector space (which can be infinite-dimensional) and A(t) is a family of endomorphism of V(i.e., a family of matrices or linear operators). By Picard’s existence and uniqueness theorem20 (particularized to systems of homogeneous linear equations), if A(t) is continuous and ∥A(t)∥is bounded on some interval I⊂Rcontaining t0, then the IVP (1) has a unique solution on I. Inmostexamplesinphysics,A(t)is(uptoafactor)theHamiltonianinaclassicalorquantummechanicalsystem,andEq.(1)corresponds to Hamilton’s or Schrödinger’s equation, respectively. The case of the Schrödinger equation will be treated in Sec. II A. It is convenient to write the solution y(t) as J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-2 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp y(t)=U(t,t0)y(t0)∀t∈I, (2) where U(t,t0) is known as the matrizant,21 evolution operator, propagator, transfer matrix, etc., depending on the context. Using the evolution operator, Eq. (1) is transformed into ∂U(t,t0) ∂t=A(t)U(t,t0), U(t0,t0)=IV, (3) with IVbeing the identity automorphism of V. For Vfinite-dimensional, it is easily checked that U(t,t0) is an automorphism of Vfor all t,t0∈I, since we have that (see, for instance, Ref. 22) detU(t,t0)=detU(t0,t0)exp(∫t t0tr(A(s))ds)=exp(∫t t0tr(A(s))ds)(4) since detU(t0,t0)=1 and A(t) is bounded on I. The infinite-dimensional case will be discussed in Sec. II A. The family of evolution operators verifies the groupoid property, U(t2,t1)U(t1,t0)=U(t2,t0), U(t1,t0)=U(t0,t1)−1(5) ∀t0,t1,t2∈I. See Appendix A for a discussion of the groupoid property in the context of the Wei–Norman factorization. Only in the time-independent case A(t)=A,∀t∈I, we find that U(t,t0)=U(t−t0) and U(t) defines a 1-parameter group, U(t1+t2)=U(t1)U(t2), (6) with Aas the infinitesimal generator. Equation (3) is more general than Eq. (1) and has the advantage that the initial condition, U(t0,t0)=IV, is fixed. For simplicity of notation, we shall simply write U(t)≡U(t,t0) and take t0=0 in most of the cases, but we should keep in mind the dependence of U(t) on the initial time t0. In what follows, we shall assume that A(t) can be written as a finite sum, A(t)=n ∑ i=1αi(t)Xi∀t∈I, (7) where B={Xi,i=1,...,n=dimG}forms a basis of a Lie algebra G, realized as endomorphisms of V, with commutation relations given by [Xi,Xj]=n ∑ k=1CijkXk(8) and Cijk are known as the structure constants of the Lie algebra Gand the basis B. Then, the WN theorem1,2 states that the evolution operator U(t) can be factorized as U(t)=eg1(t)X1eg2(t)X2⋅⋅⋅egn(t)Xn,gi(0) =0, i=1,...,n, (9) where the functions gi(t),i=1,...,nsatisfy a set of non-linear first-order differential equations. This construction holds, in general, only in an open subinterval J⊂Icentered at t0=0 (local theorem). There are some important cases where the factorization is valid for all t∈I (global theorem), namely, for both solvable algebras and the case of 2 ×2 real matrices.2 A. Wei–Norman factorization applied to Schrödinger’s equation The quantum evolution equation analogous to Eq. (1) is given by the Schrödinger equation [note the presence of the imaginary unit here as compared to Eq. (1)], i∂ ∂t∣Ψ(t)⟩=H(t)∣Ψ(t)⟩,∣Ψ(t0)⟩=∣Ψ0⟩, (10) J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-3 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp where we have used Diracs’s ket notation, i.e., ∣Ψ(t)⟩,∣Ψ0⟩∈DH⊂H, and His a (infinite-dimensional, in general) Hilbert space. Here, H(t) is a family of essentially self-adjoint operators acting on a common dense domain DH⊂H. Under mild conditions for H(t), the Schrödinger equation has a unique solution ∣Ψ(t)⟩for any ∣Ψ0⟩∈Hvalid for all t∈R. Introducing the evolution operator Uas ∣Ψ(t)⟩=U(t,t0)∣Ψ0⟩∀t∈R, (11) this differential equation can be transformed into [note again the presence of the imaginary unit as compared to Eq. (3)] i∂U(t,t0) ∂t=H(t)U(t,t0), U(t0,t0)=IH. (12) In this infinite-dimensional case, a generalization of Eq. (4) also applies, and using the fact that H(t) is essentially self-adjoint (and the presence of the imaginary unit), we conclude that U(t,t0) is a family of unitary operators, satisfying the groupoid property (5). Only in the case H(t)=H0,∀t∈R, this family constitutes a 1-parameter group with infinitesimal generator H0. As earlier, we shall simply write U(t) for U(t,t0), but we should keep in mind the dependence on t0if the Hamiltonian is time-dependent. Assuming that the Hamiltonian H(t) can be written as H(t)=n ∑ i=1αi(t)Xi∀t∈R, (13) where, now, Xi,i=1,...,n, are essentially self-adjoint operators having the same common dense domain DH⊂H, and closing the Lie algebra G[with commutation relations (8)], the WN factorization (9) of the evolution operator U(t) can also be performed. The factorization will be valid in an interval J⊂R. The same considerations about the local or global character of the factorization also apply here since this depends (except for some particular cases such as that of 2 ×2 real matrices) on the structure constants of the Lie algebra Gand not on the particular representation (either as matrices or operators) of the Xi. III. EVOLUTION OPERATOR FOR GCK We shall now consider the specific case of the generalized Caldirola–Kanai (GCK)3,4 quantum Hamiltonian, H(t)=1 2m(t)P2+1 2m(t)ω2(t)X2, (14) where X≡xand P≡−i∂ ∂xare the usual dimensionless position and momentum operators in one dimension. Both the mass m(t) and the frequency ω(t) are taken to depend on time. This Hamiltonian appears in many physical situations, generalizing the standard harmonic oscillator [ω(t)=ω0,m(t)=m0] and the time-dependent (parametric) harmonic oscillator [m(t)=m0] appearing, for instance, in ion traps.6 The general case appears, for instance, in some cosmological models.8 Given that the Hamiltonian is time-dependent, the corresponding time evolution operator U(t) does not take the simple textbook exponential form. Instead, we shall follow the WN factorization approach for its computation, and for that purpose, we first proceed to find the Lie algebra associated with our Hamiltonian. The three operators K=1 2 ∂2 ∂x2=−P2 2, (15) V=X2 2, (16) and S=i 2D, (17) with D=1 2(XP +PX) as the dilation operator, form a basis of an su(1,1) algebra. Note that K,V, and Dare Hermitian operators, and thus, S is anti-Hermitian. The Hamiltonian in Eq. (14) is obviously an element of this algebra, H(t)=−K m(t)+m(t)ω2(t)V, (18) where the dilation element Sis not present. However, due to the commutation relations of the Lie algebra su(1,1), Swill appear in the Wei–Norman factorization. We could have considered the most general element of the Lie algebra, including a term in Sas well as in the Hamiltonian (see Refs. 9and 10 for a detailed study of this case in the WN approach). However, the dilation element Sof the Lie algebra is not realizable in most physical applications, and in any case, it can be easily removed by a gauge transformation.23 J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-4 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp The corresponding commutation relations defining the su(1,1) Lie algebra are [K,V]=2S,[K,S]=K,[V,S]=−V. (19) Invoking now the Wei–Norman ansatz (see Refs. 1and 2and Sec. II), we can express the time evolution operator U(t) as a product of three exponential operators, U(t)=eg1(t)M1eg2(t)M2eg3(t)M3, (20) where BM≡{M1,M2,M3}is a given permutation of the basis {K,V,S}. For convenience, we have labeled the indices of the gfunctions according to the ordering of the exponentials. Each exponent in egi(t)Mi,i=1,2,3 consists of a product of an unknown time-dependent function gi(t) and the corresponding basis operator Mi. Since these operators do not commute, the explicit form of the functions gdepends on the chosen permutation BM[although the expression of U(t) does not depend on it]. The advantage with this factorized form (as opposed to the exponential of a sum) is that the application of each exponential factor egi(t)Mito a ket is straightforward, even for time-dependent functions gi(t). Note that each exponential operator egi(t)Micorresponds to a one-parameter transform subgroup given in Eq. (B11) of Appendix B (see also Ref. 13). This realization constitutes a twofold representation (the metaplectic representation) of the symplectic group of canonical transformations. A. Ordering B1 For the particular ordered basis B1={V,S,K}, the expression of U(t) is chosen as U(t)=eg1(t)Veg2(t)Seg3(t)K,U(t=0) =I. (21) Applying the general description of the WN factorization method explained in Sec. II, we derive the three coupled differential equations satisfied by the functions gi(t), −ig′ 1(t)=W(t) m0g1(t)2−m0ω2(t) W(t),g1(0) =0, −ig′ 2(t)=2W(t) m0g1(t), g2(0) =0, −ig′ 3(t)=W(t) m0eg2(t),g3(0) =0, (22) where W(t)=m0 m(t), with m0=m(0). It should be stressed that, from these equations, g1(t) and g3(t) are pure imaginary, while g2(t) is real. These, together with the hermiticity of the operators K,V, and D, guarantees that the evolution operator U(t) is unitary. This turns out to be the same condition for the rest of this paper and will not be further discussed. The first equation is an uncoupled Riccati equation for g1(t), and once solved, the second and then the third equations are solved by quadratures. In Sec. VIII, we provide the explicit solutions of these equations for the usual Caldirola–Kanai case3,4 m(t)=m0e2γtand ω(t)=ω0. For general time-dependent functions m(t) and ω(t), we shall instead transform the Riccati equation into a second-order linear differential equation to gain some physical insight. With the standard change of functions,24 g1(t)=im0u′2(t) W(t)u2(t), (23) the equation for u2(t) results in u″ 2(t)−W′(t) W(t)u′2(t) + ω(t)2u2(t)=0, u2(0) =1, u′2(0) =0. (24) This is the Euler–Lagrange equation associated with the classical Hamiltonian corresponding to Eq. (14). It is the most general equation for an oscillator with time-dependent frequency and mass (or damping coefficient). See Ref. 5for the driven case and also Ref. 9in the framework of the WN approach. In terms of the function u2(t), g2(t) can be solved as g2(t)=−logu2(t)2. (25) J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-5 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp Thus, the factors eg1(t)Veg2(t)Sappearing in the operator U(t) represent a dilation by 1 u2(t)followed by a multiplication by the phase exp(im0u′ 2(t) 2W(t)u2(t)x2). Once we know g2(t), g3(t) is expressed as g3(t)=i m0∫t 0 W(t′) u2(t′)2dt′=i m0 u1(t) u2(t), (26) where u1(t)=u2(t)∫t 0 W(t′) u2(t′)2dt′,u1(0) =0, u′1(0) =1, (27) is a new, independent solution of Eq. (24). In terms of the solutions u1and u2, we have W(t)=u′1(t)u2(t)−u1(t)u′2(t). Thus, W(t) turns out to be the Wronskian of the two fundamental solutions of Eq. (24). Thefactor eg3(t)Kappearing inthe operatorU(t) represents anintegral transformknown as theFresnel propagator(or Fresnel transform) [see Appendix B, the first line in Eq. (B11)]. The Fresnel propagator is a non-point (or non-geometric) transform, i.e., it is not obtained, up to a phase, as a change of variables in the argument of the function (see Appendix B). In optics, it accounts for the propagation of waves in free space, and in quantum mechanics, it is responsible for the time evolution of a free non-relativistic particle. B. Ordering B2 If we choose a different ordering, for instance, B2={K,S,V}, the evolution operator is factorized as U(t)=eh1(t)Keh2(t)Seh3(t)V. (28) Applying again the WN factorization method, the coupled differential equations satisfied by the functions hi(t) are −ih′1(t)=−m0ω2(t) W(t)h1(t)2+W(t) m0,h1(0) =0, −ih′2(t)=2m0ω2(t) W(t)h1(t), h2(0) =0, −ih′3(t)=−m0ω2(t) W(t)e−h2(t),h3(0) =0. (29) Using again the change of functions h1(t)=−iW(t)v′ 2(t) m0ω2(t)v2(t), (30) the equation for v2(t) takes the related form v″ 2(t) + (W′(t) W(t)−2ω′(t) ω(t))v′ 2(t) + ω2(t)v2(t)=0, v2(0) =1, v′ 2(0) =0. (31) The friction coefficient in this equation, W′(t) W(t)−2ω′(t) ω(t), can be interpreted as corresponding to a time-dependent mass ˜ m(t)=m2 0ω2 0 m(t)ω(t)2, where ω0=ω(0) has been introduced for convenience. Hence, Eq. (31) is the Euler–Lagrange equation associated with the new classical Hamiltonian ˜ H(t)=1 2˜ m(t)p2+1 2˜ m(t)ω2(t)x2. (32) Note that ˜ H(t) is H(t) with the functions multiplying Kand Vinterchanged. In the particular case of constant frequency ω(t)=ω0, the new friction coefficient has the opposite sign. Therefore, Eq. (31) corresponds to the Euler–Lagrange equation for a mirror particle in the sense of the Bateman dual system.25 In terms of the function v2(t), h2(t) can be solved as h2(t)=logv2(t)2, (33) analogous to Eq. (25), but with the opposite sign. J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-6 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp In this case, the factors eh2(t)Seh3(t)Vappearing in the operator U(t) represent multiplication by the phase exp(−iW(t)v′ 2(t) m0ω2(t)v2(t)x2) followed by a dilation by v2(t). Once we know h2(t), h3(t) is expressed as h3(t)=−im0∫t 0 ω2(t′) W(t′)v2(t′)2dt′=−im0ω2 0 v1(t) v2(t), (34) where v1(t)=v2(t) ω2 0∫t 0 ω(t′)2 W(t′)v2(t′)2dt′,v1(0) =0, v′ 1(0) =1, (35) is a new solution of Eq. (31). In terms of the solutions v1and v2, we have that v′ 1(t)v2(t)−v1(t)v′ 2(t)=ω(t)2 W(t)is the new Wronskian of the two fundamental solutions of Eq. (31). Out of the other four remaining ordered basis, two of them lead to similar results to the basis B1with the same Euler–Lagrange equation (24), while the other two are similar to B2with the same Euler–Lagrange equation (31) and will not be further discussed here [see Ref. 26 for a detailed study of the Wei–Norman method for the group SL(2,R), where the same basis is considered and the six orderings are discussed for the purpose of obtaining non-linear superposition principles]. IV. GENERAL FRESNEL TRANSFORM In this section, we consider the possibility of relating a GCK system, with frequency ω(t) and mass m(t) to the free particle. That is, we are interested in a unitary transformation F(t), called the General Fresnel transform (GFT), from the Hilbert space Htof solutions to the Schrödinger equation for the GCK system (14) to the corresponding Hilbert space H0 tfor the free particle H0=−1 m0K, (36) wherem0=m(0)as inSec.III.We takethenomenclaturefrom Ref.14,wherethey denoteagenerallinear integraltransform(see Appendix B) with the integral kernel given by (B5) by a general Fresnel transform since it contains the Fresnel propagator as a particular case. Denoting by U(t) and U0(t) the evolution operators for the GCK system and the free particle, respectively, the following diagram is commutative and all operators appearing in it are unitary: where G(t)=F(t)−1. From the diagram, it can be immediately seen that G(t)=U(t)U0(t)−1. The corresponding differential equation for G(t) is i∂G(t) ∂tG(t)−1=H(t)−G(t)H0G(t)−1,G(0) =I, (37) generalizing the Schrödinger equation for the evolution operator [Eq. (12)]. This equation can also be solved by using the WN factorization technique since both H(t) and H0belong to the same su(1,1) algebra. Using the same basis as in Sec. III, for the specific ordering B1, the expression of G(t) is chosen as G(t)=ef1(t)Vef2(t)Sef3(t)K, (38) where the functions fi(t),i=1,2,3 verify the new non-linear equations −if′1(t)=W(t) m0f1(t)2−m0ω2(t) W(t), −if′2(t)=2W(t) m0f1(t), −if′3(t)=W(t) m0ef2(t)−1 m0. (39) J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-7 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp The equation for f3has an extra term −1 m0as compared to Eq. (22), reflecting the fact that we are now solving for the general Fresnel transform [Eq. (37)], instead of the evolution operator U(t) [Eq. (12)]. As in the case of the evolution operator, we shall transform the present Riccati equation into a second-order linear differential equation to gain some physical insight. With the standard change of functions,24 f1(t)=im0u′2(t) W(t)u2(t), (40) the equation for u2(t) is u″ 2(t)−W′(t) W(t)u′2(t) + ω(t)2u2(t)=0, u2(0) =1, u′2(0) =0, (41) equivalent to Eq. (24). As above, this is the Euler–Lagrange equation associated with the classical Hamiltonian corresponding to Eq. (14). In terms of the function u2(t), g2(t) can be solved as f2(t)=−logu2(t)2. (42) Once we know f2(t), f3(t) is expressed as f3(t)=i m0∫t 0(W(t′) u2(t′)2−1)dt′=i m0(u1(t) u2(t)−t), (43) where u1(t)=u2(t)∫t 0 W(t′) u2(t′)2dt′,u1(0) =0, u′1(0) =1, (44) isanewsolutionofEq.(41).Intermsofthesolutionsu1andu2,wehaveW(t)=u′1(t)u2(t)−u1(t)u′2(t),theWronskianofthetwofundamental solutions of Eq. (41). Note the presence of the term −it m0in Eq. (43), as compared to Eq. (26), whose origin is the abovementioned extra term appearing in the equation for g3in Eq. (39). In addition, the general Fresnel transform can be factorized in a different basis and/or ordering. In particular, it can be factorized in the ordering given by B2, with results similar to those of Sec. III B. The unitary transformation G(t) maps the free particle into an arbitrary GCK system. Using the explicit form of these solutions, we confirm that G(t)=U(t)U0(t)−1for this particular case. Thegeneral Fresneltransformcan befurthergeneralized toatransformation relatinganarbitrary GCKsystem totheharmonic oscillator or even relating two arbitrary GCK systems. However, in any case, the general Fresnel transform will be a non-point transformation (see Appendix B) since it will always include an exponential factor ef3(t)K(the Fresnel propagator), which is a non-trivial integral operator in this representation. It is desirable to build a transform whose factorization does not involve the Fresnel propagator, being therefore a point transformation. This will be considered in Sec. V. V. ARNOLD TRANSFORM In this section, we consider the possibility of relating two different GCK systems, namely, system 1 with frequency ω1(t) and mass m1(t) and system 2 with frequency ω2(t) and mass m2(t), through a point transformation, with the pay-off of an additional diffeomorphism in time. That is, we are interested in a transformation A(t) [and its inverse B(t)≡A(t)−1] from system 1 to system 2 such that the following diagram is commutative and all operators appearing in it are unitary: In this diagram, H(1) t1is the Hilbert space of solutions to the Schrödinger equation for system 1 at time t1and H(2) t2is the Hilbert space of solutions to the Schrödinger equation for system 2 at time t2.U1(t1) and U2(t2) are the corresponding evolution operators for systems 1 and 2. At the bottom line of this diagram, H0 t1=t2=0represents the identical Hilbert spaces H(1) t1=0≡H(2) t2=0. J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-8 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp With this solution, the expression of gi(t) are easily computed using (51), g1(t)=−ie2γtsin(ωdt)(γ(ω2 0+Ω2)sin(ωdt)+ωd(ω2 0−Ω2)cos(ωdt)) (γ2+Ω2)sin2(ωdt)+γωdsin(2ωdt)+ω2 dcos2(ωdt), g2(t)=−log⎛ ⎝e−2γt((γ2+Ω2)sin2(ωdt)+γωdsin(2ωdt)+ω2 dcos2(ωdt)) ω2 d⎞ ⎠, g3(t)=0, (77) where the last function is zero by construction (point transformation). In this case, the information is encoded in the diffeomorphism in time appearing in Eq. (45). It should be stressed that the solution b(t) to the generalized Ermakov equation never vanishes; therefore, the functions gihave no singularities (except at infinity). Thus, the Arnold transform to the harmonic oscillator has a global character; see Sec. VII. See also Ref. 30 for a detailed discussion of this case. D. Arnold transform to the free particle In the case of the Arnold transform to the free particle, the expression of the gi(t) is g1(t)=−im0ω2 0e2γtsin(ωdt) γsin(ωdt)+ωdcos(ωdt), g2(t)=−log(e−γt ωd(γsin(ωdt) + ωdcos(ωdt))2 , g3(t)=0, (78) where, again, the last function is zero by construction. As in the case of the evolution operator or the Fresnel transform, the gidiverges for the same finite values tk. In fact the Arnold transform to the free particle, the GFT, and the evolution operator coincide when Ψ0(x)=ψ(x,0); see Sec. VII. See also Refs. 5and 31 for a discussion of this case. IX. CONCLUSIONS In this paper, we have applied the Wei–Norman method, with SU(1,1) algebra, to the GCK oscillator, obtaining a factorization for the evolutionoperator in termsof functions gi(t)appearing in theexponential factorsthat can beexpressed interms ofthe solutions ofthe Euler– Lagrange equations associated with the classical version of the GCK Hamiltonian or a mirror version of it (with inverted mass), depending on the chosen ordered basis of the Lie algebra. In this way, we provide a semiclassical interpretation to the WN method, building the evolution operator in terms of classical solutions. We also provide factorizations, by means of a modified WN method, for other integral transforms such as the Fresnel transform and the Arnold transforms and their generalizations, which relate different GCK systems in a unitary way. The Arnold transform is characterized by the condition of being a point transformation (i.e., a local transform), implying that its factorization involves only two factors. The pay-off for this simplicity is the need for an extra reparameterization in time, implying that the Arnold transform maps solutions of one system into the other but with different, although related, evolution times. One of the problems with factorization techniques such as the WN method is the impossibility of obtaining the factorization for certain time values ts, where all the functions gidiverge. These correspond to zeros of u2(t) and are usually denoted in the literature focal points or caustics. We have been able to obtain the expression of the evolution operators at the caustics using a fine-tuning analysis of the limiting process when tapproaches ts, obtaining that the evolution operator involves the Fourier transform of the initial state, followed by a rescaling and a local phase. In addition to this, there is a parity transformation and a phase jump when crossing the caustic. The example of the CK oscillator (with constant frequency and damping coefficient), as the simplest example of time-dependent Hamiltonian with SU(1,1) symmetry, is thoroughly discussed. The techniques developed in this paper can also be applied to other groups, such as SU(2), with applications in the case of a spin in the presence of a time-dependent magnetic field32 or in photonic systems such as waveguide arrays with a z-dependent refraction index and couplings.33 It would also be interesting to generalize our construction to other cases where the WN method is not directly applicable but where we can still use the interaction picture,34 apply a mean field approximation,35 or a perturbative treatment. A further interesting study would be the construction of Wigner functions using the WN method, thus making a connection between the semiclassical description provided by the Wigner functions and the one provided in this paper. J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-15 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp ACKNOWLEDGMENTS J.G. thanks the Spanish Ministerio de Ciencia, Innovación y Universidades for financial support (Grant Nos. FIS2017-84440-C2-2P and PGC2018-097831-B-I00). M.B. acknowledges the hospitality of the University of Jaén and the Institute Carlos I of Theoretical and Computational Physics (University of Granada). APPENDIX A: GROUPOID PROPERTY OF THE EVOLUTION OPERATOR IN THE WEI–NORMAN FORMULATION In this appendix, we shall delve into the groupoid property satisfied by the evolution operator given in Eq. (5). The main difference of the groupoid property (5) with respect to the group property (6) [apart from the obvious dependence in two parameters (t,t0) in the case of groupoid] is that the composition of two evolution operators U(t1,t0) and U(t2,t′1), U(t2,t′1)U(t1,t0), requires t′1=t1, i.e., not all elements of the groupoid can be composed. In addition, the main difference of the groupoid with respect to the semigroup is that in the semigroup, the existence of the inverse is not guaranteed for all elements, not even the existence of the identity element. The structure of a groupoid is more similar to a group than to a semigroup. Elements of a groupoid can be unitary, for instance (as in the case of the Schrödinger equation), but elements of a semigroup cannot be unitary, as it happens in the case of the evolution under the heat equation or in open systems. LetusstudytheimplicationsofthegroupoidpropertyintheWei–Normancontext,i.e.,whataretheconditionsrequiredonthefunctions giin the WN factorization in order for the evolution operator to satisfy the groupoid property (5). To simplify the notation, we shall consider the case of the GCK oscillator studied in Sec. III, where the Lie algebra is SU(1,1). We shall also restrict to the ordering B1of Sec. III A for concreteness. Consider the factorized evolution operator given in Eq. (21), which, restoring the dependence on the initial time t0, can be written as U(t,t0)=eg1(t,t0)Veg2(t,t0)Seg3(t,t0)K,U(t0,t0)=I. (A1) In this case, the groupoid property (5) for the Wei–Norman factorized evolution operator is written as U(t2,t0)=eg1(t2,t0)Veg2(t2,t0)Seg3(t2,t0)K=U(t2,t1)U(t1,t0) =eg1(t2,t1)Veg2(t2,t1)Seg3(t2,t1)Keg1(t1,t0)Veg2(t1,t0)Seg3(t1,t0)K. (A2) It can be shown that this equation implies the following restriction for the functions gi: g1(t2,t0)=g1(t2,t1) + eg2(t2,t1)g1(t1,t0) 1−g1(t1,t0)g3(t2,t1), g2(t2,t0)=g2(t2,t1) + g2(t1,t0)−log[1−g1(t1,t0)g3(t2,t1)]2, g3(t2,t0)=g3(t1,t0) + eg2(t1,t0)g3(t2,t1) 1−g1(t1,t0)g3(t2,t1). (A3) It should be stressed that these equations are compatible with the unitarity conditions mentioned in Sec. III A, namely, that g1and g3 are pure imaginary, while g2is real. In addition, the condition for the inverse in (5) implies g1(t0,t1)=−e−g2(t1,t0)g1(t1,t0) 1−e−g2(t1,t0)g1(t1,t0)g3(t1,t0), g2(t0,t1)=−g2(t1,t0)−log[1−e−g2(t1,t0)g1(t1,t0)g3(t1,t0)]2, g3(t0,t1)=−e−g2(t1,t0)g3(t1,t0) 1−e−g2(t1,t0)g1(t1,t0)g3(t1,t0). (A4) Equations (A3) are of functional type; therefore, the question arises if these functional equations are equivalent to the WN equations (22) (under the assumption of differentiability or at least continuity of the gi). The answer is affirmative, but some minor modifications should be made to Eq. (22) to make explicit the dependence on t0, J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-16 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp −ig′ 1(t,t0)=1 m(t)g1(t,t0)2−m(t)ω2(t), g1(t0,t0)=0, −ig′ 2(t,t0)=2 m(t)g1(t,t0), g2(t0,t0)=0, −ig′ 3(t,t0)=1 m(t)eg2(t,t0),g3(t0,t0)=0, (A5) where the prime indicates the derivative with respect to the first argument. We restore the use of m(t) instead of W(t) in the equations since now m0=m(0) does not make sense in this setting. Now, we are going a step forward, and we calculate the initial velocities [using the initial conditions in Eq. (A5)], g′ 1(t0,t0)=−im(t0)ω2(t0), g′ 2(t0,t0)=0, g′ 3(t0,t0)=i m(t0). (A6) These conditions are clearly valid for any value of t0; therefore, we can rewrite Eqs. (A5) as g′ 1(t,t0)=g′ 3(t,t)g1(t,t0)2+g′ 1(t,t), g1(t0,t0)=0, g′ 2(t,t0)=2g′ 3(t,t)g1(t,t0), g2(t0,t0)=0, g′ 3(t,t0)=g′ 3(t,t)eg2(t,t0),g3(t0,t0)=0. (A7) If now we compute the derivatives g′i(t,t0) as g′ i(t,t0)=limh→0gi(t+h,t0)−gi(t,t0) hand express gi(t+h,t0) in terms of gj(t+h,t) and gj(t,t0) using Eq. (A3), we arrive to Eqs. (A7) [using also that g′2(t,t)=0 from Eq. (A6)]. For completeness, we provide the expression of the WN functions gi(t,t0) in terms of the solutions of the classical equations of motion, generalizing Eqs. (23), (25), and (26). First of all, we should obtain the solutions of Eqs. (24) and (27) but by satisfying the same initial conditions at arbitrary t0instead of t0=0. Denoting by u1(t,t0) and u2(t,t0) those solutions, they are related to u1(t) and u2(t) by a canonical transformation, (u1(t,t0) u2(t,t0))=1 W(t0)(u2(t0)−u1(t0) −u′2(t0)u′1(t0))(u1(t) u2(t)). (A8) Now, it can be checked that the following functions g1(t,t0)=im(t)u′2(t,t0) u2(t,t0), g2(t,t0)=−logu2(t,t0)2, g3(t,t0)=i m(t0) u1(t,t0) u2(t,t0), (A9) satisfy Eq. (A5). APPENDIX B: LINEAR INTEGRAL TRANSFORMS AND ITS RELATION TO LINEAR CANONICAL TRANSFORMATIONS IN PHASE SPACE Linear integral transforms13,14 are operators characterized by an integral kernel, ˆ CΨ(x)=∫∞ −∞C(x,x′)Ψ(x′)dx′, (B1) where C(x,x′) is the integral kernel of the transform ˆ C. A well-known example of this kind of transform is Fourier Transform (FT), where the integral kernel is CFourier(x,x′)=1 √2πe−ix′x, with its generalizations to the Fractional Fourier Transform (FrFT), with kernel J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-17 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp CFrFT(x,x′)=1 √2πisin θei(x2+x′2)cos θ−2xx′ 2 sin θ. (B2) In the previous equation, choosing θ=ωt, we recover the propagator describing time evolution in a harmonic oscillator, Cosc(x,x′,t)=√mω 2πihsin(ωt)eimω  h(x2+x′2)cos(ωt)−2xx′ 2 sin(ωt), (B3) where the dimensional constants (m,ω, and h) have been restored. Another important integral transform, with applications in optics, is the Fresnel Transform, with kernel CFresnel(x,x′,z)=1 iλzeiπ λz(x−x′)2, (B4) describing propagation in free space along the zdirection of waves of wavelength λ. A natural generalization of these transforms consists in considering the most general quadratic polynomial in xand x′in the exponent of the integral kernel, thus leading to what is known as linear canonical transform13 or Generalized Fresnel Transform (GFT),14 with kernel CM(x,x′)=1 √2πibeiax′2−2xx′+dx2 2b, (B5) which is the exponential of a quadratic form related to the matrix M=(a b c d), (B6) satisfying det(M)=ad −bc =1. Thus, M∈SL(2,R)≈Sp(1,R). Note that cdoes not appear in the expression of the kernel, given that c=ad−1 b from the condition on the determinant. Equation (B5) is not well defined in the case b=0, i.e., for lower triangular matrices. In this case, d=1 a≠0, and the integral kernel can be written as13 CM(b=0)(x,x′)=1 √aeic 2ax′2δ(x−x′/a). (B7) The resulting integral transformation in this case is ˆ CM(b=0)Ψ(x)=1 √aeic 2ax2Ψ(x/a) (B8) and is called geometric or point transformation. A point transformation in phase space is a canonical transformation (q,p)→( Q,P) induced by a coordinate transformation, i.e., if  Q= Q(q,t), then P=J(q,t)−1p, where J(q,t) is the Jacobian of the coordinate transformation. A point transformation does not involve an integral (and it is therefore local), corresponding to a dilation by 1/afollowed by a multiplication by a phase quadratic in x. The factor 1 √amultiplying the function ensures the unitarity of the dilation. Note that the GFT transform is a unitary transformation with the usual L2(R) scalar product. The action on operators Xand Pby this unitary transformation is ˆ CMXˆ C† M=dX −bP ≡X′, ˆ CMPˆ C† M=−cX +aP ≡P′. (B9) Thus, (X′ P′)=M−1(X P), and therefore, ˆ CMconstitutes a unitary twofold representation of the group SL(2,R)≈Sp(1,R), known as the metaplectic representation. 1. One-parameter transform subgroups Since the GFT transform constitutes a unitary representation of the SL(2,R) group of linear canonical transformations in phase space, we can use well-known factorization formulas in terms of uniparametric subgroups. Let us define the subgroups Mf=(1b 0 1),Mg=(1 0 c1),Md=⎛ ⎝a0 01 a⎞ ⎠. (B10) J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-18 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp Explicitly, ˆ CMfΨ(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 transformations are well known in the optics literature. The transformation generated by Mfis known as the Fresnel propagator in free space (in quantum mechanics, it corresponds to the evolution operator of a free Galilean particle), the transformation generated by Mg isknown as thequadrature phase operator(or Gauss–Weierstrass operator), and thetransformation generated byMdis known asthe dilation operator (in quantum mechanics, it corresponds to a squeezing operator). 2. Infinitesimal generators of integral transforms It is useful to find the differential operators that generate the integral transform for each one of the uniparametric subgroups, in the sense that ˆ CM(t)Ψ(x)=etNΨ(x). (B12) The differential operator Ncan be computed by differentiation with respect to the parameter tat t=0 (assuming that t=0 corresponds totheidentitytransform).Foreachoneoftheuniparametricsubgroups discussed inAppendixB 1, thecorrespondinginfinitesimal generators are13 Nf=i 2 ∂2 ∂x2=iK, Ng=i 2x2=iV, Nd=−(x∂ ∂x+1 2)=−2S, (B13) where we have taken b=t,c=t, and a=etin Eq. (B10). Thus, up to constants, they coincide with the operators K,V, and Sdefined in Eqs. (15)–(17), showing that operators such as the evolution operator, the general Fresnel transform, or the Arnold transform are, in fact, linear integral transforms that can be written as the product of the uniparametric subgroups of integral transforms discussed in Appendix B 1. REFERENCES 1J. Wei and E. Norman, “Lie algebraic solution of linear differential equations,” J. Math. Phys. 4, 575–581 (1963). 2J.WeiandE.Norman,“Onglobalrepresentationsofthesolutionsoflineardifferentialequationsasaproductofexponentials,”Proc.Am.Math.Soc.15,327–334(1964). 3P. Caldirola, “Forze non conservative nella meccanica quantistica,” Il Nuovo Cimento 18, 393–400 (1941). 4E. Kanai, “On the quantization of the dissipative systems,” Prog. Theor. Phys. 3, 440–442 (1948). 5V. Aldaya, F. Cossío, J. Guerrero, and F. F. López-Ruiz, “The quantum Arnold transformation,” J. Phys. A: Math. Theor. 44, 065302 (2011). 6G. Harari, Y. Ben-Aryeh, and A. Mann, “Propagator for the general time-dependent harmonic oscillator with application to an ion trap,” Phys. Rev. A 84, 062104 (2011). 7B. M. Rodríguez-Lara, P. Aleahmad, H. M. Moya-Cessa, and D. N. Christodoulides, “Ermakov–Lewis symmetry in photonic lattices,” Opt. Lett. 39, 2083 (2014). 8L. Parker, “Quantized fields and particle creation in expanding universes. I,” Phys. Rev. 183, 1057 (1969). 9¸S. A. Büyüka¸sık and Z. Çayiç, “Exactly solvable Hermite, Laguerre, and Jacobi type quantum parametric oscillators,” J. Math. Phys. 57, 122107 (2016). 10¸S. A. Büyüka¸sık, “Squeezing and resonance in a generalized Caldirola–Kanai type quantum parametric oscillator,” J. Math. Phys. 59, 082104 (2018). 11A. Palma and I. Pedraza, “On the time-dependent solutions of the Schrödinger equation,” in Topics in the Theory of Chemical and Physical Systems edited by S. Lahmar, J. Maruani, S. Wilson, and G. Delgadp-Barrio (Springer, 2007), pp. 147–150. 12A. Palma, M. Villa, and L. Sandoval, “On the time-dependent solutions of the Schrödinger’s equation. II. The one-mode field perturbed harmonic oscillator,” Int. J. Quantum Chem. 112, 2441–2443 (2012). 13K. B. Wolf, Integral Transforms in Science and Engineering (Springer, Boston, 1979). 14D. F. V. James and G. S. Agarwal, “The generalized Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996). 15P. A. Horváthy, “The Maslov correction in the semiclassical Feynman integral,” Open Physics Cent. Eur. J. Phys. 9, 1 (2011). 16B. K. Cheng, “Exact evaluation of the propagator for the damped harmonic oscillator,” J. Phys. A: Math. Gen. 17, 2475 (1984). 17J. Milnor, Morse Theory, Annals of Mathematics Studies Vol. 51 (Princeton University Press, 1963). 18J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005). 19A. E. Siegman, Lasers (University Science Books, California, 1986). 20E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (Tata Mcgraw-Hill Publishing Co., Ltd., New Delhi, 1955). J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-19 Published under license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp 21H. F. Baker, “On the integration of linear differential equations,” Proc. London Math. Soc. s1-35, 334–378 (1902). 22S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “The Magnus expansion and some of its applications,” Phys. Rep. 470, 151–238 (2009). 23J. Guerrero and F. F. López-Ruiz, “The quantum Arnold transformation and the Ermakov–Pinney equation,” Phys. Scr. 87, 038105 (2013). 24N. Kh. Ibraglmov, “Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie),” Russ. Math. Surv. 47, 89 (1992). 25J. Guerrero, F. F. López-Ruiz, V. Aldaya, and F. Cossío, “Symmetries of the quantum damped harmonic oscillator,” J. Phys. A: Math. Theor. 45, 475303 (2012). 26J. F. Cariñena, G. Marmo, and J. Nasarre, “The non-linear superposition principle and the Wei–Norman method,” Int. J. Mod. Phys. A 13, 3601–3627 (1998). 27S. Wang, H.-Y. Fan, and H.-C. Yuan, “Fresnel operator, squeezed state and Wigner function for Caldirola–Kanai Hamiltonian,” Mod. Phys. Lett. A 26, 1433 (2011). 28H.-y. Fan and L.-y. Hu, “Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method,” Front. Phys. 7, 261–310 (2012). 29S. Cruz y Cruz and Z. Gress, “Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium,” Ann. Phys. 383, 257–277 (2017). 30J. Guerrero and F. F. López-Ruiz, “On the Lewis–Riesenfeld (Dodonov–Man’ko) invariant method,” Phys. Scr. 90, 074046 (2015). 31J. Guerrero, V. Aldaya, F. F. López-Ruiz, and F. Cossío, “Unfolding the quantum Arnold transformation,” Int. J. Geom. Methods Mod. Phys. 09, 1260011 (2012). 32M. Enríquez and S. Cruz y Cruz, “Exactly solvable one-qubit driving fields generated via nonlinear equations,” Symmetry 10, 567 (2018). 33B. M. Rodríguez-Lara, R. El-Ganainy, and J. Guerrero, “Symmetry in optics and photonics: A group theory approach,” Sci. Bull. 63, 244 (2018). 34R. Román-Ancheyta, M. Berrondo, and J. Récamier, “Parametric oscillator in a Kerr medium: Evolution of coherent states,” J. Opt. Soc. Am. B 32, 1651 (2015). 35M. Berrondo and J. Récamier, “Dipole induced transitions in an anharmonic oscillator: A dynamical mean field model,” Chem. Phys. Lett. 503, 180 (2011). J. Math. Phys. 61, 082107 (2020); doi: 10.1063/1.5143586 61, 082107-20 Published under license by AIP Publishing