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Presentation used to defend the PhD thesis: "Lie systems and applications to Quantum Mechanics", held in Zaragoza Spain on 23th October 2009.

Presentation used to defend the PhD thesis: "Lie systems and applications to Quantum Mechanics", held in Zaragoza Spain on 23th October 2009.

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Thesis defence Thesis defence Presentation Transcript

  • Introduction Geometric theory of Lie systems The theory of Quasi-Lie schemes Conclusions Lie systems and applications to Quantum Mechanics Javier de Lucas Araujo November 29, 2009 Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Geometric theory of Lie systems History of Lie systems The theory of Quasi-Lie schemes Conclusions Lie’s work About one century ago, Lie investigated those linear homogeneous systems of ordinary differential equations n dx j = ajk (t)x k , j = 1, . . . , n, dt k=1 admitting their general solution to be written as n j x j (t) = λk x(k) (t), j = 1, . . . , n, k=1 with {x(1) (t), . . . , x(n) (t)} being a family of linear independent particular solutions and {λ1 , . . . , λn } a set of real constants. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Geometric theory of Lie systems History of Lie systems The theory of Quasi-Lie schemes Conclusions Lie noticed that any non-linear change of variables x → y transforms the previous linear system into a non-linear one of the form dy j = X j (t, y ), j = 1, . . . , n, (1) dt whose solution could be expressed non-linearly as y j (t) = F j (y(1) (t), . . . , y(n) (t), λ1 , . . . , λn ), (2) with {y(1) (t), . . . , y(n) (t)} being a family of certain particular solutions for (1). He called the expressions of the above kind superposition rules. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Geometric theory of Lie systems History of Lie systems The theory of Quasi-Lie schemes Conclusions Lie characterized those non-autonomous systems of first-order differential equations admitting the general solution to be written in terms of certain families of particular solutions and a set of constants. In his honour, these systems are called nowadays Lie systems and the expressions of the form (2) are still called superposition rules. Many work have been done since then and many applications and developments for this theory have been done by many authors. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Non-autonomous systems and vector fields In the modern geometrical lenguage, we describe any non-autonomous system on Rn , written in local coordinates dx j = X j (t, x), j = 1, . . . , n, (3) dt by means of the t-dependent vector field n ∂ X (t, x) = X j (t, x) , (4) ∂x j j=1 whose integral curves are those given by the above system. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Definition The system (3) is a Lie system if and only if the t-dependent vector field (4) can be written r X (t) = bα (t)Xα , (5) α=1 where the vector fields Xα close on a finite-dimensional Lie algebra of vector fields V , i.e. there exist r 3 real constants cαβγ such that [Xα , Xβ ] = cαβγ Xγ , α, β = 1, . . . , r . The Lie algebra V is called a Vessiot-Guldberg Lie algebra of vector fields of the Lie system (5). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Examples of Lie systems Linear homogeneous and inhomogeneous systems of differential equations. Riccati equations and Matrix Riccati equations. Other non-autonomous systems related to famous higher-order differential equations: 1 Time-dependent harmonic oscillators. 2 Milne-Pinney equations. 3 Ermakov systems. 4 Time-dependent Lienard equations. The study of the previous systems related to higher-order differential equations is developed in this work by the first time and, furthermore, many applications to Physics have been found.
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Definition The system (3) is said to admit a superposition rule, if there exists a map Φ : U ⊂ Rn(m+1) → Rn such that given a generic family of particular solutions {x(1) (t), . . . , x(m) (t)} and a set of constants {k1 , . . . , kn } , its general solution x(t) can be written as x(t) = Φ(x(1) (t), . . . , x(m) (t), k1 , . . . , kn ). Lie’s theorem Lie proved that any Lie system (5) on Rn admits a superposition rule in terms of m generic particular solutions with r ≤ mn. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Superposition Rule for Riccati equations We asserted that the Riccati equation is a Lie system and therefore it admits a superposition rule. This is given by x1 (x3 − x2 ) − kx2 (x3 − x1 ) x= . (x3 − x2 ) − k(x3 − x1 ) Taken three different solutions of the Ricatti equation x(1) (t), x(2) (t), x(3) (t) and a constant k ∈ R ∪ {∞} this expression allows us to get the general solution. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Given a Lie system (5), it admits a superposition rule x = Φ(x(1) , . . . , x(m) , k1 , . . . , kn ). where the map Φ can be inverted on the last n variables to give rise to a new map (k1 , . . . , kn ) = Ψ(x, x(1) , . . . , x(m) ), Differentiating, we get m n j ∂Ψj X (t, x )Ψ ≡ ˜ X i (t, x(β) (t)) i = 0, j = 1, . . . , n. β=0 i=1 ∂x(β) Hence the functions {Ψj | j = 1, . . . , n} are first-integrals for the distribution D spanned by the vector fields X (t, x ). Note that for ˜ the sake of simplicity we call x(0) the variable x.
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Definition n i Given a t-dependent vector field X (t, x(0) ) = i=1 X (t, x(0) )∂x(0) , i on Rn , we call prolongation X of X to the manifold Rn(m+1) to the vector field m n ∂Ψj X (t, x ) ≡ ˜ X i (t, x(β) (t)) i . β=0 i=1 ∂x(β) Under certain conditions, n first-integrals for the prolongated t-dependent vector fields allow us to get the superposition rule. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Lie systems and equations in Lie groups Lemma Given any Lie algebra V of complete vector fields on Rn and a connected Lie group G with Te G V , there exists an effective, up to a discrete set of points, action ΦV ,G : G × Rn → Rn such that their fundamental vector fields are those in V . This facts implies important consequences in the theory of Lie systems. As we next show, it provides a way to reduce the study of Lie systems to a particular family of them. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Proposition Given the Lie system X (t) = n bα (t)Xα related to a Lie α=1 Vessiot Gulberg Lie algebra V , we can associate it with the equation n R g =− ˙ bα (t)Xα (g ), g (0) = e, α=1 in a connected Lie group G with Te G V and with the Xα R right-invariant vector fields clossing on the same commutation relations as the Xα . Moreover, the corresponding action ΦV ,G : G × Rn → Rn determines that the general solution for X is x(t) = ΦV ,G (g (t), x0 ). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Integrability of Riccati equations Fixing the basis {∂x , x∂x , x 2 ∂x } of vector fields on R for a Vessiot–Gulberg Lie algebra V for Riccati equations, each Riccati equation can be considered as a curve (b0 (t), b1 (t), b2 (t)) in R3 . ¯ A curve A of the group G of smooth curves in G = SL(2, R) transforms every curve x(t) in R into a new curve x (t) in R ¯ given by x (t) = ΦV ,G (A(t), x(t)). Correspondingly, the t-dependent change of variables ¯ x (t) = ΦV ,G (A(t), x(t)) transforms a Riccati equation with coefficients b0 , b1 , b2 into a new Riccati equation with new t-dependent coefficients, b0 , b1 , b2 . Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Theorem The group G of curves in a Lie group G associated with a Lie system, here SL(2, R), acts on the set of these Lie systems, here Riccati equations. The group G also acts on the left on the set of curves in ¯ SL(2, R) by left translations, i.e. a curve A(t) transforms the ¯ curve A(t) into a new one A (t) = A(t)A(t). If A(t) is a solution of the equation in SL(2, R) related to a Riccati equation, then the new curve A (t) satisfies a new equation in SL(2, R) but with a different right hand side a (t) associated with a new Riccati equation. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Lie group point of view The relation between both curves in sl(2, R) is ¯ ¯ ˙ a (t) = A(t)a(t)A−1 (t) + A(t)A−1 (t). ¯ ¯ If the right hand is in a solvable subalgebra of sl(2, R) the system can be integrated. J.F. Cari˜ena, J. de Lucas and A. Ramos, A geometric approach to n integrability conditions for Riccati equations, Electr. J. Diff. Equ. 122, 1–14 (2007). ˙ ¯ If not, putting apart A(t), the previous equation becomes a Lie system related to a Vessiot–Gulberg Lie algebra isomorphic to sl(2, R) ⊕ sl(2, R). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Some restrictions on the solutions of this system make it to become a solvable Lie system. On one hand, this restriction make the system not having always solution and some integrability conditions arise. On the other hand, this system allows to characterize many integrable families of Riccati equations. J.F. Cari˜ena and J. de Lucas, Lie systems and integrability n conditions of differential equations and some of its applications, Proceedings of the 10th international conference on differential geometry and its applications, World Science Publishing, Prague, (2008). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics SODE Lie systems Definition Any system x i = F i (t, x, x), with i = 1, . . . , n, can be studied ¨ ˙ through the system corresponding for the t-dependent vector field n X (t) = (v i ∂x i + F i (t, x, v )∂v i ). i=1 We call SODE Lie systems those SODE for which X is a Lie system. J.F Cari˜ena, J. de Lucas and M.F. Ra˜ada. Recent Applications n n of the Theory of Lie Systems in Ermakov Systems, SIGMA 4, 031 (2008). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Generalization of the theory of SODEs Even if here we are dealing with second-order differential equations, this procedure can be applied to any higher-order differential equation. The study of superposition rules for second-order differential equations is new, only it is cited once by Winternitz. There exist a big number of systems of higher “Lie systems”. Example of SODEs Time-dependent frequency harmonic oscillators. Milne–Pinney equations. Ermakov systems. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Harmonic Oscillators and Milne–Pinney equation The Milne–Pinney equation with t-dependent frequency is x = −ω 2 (t)x + kx −3 can be associated with the following system ¨ of first-order differential equations   x = v, ˙  v = −ω 2 (t)x + k . ˙ x3 This system describes the integral curves for the t-dependent vector field ∂ k ∂ X (t) = v + −ω 2 (t)x + 3 . ∂x x ∂v Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics This is a Lie system because X can be written as X (t) = L2 − ω 2 (t)L1 , where the vector fields L1 and L2 are given by ∂ ∂ k ∂ 1 ∂ ∂ L1 = x , L2 = v + 3 , L3 = x −v , ∂v ∂x x ∂v 2 ∂x ∂v which are such that [L1 , L2 ] = 2L3 , [L3 , L2 ] = −L2 , [L3 , L1 ] = L1 Therefore, the Milne–Pinney equation is a Lie system related to a Vessiot-Gulberg Lie algebra isomorphic to sl(2, R). If k = 0 we obtain the same result for t-dependent frequency harmonic oscillators. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics As the Milne–Pinney equation is a Lie system, the theory of Lie systems can be applied and, for example, a new superposition rule can be obtained: 1/2 2 2 4 4 2 2 x =− λ1 x1 + λ2 x2 ±2 λ12 (−k(x1 + x2 ) + I3 x1 x2 ) . (6) J.F. Cari˜ena and J. de Lucas, A nonlinear superposition rule for n the Milne–Pinney equation, Phys. Lett. A 372, 5385–5389 (2008). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Also, if we apply the theory of integrability to the Milne–Pinney equation or the harmonic oscillator, we can many integrable particular cases, as the Caldirola-Kanai Oscillator. Moreover, many new results and integrable cases can be obtained. J.F. Cari˜ena and J. de Lucas, Integrability of Lie systems and n some of their applications in Physics, J. Phys. A 41, 304029 (2008). J.F. Cari˜ena, J. de Lucas and M.F. Ra˜ada, Lie systems and n n integrability conditions for t-dependent frequency harmonic oscillators, to appear in the Int. J. Geom. Methods in Mod. Phys. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Mixed superposition rules Definition We say that a mixed superposition rule is an expression allowing us to obtain the general solution of a system in terms of a family of particular solutions of other differential equations. Theorem Lie systems related to t-dependent vector fields of the form X (t) = r bα (t)Xα with vector fields Xα closing on the same α=1 commutation relations, i.e. [Xα , Xβ ] = cαβγ Xγ , cαβγ ∈ R, admit a mixed superposition rule. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Mixed superposition rules The Milne–Pinney equation and the harmonic oscillator The Milne–Pinney equation and the harmonic oscillator in first-order   x = v, ˙ x = v, ˙ k 2  v = −ω (t)x + , ˙ v = −ω 2 (t), ˙ x3 can be related to the vector fields X (t) = L2 − ω 2 (t)L1 for arbitrary k and k = 0. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Mixed superposition rules Moreover, the chosen basis of fundamental vector fields closes on the same commutation relations. Therefore, there exists a mixed superposition rule relating both of them. It can be seen to be given by √ 2 1/2 x= I2 y 2 + I1 z 2 ± 4I1 I2 − kW 2 yz . |W | Furthermore, a certain Riccati equation satisfies the same conditions, therefore, it can be obtained the new mixed superposition rule (C1 (x1 − x2 ) − C2 (x1 − x3 ))2 + k(x2 − x3 )2 x= . (C2 − C1 )(x2 − x3 )(x2 − x1 )(x1 − x3 )
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Quantum Lie systems Consider the below facts Any (maybe infinite-dimension) Hilbert space H can be considered as a real manifold. As there exists a isomorphism Tφ H H we get that T H H ⊕ H. Each operator T on a Hilbert space H can be used to define a map X A : ψ ∈ H → (ψ, Aψ) ∈ T H Any t-dependent operator A(t) can be related at any time t as a vector field X A(t) . The X A(t) is a t-dependent vector field on a maybe infinite dimensional manifold H. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Geometric Schrodinger equation Given a t-dependent hermitian Hamiltonian H(t), the associated Schr¨dinger equation ∂t Ψ = −iH(t)Ψ describes the integral curves o for the t-dependent vector field X −iH(t) . Theorem Given two skew self-adjoint operators A, B associated with two vector fields X A and X B , we have that [X A , X B ] = −X [A,B] . Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Definition We say that a t-dependent hermitian Hamiltonian H(t) is a Quantum Lie system if we can write r H(t) = bα (t)Hα , α=1 with the Hα hermitian operators such that the iHα satisfy that there exist r 3 real constants such that r [iHα , iHβ ] = cαβγ iHγ , α, β = 1, . . . , r . γ=1 Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Note that if H(t) is a Quantum Lie system, we can write r X (t) ≡ X −iH(t) = bα (t)Xα , α=1 where Xα = X −iHα , and r [Xα , Xβ ] = cαβγ Xγ , α, β = 1, . . . , r . γ=1 Schr¨dinger equations are the equations of the integral curves for a o t-dependent vector field satisfying the analogous relation to Lie systems but in a maybe infinte-dimensional manifold H. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Flows of “operational” vector fields Note that the vector fields X A admit the flows Fl : (t, ψ) ∈ R × H → Flt (ψ) = exp(tA)(ψ) ∈ H. Action of a quantum Lie system In case that H(t) is a quantum Lie system, there exist certain vector fields {Xα | α = 1, . . . , r } closing on a finite-dimensional Lie algebra V and we can define an action ΦV ,G : G × H → H, with Te G V , satisfying that for a basis of Te G given by {aα | α = 1, . . . , r } closing on the same commutation relations than the Xα , we get d ΦV ,G (exp(−taα ), ψ) = Xα (ψ), dt t=0 that is, the Xα are the fundamental vector fields related to the aα .
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Reduction to equations in Lie groups In case that H(t) is a Quantum Lie system, the related action ΦG ,V : G × H → H allows to relate the Schr¨dinger equation o related to H(t) with an equation n R g =− ˙ bα (t)Xα (g ), g (0) = e, α=1 in a connected Lie group G with Te G V and with R Xα (g ) = Rg ∗e aα and aα a basis of Te G . Moreover, the general solution for X is Ψt = ΦV ,G (g (t), Ψ0 ). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics Spin Hamiltonians Consider the t-dependent Hamiltonian H(t) = Bx (t)Sx + By (t)Sy + Bz (t)Sz , with Sx , Sy and Sz being the spin operators. Note that the t-dependent Hamiltonian H(t) is a quantum Lie system and its Schr¨dinger equation is o dψ = −iBx (t)Sx (ψ) − iBy (t)Sy (ψ) − iBz (t)Sz (ψ), dt Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics that can be seen as the differential equation determinating the integral curves for the t-dependent vector field X (t) = Bx (t)X1 + By (t)X2 + Bz (t)X3 , where (X1 )ψ = −iSx (ψ), (X2 )ψ = −iSy (ψ), (X3 )ψ = −iSz (ψ). Therefore our Schrodinger equation is a Lie system related to a quantum Vessiot-Guldberg Lie algebra isomorphic to su(2). Another example of Quantum Lie systems is given by the family of time dependent Hamiltonians H(t) = a(t)P 2 + b(t)Q 2 + c(t)(QP + PQ) + d(t)Q + e(t)P + f (t)I . Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Fundamentals Introduction Superpositions and connections Geometric theory of Lie systems Lie systems and Lie groups The theory of Quasi-Lie schemes Integrability of Riccati equations Conclusions SODEs Quantum Mechanics The theory of Lie systems allow us to investigate all these systems to obtain exact solutions, see J.F. Cari˜ena, J. de Lucas and A. Ramos, A geometric approach to n time evolution operators of Lie quantum systems. Int. J. Theor. Phys. 48 1379–1404 (2009). or analyse integrability conditions in Quantum Mechanics, see J.F. Cari˜ena and J. de Lucas, Integrability of Quantum Lie n systems. accepted for publication in the Int. J. Geom. Methods Mod. Phys. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Definition of quasi-Lie scheme Definition Let W , V be non-null finite-dimensional real vector spaces of vector fields on a manifold N. We say that they form a quasi-Lie scheme S(W , V ), if the following conditions hold: 1 W is a vector subspace of V . 2 W is a Lie algebra of vector fields, i.e. [W , W ] ⊂ W . 3 W normalises V , i.e. [W , V ] ⊂ V . Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Quasi-Lie schemes are tools to deal with non-autonomous systems of first-order differential equations. Let us describe those systems that can be described by means of a scheme. Definition We say that a t-dependent vector field X is in a quasi-Lie scheme S(W , V ), and write X ∈ S(W , V ), if X belongs to V on its domain, i.e. Xt ∈ V|NtX . Lie systems and Quasi-Lie schemes Given a Lie system related to a Vessiot-Guldber Lie algebra of vector fields V , then S(V , V ) is a quasi-Lie scheme. Hence, Lie systems can be studied through quasi-Lie schemes. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Abel equations Abel equations Abel differential equations of the first kind are of the form x = f3 (t)x 3 + f2 (t)x 2 + f1 (t)x + f0 (t). ˙ Consider the linear space of vector fields V spanned by the basis ∂ ∂ ∂ ∂ X0 = , X1 = x , X2 = x 2 , X3 = x 3 , ∂x ∂x ∂x ∂x and define the Lie algebra W ⊂ V as W = X0 , X1 . Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Moreover, as [X0 , X2 ] = 2X1 , [X0 , X3 ] = 3X2 , [X1 , X2 ] = X2 , [X1 , X3 ] = 2X3 , then [W , V ] ⊂ V and S(W , V ) is a scheme. Finally, the Abel equation can be described through this scheme because 3 X (t, x) = fα (t)Xα (x), α=0 and thus X ∈ S(W , V ). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Symmetries of a scheme Definition We call symmetry of a scheme S(W , V ) to any t-dependent change of variables transforming any X ∈ S(W , V ) into a new X ∈ S(W , V ). We have charecterized different kinds of infinite-dimensional groups of transformations of a scheme: The group of the scheme, G(W ). The extended group of the scheme, Ext(W ). Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Theorem Given a scheme S(W , V ) and an element g ∈ Ext(W ), then for any X ∈ S(W , V ) we get that g X ∈ S(W , V ). Definition Given a scheme S(W , V ) and a X ∈ S(W , V ), we say that X is a quasi-Lie system with respect to this scheme if there exists a symmetry of the scheme such that g X is a Lie system. Theorem Every Quasi-Lie system admits a t-dependent superposition rule. J.F. Cari˜ena, J. Grabowski and J. de Lucas, Quasi-Lie systems: n theory and applications, J. Phys. A 42, 335206 (2009).
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions We can characterise those non-autonomous systems of first-order differential equations admitting a t-dependent superposition rule. Such systems are called Lie families. Theorem A system (1) admits a t-dependent superposition rule if and only if n ¯ X = ¯ bα (t)Xα , α=1 where the Xα satisfy that there exist n3 functions fαβγ (t) such that n ¯ ¯ [Xα , Xβ ] = ¯ fαβγ (t)Xγ , α, β = 1, . . . , n. γ=1 Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions The applications of quasi-Lie systems and Lie families are very broad, we can just say some of them Dissipative Milne–Pinney equations Emden equations Non-linear oscillators Mathews–Lakshmanan oscillators Lotka Volterra systems Etc. Moreover, they can be applied to Quantum Mechanics and PDEs also. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Dissipative Ermakov systems are 1 x = a(t)x + b(t)x + c(t) ¨ ˙ . x3 As a first-order differential equation it can be written x ˙ = v, v ˙ = a(t)v + b(t)x + c(t) x13 . Consider the space V spanned by the vector fields: ∂ ∂ 1 ∂ ∂ ∂ X1 = v , X2 = x , X3 = 3 , X4 = v , X5 = x . ∂v ∂v x ∂v ∂x ∂x Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Take the three-dimensional Lie algebra W ⊂ V generated by the vector fields ∂ ∂ ∂ Y1 = X1 = v , Y2 = X2 = x , Y3 = X5 = x . ∂v ∂v ∂x What is more, as [Y1 , X3 ] = −X3 , [Y1 , X4 ] = X4 , [Y2 , X3 ] = 0, [Y2 , X4 ] = X5 − X1 , [Y3 , X3 ] = 0, [Y3 , X4 ] = −3X3 . Therefore [W , V ] ⊂ V and S(W , V ) is a quasi-Lie scheme. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Moreover, the above system describes the integral curves for the t-dependent vector field X (t) = a(t)X1 + b(t)X2 + c(t)X3 + X4 , Therefore X ∈ S(W , V ). The corresponding set of t-dependent diffeomorphisms of TR related to elements of Ext(W ) reads x = γ(t)x , , α(t) = 0, γ(t) = 0. v = α(t)v + β(t)x , Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions The transformed system of differential equations obtained through our scheme is  dx  dt = β(t)x + α(t)v ,   dv b(t) β(t) β 2 (t) ˙ β(t) dt = α(t) + a(t) α(t) − α(t) − α(t) x a(t) − β(t) − α(t) v + c(t) 13 . ˙  +   α(t) α(t) x Note that if β(t) = 0 this system is associated with d 2x dx 1 = a(t) + b(t) x + c(t) 3 . dt 2 dt x Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Properties of Quasi-Lie schemes Geometric theory of Lie systems Lie families The theory of Quasi-Lie schemes Dissipative Milne-Pinney equations Conclusions Related Lie systems Thus, the resulting transformation is determined by α(t) = c(t) k and β(t) = 0, for a certain constant k. As a consequence, we obtain a t-dependent superposition rule √ 2 1/2 x(t) = ¯2 ¯2 I2 x1 (t) + I1 x2 (t) ± 4I1 I2 − kW 2 x1 (t)¯2 (t) ¯ x , W with x1 , x2 solutions for the differential equation ¯ ¯ x = a(t)x + b(t)x. ¨ ˙ Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Geometric theory of Lie systems The theory of Quasi-Lie schemes Conclusions Advances in Lie systems Let us schematize some developments obtained during this work: Usual Lie systems Lie systems in this work First-order Higher-order Finite-dimensional Infinite-dimensional manifolds manifolds (QM) Ordinary differential Partial differential equations equations Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics
  • Introduction Geometric theory of Lie systems The theory of Quasi-Lie schemes Conclusions Other advances 1 The theory of integrability of Lie systems have been developed. It has been shown that Matrix Riccati equations can be used to analyse the integrability of Lie systems. 2 The theory of Quantum Lie systems has been analysed. Many integrable cases have been understood geometrically and new ones have been provided. These methods allow us to get integrable models to analyse Quantum Systems. 3 The theory of Lie systems have been applied to analyse higher order differential equations. The theory of integrability has been aplied here and many results have been obtained and explained. We have found many applications to Physics of many Lie systems 4 The theory of Quasi-Lie schemes has been started. The number of applications of this theory is very broad and the number of systems analysed with it is still very small. Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics