thesis defence

IntroductionGeometric theory of Lie systemsThe theory of Quasi-Lie schemes

Conclusions

Lie systems and applications to QuantumMechanics

Javier de Lucas Araujo

November 29, 2009

Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics


Conclusions

History of Lie systems

Lie’s work

About one century ago, Lie investigated those linear homogeneoussystems of ordinary differential equations

dx j

dt=

n∑k=1

ajk(t)xk , j = 1, . . . , n,

admitting their general solution to be written as

x j(t) =n∑

k=1

λkx j(k)(t), j = 1, . . . , n,

with {x(1)(t), . . . , x(n)(t)} being a family of linear independentparticular solutions and {λ1, . . . , λn} a set of real constants.



Conclusions


Lie noticed that any non-linear change of variables x → ytransforms the previous linear system into a non-linear one of theform

dy j

dt= X j(t, y), j = 1, . . . , n, (1)

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 particularsolutions for (1). He called the expressions of the above kindsuperposition rules.



Conclusions


Lie characterized those non-autonomous systems of first-orderdifferential equations admitting the general solution to bewritten in terms of certain families of particular solutions anda set of constants.

In his honour, these systems are called nowadays Lie systemsand the expressions of the form (2) are still calledsuperposition rules.

Many work have been done since then and many applicationsand developments for this theory have been done by manyauthors.



Conclusions

FundamentalsSuperpositions and connectionsLie systems and Lie groupsIntegrability of Riccati equationsSODEsQuantum Mechanics

Non-autonomous systems and vector fields

In the modern geometrical lenguage, we describe anynon-autonomous system on Rn, written in local coordinates

dx j

dt= X j(t, x), j = 1, . . . , n, (3)

by means of the t-dependent vector field

X (t, x) =n∑

j=1

X j(t, x)∂

∂x j, (4)

whose integral curves are those given by the above system.



Conclusions


Definition

The system (3) is a Lie system if and only if the t-dependentvector field (4) can be written

X (t) =r∑

α=1

bα(t)Xα, (5)

where the vector fields Xα close on a finite-dimensional Lie algebraof vector fields V , i.e. there exist r 3 real constants cαβγ such that

[Xα,Xβ] = cαβγXγ , α, β = 1, . . . , r .

The Lie algebra V is called a Vessiot-Guldberg Lie algebra ofvector fields of the Lie system (5).


Examples of Lie systems

Linear homogeneous and inhomogeneous systems ofdifferential equations.

Riccati equations and Matrix Riccati equations.

Other non-autonomous systems related to famoushigher-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-orderdifferential equations is developed in this work by the first timeand, furthermore, many applications to Physics have been found.


Conclusions


Definition

The system (3) is said to admit a superposition rule, if thereexists a map Φ : U ⊂ Rn(m+1) → Rn such that given a genericfamily of particular solutions {x(1)(t), . . . , x(m)(t)} and a set ofconstants {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 superpositionrule in terms of m generic particular solutions with r ≤ mn.



Conclusions


Superposition Rule for Riccati equations

We asserted that the Riccati equation is a Lie system and thereforeit admits a superposition rule. This is given by

x =x1(x3 − x2)− kx2(x3 − x1)

(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 allowsus to get the general solution.


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 giverise to a new map

(k1, . . . , kn) = Ψ(x , x(1), . . . , x(m)),

Differentiating, we get

X (t, x)Ψj ≡m∑β=0

n∑i=1

X i (t, x(β)(t))∂Ψj

∂x i(β)

= 0, j = 1, . . . , n.

Hence the functions {Ψj | j = 1, . . . , n} are first-integrals for thedistribution D spanned by the vector fields X (t, x). Note that forthe sake of simplicity we call x(0) the variable x .


Conclusions


Definition

Given a t-dependent vector field X (t, x(0)) =∑n

i=1 X i (t, x(0))∂x i(0)

,

on Rn, we call prolongation X of X to the manifold Rn(m+1) to thevector field

X (t, x) ≡m∑β=0

n∑i=1

X i (t, x(β)(t))∂Ψj

∂x i(β)

.

Under certain conditions, n first-integrals for the prolongatedt-dependent vector fields allow us to get the superposition rule.



Conclusions


Lie systems and equations in Lie groups

Lemma

Given any Lie algebra V of complete vector fields on Rn and aconnected Lie group G with TeG ' V , there exists an effective, upto a discrete set of points, action ΦV ,G : G × Rn → Rn such thattheir fundamental vector fields are those in V .

This facts implies important consequences in the theory of Liesystems. As we next show, it provides a way to reduce the study ofLie systems to a particular family of them.



Conclusions


Proposition

Given the Lie system X (t) =∑n

α=1 bα(t)Xα related to a LieVessiot Gulberg Lie algebra V , we can associate it with theequation

g = −n∑

α=1

bα(t)X Rα (g), g(0) = e,

in a connected Lie group G with TeG ' V and with the X Rα

right-invariant vector fields clossing on the same commutationrelations as the Xα. Moreover, the corresponding actionΦV ,G : G × Rn → Rn determines that the general solution for X is

x(t) = ΦV ,G (g(t), x0).



Conclusions


Integrability of Riccati equations

Fixing the basis {∂x , x∂x , x2∂x} of vector fields on R for a

Vessiot–Gulberg Lie algebra V for Riccati equations, each Riccatiequation 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 Rgiven by x ′(t) = ΦV ,G (A(t), x(t)).

Correspondingly, the t-dependent change of variablesx ′(t) = ΦV ,G (A(t), x(t)) transforms a Riccati equation withcoefficients b0, b1, b2 into a new Riccati equation with newt-dependent coefficients, b′0, b

′1, b′2.



Conclusions


Theorem

The group G of curves in a Lie group G associated with a Liesystem, here SL(2,R), acts on the set of these Lie systems, hereRiccati equations.

The group G also acts on the left on the set of curves inSL(2,R) by left translations, i.e. a curve A(t) transforms thecurve 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 aRiccati equation, then the new curve A′(t) satisfies a newequation in SL(2,R) but with a different right hand side a′(t)associated with a new Riccati equation.



Conclusions


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 systemcan be integrated.

J.F. Carinena, J. de Lucas and A. Ramos, A geometric approach tointegrability conditions for Riccati equations, Electr. J. Diff. Equ.122, 1–14 (2007).

If not, putting apart ˙A(t), the previous equation becomes a Liesystem related to a Vessiot–Gulberg Lie algebra isomorphic tosl(2,R)⊕ sl(2,R).



Conclusions


Some restrictions on the solutions of this system make it tobecome a solvable Lie system.

On one hand, this restriction make the system not havingalways solution and some integrability conditions arise.

On the other hand, this system allows to characterize manyintegrable families of Riccati equations.

J.F. Carinena and J. de Lucas, Lie systems and integrabilityconditions of differential equations and some of its applications,Proceedings of the 10th international conference on differentialgeometry and its applications, World Science Publishing, Prague,(2008).



Conclusions


SODE Lie systems

Definition

Any system x i = F i (t, x , x), with i = 1, . . . , n, can be studiedthrough the system corresponding for the t-dependent vector field

X (t) =n∑

i=1

(v i∂x i + F i (t, x , v)∂v i ).

We call SODE Lie systems those SODE for which X is a Liesystem.

J.F Carinena, J. de Lucas and M.F. Ranada. Recent Applicationsof the Theory of Lie Systems in Ermakov Systems, SIGMA 4, 031(2008).



Conclusions


Generalization of the theory of SODEs

Even if here we are dealing with second-order differentialequations, this procedure can be applied to any higher-orderdifferential equation.

The study of superposition rules for second-order differentialequations 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.



Conclusions


Harmonic Oscillators and Milne–Pinney equation

The Milne–Pinney equation with t-dependent frequency isx = −ω2(t)x + kx−3 can be associated with the following systemof first-order differential equations x = v ,

v = −ω2(t)x +k

x3.

This system describes the integral curves for the t-dependentvector field

X (t) = v∂

∂x+

(−ω2(t)x +

k

x3

)∂

∂v.



Conclusions


This is a Lie system because X can be written asX (t) = L2 − ω2(t)L1, where the vector fields L1 and L2 are givenby

L1 = x∂

∂v, L2 = v

∂

∂x+

k

x3

∂

∂v, L3 =

1

2

(x∂

∂x− v

∂

∂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 aVessiot-Gulberg Lie algebra isomorphic to sl(2,R). If k = 0 weobtain the same result for t-dependent frequency harmonicoscillators.



Conclusions


As the Milne–Pinney equation is a Lie system, the theory of Liesystems can be applied and, for example, a new superposition rulecan be obtained:

x = −(λ1x2

1 + λ2x22 ± 2

√λ12(−k(x4

1 + x42 ) + I3 x2

1 x22 )

)1/2

. (6)

J.F. Carinena and J. de Lucas, A nonlinear superposition rule forthe Milne–Pinney equation, Phys. Lett. A 372, 5385–5389 (2008).



Conclusions


Also, if we apply the theory of integrability to the Milne–Pinneyequation or the harmonic oscillator, we can many integrableparticular cases, as the Caldirola-Kanai Oscillator. Moreover, manynew results and integrable cases can be obtained.

J.F. Carinena and J. de Lucas, Integrability of Lie systems andsome of their applications in Physics, J. Phys. A 41, 304029(2008).J.F. Carinena, J. de Lucas and M.F. Ranada, Lie systems andintegrability conditions for t-dependent frequency harmonicoscillators, to appear in the Int. J. Geom. Methods in Mod. Phys.



Conclusions


Mixed superposition rules

Definition

We say that a mixed superposition rule is an expression allowingus to obtain the general solution of a system in terms of a familyof particular solutions of other differential equations.

Theorem

Lie systems related to t-dependent vector fields of the formX (t) =

∑rα=1 bα(t)Xα with vector fields Xα closing on the same

commutation relations, i.e. [Xα,Xβ] = cαβγXγ , cαβγ ∈ R, admit amixed superposition rule.



Conclusions



The Milne–Pinney equation and the harmonic oscillator

The Milne–Pinney equation and the harmonic oscillator infirst-order x = v ,

v = −ω2(t)x +k

x3,

{x = v ,

v = −ω2(t),

can be related to the vector fields X (t) = L2 − ω2(t)L1 forarbitrary k and k = 0.



Moreover, the chosen basis of fundamental vector fields closes onthe same commutation relations. Therefore, there exists a mixedsuperposition rule relating both of them. It can be seen to begiven by

x =

√2

|W |

(I2y 2 + I1z2 ±

√4I1I2 − kW 2 yz

)1/2.

Furthermore, a certain Riccati equation satisfies the sameconditions, therefore, it can be obtained the new mixedsuperposition rule

x =

√(C1(x1 − x2)− C2(x1 − x3))2 + k(x2 − x3)2

(C2 − C1)(x2 − x3)(x2 − x1)(x1 − x3).


Conclusions


Quantum Lie systems

Consider the below facts

Any (maybe infinite-dimension) Hilbert space H can beconsidered as a real manifold.

As there exists a isomorphism TφH ' H we get thatTH ' H⊕H.

Each operator T on a Hilbert space H can be used to define amap X A : ψ ∈ H → (ψ,Aψ) ∈ THAny t-dependent operator A(t) can be related at any time tas a vector field X A(t).

The X A(t) is a t-dependent vector field on a maybe infinitedimensional manifold H.



Conclusions


Geometric Schrodinger equation

Given a t-dependent hermitian Hamiltonian H(t), the associatedSchrodinger equation ∂tΨ = −iH(t)Ψ describes the integral curvesfor the t-dependent vector field X−iH(t).

Theorem

Given two skew self-adjoint operators A,B associated with twovector fields X A and X B , we have that

[X A,X B ] = −X [A,B].



Conclusions


Definition

We say that a t-dependent hermitian Hamiltonian H(t) is aQuantum Lie system if we can write

H(t) =r∑

α=1

bα(t)Hα,

with the Hα hermitian operators such that the iHα satisfy thatthere exist r 3 real constants such that

[iHα, iHβ] =r∑

γ=1

cαβγ iHγ , α, β = 1, . . . , r .



Conclusions


Note that if H(t) is a Quantum Lie system, we can write

X (t) ≡ X−iH(t) =r∑

α=1

bα(t)Xα,

where Xα = X−iHα , and

[Xα,Xβ] =r∑

γ=1

cαβγXγ , α, β = 1, . . . , r .

Schrodinger equations are the equations of the integral curves for at-dependent vector field satisfying the analogous relation to Liesystems but in a maybe infinte-dimensional manifold H.


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 certainvector fields {Xα |α = 1, . . . , r} closing on a finite-dimensional Liealgebra V and we can define an action ΦV ,G : G ×H → H, withTeG ' V , satisfying that for a basis of TeG given by{aα |α = 1, . . . , r} closing on the same commutation relationsthan the Xα, we get

d

dt

∣∣∣∣t=0

ΦV ,G (exp(−taα), ψ) = Xα(ψ),

that is, the Xα are the fundamental vector fields related to the aα.


Conclusions


Reduction to equations in Lie groups

In case that H(t) is a Quantum Lie system, the related actionΦG ,V : G ×H → H allows to relate the Schrodinger equationrelated to H(t) with an equation

g = −n∑

α=1

bα(t)X Rα (g), g(0) = e,

in a connected Lie group G with TeG ' V and withX Rα (g) = Rg∗eaα and aα a basis of TeG . Moreover, the general

solution for X isΨt = ΦV ,G (g(t),Ψ0).



Conclusions


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 thet-dependent Hamiltonian H(t) is a quantum Lie system and itsSchrodinger equation is

dψ

dt= −iBx(t)Sx(ψ)− iBy (t)Sy (ψ)− iBz(t)Sz(ψ),



Conclusions


that can be seen as the differential equation determinating theintegral 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 aquantum Vessiot-Guldberg Lie algebra isomorphic to su(2).

Another example of Quantum Lie systems is given by the family oftime dependent Hamiltonians

H(t) = a(t)P2 +b(t)Q2 +c(t)(QP +PQ)+d(t)Q +e(t)P + f (t)I .



Conclusions


The theory of Lie systems allow us to investigate all these systemsto obtain exact solutions, see

J.F. Carinena, J. de Lucas and A. Ramos, A geometric approach totime evolution operators of Lie quantum systems. Int. J. Theor.Phys. 48 1379–1404 (2009).

or analyse integrability conditions in Quantum Mechanics, see

J.F. Carinena and J. de Lucas, Integrability of Quantum Liesystems. accepted for publication in the Int. J. Geom. MethodsMod. Phys.



Conclusions

Properties of Quasi-Lie schemesLie familiesDissipative Milne-Pinney equations

Definition of quasi-Lie scheme

Definition

Let W ,V be non-null finite-dimensional real vector spaces ofvector fields on a manifold N. We say that they form a quasi-Liescheme 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 .



Conclusions


Quasi-Lie schemes are tools to deal with non-autonomous systemsof first-order differential equations. Let us describe those systemsthat can be described by means of a scheme.

Definition

We say that a t-dependent vector field X is in a quasi-Lie schemeS(W ,V ), and write X ∈ S(W ,V ), if X belongs to V on itsdomain, i.e. Xt ∈ V|NX

t.

Lie systems and Quasi-Lie schemes

Given a Lie system related to a Vessiot-Guldber Lie algebra ofvector fields V , then S(V ,V ) is a quasi-Lie scheme. Hence, Liesystems can be studied through quasi-Lie schemes.



Conclusions


Abel equations

Abel equations

Abel differential equations of the first kind are of the form

x = f3(t)x3 + f2(t)x2 + f1(t)x + f0(t).

Consider the linear space of vector fields V spanned by the basis

X0 =∂

∂x, X1 = x

∂

∂x, X2 = x2 ∂

∂x, X3 = x3 ∂

∂x,

and define the Lie algebra W ⊂ V as

W = 〈X0,X1〉.



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 Abelequation can be described through this scheme because

X (t, x) =3∑

α=0

fα(t)Xα(x),

and thus X ∈ S(W ,V ).



Conclusions


Symmetries of a scheme

Definition

We call symmetry of a scheme S(W ,V ) to any t-dependentchange of variables transforming any X ∈ S(W ,V ) into a newX ′ ∈ S(W ,V ).

We have charecterized different kinds of infinite-dimensionalgroups of transformations of a scheme:

The group of the scheme, G(W ).

The extended group of the scheme, Ext(W ).


Theorem

Given a scheme S(W ,V ) and an element g ∈ Ext(W ), then forany X ∈ S(W ,V ) we get that gFX ∈ S(W ,V ).

Definition

Given a scheme S(W ,V ) and a X ∈ S(W ,V ), we say that X is aquasi-Lie system with respect to this scheme if there exists asymmetry of the scheme such that gFX is a Lie system.

Theorem

Every Quasi-Lie system admits a t-dependent superposition rule.

J.F. Carinena, J. Grabowski and J. de Lucas, Quasi-Lie systems:theory and applications, J. Phys. A 42, 335206 (2009).


Conclusions


We can characterise those non-autonomous systems of first-orderdifferential 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

X =n∑

α=1

bα(t)Xα,

where the Xα satisfy that there exist n3 functions fαβγ(t) such that

[Xα, Xβ] =n∑

γ=1

fαβγ(t)Xγ , α, β = 1, . . . , n.



Conclusions


The applications of quasi-Lie systems and Lie families are verybroad, 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 PDEsalso.



Conclusions


Dissipative Ermakov systems are

x = a(t)x + b(t)x + c(t)1

x3.

As a first-order differential equation it can be written{x = v ,v = a(t)v + b(t)x + c(t) 1

x3 .

Consider the space V spanned by the vector fields:

X1 = v∂

∂v, X2 = x

∂

∂v, X3 =

1

x3

∂

∂v, X4 = v

∂

∂x, X5 = x

∂

∂x.



Conclusions


Take the three-dimensional Lie algebra W ⊂ V generated by thevector fields

Y1 = X1 = v∂

∂v, Y2 = X2 = x

∂

∂v, Y3 = X5 = x

∂

∂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.



Conclusions


Moreover, the above system describes the integral curves for thet-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-dependentdiffeomorphisms of TR related to elements of Ext(W ) reads{

x = γ(t)x ′,v = α(t)v ′ + β(t)x ′,

, α(t) 6= 0, γ(t) 6= 0.



Conclusions


The transformed system of differential equations obtained throughour scheme is

dx ′

dt = β(t)x ′ + α(t)v ′,dv ′

dt =(

b(t)α(t) + a(t)β(t)

α(t) −β2(t)α(t) −

β(t)α(t)

)x ′

+(

a(t)− β(t)− α(t)α(t)

)v ′ + c(t)

α(t)1

x ′3 .

Note that if β(t) = 0 this system is associated with

d2x ′

dt2= a(t)

dx ′

dt+ b(t) x ′ + c(t)

1

x ′3.



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, weobtain a t-dependent superposition rule

x(t) =

√2

W

(I2x2

1 (t) + I1x22 (t)±

√4I1I2 − kW 2 x1(t)x2(t)

)1/2,

with x1, x2 solutions for the differential equation

x = a(t)x + b(t)x .



Conclusions

Advances in Lie systems

Let us schematize some developments obtained during this work:

Usual Lie systems

First-order

Finite-dimensionalmanifolds

Ordinary differentialequations

Lie systems in this work

Higher-order

Infinite-dimensionalmanifolds (QM)

Partial differentialequations



Conclusions

Other advances

1 The theory of integrability of Lie systems have been developed. It has been shown that Matrix Riccatiequations 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 understoodgeometrically and new ones have been provided. These methods allow us to get integrable models toanalyse Quantum Systems.

3 The theory of Lie systems have been applied to analyse higher order differential equations. The theory ofintegrability has been aplied here and many results have been obtained and explained. We have foundmany 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 broadand the number of systems analysed with it is still very small.


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