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# Thesis defence

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

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### Thesis defence

1. 1. 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
2. 2. 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
3. 3. 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
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. 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.
8. 8. 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
9. 9. 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
10. 10. 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.
11. 11. 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
12. 12. 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
13. 13. 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
14. 14. 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
15. 15. 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
16. 16. 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
17. 17. 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
18. 18. 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
19. 19. 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
20. 20. 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
21. 21. 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
22. 22. 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
23. 23. 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
24. 24. 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
25. 25. 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
26. 26. 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 )
27. 27. 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
28. 28. 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
29. 29. 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
30. 30. 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
31. 31. 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 .
32. 32. 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
33. 33. 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
34. 34. 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
35. 35. 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
36. 36. 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
37. 37. 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
38. 38. 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
39. 39. 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
40. 40. 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
41. 41. 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).
42. 42. 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
43. 43. 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
44. 44. 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
45. 45. 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
46. 46. 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
47. 47. 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
48. 48. 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
49. 49. 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
50. 50. 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