Picard–Vessiot theory and integrability

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Picard–Vessiot theory and integrability Juan J. Morales-Ruiz ∗ Technical University of Madrid, Spain

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Article history: Received 12 February 2014 Received in revised form 6 June 2014 Accepted 2 July 2014 Available online xxxx

abstract We survey some recent applications of the Differential Galois Theory of linear differential equations to the integrability (or solvability) of Dynamical Systems and Spectral Problems. © 2014 Elsevier B.V. All rights reserved.

MSC: primary 12H05 secondary 32S65 Keywords: Differential Galois Theory Picard–Vessiot theory Integrability of dynamical systems Schrödinger equation Riccati equation Liouvillian solution

0. Introduction This survey addresses a Galoisian approach to the integrability of two topics: the non-integrability criteria for complex analytical finite dimensional Hamiltonian systems and the relatively new approach to the solvability–integrability of some spectral problems. Integrability here means more or less what people understand by ‘‘solvable in closed form’’, and the concrete meaning of what kind of solvability we consider here will be made precise along the paper. Thus, after the necessary results of Picard–Vessiot theory, the second section of this paper is devoted to the non-integrability results by means of the variational equations around a particular solution. As I already wrote a joint survey with Ramis about this topic four years ago [1], I only consider here the new results not included in the said survey. Even with the above limitation in mind, we essentially only overview the main ideas without entering into the details, because most of the papers are considerably technical and it is not easy to give very precise concise statements. The third section is devoted to the integrability of spectral problems. It is mainly based on my joint paper with Acosta and Weil [2], but in Section 3.3 a new Galoisian interpretation of a scattering problem is given for a concrete family of potentials. This topic is connected with Galoisian interpretations of families of Special Functions. In fact, from the research work in the area along many years, we know the relevance of the ‘‘botanic’’ of higher transcendental functions in this field. I decided to write a survey on it to spread this new point of view for the researches interested in these mathematical-physics problems. I apologize that, due to space limitations, this survey will not cover all existing applications of Differential Galois Theory to integrability. In particular, I will not review recent applications of the Picard–Vessiot theory to the integrability of polynomial fields on the plane (see [3,4]), neither the interesting new applications to integrability of the non-linear differential Galois theory of Malgrange and Umemura (works by Casale and others, see Section 1 for some references).

∗

Tel.: +34 91 336 66 66. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.geomphys.2014.07.006 0393-0440/© 2014 Elsevier B.V. All rights reserved.

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The interested reader can find another nice survey, on several problems included in this paper, as well as other problems not reviewed here, like the application of the Picard–Vessiot theory to the Ising model, in the Habilitation Thesis of Weil [5]. Added in Proof: After this paper was written, I have noticed that Ramis (private communication) also obtained some years ago a Galoisian interpretation of the scattering problem studied in Section 3.3. 1. Picard–Vessiot theory For convenience of the reader, we give a brief account of useful terminology and results of the Galois theory of linear differential equations, also called the Picard–Vessiot theory. This is essentially the only kind of differential Galois theory considered along this paper. As the Galois group of a linear differential equation is an algebraic group, we start with a sketch about algebraic groups. Standard references on algebraic groups are [6,7]. A linear algebraic group G (over C) is a subgroup of GL(m, C) whose matrix coefficients satisfy polynomial equations over C. It has both the structure of a non-singular algebraic variety and that of a group, both structures being compatible in that the group operation and inversion are morphisms of algebraic varieties. We note that in a linear algebraic group there are two different topologies: the Zariski topology, where the closed sets are the algebraic sets, and the usual Hausdorff topology. In particular, an algebraic group is a complex analytical Lie group and we can consider its Lie algebra. Therefore the dimension of G is the dimension of the Lie algebra of G. Given a linear algebraic group, the maximal connected subgroup G0 which contains the identity is an algebraic group called the identity component of G. For those familiar with Lie theory, and since G is a complex analytical Lie group due to satisfying algebraic equations, the underlying group of G0 coincides with the identity component of G considered as a complex analytical Lie group. We remark that an algebraic linear (or affine) group G is usually defined as an affine algebraic variety with a group structure, with the above compatibility condition as to group multiplication and taking inverses. Then, there is a rational faithful representation of G as a closed subgroup of GL(m, C), for some m, and we obtain the equivalence with our definition. It is clear that the classical linear complex groups are linear algebraic groups. For instance SL(n, C), SO(n, C) (rotation group) and Sp(n, C) ⊂ GL(2n, C) (symplectic group) are linear algebraic groups, since they are defined by polynomial identities. Proposition 1.1. The identity component G0 of a linear algebraic group G is a closed (with respect to the above two topologies) normal subgroup of G of finite index and is connected with respect to the above two topologies. Furthermore G/G0 is a finite group given by the classes of the irreducible connected components of G. We note that by the above proposition G0 is also a linear algebraic group and the Lie algebra of G, Lie(G) = G and G is connected if, and only if, G = G0 . The characterization of the connected solvable linear algebraic groups is given by the Lie–Kolchin theorem. Theorem 1.2 (Lie–Kolchin Theorem). A connected linear algebraic group is solvable if, and only if, it is conjugate to a triangular group. The algorithmic methods in Picard–Vessiot theory are mainly based on the classification of the algebraic subgroups of a given algebraic group. In particular, Kovacic algorithm in Appendix A is based on the following theorem. Theorem 1.3. Let G be an algebraic subgroup of SL(2, C). Then one of the following four cases can occur. (1) G is triangularizable. (2) G is conjugate to a subgroup of the infinite dihedral group (also called meta-abelian group) and case (1) does not hold. (3) Up to conjugation G is one of the following finite groups: Tetrahedral group, Octahedral group or Icosahedral group, and cases (1) and (2) do not hold. (4) G = SL(2, C). Given a subset S ⊂ GL(n, C), let M be the group generated by S and let G be the Zariski closure of the group M, i.e., the smallest linear algebraic subgroup which contains M. We say that G is topologically generated by M. Now we introduce the Picard–Vessiot theory. At the end of the nineteenth century, Picard ([8,9], [10, Chapitre XVII]) and Vessiot [11], created and developed a Galois theory for linear differential equations. This field of study, henceforth called Picard–Vessiot theory, was continued from the forties to the sixties of the twentieth-century by Kolchin, through the introduction of the modern algebraic abstract terminology and the obtention of new important results. Two references about the Picard–Vessiot theory are [12,13]. We partially follow [1]. For a nice introduction see [14,15]. As we shall see, within differential Galois theory there is a very nice concept of ‘‘integrability’’, i.e., solutions in closed form. Furthermore, all information about the integrability of the equation is coded in the identity component of the Galois group: a linear equation is integrable if, and only if, the identity component of its Galois group is solvable. We only review the necessary definitions and results of the Picard–Vessiot theory in order to understand the applications to integrability. For more information see [12].

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A differential field K is a field with a derivative (or derivation) ∂ = ′ , i.e., an additive mapping satisfying the Leibniz rule. From now on we will assume that K = M (Γ ), the meromorphic functions over a connected Riemann surface Γ . The reason for this notation will be clear below: Γ − Γ will be the set of singular points of the linear differential equation, i.e., poles d of the coefficients with dx as derivation, x being a local coordinate over the Riemann surface Γ . A particular classical case is 1 when K = C(x) = M (P ) is the field of rational functions, i.e., the field of meromorphic functions over the Riemann sphere P1 . Another interesting example for the applications is a field of elliptic functions. We can define differential subfields and differential extensions in a direct way by requiring that inclusions commute with the derivation. Analogously, a differential automorphism in K is an automorphism commuting with the derivative. Let y′ = Ay,

A ∈ Mat (m, K )

(1.1)

be a system of linear differential equations. We now proceed to associate to (1.1) the so-called Picard–Vessiot extension of K . The Picard–Vessiot extension L of (1.1) is an extension of K , such that if φ1 , . . . , φm is a ‘‘fundamental’’ system of solutions of Eq. (1.1) (i.e., linearly independent over C), then L = K (φij ) (rational functions in K in the coefficients of the ‘‘fundamental’’ matrix Φ = (φ1 · · · φm )). This is the extension of K generated by K together with φij . We observe that L is a differential field (by (1.1)). The existence and unicity of the Picard–Vessiot extensions was proven by Kolchin. In the analytical case K = M (Γ ), ′ = d/dx, this result is essentially the existence and uniqueness theorem for linear differential equations. As in classical Galois theory of algebraic equations, we define the Galois group of (1.1), G := GalK (L) = Gal(L/K ), as the group of all the (differential) automorphisms of L leaving the elements of K fixed. Then one of the main results of the theory is that the Galois group of (1.1) is faithfully represented as an algebraic linear group over C, the representation is given by the action σ ∈ G,

σ (Φ ) = Φ Bσ , Bσ ∈ GL(m, C). Furthermore, by a classical theorem credited to Schlesinger, the relation between the monodromy and the Galois group is as follows. Let Γ − Γ be the set of singular points of the equation i.e., the poles of the coefficients on Γ . We recall that the monodromy group of the equation is the subgroup of the linear group defined as the image of a representation of the fundamental group π1 (Γ ) into the linear group GL(m, C). This representation is obtained by analytical continuation of the solutions along the elements of π1 (Γ ). The monodromy group M is contained in the Galois group G and if the equation is Fuchsian (i.e., it has regular singular points only), then M is Zariski dense in G, see for instance [12]. In particular, this implies that for Fuchsian differential equations the Galois group is solvable or commutative, if, an only if, the monodromy group is solvable or commutative, respectively. In the general case, Ramis found a generalization of the above and, for example, he showed that the Stokes matrices associated to an irregular singularity belong to the Galois group, see [16]. We call a linear differential equation integrable if there exists a chain of differential extensions K1 := K ⊂ K2 ⊂ · · · ⊂ Kr := L, where each extension is given by the adjunction of one element a, Ki ⊂ Ki+1 = Ki (a, a′ , a′′ , . . .), such that a satisfies one of the following conditions: (i) a′ ∈ Ki , (ii) a′ = ba, b ∈ Ki , (iii) a is algebraic over Ki . Then, it can be proven that a linear differential equation is integrable if, and only if, the identity component G0 of the Galois group is a solvable group. In particular, if G0 is commutative, the equation is integrable. The usual terminology for integrable linear equations is that the associated Picard–Vessiot extension is a Liouville extension [14,12]. We prefer to use a terminology in agreement with our dynamical approach and with the creators of the theory [11]. The above definition of integrability was essentially introduced by Liouville, he called it ‘‘integration by explicit finite quantities’’ [17]. Furthermore for second order reduced linear differential equations y′′ = ry,

(1.2)

being r = r (x) a polynomial, he obtained a characterization of this integrability: Eq. (1.2) is integrable, if the associated Riccati equation

θ ′ = r − θ 2,

θ=

y′

(1.3) y has an algebraic solution over K = C(x). Today this result has been extended for differential fields K of characteristic zero and with algebraic closed field of constants [12,14]. As it is relevant for integrability problems, we will state next a generalization of the Liouville result for general linear complex analytic differential systems. Thus we are looking for a characterization of the integrability of these systems, using a suitable associated system of non-linear differential equations. We follow [18]. Let G be a complex algebraic group over C, K a differential field containing C and GK := the group G with coefficients in K . We want to characterize the integrability of the system y′ = A(x)y,

A(x) ∈ Lie(G) ⊗ K ,

being G a complex linear algebraic group.

(1.4)

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Let Φ (x) be a fundamental matrix of the above equation. Recall that, the gauge transformations are transformations of this fundamental matrix, g (x) ∈ GK ,

Φ (x) = g (x)Ψ (x). The transformed equation z′ = g [A](x)z,

g [A] = g −1 Ag − g −1 g ′ ,

(1.5)

has Ψ (x) as a fundamental matrix. We remark that we only consider here gauge transformations with coefficients that remain in the differential field K . When K is a field of meromorphic functions, we can interpret this kind of gauge transformations as a change of frame in the meromorphic connection defined by the linear differential equation. As a first example, on suitable differential fields, we can read the d’Alembert classical reduction of order, when a particular solution is given as a gauge transformation. From a linear system as above y′ = A(x)y,

A(x) ∈ Lie(G) ⊗ K ,

(1.6)

is it possible to reduce the group G to an algebraic subgroup H ⊂ G by means of a gauge change?, i.e., is there a g (x) ∈ GK such that g [A] ∈ Lie(H ) ⊗ K ? The answer is yes, but maybe we have to consider Eq. (1.6) defined over a suitable algebraic extension of K . Theorem 1.4 (Kolchin–Kovacic). Let H be the Galois group of Eq. (1.6). Then (1) H ⊂ G. (2) There exists an algebraic extension  K of K , and a gauge transform g (x) ∈ G K , such that Eq. (1.6) is transformed to an equation over H: g [A] ∈ Lie(H ) ⊗  K. A related result is that the degree of transcendence of the Picard–Vessiot extension L = K (φij ) of the differential system (1.6) coincides with the dimension of the Galois group: degtr(L/K ) = dim(Gal(L/K )) = dim(Lie(Gal(L/K ))).

(1.7)

Let G be a complex algebraic group, the (complete) flag manifold of G is the homogeneous space Flag(G) = G/B, being B a Borel subgroup of G (a maximal connected solvable subgroup of G). Then our linear system y′ = A(x)y,

A(x) ∈ Lie(G) ⊗ K ,

(1.8)

is projected to a non-linear system on the flag manifold of G: the flag differential system associated to (1.8). The flag system is an example of a Lie–Vessiot system, differential systems with superposition principles, i.e., the general solution is obtained from a finite number of particular solutions and several constants. Other examples of Lie–Vessiot systems are Riccati scalar equations, projective Riccati equations, matrix Riccati equations associated to projections on Grassmannians of the linear system or, even more general, arbitrary flag systems associated to a linear system. A nice Galois theory can be formulated for general Lie–Vessiot systems ([18], see also [19]). Examples of flag systems (1) G = GL(n, C): a fundamental matrix solution Φ (x) has a LU type factorization

Φ = WΨ , W : lower triangular with 1 over the diagonal, Ψ : upper triangular. The coordinates of W = W (x), wij , i > j, satisfy a non-linear polynomial (in the wij coordinates) differential system: the flag system. Gauge interpretation: W (x) gauge transform over a suitable differential field. (2) G = SO(3, C):

 ′

y1 y′2  = y′3

0 −a(x) −b ( x )



a(x) 0 −c (x)

b(x) c (x) 0

y1 y2 y3

  .

Flag system:

w′ =

−b − ic 2

− iaw +

ic − b 2

w2 (Riccati type).

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(3) Example 3. G = SL(2, C): y′′ = r (x)y. Flag system: θ ′ = r (x) − θ 2 (Riccati associated equation). This is the context of Liouville theorem. We have the following generalization of the Liouville result. Theorem 1.5. A necessary and sufficient condition for the integrability of equation y′ = A(x)y,

A(x) ∈ Lie(G) ⊗ K ,

is that the associated flag system has an algebraic solution over the differential field K . For the generic equation G = GL(n, C) (ie, A ∈ Mat (n, K )) the proof is easy and based on the following facts: – A fundamental matrix has a LU type factorization:

Φ = WΨ . – – – –

W is a gauge transformation, solution of the flag system. The transformed system with fundamental matrix Ψ is triangular. The Galois group of a triangular linear system is solvable. Application of the Kolchin–Kovacic reduction.

The proof in the general situation is similar but more involved. We finish this section with the algebrization procedure, very important in the applications. For some concrete families of it is possible, to replace the original differential equation over a Riemann surface, by a new differential equation over the Riemann sphere P1 (i.e., with rational coefficients) by a change of the independent variable. This equation on P1 is called the algebraic form of the equation. In a more general way we will consider the effect of a finite ramified covering on the Galois group of the original differential equation. Theorem 1.6 ([20], See also [21]). Let Γ be a Riemann surface, d dx

y = A(x)y,

A ∈ Mat (m, C(x))

(1.9)

a linear differential equation on P1 and x : Γ → P1 , x = x(t ) a finite ramified covering of P1 (t a local parameter in Γ ). Let d dt

y = x∗ (A)(t )y,

x∗ (A) ∈ Mat (m, M (Γ ))

(1.10)

be the pull-back of Eq. (1.9) by x (i.e., the equation obtained by the change of variables x = x(t )). Then the identity components of the Galois groups of Eqs. (1.9) and (1.10) are the same. We say that a linear differential equation d dt

y = B(t )y,

B ∈ Mat (m, M (Γ ))

(1.11)

is algebrizable if it is the pull-back of a linear differential equation (1.9) with rational coefficients. In order to apply Kovacic’s algorithm it is important to know whether a given second order linear differential equation is algebrizable. An algorithm to algebrize equations of the type y¨ = r (t )y is proposed in [22]. We say that a change of variable x = x(t ) is Hamiltonian if and only if (x(t ), x˙ (t )) is a solution curve of the autonomous 1-degree of freedom Hamiltonian system H = H (x, x˙ ) =

x˙ 2 2

+ V (x).

Proposition 1.7 (Algebrization Algorithm [22]). The differential equation y¨ = r (t )y

(1.12)

is algebrizable through a Hamiltonian change of variable x = x(t ) if, and only if, there exist f , α such that

α′ , α

f

α

∈ C(x),

where f (x(t )) = r (t ),

α(x) = 2(h − V (x)) = x˙ 2 .

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Furthermore, the algebraic form of the equation y¨ = r (t )y is y′′ +

f 1 α′ ′ y − y = 0, 2 α α

(1.13)

where ′ = d/dx. From the above we know that when r (t ) belong to the field of meromorphic functions over Γ , M (Γ ), then the identity component of the Galois group is preserved by the above change of variables x = x(t ). We would like to point out that in the last years a new non-linear differential Galois theory [23–29] as well as a Parametric Picard–Vessiot theory has come into being [30]. We will not review these works along this survey. 2. Integrability of Hamiltonian systems As a survey has been published about this topic in 2010 [1], I only mention here some new results from then until now. Even with the above in mind, it would not be possible for me to give a complete account of this topic. 2.1. General theorems For the sake of completeness we recall following [1] the general theorems for obstructions to meromorphic integrability of complex analytical Hamiltonian systems x˙ i = ∂ H /∂ yi ,

y˙ i = −∂ H /∂ xi ,

i = 1, . . . , n.

We recall here the definition of integrability for Hamiltonian systems. One says that XH = (∂ H /∂ yi , −∂ H /∂ xi )i = 1, . . . , n, is completely integrable or Liouville integrable if there are n functions f1 = H, f2 , . . . , fn , such that (1) they are functionally independent i.e., the 1-forms dfi i = 1, 2, . . . , n, are linearly independent over a dense open set U ⊂ M, U¯ = M; (2) they form an involutive set, {fi , fj } = 0, i, j = 1, 2, . . . , n. We recall that in canonical coordinates the Poisson bracket has the classical expression

{f , g } =

n  ∂f ∂g ∂f ∂g − . ∂ y ∂ x ∂ xi ∂ yi i i i=1

We remark that in virtue of item (2) above the functions fi , i = 1, . . . , n are first integrals of XH . It is very important to be precise regarding the degree of regularity of these first integrals. Here we assume that the first integrals are meromorphic. Unless otherwise stated, this is the only type of integrability of Hamiltonian systems that we consider in the next pages. Sometimes, to recall this fact we shall talk about meromorphic (complete) integrability. Given a dynamical system, z˙ = X (z ),

(2.1)

with a particular integral curve z = φ(t ), at the end of the nineteenth century Poincaré introduced the variational equation (VE) along z = φ(t ),

ξ˙ = X ′ (φ(t ))ξ ,

(2.2)

as the fundamental tool to study the behavior of (2.1) in a neighborhood φ(t ). Eq. (2.2) describes the linear part of the flow of (2.1) along z = φ(t ). Now we can write the variational equations along a particular integral curve z = φ(t ) of the vector field XH

ξ˙ = XH′ (φ(t ))ξ .

(2.3)

Using the linear first integral dH (z (t )) of the variational equation it is possible to reduce this variational equation and to obtain the so-called normal variational equation which, in suitable coordinates, can be written as a linear Hamiltonian system

η˙ = JS (t )η, where, as usual,

 J =

0 −I



I 0

is the standard matrix of the symplectic form of dimension 2(n − 1).

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More generally, if, including the Hamiltonian, there are m meromorphic first integrals independent over Γ and in involution, we can reduce the number of degrees of freedom of the variational equation (2.3) by m and obtain the normal variational equation (NVE) which, in suitable coordinates, can be written as a 2(n − m)-dimensional linear system

η˙ = JS (t )η,

(2.4)

where now J is the matrix of the symplectic form of dimension 2(n − m). For more details about the reduction to the NVE, see [20] (or [21]). Theorem 2.1 ([20], See also [21]). Assume a complex analytic Hamiltonian system is meromorphically completely integrable in a neighborhood of the integral curve z = φ(t ). Then the identity components of the Galois groups of the variational equations (2.3) and of the normal variational equations (2.4) are commutative groups. We remark that it is a typical version of several possible theorems. In some cases it is interesting to add to the manifold M some points at infinity; thus we suppose that we are in the following situation: M is an open subset of a complex manifold M, M \ M is an hypersurface (which is by definition the hypersurface at infinity), the two-form ω on M defining the symplectic structure extends meromorphically on M and the vector field XH extends meromorphically on M. In such a case, when (2.3) has irregular singular points at infinity, we only obtain obstructions to the existence of first integrals which are meromorphic along Γ , i.e., also at the points at infinity of Γ ; for example, for rational first integrals when M is a projective manifold. From a dynamical point of view, the singular points of the variational equation (2.3), Γ − Γ , correspond to equilibrium points, meromorphic singularities of the Hamiltonian field or points at infinity. Theorem 2.2 ([20], See also [21]). Consider a complex symplectic manifold (M , ω), which is an open subset of a complex manifold M, M \ M being an hypersurface and ω admitting a meromorphic extension on M. Let XH be a meromorphic vector field on M which is analytic and Hamiltonian on M. If the system z˙ = XH (z ) is meromorphically integrable in a neighborhood in M of some integral curve Γ with first integrals which extend into meromorphic functions on a neighborhood of Γ , then the identity component of the Galois groups of (2.3) and (2.4) (interpreted as differential equations on Γ ) are commutative groups. In particular, let M be an open domain of a symplectic complex space and assume the points at infinity of (2.3) (or (2.4)) are irregular singular points and the identity component of the Galois group of (2.3) (or (2.4)) is not commutative, then the Hamiltonian system is not integrable by rational first integrals. Theorem 2.1 (and 2.2) has been generalized to higher order variational equations VE k along Γ , with k > 1 (the solutions of these equations are the quadratic, cubic, etc. contributions to the flow of the Hamiltonian system along the particular solution z = φ(t ) = φ(z0 , t )), VE1 being Eq. (2.2) [31]. The ‘‘fundamental’’ solution of VE k of a dynamical system (2.1) is given by

(φ (1) (t ), φ (2) (t ), . . . , φ (k) (t )), where

φ(z , t ) = φ(z0 , t ) + φ (1) (t )(z − z0 ) + · · · +

1 (k) φ (t )(z − z0 )k + · · ·

k!

the Taylor series up to order k of the flow φ(z , t ) with respect to the variable z at the point (z0 , t ). That is, φ (k) (t ) = ∂k φ(z0 , t ). ∂ zk

The initial conditions are φ (1) (0) = Idm and φ (j) (0) = 0 for all j > 1. We stress that, in contrast to some defini-

tions, we do not consider the differential equations for φ (k) , but for (φ (1) , φ (2) , . . . , φ (k) ), as variational equation of order k. The variational equation VE k is not linear, but it is in fact equivalent to a linear differential equation: there exists a linear differential equation LVE k with coefficients in the field of meromorphic functions over Γ such that the differential extensions generated by the solutions of VE k coincide with the Picard–Vessiot extensions of LVE k . Then we can consider the Galois group Gk of VE k , Gk = Gal(VE k ), i.e., of the LVE k . For simplicity we shall denote both VE k and LVE k by VE k . Furthermore, the singular points of the equations VE k are the same as the first order variational equation VE 1 and a singular point of VE k (k ≥ 1) is irregular if and only if it is irregular for VE 1 . Although in order to obtain the main theoretical results it is convenient to work with the jet formalism (see [31]), from a computational practical point of view, the higher order variational equations can be obtained using the small parameter method of the masters (Poincaré, Liapunov, . . . ). Hence, we expand the general solution of the non-linear equation (2.1) along the particular solution φ(t )

ξ (2) 2 ξ (k) k z (t ) − φ(t ) = ξ (1) ε + ε + ··· + ε + ···, 2! k! being ε a small parameter. Introducing the above in Eq. (2.1) and equating the same powers of ε , using the fact that X (z ) = X (φ(t )) + X ′ (φ(t ))(z (t ) − φ(t )) +

1 2!

X (2) (φ(t ))(z (t ) − φ(t ))2 + · · · +

1 (k) X (φ(t ))(z (t ) − φ(t ))k + · · · , k!

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we obtain the variational equation of order k, VE k ,

ξ˙ (j) = X ′ (φ(t ))ξ (j) + Pj (ξ (1) , . . . , ξ (j−1) ),

j = 1, . . . , k,

(2.5) (1)

(j−1)

where P1 ≡ 0 (i.e., VE 1 becomes (2.2)) and Pj is polynomial of degree j in ξ , . . . , ξ , with meromorphic coefficients over Γ (i.e., in the same differential field of coefficients of the first order variational equation). The initial conditions of (2.5) are (ξ0(1) , ξ0(2) , . . . , ξ0(k) ) = (Idm , 0, . . . , 0). We observe that the solution of (2.5) is obtained from the solution of the first order variational equation by successive applications of the constants variation method of Lagrange: we substitute the solutions (2) of VE 1 in P2 (ξ1 ), we solve the second of the equations of VE 2 , i.e., the equation in ξ (2) (with initial conditions ξ0 = 0) and we substitute all the above solutions of VE 2 in P3 (ξ1 , ξ2 ), and so on. In particular, (2.5) is integrable if, and only if, VE 1 is integrable. For instance, the third order variational equation, VE 3 , is explicitly given by the system d dt d dt d dt

ξ j ,k =



∂i Xj ξi,k ,

i

ξj,k1 k2 =



∂i Xj ξi,k1 k2 +

i1 ,i2

i

ξj,k1 k2 k3 =





∂i21 ,i2 Xj ξi1 ,k1 ξi2 ,k2 , (2.6)

∂i Xj ξi,k1 k2 k3 +

i1 ,i2

i

+





∂

ξ

∂i21 ,i2 Xj ξi1 ,k1 k2 ξi2 ,k3 +

ξ

ξ

3 i1 ,i2 ,i3 Xj i1 ,k1 i2 ,k2 i3 ,k3

i1 ,i2 ,i3

 i1 ,i2

∂i21 ,i2 Xj ξi1 ,k1 k3 ξi2 ,k2 +

 i1 ,i2

∂i21 ,i2 Xj ξi1 ,k1 ξi2 ,k2 k3

,

where ξ (1) = (ξj,k ), ξ (2) = (ξj,k1 k2 ), ξ (3) = (ξj,k1 k2 k3 ), ξj,k := ∂∂z φi (z0 , t ), ξj,k1 k2 := ∂ z ∂ ∂ z φi (z0 , t ), ξj,k1 k2 k3 := ∂ z ∂∂z ∂ z k k1 k2 k1 k2 k3 φi (z0 , t ), j = k = k1 = k2 = k3 = 1, . . . , m. We remark that the matrix (ξj,k ) is in fact the fundamental matrix of Eq. (2.3). Now we are going to describe the practical method of linearization of the equations VE k . The problem is to find a system of linear equations for 2

3

(ξ (1) (t ), ξ (2) (t ), . . . , ξ (k) (t )) equivalent to VE k . It is enough to write the equations satisfied by the monomials appearing in Pj . This is the content of the next lemma. Lemma 2.3. Let z ∈ Cq . Assume the components (z1 , . . . , zq ) of z satisfy linear homogeneous differential equations z˙i =

q

aij (t )zj . Then the monomials z k := differential equations. j =1

k

q

i=1

zi i of order |k| = k1 + · · · + kq satisfy also a system of linear homogeneous

Proof. Let k = (k1 , . . . , kq ) a multi-index of non-negative integers. Then d dt

k

z =

q 

 k j −1 kj zj

j =1

q 

ajr zr

q 

 k zi i

,

(2.7)

i=1,i̸=j

r =1

the right hand side being also homogeneous of degree |k| in z.



We observe that the above lemma is nothing other than the pull-back to the symmetric fiber bundle, S k (Cq ), of the q connection associated to the linear differential equation z˙i = j=1 aij (t )zj (see [21], Section 2.3). We will denote the coefficient matrix of the differential system obtained by this pull-back by symmk (A), being (aij ) = A. Then, in order to linearize VE k , after the last equation corresponding to VE k we can supplement the system of linear   differential equations with the equations for the components of (ξ (i1 ) )m1 , (ξ (i2 ) )m2 , . . . , (ξ (is ) )ms . Then the structure of the coefficient matrix Ak of the linearized VE k will be block lower triangular

 Ak =

symk (A1 ) Bk

0 Ak−1



,

(2.8)

being A1 , Ak−1 the matrices of the first and k − 1 order variational equations, respectively. For more details about this linearization see [31]. Since we can now consider the equations VE k as linear differential equations, we can talk about their Picard–Vessiot extensions and about their Galois groups Gk . Theorem 2.4 ([31]). Assume that a complex analytical Hamiltonian system is integrable by meromorphic first integrals in a neighborhood of the integral curve z = φ(t ). Then the identity components (Gk )0 , k ≥ 1, of the Galois groups of the variational equations along Γ are commutative.

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Theorem 2.1 has important consequences for the integrability of Hamiltonian systems with homogeneous potentials [32]. Consider an n-degrees-of-freedom Hamiltonian system with Hamiltonian H (x, y) = T + V =

1 2

(y21 + · · · + y2n ) + V (x1 , . . . , xn ),

(2.9)

V being a complex homogeneous function of non-zero integer degree k and 2 ≤ n. From the homogeneity of V , it is possible to obtain an invariant plane x = z (t )c, y = z˙ (t )c, where z = z (t ) is a solution of the (scalar) hyperelliptic differential equation z˙ 2 =

2 k

(1 − z k )

(where we assume case k ̸= 0), and c = (c1 , c2 , . . . , cn ) is a solution of the equation c = V ′ (c).

(2.10)

This is our particular solution Γ along which we compute the variational equation VE and the normal variational equation NVE. We shall call these the homothetical solutions of the Hamiltonian system (2.9) and denote solutions of (2.10) as homothetical points. In most of the references about the integrability of the homogeneous potentials the solutions of (2.10) are called Darboux points. The VE along Γ is given in the temporal parametrization by

η¨ = −z (t )k−2 V ′′ (c)η. Assume V ′′ (c) is diagonalizable. Assuming that the Hessian matrix V ′′ (c) is diagonalizable, it is possible to express the VE as a direct sum of second order equations

η¨ i = −z (t )k−2 λi ηi ,

i = 1, 2, . . . , n,

where we keep η for the new variable, λi being the eigenvalues of the matrix V ′′ (c). We call these eigenvalues Yoshida coefficients. One of the above second order equations is the tangential variational equation, say, the equation corresponding to λn = k − 1. This equation is trivially solvable, whereas the NVE is an equation in the variables ξ := (η1 , . . . , ηn−1 ) := (ξ1 , . . . , ξn−1 ), i.e.,

ξ¨ = −z (t )k−2 diag (λ1 , . . . , λn−1 )ξ . Now, following Yoshida [33], we consider the change of variable x =: z (t )k . Thanks to the symmetries of this problem, we obtain as NVE a system of independent hypergeometric differential equations in the new independent variable x x(1 − x)

d2 ξ dx2

 +

k−1 k

−

3k − 2 2k

 x

dξ dx

+

λi 2k

ξ = 0,

i = 1, 2, . . . , n − 1.

(ANVEi )

We remark that this change is an algebrization of the NVE and we know that the identity component of the Galois group of this equation is preserved by this change. Each of these equations ANVEi , corresponding to the Yoshida coefficient λi , is part of the system called the algebraic normal variational equation ANVE. As a matters of fact, the ANVE splits into a system of n − 1 independent equations ANVEi , i = 1, . . . , n − 1. Each one of the above ANVEi is an hypergeometric equation with three regular singular points at x = 0, x = 1 and x = ∞. By Theorem 1.6 the identity component of the Galois Group of the NVE is the same as the identity component of the Galois Group of the ANVE. Then, by applying Kimura’s theorem (Theorem B.1, Appendix B) it is obtained the following: Theorem 2.5 ([32], See also [21]). Let XH be a Hamiltonian system given by (2.9) and c an homothetical point such that V ′′ (c) is diagonalizable. If XH is meromorphically completely integrable, then each pair (k, λi ) matches one of the following items (p being

10

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an arbitrary integer):

λ

k

λ

k

2

1

k

p + p (p − 1)

k 2

10

−3

2

2

arbitrary z ∈ C

11

3

1 − 24 +

1 24

(2 + 6p)2

3

−2

arbitrary z ∈ C

12

3

1 − 24 +

1 24

3

4

−5

13

3

1 − 24 +

1 24

6

5

−5

14

3

1 − 24 +

1 24

 12

49 40

−

49 40

1 40

−

1 40

 10

+ 10p

3

2

(4 + 10p)2 2

25 24

−

1 24

6

−4

9 8

−

 1 4 8

3

+ 4p

15

4

7

−3

−

(2 + 6p)2 2 3 + 6p 2

5

−3

1 24 1 24

16

8

25 24 25 24

17

5

9 − 40 +

9

−3

25 24

1 24

6

2

18

k

1 2

− −

5

+ 6p

− 18 + 9 + − 40

 k−1 k

 12

+ 6p

5

2

5

5

 1 4 8

3

1 10 40 3 1 4 40



2 + 6p 2 + 6p + 6p

+ 4p

(2.11)

2

2

2 + 10p

( + 10p)2

+ p (p + 1) k



This theorem is a generalization of a necessary condition of integrability obtained by Yoshida using Ziglin’s approach [33]. Hence, in order to prove the non-integrability of a given Hamiltonian system with a homogeneous potential: (i) we find the homothetical points, solutions ci of the equation c = V ′ (c) (ii) we prove that for some of the ci in (i), at least one of the eigenvalues of V ′′ (ci ) is not inside the table (2.5). With the above in mind we survey now recent contributions related in some way to Theorems 2.1 and 2.4. 2.2. New tools Here we review two new theoretical results connected in some way with the general non-integrability theorems. 2.2.1. Effective Kolchin–Kovacic reduction These results are mainly motivated by the Ph.D. Thesis of Aparicio-Monforte [34]. Despite of the theoretical relevance of Kolchin–Kovacic reduction (Theorem 1.4), as far as we know this reduction has barely not been used for the effective computation of the Galois group of concrete families of linear differential equations. A previous paper by Tsygvintsev that seems related in some way to the above reduction, using the linear structure of the Lie algebra of coefficients to study the monodromy group of a specific family of variational equations arising in the three-bodyproblem [35]. However, a new approach to this reduction problem is being formulated in connection with the integrability of Hamiltonian systems by Aparicio-Monforte, Barkatou, Compoint, Simon and Weil ([36–41], see also [34]) as we will review now. As the precise results are technical we only sketch the main ideas. The starting problem is to reduce the higher order variational equation of order k around a solution of a dynamical system and, in particular, apply it to the non-integrability of Hamiltonian systems. The method is iterative and mainly based on the following facts: (1) The so-called Wei–Norman decomposition of the matrix A = A(x) of system (1.1), A=

r 

ai (x)Mi ,

(2.12)

i=1

being a1 , . . . , ar a basis of the complex space generated by the entries of A. (2) The structure of the (linearized) variational equations (2.8). (3) The existence of a partial reduction for the diagonal block part of the matrix (2.8), provided the variational Ak−1 is already in reduced form. (4) Using another gauge transformation obtain as many zeros as possible for the lower block diagonal coefficients, preserving the diagonal blocks (they are already in reduced form).

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(5) Verify that no more reductions are possible, i.e., the obtained differential system is actually the Kolchin–Kovacic reduced system. Variants of the above method have been applied to obtain the non-integrability of several systems: degenerated Hénon–Heiles system [38], Swinging Atwood Machine [40], Friedmann–Robertson–Walker cosmological model [41], etc. I only make some remarks about the Hénon–Heiles system. Using the third order variational equations around a suitable particular integral curve, it was proved in [31] the non-integrability of the degenerated Hénon–Heiles system H =

1 2

1

1

1

1

2

2

3

2

(y21 + y22 ) + x21 + x22 + x31 + x1 x22 .

(2.13)

Now the same result can be obtained with the above systematic reduction method, because the dimension of the Galois group of the first order variational equation is one (and then item (4) of the above method works well) and as the Lie algebra of the final reduced form of the third order variational equation is not abelian, only remains to prove that this reduced form is actually the Kolchin–Kovacic one (i.e., no more reductions are possible). This is achieved using the fact that the degree of transcendence of the Picard–Vessiot extension is equal to the dimension of the Galois group formula (1.7). We mention that a problem related with the Kolchin–Kovacic reduction for the variational equations is some kind of a formal inverse of Theorem 2.4: assuming that the Galois group of the variational equation is abelian at any order, is it possible to construct a formal first integral around the particular solution? To this problem are devoted papers [37,41]. I would like also to remark that for reductive groups a nice invariant theory exists, this fact allows a very efficient Kolchin–Kovacic reduction for these groups [37]. For all of the above see also [5]. 2.2.2. Extension to non Hamiltonian dynamical systems Ayoul and Zhung proposed in 2010 an extension of Theorem 2.4 to non-Hamiltonian dynamical systems [42]. We review their work. Let X be a vector field over a manifold M of dimension n defining the dynamical system x˙ i = Xi (x1 , . . . , xn ),

i = 1, . . . , n.

(2.14)

The lift to the cotangent bundle is the Hamiltonian vector field Xˆ = XH over T M defined by the Hamilton function ∗

H (x1 , . . . , xn , y1 , . . . , yn ) =

n 

yi Xi (x1 , . . . , xn ).

i=1

We observe that this lift of X is the field obtained by duality on the first prolongation



n n   ∂ Xn ∂ X1 X1 , . . . , Xn , x˙ j ,..., x˙ j ∂ xj ∂ xj j =1 j=1



of X along its integral curves. Thus, in some sense any dynamical system can be considered a Hamiltonian system. We say that the non-Hamiltonian vector field X = (X1 , . . . , Xn ) on M (or the associated dynamical system (2.14)) is integrable if there are r vector fields Y1 , . . . , Yr and s functions f1 , . . . , fs , such that: (1) (2) (3) (4)

r + s = n, Y1 = X , [Yi , Yj ] = 0, i, j = 1, . . . , r, Yi (fk ) = 0, i = 1, . . . , r, k = 1, . . . , s, i.e., the functions f1 , . . . , fs are first integrals of all the fields.

It is not clear to me whether this notion of integrability is the most general one, but it includes others definitions, for instance, the complete integrability of Hamiltonian systems and the ‘‘naive’’ concept of integrability: the existence of n − 1 first integrals for the field X . As in the Hamiltonian case, we will talk about meromorphic integrability if the functions and fields above are meromorphic. Now let x = φ(t ) be a particular integral curve of a complex analytical dynamical system (2.14), defining a Riemann surface Γ , and the variational equations or order k ≥ 1 around this particular solution. Then Theorem 2.4 is valid mutatis mutandis to this non-Hamiltonian situation: Theorem 2.6 ([42]). Assume that a complex analytical dynamical system (2.14) is meromorphically integrable in a neighborhood of the integral curve x = φ(t ). Then the identity components (Gk )0 , k ≥ 1, of the Galois groups of the variational equations along Γ are commutative. Key points for proving this theorem are the following: – A necessary condition for the integrability of X is the complete integrability of the cotangent lift Xˆ . – The Galois groups of the variational equations Gk are the same as the Galois groups of the variational equations of the cotangent lift.

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2.2.3. Non-meromorphic first integrals In two papers Combot extends the applicability of Theorems 2.1 and 2.4 considering not meromorphic first integrals. Paper [43] is devoted to study Theorems 2.1 and 2.4 when the potential is algebraic (but not meromorphic) in the ordinary configuration space, as happens in Celestial Mechanics. Hence a direct application of the above theorems no longer works. Thus the theorems for meromorphic integrability are extended by considering a suitable algebraic variety where the potential becomes meromorphic. This result is applied to obtain a completely rigorous proof of the non-existence of an additional meromorphic first integral for the 3-body problem, in this way it is filled a technical gap in the Tsygvintsev, Morales-Ruiz and Simon previous results on the non-integrability of several body problems in Celestial Mechanics (see [1] and references therein). Combot also studied the non-integrability of the Hamiltonian system defined the gravitational dynamics of a triaxial Riemann ellipsoid of liquid [44]. The Hamiltonian is quite not ‘‘standard’’, H (x1 , x2 , y1 , y2 ) = r

being r =





y21 +

x42 y22 x42

+r



+∞

 + 0



z+

4 x22

α dz  z2

+ rz +

x22

,

(2.15)

4

x21 + x22 . The dependence of the Hamiltonian with respect to the position variables x1 , x2 , is given through

a family of elliptic integrals (in fact these elliptic integrals can be expressed by the complete elliptic functions integrals of the first and second type) and, as this dependence is not meromorphic (and even not algebraic, i.e., paper [43] is neither applicable), it is necessary to extend non-integrability Theorems 2.1 and 2.4 to this new situation. After extending Theorem 2.1, it is proved in the said paper that the Hamiltonian system defined by (2.15) has not an additional first integral with some specific regularity properties. We remark that, due to the more ‘‘transcendental’’ nature of the Hamiltonian, it will not be possible to apply Kovacic algorithm to the (normal) variational equation, because now there are, generically, basic obstructions to its algebrization. In my opinion this paper could be connected in some way with one of the Open Problems that I proposed in the paper [45]. 2.3. Homogeneous potentials and related problems After the deep results mainly obtained by Maciejewski and Prybylska for homogeneous potentials, already survey in [1], new relevant achievements have been obtained in the last four years. We survey some of them. We start with some papers by Combot. These results coming essentially from his Ph.D. Thesis [46]. In [47] obstructions to integrability were obtained for potentials of degree −1, using second order variational equations, assuming that the eigenvalues of the Hessian along a homothetic solution satisfy the necessary integrability conditions of (2.10). In this way, an effective theorem is given by means of a certain identity to be satisfied by the derivatives up to order 3 of the potential at the Darboux point. In some sense, this result can be considered as a natural (but more involved) continuation up to order two of the first order conditions in (2.10) (given by the eigenvalues of the Hessian). The said condition on the potential is related (as frequently happens) to a residue in the second order variational equation, i.e., to the non-existence of a suitable logarithmic term. The power of this result is illustrated with some applications; for instance, to obtain a new proof of a result of Duval and Maciejewski [48] about integrability conditions, whenever the Hessian matrix of the potential is not diagonalizable. The paper [49] was devoted to ‘‘generic’’ potentials of degree −1 on the plane. In this case we only need to study one eigenvalue of the table (2.11) in 2.5 (corresponding to an additional first integral). The main result of the above paper is to obtain all the integrability cases of a very broad family (generic in some sense) of potentials. In the proof it is considered the total set of higher order variational equations. A key point in the analysis is the existence of an upper bound for the possible admissible eigenvalues given by table (2.11) (first order obstructions), which in some cases is related with the universal global constraints of Maciejewski and Pryzbylska (see [1] and references therein). In a joint paper with Koutschan [50], Combot studied a ‘‘degenerated’’ family of potentials of degree −1 on the plane. This case is not generic, the methods of paper [49] does not work and the study of this family presents analogous problems to the degenerated case of the Atwood machine studied by Martinez and Simó, and with some special cases found by Maciejewski and Przybylska in their analysis of the homogeneous polynomials potentials (see [1]). In this paper, new computational tools from D-finite systems are introduced and, as an application, the integrability of the potential (in polar coordinates) V (r , θ ) =

1 r

(a + beiθ + ce2iθ + de3iθ ),

has been exhaustively studied. Paper [51] addresses to the study of the non-integrability of the Celestial Mechanics N-body problem. In particular, is proved the non-integrability of the equal masses N-body problem on the plane, when the angular momentum is different from zero. This result can be considered a generalization of my previous joint work with Simon about the equal masses case (see [1]). In this situation, it is clarified the concept of integrability on the level surfaces of known first integrals. The final result is obtained by means of the analysis of a variational equation of Heun type depending on two parameters: the energy

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and the non-trivial eigenvalue of the Hessian of the potential at the Darboux point. Some Galoisian properties for Heun’s families of equations are pointed out in Appendix B. Although Combot doest not use these properties (he essentially applies Kovacic algorithm), maybe it could be possible to use these properties to obtaining an alternative proof of Combot results. Casale, Duval, Maciejewski and Prybylska studied in [52] the non-integrability of general homogeneous potentials of degree zero. They proved that a necessary condition for the complete integrability by rational first integrals is that the Hessian V ′′ (c) (over homothetical point c) must be diagonalizable with integer eigenvalues λi . The proof is (partially) based on a variational equation along the homothetical solution, transformable to a direct sum of one parameter hypergeometric confluent equations in Kummer form (B.10) y′′ +

1 2

−x x

y′ −

a x

y = 0,

a=

λi 2

.

(2.16)

Then by Theorems 2.2 and B.3, λi must be an integer, and the transcendence degree formula (1.7) plays also and important role in the proof. Maciejewski and Prybylska studied the integrability problem of potentials that are not homogeneous but given by sums of homogeneous V = V1 + V2 , [53]. In their work they used previous results by Mondejar and Yoshida about this kind of potentials, as well as Kimura’s Theorem B.1. As an application it is considered in the said paper the case where V1 is a quadratic form and the degree of V2 is greater than two. The same authors obtained non-integrability results of the so-called dumbbell three-body problem, i.e., the gravitational planar three-body problem where the distance of two of the masses is constant [54]. As the Hamiltonian is not meromorphic, it is necessary to extend the applicability of our theorems to a suitable phase space, as Combot did in [43]. The algebraic (normal) variational equation obtained has a total of seven regular-singular points over the Riemann sphere. Apparently this considerable amount of singular points increases considerably the complexity in Kovacic‘s algorithm, but the final non-integrability result is obtained by a relatively easy application of the Kovacic algorithm (the so-called necessary conditions in Kovacic’s paper [55]). Studzinski and Prybylska studied in [56] the non-integrability of homogeneous rational potentials. In particular, for two degrees of freedom, it was proved the existence of universal relations between the eigenvalues of the Hessian V ′′ (c), as it happens in the polynomial case (see [1]). In two papers [57,58] Duval and Maciejewski studied the structure of the higher variational equations for the homogeneous potentials, first for the degree of homogeneity k = ±2 and then for the rest of (integer) possible values. From Theorem 2.5 we know that for k = ±2 there are no first order obstructions to integrability. Then in the paper [57] is proved, for instance, that for k = 2 integrability is essentially equivalent to that the system can be described by a system of independent harmonic oscillators, provided a non-resonance condition is satisfied. For k = −2 it is proved that there are no obstructions to integrability using higher order variational equations and hence, no obstruction at all, by means of variational equations. Ref. [58] is mainly devoted to the study of the commutativity of the identity component G02 of the Galois group of second order variational equations, assuming commutativity for the identity component of the Galois group of the first variational equation (i.e., that there are no first order obstruction to integrability). Although the methods of paper [57] no longer work, the paper uses some results from [57], for instance, that the commutativity of G0k is characterized by means of the same property of two suitable differential subsystems of the second order variational equations. In addition, an essential technical tool here is the introduction of a group cohomological structure over G02 , clarifying the Picard–Vessiot structure of the second order variational equation associated to a commutative G02 . As it was remarked by the authors, this paper has connections with Refs. [36,47]. We remark that in [59], a comparative study is presented for the homogeneous potentials, between the necessary conditions for integrability considered in this survey and the corresponding conditions obtained using methods of blowingup the singularities of the potential (for instance, around the triple collision for some three-body-problems). 2.4. Geodesic motion As far as I know it is only very recently that the general non-integrability theorems of Section 2.1 have been applied to geodesic motion. In [60,61] Waters showed the meromorphic non-integrability of the geodesic motion on the so-called sectorial harmonic surfaces and on the surfaces xyz = 1,

x2 y2 z = 1.

The essential algorithmic tool here is again Kovacic’s algorithm. A considerable generalization of the results in [61] has been obtained by Combot and Waters in [62]: they proved the meromorphic non-integrability of the geodesic flow of families of algebraic manifolds given by z p − f (x1 , . . . , xn ) = 0,

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being f an homogeneous rational function. In particular, the geodesic flow for manifolds defined by z = λ1 xk1 + · · · + λn xkn ,

k ∈ N,

xn ym z l = 1,

n, m , l ∈ Z

is generically non-integrable. The non-integrability of the Zipoy–Voorhees space–time ds2 = −

(x − 1)2 2 (x + 1)3 (1 − y2 ) 2 (x2 − 1)2 (x + 1)4 dt + dφ + (x + 1)2 x−1 (x2 − y2 )3



dx2 x2 − 1

+

dy2 1 − y2



,

(being (x, y, φ) prolate spheroidal coordinates) has been studied by Maciejewski, Przybylska and Stachowiak [63]. The particular integral curve Γ is the straight line solution y = pφ = 0. Then, after algebrization and reduction to normal form (A.2), the normal variational equation over the Riemann sphere has six singular points and an exhaustive application of the Kovacic algorithm proved the non-integrability of the geodesic motion on this space–time. 2.5. Superstring and the AdS/CFT correspondence In the last three years a new line of applications of the non-integrability theorems of Section 2.1 is being open, in the area of string theory, relevant for the AdS/CFT (Anti de Sitter–Conformal Field Theory) correspondence (or duality). Motivated by some chaotic numerical experiments suggesting the non-integrability for some models, Basu, Pando Zayas, Stepanchuk, S.A. Tseytlin, Ghosh, D. Das, Giataganas, Zoubos, Chervonyi, Lunin and Sfetsos, in several papers applied the above theorems to the study of non-integrability of some finite degree of freedom Hamiltonian subsystems of specific models of classical (non-quantum) string field equations [64–70]. Following [66], I illustrate the idea with one of the results: the non-integrability for the so-called AdS soliton background. The field equations are defined by a Lagrangian density depending on two variables, L = L(t , σ ), being (t , σ ) the worldsheet coordinates of the string (the corresponding action is Polyakov action). By a suitable ansatz, it is possible to consider a finite dimensional Hamiltonian subsystem of three and a half degrees of freedom that, after some reductions by two ignorable coordinates, can be written as a two degrees of freedom Hamiltonian H (x, R, px , pR ) =

T ( x) d 2 a2 x 2

2

2

p2x + e−2ax (p2R − E 2 ) + α 2 e2ax R2 ,

2

T (x) := 1 − e−dax ,

(2.17)

being a, d and α parameters. As a matter of fact, the ‘‘physical’’ phase space is constrained to H = 0 and E = pt is the canonical conjugate momentum to time t (the dynamics of essentially the same Hamiltonian system is considered in [71]). The variable R represents the radius of a circle where the string is wrapped. The invariant plane is defined by x = constant = x0 , px = 0, it gives the particular integral curve Γ defined by R(t ) = A sin α t, and the normal variational equation is of Mathieu type

ξ¨ −



2E 2 d

+

2α 2 A2 d

cos 2α t



ξ = 0.

(2.18)

The non-integrability of the finite degree Hamiltonian subsystem is achieved, provided α ̸= 0 and E ̸= 0 (see Appendix B). But a remark is in order here. When we algebrize the Mathieu equation, for instance using the change R = A sin α t, like in Ref. [66] (we arrive to similar situations with other changes of variables, for instance the change made in the Appendix B), then the new independent variable is R and the point at infinite R = ∞ on the Riemann sphere is an irregular singular point of the (algebraic) normal variational equation. Hence by Theorem 2.2 we can only obtain an obstruction to the existence of an additional rational first integral. Hence this finite-dimensional Hamiltonian subsystem is non-integrable by rational first integrals. This result suggest that the global field equations are ‘‘non-integrable’’ in some sense. Sometimes a partial differential system defined by the curvature zero condition of a family of connections, depending on an spectral parameter, is called an ‘‘integrable’’ system of partial differential equations. Then it is believed that a necessary condition for the integrability of a system of partial differential equations is, that any Hamiltonian subsystem with a finite number degrees of freedom must be also integrable in any reasonable sense: lack of chaotic behavior, sufficiently regular first integrals, etc. However, with respect to Liouville complete integrability with rational (or even meromorphic) first integrals (in the complex analytical category), maybe this assumption is too strong. For instance, it is well-known that the five Painlevé transcendents are obtained as suitable reductions of the Anti-Self-Dual Yang–Mills equations, which are defined as a zero curvature condition of a family of holomorphic connections depending on a spectral parameter (see, for instance, [72]). But some of the Painlevé transcendents are non-integrable as Hamiltonian systems with suitable regularity of the first integrals: for instance, some subfamilies of Painlevé II are not integrable by means of rational first integrals (see [73,74]; similar nonintegrability results are obtained in other Painlevé families: for Painlevé VI, see [75]). A nice survey about the relevance of the integrability problems in the framework of the AdS/CFT correspondence is given in [76].

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2.6. Other problems One of the problems that prevents the applicability of the non-integrability theorems is that we need to know very well a particular integral curve Γ not reduced to an equilibrium point. Martinez–Simó and Salnikov proposed independently a computational method to avoid the search for a particular integral curve [77,78]. The idea is to compute numerically a relatively nice integral curve, i.e., an integral curve with no chaotic behavior, and to take into account that the monodromy (or Galois) group of the variational equations measures in some sense the different dynamical behavior (with respect to Γ ) of the integral curves in a neighborhood of it. For instance, following a suggestion of Ramis, in the Ref. [79] it is studied the non-integrability of the double pendulum defined by the two degrees of freedom Lagrangian L = θ˙12 + θ˙1 θ˙2 cos(θ1 − θ2 ) +

1 2

θ˙22 + 2g cos θ1 + g cos θ2 ,

for which no particular solution is explicitly known. Also Simon obtained numerical evidences of the non-integrability of a degenerated Friedmann–Robertson–Walker cosmological model in [41], using his previous paper on the tensorial representation structure of the higher order variational equations [40]. We finish this first part of the survey, with a nice result by Christov on the non-integrability of the truncated (Birkhoff–Gustafson) normal forms, up to order three, of Hamiltonian systems of three degrees of freedom around elliptic equilibrium points [80]: H = H2 + H3 + · · · , being H2 a sum of independent harmonic oscillators of frequencies ω1 , ω2 , ω3 . In the said paper it is studied the integrability for the resonant ratios 1 : 2 : ω, with ω = 1, 3, 4. As this truncated system, H2 + H3 , has H2 as another independent first integral, the problem is to study the existence of an additional first integral in involution with the above two. It is proved the non-existence of such additional (meromorphic) first integral. The method of proof for ω = 3 is interesting because, it is obtained that the (first order) variational equations have a Galois group with a solvable but not commutative identity component. As far as I know, the only previous case where we found this behavior was in proving the non-integrability of Hill’s problem [81]. 3. Some spectral problems 3.1. Discrete spectrum Here we describe some results of paper [2]. The objective is to obtain in closed form the discrete spectrum (bound states) of the one–dimensional stationary Schrödinger equation. Two essential ingredients here will be again the Algebrization procedure and Kovacic’s Algorithm. 3.1.1. Schrödinger equation We recall that in Quantum Mechanics the Hamiltonian operator is the Schrödinger (non-relativistic, stationary) operator which is given by H =−

h¯ 2 2m

∇ 2 + U (x)

and the Schrödinger equation is H ψ = E ψ , where x is the coordinate, the eigenfunction ψ is the wave function, the eigenvalue E is the energy level, U (x) is the potential or potential energy and the solutions of the Schrödinger equation are the states of the particle. We only consider the one-dimensional Schrödinger equation written as follows: H ψ = E ψ,

H =−

d2 dz 2

+ V (z ),

(3.1)

where z = x (cartesian coordinate: problem on the line) or z = r (radial coordinate: problem on the half-line) and we normalize the units such that h¯ = 2m = 1. We interpret the Schrödinger equation (3.1) here as a differential equation with a parameter E and we look for the values of the parameter E ∈ C such that Eq. (3.1) is integrable in Picard–Vessiot sense. We will denote Λ ⊂ C for such values of E for which Eq. (3.1) becomes integrable and we call Λ the algebraic spectrum of the Schrödinger equation (3.1) (see [2] for the motivation for this terminology). 3.1.2. Application of the Picard–Vessiot theory We restrict our study here to three families of equations: the harmonic oscillator, the Liouville potential and the Coulomb potential (other families are studied in the original paper).

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The harmonic oscillator The Schrödinger equation (3.1) becomes d2 ψ dx2

 =

 ω2 x2 − E ψ.

1 4

(3.2)

By means of an easy change of variable (see Appendix B) we can write Eq. (3.2) as Weber equation (B.5), d2 ψ dx2

 =



1

x2 − λ ψ,

4

with λ = E /ω, and by the integrability characterization of the Weber equation given in the said Appendix we obtain that the algebraic spectrum of (3.2) is

 En =

1 2

 + n ω,

n ∈ Z.

We remark that for natural n this gives us the classical discrete spectrum of the harmonic oscillator. The Liouville potential The potential is V (x) = e−2x . The Schrödinger equation is d2 ψ

= e−2x − E ψ. (3.3) dx2 It is possible to algebrize this equation converting it into a Bessel equation. By means of the change z = e−x , we obtain 

d2 ψ



dψ

+ (E − z 2 )ψ = 0, dz which is ‘‘almost’’ a Bessel equation. In fact, by the change z → iz, we obtain the Bessel equation: z2

z2

+z

dz 2

d2 ψ

+z

dz 2

dψ dz

+ (z 2 + E )ψ = 0.

Now E = −ν 2 and by Corollary B.4 (Appendix B) the algebraic spectrum is En = −( 21 + n)2 . As above, the algebraic spectrum contains the standard discrete spectrum. The Coulomb potential The Schrödinger equation in this case can be written as d2 ψ dr 2

 =

l(l + 1) r2

−

e2 r

+

e4



4(l + 1)2

− E ψ,

l ∈ Z.

(3.4)

This equation is the radial equation (in reduced form), obtained when we separate coordinates in the three dimensional Schrödinger equation of the motion of the electron in the hydrogen atom; e is the charge of the electron and l is a quantum number that comes from the angular part in the process of separation of variables (see any quantum mechanics book, for instance [82]). Now it is possible to transform Eq. (3.4) to a Whittaker type Eq. (B.11). Through of the change

 r →

−4 (l + 1)2 E + e4 r, l+1

falls in a Whittaker differential equation with the parameters given by

κ= 

e2 (l + 1)

−4 ( l + 1 ) E + e 4 2

,

1

µ=l+ . 2

Applying Martinet–Ramis theorem (Theorem B.3 of Appendix B) we can impose ±κ ± µ half integer. Then the algebraic spectrum is given by Eln =

e4 4(l + 1)2

λln ,

being

 λln ∈ 1 −



l+1 l+1+n

2





: n ∈ Z+ ∪ 1 −



l+1 l−n

2

 : n ∈ Z+ .

This algebraic spectrum includes again the standard analytic discrete spectrum, obtained in Quantum Mechanics. In [2] the algebraic spectrum is obtained by a direct application of Kovacic’s algorithm.

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Remark 3.1. We computed the discrete spectrum, but by means of the Picard–Vessiot theory it is also possible to compute in closed form the corresponding eigenfunctions: Kovacic algorithm gives us a particular Liouvillian solution (and hence, the general solution) for the values of the algebraic spectrum. Only three potentials have been considered here but, as far a we know, all the known exactly solvable potentials can be handled by Picard–Vessiot theory, obtaining the discrete spectrum and the corresponding eigenfunctions (bound states) in a similar way to the examples above [2]. 3.2. Darboux transformations We denote ψ(V0 , E ) the solution of the Schrödinger equation

 −

d2 dx2



+ V0 (x) ψ = E ψ.

(3.5)

Let ψ(V0 , E0 ) = ψ(V0 , E0 )(x) be a particular solution of the above equation. Then the transformed Schrödinger equation

 −

d2 dx2

 + V1 (x) ψ = E ψ,

(3.6)

where V1 (x) = V0 (x) − 2

d2 dx2

log V0 (z ),

has solution W (ψ(V0 , E ), ψ(V0 , E0 ))

ψ(V1 , E ) =

ψ(V0 , E0 )

,

for any value E ̸= E0 (W will denote the Wronskian). This is a Darboux transformation for the Schrödinger equation. We observe that if we know explicitly a particular solution of the initial equation for a concrete value of the energy, from the solutions of this equation for any other value of E, we obtain the solutions of the transformed equation, for the same value of E ([83,84], see also the monograph [85]). The Darboux transformation is related to the so-called factorization method in Quantum Mechanics and with the very related topics of Dirac creation and annihilation operators and Supersymmetric Quantum Mechanics (see [2,85] and references therein). Furthermore it explains the hidden mechanism in most of the problems for which the eigenvalues and corresponding bound states are founded in closed form: by a suitable Darboux transformation from the fundamental bound states, the potential preserve its form and it is only shifted by a constant, in an analogous way to the harmonic oscillator. These potentials are called Shape Invariant Potentials. A crucial point for the application of the Picard–Vessiot theory is that the transformed potential and the transformed solution, can be also expressed through a logarithmic derivative of the solution ψ(V0 , E ) V1 = V0 − 2

d dx

log ψ(V0 , E0 ),

ψ(V1 , E ) =



d dx

−

d dx

 log ψ(V0 , E0 ) ψ(V0 , E ).

Now we recall that the logarithmic derivative is a solution of the Riccati associate equation and the integrability of a second order equation is equivalent to the existence of an algebraic solution of the Riccati equation (Liouville theorem: see Section 1). Then it is possible to prove the invariance of the identity component of the Galois group by the Darboux transformations defined by algebraic solutions of the associate Riccati equation. For more information see [2]. 3.3. Quantum mechanics scattering on the line This subsection is motivated by a joint work with Peris some years ago ([86], see also, Chapter 7 of [21]) about a Galoisian interpretation, in the framework of the non-integrability theorems of Section 2.1, of a paper of Grotta-Ragazzo on the chaotic behavior of two degrees of freedom Hamiltonian systems, with an homoclinic orbit contained in a invariant plane [87]. One of the consequences of [86,87] is that, under some assumptions related with the fact that the Schrödinger equation on the line (3.1) has regular singular points at the infinite, the scattering problem is reflectionless, i.e. the reflection coefficient is zero for some value of the spectral parameter (the value of the spectral parameter corresponds to the frequency of oscillation at the equilibrium point in the homoclinic orbit in the Hamiltonian model), if and only if the identity component of the Galois group is commutative for this value of the spectral parameter. We illustrate our Galoisian approach to the scattering problem with a new family of examples, the Rosen–Morse potential, that is interesting for the direct as well as for the inverse scattering problem. As far as we know, no previous Galoisian studies about this family can be found in the literature. The potential is V (x) =

u cosh2 α x

,

(3.7)

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being u and α non-zero real parameters. It is convenient to normalize the parameters, by means of the change of independent variable x → α x. Then using the same notation x for the new independent variable, the Schrödinger equation becomes d2

 −

dx2

+



A cosh2 x

ψ = k2 ψ,

(3.8)

being A = u/α 2 a real parameter and λ = E /α 2 := k2 , as usual in scattering theory. Some people call Rosen–Morse to the potential (3.7), but in fact the potential studied by Rosen and Morse in their 1933 paper [88] was more general: V (x) =

u cosh2 α x

+ v tanh α x,

(3.9)

being u,v and α real parameters, with u < 0 and α ̸= 0. The study of the scattering problem for the potential (3.7) is also studied in the book [82] where, in particular, the reflectionless case is pointed out (Section 25, Problem 4). Two standard mathematical references about the inverse quantum mechanical scattering problem on the line are [89,90], where the direct (or forward) scattering problem is also described. The potential (3.7) belongs to the Fadeev class of potentials: potentials V (x) which satisfy ∞



|V (x)|(1 + |x|)dx < ∞. −∞

For this class of potentials the number of eigenvalues (discrete spectrum, corresponding to bound states) of the Schrödinger equation

 −

d2 dx2

 + V (x) ψ = k2 ψ,

(3.10)

is finite λi = k2i , k = iκi , i = 1, 2, . . . , n, κi > 0, being the continuous spectrum λ ≥ 0, i.e., the real values of k. We remark that there are some potentials in this class with no bound states. Then the scattering is described by the unimodular matrix S (k) =



a(k) b(k)

b¯ (k) . a¯ (k)



(3.11)

The reflection and transmission coefficients are r (k) =

b(k) a(k)

,

t (k) =

1 a(k)

,

respectively. Moreover the discrete spectrum is given by the zeros of a(k) on the imaginary positive axis of the complex plane k: a(iκj ) = 0, κj > 0, j = 1, 2 . . . , n. By definition, the reflectionless potentials are the potentials V (x) such that r (k) is identically zero, i.e., b(k) identically zero. For terminology and notation we essentially follow [90], they are slightly different in the Ref. [89]. In physics the reflection and transmission coefficients are usually defined as the modulus of the reflection and transmission coefficients considered here. For real k (i.e., k in the continuous spectrum), there are two bases of solutions (ϕ+ , ϕ− ) and (ψ+ , ψ− ) of Eq. (3.10), with the following asymptotic behavior of points at the infinity

(ϕ+ (x, k), ϕ− (x, k)) ∼ (eikx , e−ikx ), for x → −∞ (ψ+ (x, k), ψ− (x, k)) ∼ (eikx , e−ikx ), for x → +∞. The scattering matrix is the transition matrix between the above two bases of solutions

(ψ+ , ψ− ) = (ϕ+ , ϕ− )S (k). Hence we have the following relation t (k)ψ+ (x, k) = ϕ+ (x, k) + r (k)ϕ− (x, k).

(3.12)

By means of the asymptotic behavior of the solutions, Eq. (3.12) can be interpreted as an incident plane wave coming from x = −∞, ϕ+ (x, k) ∼ eikx , that is transmitted as t (k)ψ+ (x, k) ∼ t (k)eikx (for x → +∞) and reflected as r (k)ϕ− (x, k) ∼ r (k)e−ikx (for x → −∞). The scattering of an incident wave coming from x = +∞ can be described in an analogous way: the scattering matrix is given by S −1 =



a¯ −b

−b¯ a



,

being the transmission and reflection coefficients t˜ =

1 a

= t,

r˜ = −

b¯ a

=−

t r¯ t¯

.

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In the Ref. [89] the coefficient t = t˜ is denoted by T , while r and r˜ are denoted by R2 and R1 , respectively. The prescription in [90] is also slightly different: the scattering matrix S (k) described here corresponds to S −1 (k) in [90]. Now we come back to the potential V (x) =

A cosh2 x

,

in Eq. (3.8). The transmission and reflection coefficients, as well as the discrete spectrum, are explicitly computed in [82]. We sketch the main steps of this computation. By means of the change z √ = tanh x we algebrize the equation, obtaining an associated Legendre equation (B.2) with µ2 = −k2 and ν = 1/2(−1 + 1 − 4A), complex parameters. For convenience, we choose µ = −ik. Now the potential can be expressed as Vν (x) = −

ν(ν + 1) cosh2 x

.

The points x = ±∞ in (3.8) correspond to z = ±1 in Eq. (B.2). The coefficients of transmission and reflection are obtained by means of the Kummer transition (or continuation) formula (B.9) for the Gauss hypergeometric Eq. (B.3) related to the associated Legendre equation (B.2), because the singular points 0 and 1 in (B.3) correspond to singularities 1 and −1 in (B.2), respectively. To be more precise, the scattering matrix S (k) is equal to the Kummer transition matrix (B.9) for Eq. (B.3). Thus

Γ (1 − ik)Γ (−ik) , Γ (−ik − ν)Γ (−ik + ν + 1) Γ (1 − ik)Γ (ik) b(k) = φ(−ik − ν, −ik + ν − 1, −ik + 1) = . Γ (1 + ν)Γ (−ν) a(k) = φ(1 − ν, ν, 1 − ik) =

(3.13)

We will study the discrete spectrum and the reflectionless potentials: (a) Discrete spectrum. For a given potential Vν , the discrete spectrum is the set of zeros of a(k), over the imaginary positive axis, k = iκj , κj > 0, j = 1, . . . , n. We know from the general theory that we have to assume ν(ν + 1) > 0 and without loss of generality we can consider ν > 0. The zeros of a(k) are either poles of Γ (−ik − ν) or poles of Γ (−ik + ν + 1), i.e., for −ik ± ν ∈ N. As we are looking for solutions of k over the imaginary positive axis, k = iκ , κ > 0, necessarily κ ± ν ∈ −N. Thus, ν must be real and, taking into account the expression of the potential Vν , we can assume ν > 0 (this is in agreement with the general spectral theory for the Schrödinger equation: bound states are associated to potential wells). Hence, the discrete spectrum is given by either

{iκ : −κ + ν ∈ N} = {iκj : κj = ν − j, j = [ν], . . . , 1, 0},

(3.14)

when ν is not a natural number ([ν] denoted the integer part of ν ), or

{iκj : κj = 1, . . . , n},

(3.15)

when ν = n ∈ N. (b) Reflectionless potentials r (k) ≡ 0. These are the potentials Vν , such that b(k) ≡ 0. Thus, the values of ν are either the poles of Γ (1 + ν), or the poles of Γ (−ν), i.e., for ν = n ∈ N \ {0} (assuming as above that ν > 0). We remark that for these potentials the function a = a(k) is a rational function: a(k) =

=

Γ (1 − ik)Γ (−ik) Γ (−ik − n)Γ (−ik + n + 1) n (−ik − 1) · · · (−ik − n)Γ (−ik − n)Γ (1 − ik)  k − iκj = , (n − ik) · · · (1 − ik)Γ (1 − ik)Γ (−ik − n) k + iκj j =1

with κj = j. The integrability in Picard–Vessiot sense of (a) and (b) is now straightforward. The Schrödinger equation (3.8) is transformed by algebrization into an associated Legendre equation (B.2), and the integrability of this equation is characterized in Proposition B.2. Both cases (a) and (b) fall in Case (1) of Proposition B.2, the solutions are given through some (special) Jacobi polynomials. In particular, as in the examples of Section 3.1, the discrete spectrum µ = κj is included in the algebraic spectrum. The simplest non-trivial reflectionless potential (3.7) is for ν = 1: V (x) =

−2α 2 . cosh2 α x

This is the well-known potential related to the one-soliton solution of the Korteweg de Vries equation: u(x, t ) =

−2α 2 . cosh2 α(x − 4α 2 t − x0 )

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Rosen and Morse in [88] were interested in the discrete spectrum of the potential (3.9) and they obtain its bound states in terms of the Jacobi polynomials (see Appendix B), because by algebrization the Schrödinger equation is again reduced to an hypergeometric equation. Thus, we can also handle its discrete spectrum by means of the integrability in Picard–Vessiot sense: we fall in case (1) of Kimura’s theorem (see Appendix B). An alternative way to arrive to the same result (explicit computation of the discrete spectrum and bound states) is through Darboux transformations: it is well-known that the potential (3.9) has the shape-invariant property. Of course, the same remarks are in order for the potential (3.7) to which reduces (3.9) for v = 0. 3.4. Other spectral problems Now we will give some references about other spectral problems that have been studied in the framework of the Picard–Vessiot theory. This list of references is not pretended to be exhaustive. In [91] Fauvet, Ramis, Richard-Jung and Thomann, made numerical experiments for computing the Stokes matrices at infinity of the prolate spheroidal wave equation: a singular two-parameter family of Sturm–Liouville operators. This numerical computations suggested that the discrete spectrum correspond, again, to integrable cases in Picard–Vessiot sense. In fact, it is proved in the said paper that the local Galois group at infinity is solvable if and only if, the spectral parameter is an eigenvalue. This statement has sense, because the point at the infinity is an irregular singular point, and the zeros of the Stokes multipliers as functions of the spectral parameters are the discrete spectrum. We observe the analogy with the zeros of a(k) in Section 3.3. Stachowiak and Przybylska in [92] gave a Picard–Vessiot integrability approach to the discrete spectrum of some families of Dirac equations on the line. These authors, in another joint paper with Maciejewski [93], studied the spectrum of the Ravi model, a system of two linear differential equations modeling some physical quantum systems. As in our approach to the integrability of the Schödinger equation in one dimension, classical special functions (mainly Heun’s and Whittaker’s families), as well as Kovacic’s algorithm play an important role in the said papers. In the paper [94], Blazquez and Yagasaki also studied some families of Sturm–Liouville problems, which reduce by algebrization to problems with regular singular points. In particular, as in my joint paper with Peris [86], it is assumed that the Sturm–Liouville operator reduces by algebrization to an operator with regular singular points. The main result, in agreement with all of the above, is that the values of the discrete spectrum correspond to values for which the equation is integrable in Picard–Vessiot sense. In the last eighty years several people made contributions towards a Galosian interpretation of the so-called algebrogeometric operators: algebras of commuting ordinary differential operators (we do not reference here the case of partial differential operators). The study of these operators starts in the works by Burchnall, Chaundy in the twenties of the 20th century. More recently, motivated by the connection with the integrable partial differential equations, these class of operators were studied by Krichever, Novikov and others. Braverman, Etingof, Gaitsgory and Rains (see [95] and references therein) essentially obtained that, under some assumptions, the Galois group of algebro-geometric operators must be an algebraic torus for generic values of the spectral parameter. Krigorenko also obtained Galoisian results for second order operators [96]. For algebro-geometric Schrödinger operators (given by finite-gap potentials), Yu.V. Breznev in three papers [97–99] made an exhaustive Galoisian study of these operators. The starting point of Brezhnev is a formula due to Drach for the diagonal of the kernel of the resolvent (Green function). Acknowledgments This research has been partially supported by the Spanish MINECO-FEDER Grants MTM2009-06973, MTM2012-31714. I am also indebted to Eva Miranda and Vladimir Matveev for suggesting me to write this survey. I would also like to thank the colleagues from our Integrability Seminar in Madrid for many fruitful discussions: Rafael Hérnandez-Heredero, Sonia Jiménez-Verdugo, José Rojo, Sonia L. Rueda and Maria-Angeles Zurro. Appendix A. Kovacic algorithm A classical fact is the reduction of a general second order linear differential equation z ′′ + pz ′ + qz = 0,

(A.1)

to y′′ = ry,

(A.2)

obtained by the change of variables y = ξ exp(− r =

p2 4

+

p′ 2

− q.

1 2



pd), being (A.3)

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Eq. (A.2) is the reduced form of the general second order Eq. (A.1) and it is clear that the (A.1) is integrable if and only if (A.2) is integrable. Kovacic’s algorithm is devoted to the integrability of (A.2) when r = r (x) is a rational function with complex coefficients. The algorithm is based on the classifications of the algebraic subgroups of SL(2, C) given by Theorem 1.3 and on the Liouville theorem characterization of the integrability of Eq. (A.2) by means of the associated Riccati equation (see Section 1). For more details see [55]. Although improvements for this algorithm are given in [100,101], we follow the original version given by Kovacic in [55]. Each case in Kovacic algorithm is related to each one of the algebraic subgroups of SL(2, C) given by Theorem 1.3 and its associated Riccati equation

θ′ = r − θ2 =

r −θ

√

r +θ ,

 √

θ=



y′ y

.

According to Theorem 1.3 we obtain four cases. Only for cases 1, 2 and 3 one can solve in closed form the differential equation (A.2) and case 4 correspond to non integrability. Kovacic algorithm can possibly provide one solution (y1 ), so the second one (y2 ) can be got through



dx

y2 = y1

y21

.

(A.4)

Notations. For s r = , s, t ∈ C[x], t we use: (1) (2) (3) (4)

Denote by Γ ′ be the set of (finite) poles of r, Γ ′ = {c ∈ C : t (c ) = 0}. Denote by Γ = Γ ′ ∪ {∞}. By the order of r at c ∈ Γ ′ , ◦(rc ), we mean the multiplicity of c as a pole of r. By the order of r at ∞, ◦ (r∞ ), we mean the order of ∞ as a zero of r. That is ◦ (r∞ ) = deg (t ) − deg (s).

The four cases √  √  √ √ Case 1. In this case r c and r ∞ means the Laurent series of r at c and the Laurent series of r at ∞ respectively. + − Furthermore, we define ε(p) as follows: if p ∈ Γ , then ε (p) ∈ {+, −}. Finally, the complex numbers αc+ , αc− , α∞ , α∞ will be defined in the first step. If the differential equation has not poles it only can fall in this case. Step 1. Search for each c ∈ Γ ′ and for ∞ the corresponding situation as follows:

(c0 ) If ◦ (rc ) = 0, then √  r c = 0,

αc± = 0.

(c1 ) If ◦ (rc ) = 1, then √  r c = 0,

αc± = 1.

(c2 ) If ◦ (rc ) = 2, and r = · · · + b(x − c )−2 + · · · ,

√

√  r

c

= 0,

αc± =

1±

then

1 + 4b 2

.

(c3 ) If ◦ (rc ) = 2v ≥ 4, and r = (a (x − c )−v + · · · + d (x − c )−2 )2 + b(x − c )−(v+1) + · · · ,

√  r

c

= a (x − c )−v + · · · + d (x − c )−2 ,

(∞1 ) If ◦ (r∞ ) > 2, then √  r ∞ = 0,

+ α∞ = 0,

− α∞ = 1.

(∞2 ) If ◦ (r∞ ) = 2, and r = · · · + bx2 + · · ·, then √ √  1 ± 1 + 4b ± r ∞ = 0, α∞ = . 2

then

  1 b ± +v . αc± = 2

a

22

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(∞3 ) If ◦ (r∞ ) = −2v ≤ 0, and r = (axv + · · · + d) + bxv−1 + · · · , 2

r ∞ = axv + · · · + d,

√ 

then

± and α∞ =

1 2



 b ± −v . a

Step 2. Find D ̸= ∅ defined by





m ∈ Z+ : m =

D=

ε(∞) α∞

−



ε(c )

αc , ∀ (ε (p))p∈Γ

.

c ∈Γ ′

If D = ∅, then we should start with the case 2. Now, if #D > 0, then for each m ∈ D we search ω ∈ C(x) such that

ω = ε (∞)

√ 

r ∞+

  √  ε (c ) r c + αcε(c ) (x − c )−1 .

c ∈Γ ′

Step 3. For each m ∈ D, search for a monic polynomial Pm of degree m with ′′ Pm + 2ωPm′ + (ω′ + ω2 − r )Pm = 0.

If one successes then y1 = Pm e ω is a solution of the differential equation (A.2). Else, Case 1 cannot hold. Case 2. Search for each c ∈ Γ ′ and for ∞ the corresponding situation as follows. 

Step 1. Search for each c ∈ Γ ′ and ∞ the sets Ec ̸= ∅ and E∞ ̸= ∅. For each c ∈ Γ ′ and for ∞ we define Ec ⊂ Z and E∞ ⊂ Z as follows:

(c1 ) If ◦ (rc ) = 1, then Ec = {4}. (c2 ) If ◦ (rc ) = 2, and r = · · · + b(x − c )−2 + · · · , then   √ Ec = 2 + k 1 + 4b : k = 0, ±2 . (c3 ) If ◦ (rc ) = v > 2, then Ec = {v}. (∞1 ) If ◦ (r∞ ) > 2, then E∞ = {0, 2, 4}. (∞2 ) If ◦ (r∞ ) = 2, and r = · · · + bx2 + · · ·, then   √ E∞ = 2 + k 1 + 4b : k = 0, ±2 . (∞3 ) If ◦ (r∞ ) = v < 2, then E∞ = {v}. Step 2. Find D ̸= ∅ defined by

 m ∈ Z+ : m =

D=

1 2



 e∞ −



ec

 , ∀ep ∈ Ep , p ∈ Γ

.

c ∈Γ ′

If D = ∅, then we should start the case 3. Now, if #D > 0, then for each m ∈ D we search a rational function θ defined by

θ=

1 2

c ∈Γ ′

ec x−c

.

Step 3. For each m ∈ D, search a monic polynomial Pm of degree m, such that ′′′ Pm + 3θ Pm′′ + (3θ ′ + 3θ 2 − 4r )Pm′ + θ ′′ + 3θ θ ′ + θ 3 − 4r θ − 2r ′ Pm = 0.





If Pm does not exist, then Case 2 cannot hold. If such a polynomial is found, set φ = θ + P ′ /P and let ω be a solution of

ω2 + φω +

 1 ′ φ + φ 2 − 2r = 0.

2

Then y1 = e ω is a solution of the differential equation (A.2). Case 3. Search for each c ∈ Γ ′ and for ∞ the corresponding situation as follows: 

Step 1. Search for each c ∈ Γ ′ and ∞ the sets Ec ̸= ∅ and E∞ ̸= ∅. For each c ∈ Γ ′ and for ∞ we define Ec ⊂ Z and E∞ ⊂ Z as follows:

(c1 ) If ◦ (rc ) = 1, then Ec = {12}. (c2 ) If ◦ (rc ) = 2, and r = · · · + b(x − c )−2 + · · ·, then   √ Ec = 6 + k 1 + 4b : k = 0, ±1, ±2, ±3, ±4, ±5, ±6 .

J.J. Morales-Ruiz / Journal of Geometry and Physics (

(∞) If ◦ (r∞ ) = v ≥ 2, and r = · · · + bx2 + · · ·, then   12k √ E∞ = 6 + 1 + 4b : k = 0, ±1, ±2, ±3, ±4, ±5, ±6 , n

)

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23

n ∈ {4, 6, 12}.

Step 2. Find D ̸= ∅ defined by

 m ∈ Z+ : m =

D=



n

 e∞ −

12



ec

 , ∀ep ∈ Ep , p ∈ Γ

.

c ∈Γ ′

In this case we start with n = 4 to obtain the solution, afterwards n = 6 and finally n = 12. If D = ∅, then the differential equation has not Liouvillian solution because it falls in the case 4. Now, if #D > 0, then for each m ∈ D with its respective n, search a rational function

θ=

n  12

c ∈Γ ′

ec x−c

,

and a polynomial S defined as S=



(x − c ).

c ∈Γ ′

Step 3. Search for each m ∈ D, with its respective n, a monic polynomial Pm = P of degree m, such that its coefficients can be determined recursively by P−1 = 0,

Pn = −P ,

Pi−1 = −SPi′ − (n − i) S ′ − S θ Pi − (n − i) (i + 1) S 2 rPi+1 ,





where i ∈ {0, 1 . . . , n − 1, n}. If P does not exist, then the differential equation has not Liouvillian solution because it falls in Case 4. Now, if P exists search ω such that n 

SiP

i =0

(n − i)!

ω i = 0,

then a solution of the differential equation (A.2) is given by y = e ω, 

where ω is solution of the previous polynomial of degree n. Appendix B. Some special functions B.1. Hypergeometric families B.1.1. Hypergeometric equation The general hypergeometric (or Riemann) equation is the most general second order linear differential equation over the Riemann sphere with three regular singular points at x = x1 , x2 , x3 . It is given by d2 y dx2

 +

3  1 − αi − α ′ i

i =1

x − xi



dy dx

+

  αi α ′ (xi − xj )(xi − xk ) i

x − xi

y

(x − x1 )(x − x2 )(x − x3 )

= 0,

where the second sum in the above formula is over the three circular permutations (i, j, k) of (1, 2, 3) and (αi , αi′ ), are the

exponents at the singular points xi , i = 1, 2, 3 and they must satisfy the Fuchs relation i=1 (αi + αi′ ) = 1. The standard Gauss hypergeometric equation is one of its reduced forms, with singular points at x = 0, x = 1 and x = ∞,

3

d2 y dx2

+

c − (a + b + 1)x dy x(1 − x)

dx

−

ab x(1 − x)

y = 0,

(B.1)

where a, b, c are parameters, with exponents (0, 1 − c ), (0, c − a − b) and (a, b), at x = 0, x = 1 and x = ∞, respectively. Now, we state Kimura’s Theorem that provides necessary and sufficient conditions for the integrability of the hypergeometric equation. This characterization is given in function of the difference of exponents, λ = α1 −α1′ , µ = α2 −α2′ and ν = α3 − α3′ .

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Theorem B.1 (Kimura, [102]). The hypergeometric equation is integrable if and only if either (i) At least one of the four numbers λ + µ + ν , −λ + µ + ν , λ − µ + ν , λ + µ − ν is an odd integer, or (ii) The numbers λ or −λ, µ or −µ and ν or −ν belong (in an arbitrary order) to some of the following fifteen families 1

1/2 + l

1/2 + m

arbitrary complex number

2

1/2 + l

1/3 + m

1/3 + q

3

2/3 + l

1/3 + m

1/3 + q

4

1/2 + l

1/3 + m

1/4 + q

5

2/3 + l

1/4 + m

1/4 + q

6

1/2 + l

1/3 + m

1/5 + q

7

2/5 + l

1/3 + m

1/3 + q

l + m + q even

8

2/3 + l

1/5 + m

1/5 + q

l + m + q even

9

1/2 + l

2/5 + m

1/5 + q

l + m + q even

10

3/5 + l

1/3 + m

1/5 + q

l + m + q even

11

2/5 + l

2/5 + m

2/5 + q

l + m + q even

12

2/3 + l

1/3 + m

1/5 + q

l + m + q even

13

4/5 + l

1/5 + m

1/5 + q

l + m + q even

14

1/2 + l

2/5 + m

1/3 + q

l + m + q even

15

3/5 + l

2/5 + m

1/3 + q

l + m + q even

l + m + q even l + m + q even

Here l, m and q are integers. B.1.2. Legendre equation The associated Legendre equation

µ2 (1 − z )y − 2zy + ν(ν + 1) − 1 − z2 ′′

2

′





y = 0.

(B.2)

This equation is a hypergeometric equation with singular points at z = 1, −1, ∞. The difference of exponents is µ, µ and 2ν + 1 and by Kimura’s theorem we can obtain the following (see [3]). Proposition B.2. Legendre equation (B.2) is integrable if and only if, either (1) µ ± ν ∈ Z or ν ∈ Z, or (2) (µ, ν, µ + ν) belong to one of the following seven families

( a)

µ Z+

1 2

(b)

Z±

1 3

1 Z 2

±

1 3

Z+

1 6

(c )

Z±

2 5

1 Z 2

±

1 5

Z+

1 0

(d)

Z±

1 3

1 Z 2

±

2 5

Z+

1 10

(e)

Z±

1 5

1 Z 2

±

2 5

Z+

1 10

(f )

Z±

2 5

1 Z 2

±

1 3

Z+

1 6

Case

ν

µ+ν

C

We remark that cases (1) and (2)(a) were classically known (see, for instance, [103], Chapter VI, Exercise 31). By means of the change of variables y = (1 − z 2 )µ/2 w,

1

(1 − z ), 2 Eq. (B.2) is transformed in a two-parameter subfamily of Gauss hypergeometric equation: dw + c − (a + b + 1)x − ab w = 0, dx2 dx with a = µ − ν, b = µ + ν + 1, c = µ + 1. x(1 − x)

d2 w

x=

(B.3)

J.J. Morales-Ruiz / Journal of Geometry and Physics (

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Let F be the function defined by the hypergeometric series F (a, b, c ; x) = 1 +

ab c

x + ···,

(B.4)

convergent for |x| < 1. Then for ν − µ ∈ N the solution

w(x) = F (µ − ν, µ + ν + 1, µ + 1; x) of (B.3) is a polynomial of degree ν − µ (essentially a particular case of Jacobi polynomials). For ν ∈ N, by means of some suitable transformation, also a solution of (B.3) can be reduced to a hypergeometric function (B.4) with a = −ν and hence to Jacobi polynomials. The so-called associated Legendre polynomials arise as particular solutions of (B.2) when both µ and ν are natural numbers, with µ ≤ ν : dµ y(z ) = Pνµ (z ) = (−1)µ (1 − z 2 )µ/2 µ Pν (z ), dz being Pµ the Legendre polynomials (see later). As a matter of fact, Pνµ (z ) are true polynomials for µ even. We remark that case (1) of Proposition B.2 correspond to Case 1 of Kovacic algorithm, and the above polynomials related in some way to (special) Jacobi polynomials are essentially the polynomials obtained in Case 1 (see Step 3) of the algorithm. B.1.3. Weber equation By a direct application of Kovacic’s algorithm it is possible to prove that for r (x) a polynomial of degree m, r (x) = Pm (x), a necessary condition of integrability is that the degree must be even, m = 2n. The first non-trivial case to study is for second degree polynomials r (x) = (ax + d)2 + b. By an affine change of variable in x we obtain the Weber equation y′′ −



1 4



x2 − λ y = 0.

λ ∈ C.

(B.5)

By applying Kovacic’s algorithm, it is proved that (B.5) is integrable if and only if λ = 1/2 + n, with n ∈ Z, as Kovacic himself obtained in [55]. B.1.4. Families of orthogonal polynomials Recall that the hypergeometric equation, including confluences, is a particular case of the differential equation y′′ +

L ′ λ y + y = 0, Q Q

λ ∈ C, L = a0 + a1 x, Q = b0 + b1 x + b2 x2 .

(B.6)

It is well known (see, for example, [104]) that classical orthogonal and Bessel polynomials are solutions of Eq. (B.6), for suitable values of aj , bj and λ. Namely,

• • • • • • • • •

Hermite Hn , Chebyshev of first kind Tn , Chebyshev of second kind Un , Legendre Pn , Laguerre Ln , (m) associated Laguerre Ln , (m) Gegenbauer Cn , (m,ν) Jacobi Pn Bessel Bn ,

where Family Hn Tn Un Pn Ln (m)

Q 1 1 − x2 1 − x2 1 − x2 x

Ln

x

Cn

(m)

1 − x2

Pn

(m,ν)

1 − x2 x2

Bn

L

λ

−2x −x −3x −2x 1−x m+1−x −(2m + 1)x ν − m − (m + ν + 2)x 2(x + 1)

2n n2 n(n + 2) n(n + 1) n n n(n + 2m) n(n + 1 + m + ν) −n(n + 1)

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Integrability conditions and solutions of Eq. (B.6) can be obtained applying Kovacic algorithm (Case 1 of the algorithm). Legendre and Hermite polynomials correspond to one of the integrability cases already described in Proposition B.2 and to Weber equation (B.5) for natural n, respectively, the connection is made explicit by a suitable change of variable. For the Gauss hypergeometric equation (B.1), we give the Kummer continuation (or transition) formula for the analytic prolongation between x = 0 and x = 1 for the space of solutions (see [103], Section 26). This relation is useful for the computation of the scattering matrix, when the Schrödinger is reduced to an hypergeometric equation by means of algebrization. We consider the following bases of the space of solutions at x = 0 and x = 1. At x = 0 the basis of solutions is (0)

(0)

y1 = F (a, b, c ; x),

y2 = x1−c F (a − c + 1, b − c + 1, 2 − c ; x),

(B.7)

being, as above, F the function defined by the hypergeometric series F (a, b, c ; x) = 1 +

ab c

x + ···,

convergent for |x| < 1. Analogously, at x = 1 we have the basis (1)

y1 = F (a, b, a + b − c + 1; 1 − x),

(1)

y2 = (1 − x)c −a−b F (c − a, c − b, c − a − b + 1; 1 − x).

(B.8)

The transition matrix between x = 0 and x = 1, expressing the analytical prolongation of the basis of solutions between the above points, is given by the matrix T = T (a, b, c ) ∈ GL(2, C), such that

(y(10) , y(20) ) = (y(11) , y(21) ) T . As was proved by Kummer, the matrix T is explicitly given by

 T =

φ(a, b, c ) φ(c − a, c − b, c )

 φ(a − c + 1, b − c + 1, 2 − c ) , φ(1 − a, 1 − b, 2 − c )

(B.9)

where φ is the following rational expression of gamma functions

φ(a, b, c ) =

Γ ( c ) Γ ( c − a − b) . Γ (c − a)Γ (c − b)

B.1.5. Confluent hypergeometric The confluent hypergeometric equation is a degenerate form of the hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The following are two classical forms.

• Kummer’s form c−x ′ a y′′ + y − y = 0, x

x

a, c ∈ C

(B.10)

at x = 1 • Whittaker’s form   1 κ 4µ2 − 1 ′′ y = − + y, 2 4

x

(B.11)

4x

where the parameters of the two equations are linked by κ = 2c − a and µ = 2c − 21 . Furthermore, by the formula (A.3), Whittaker’s equation is the reduced form of Kummer’s equation (B.10). The Galoisian structure of these equations has been deeply studied in [16,100]. Theorem B.3 (Martinet–Ramis, [16]). (i) Kummer’s differential equation (B.10) is integrable if and only if either, −a ∈ N,or a − 1 ∈ N, or a − c ∈ N, or 1 + c − a ∈ N. (ii) Whittaker’s differential equation (B.11) is integrable if and only if either, κ +µ ∈ 21 + N, or κ −µ ∈ 21 + N, or −κ +µ ∈ 21 + N, or −κ − µ ∈

1 2

+ N.

Bessel’s equation is a particular case of the confluent hypergeometric equation and is given by y′′ +

1 ′ x 2 − n2 y + y = 0. x x2

(B.12)

Under a suitable transformation, the reduced form of Bessel’s equation is a particular case of Whittaker’s equation (B.11). Corollary B.4. Bessel’s differential equation (B.12) is integrable if and only if n ∈

1 2

+ Z.

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B.2. Heun’s families B.2.1. Heun’s equation Heun’s equation is the generic differential equation with four regular singular points. By placing the singularities at 0, 1, c and ∞ (the situation of the singularities is irrelevant for integrability in the context of Picard–Vessiot theory), in its reduced form, Heun’s equation is y′′ = r (x)y, where r (x) =

A

B

C

D

E

F

+ , (x − 1)2 (x − c )2 αβ αγ αβ βγ δη(h − 1) A=− − + , B= − − , 2 2c c 2 2(c − 1) c−1    βγ δη(c − h) α α β β αγ + − , D= −1 , E= −1 , C = 2c 2(c − 1) c (c − 1) 2 2 2 2  γ γ F = − 1 , with α + β + γ − δ − η = 1. x

+

x−1

+

x−c δηh

+

x2

+

(B.13)

2 2 To our purposes we write the determinant Πd+1 (a, b, u, v, ξ , w) as in [100]:

 w  dξ w + 1   0   0    ..  .   0  

u

v (d − 1)ξ

0

0

0

2(u + b) w + 2(v + a) (d − 2)ξ

0 0

0 0

3(u + 2b) w + 3(v + 2a)

4(u + 3b)

... ...

... ...

... ... ... ...

0

2ξ 0

... w + (d − 1)(v + (d − 2)a) ξ

0 0

     0   0  .     d(u + (d − 1)b)   w + d(v + (d − 1)a)

B.2.2. Mathieu equation The Mathieu equation is d2 y

− (a + b cos ωt )y = 0, dt 2 being a, b and ω. Rescaling the time, it is possible to reduce to the case ω = 1, provided ω ̸= 0. With the change x = cos t we can algebrize (B.14):

(B.14)

d2 y

dy − x − (a + bx)y = 0. dx2 dx This is in fact a particular family of the confluent Heun’s equation. It has three singular points, ±1 regular singular ones and ∞ an irregular singular point. It is possible to obtain Eq. (B.14) as a confluence mechanism in (B.13), between one of the three singular points in the finite complex plane and the point at infinite, that becomes now an irregular singular point. Then by applying Kovacic’s algorithm we obtain that for b ̸= 0 and ω ̸= 0, Eq. (B.14) is non-integrable [22].

(1 − x2 )

B.2.3. Biconfluent Heun The equation y′′ =



x2 + δ 1 x +

δ12 4

− δ2 +

δ3 2x

+

δ02 − 1 4x2



y,

(B.15)

is the well known biconfluent Heun equation, which has been deeply analyzed by Duval and Loday-Richaud in [100, p. 236]. Theorem B.5 ([100]). The biconfluent Heun equation (B.15) has Liouvillian solutions if and only if it falls in Case 1 of Kovacic algorithm and one of the following conditions is fulfilled. (1) δ02 = 1, δ3 = 0 and δ2 ∈ 2Z + 1. (2) δ02 = 1, δ3 ̸= 0 and δ2 ∈ 2Z∗ + 1 with |δ2 | ≥ 3, and if ε = sign δ2 , then

  δ3 Π(|δ2 |−1)/2 0, 1, 2, εδ1 , −2ε, εδ1 − = 0. 2

(3) δ0 ̸= ±1, ±δ0 ± δ2 ∈ 2Z and if ε0 , ε∞ ∈ {±1} are such that ε∞ δ2 − ε0 δ0 = 2d∗ ∈ 2N∗ then ∗

  1 Πd∗ 0, 1, 1 + ε0 δ0 , ε∞ δ1 , −2ε∞ , (ε∞ δ1 (1 + ε0 δ0 ) − δ3 ) = 0. 2

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B.2.4. Lamé equation Although it is not explicitly used along the text, we included here some Galoisian results about the Lamé equation, as it is a classical subfamily of the Heun family and moreover it is relevant in some of the cited references, like in the study of finite-gap potentials. The algebraic form of the Lamé Equation is [103,105] d2 y dx2

+

f ′ (x) dy 2f (x) dx

−

n(n + 1)x + B f (x)

y = 0,

(B.16)

where f (x) = 4x3 − g2 x − g3 , with n, B, g2 and g3 parameters such that the discriminant of f , ∆ = 27g32 − g23 , is non-zero and, therefore, it has no multiple roots. This equation is a Fuchsian differential equation with four singular points over the Riemann sphere: the roots e1 , e2 , e3 of f and the point at infinity. Thus, the Lamé equation is a subfamily of Heun’s family. The mutually-exclusive known cases of solutions in closed form of the Lamé equation (B.16) are the following. (i) The Lamé–Hermite case (see [103,105]). We have n ∈ N and arbitrary parameters B, g2 and g3 . (ii) The Brioschi–Halphen–Crawford case (see [106,103]). We have n ∈ N such that m := n + 21 ∈ N and parameters B, g2 and g3 satisfying an algebraic condition Qm (g2 /4, g3 /4, B) = 0, where Qm ∈ Z[g2 /4, g3 /4, B] is a polynomial of degree m in B, known as the Brioschi determinant. (iii) The Baldassarri case (see [107]). One asks n to satisfy that n + 21 ∈ 13 Z ∪ 14 Z ∪ 15 Z − Z besides some additional (involved) algebraic restrictions on the other parameters. It is possible to prove that the only integrable cases of the Lamé equation are cases (i)–(iii) above, see [21]. In (ii) and (iii) the general solution of (B.16) is algebraic and the Galois group is finite. Case (i) splits in the following two subcases [103, 105]. (i.1) The Lamé case. For a fixed integer n, this equation admits a solution (called Lamé function) of the form E (x) =

3 

(x − ei )ki Pm (x),

(B.17)

i=1

being Pm a monic polynomial of degree m = n/2 − (k1 + k2 + k3 ) and ki ∈ 0, 21 , i = 1, 2, 3. Since m ∈ N, eight different possibilities regarding n appear: If n is even, we have k1 = k2 = k3 = 0 or just one zero ki ; if n is odd, we could have all non-zero ki ’s or combinations with exactly one non-zero ki . All these possibilities give rise to classes of Lamé functions. Concerning parameter B, it must be one of the m + 1 different roots B1 , . . . , Bm+1 of certain irreducible polynomial of degree m + 1, with all its roots real and simple [106]. Furthermore, the numbers Bi are reals. (i.2) The Hermite case. Here we are not in case (i.1) and n is an arbitrary natural number. We also fall in case 1 of Kovacic algorithm, but with a diagonal Galois group.





Remark B.6. We notice that the polynomial Pm in (i.1) satisfies a second order linear differential equation similar to the one that appears in the first case of Kovacic algorithm. As a matter of fact, it is possible to obtain the above passing into normal form and applying Kovacic algorithm. Therefore the second linear independent solution is not algebraic and the associated Riccati equation has no rational first integral. References [1] J. Morales-Ruiz, J.-P. Ramis, Integrability of dynamical systems through differential Galois theory: a practical guide, in: P.-B. Acosta-Humánez, F. Marcellán (Eds.), Differential Algebra, Complex Analysis and Orthogonal Polynomials, in: Contemp. Math., vol. 509, Amer. Math. Soc., Providence, RI, 2010, pp. 143–220. [2] P. Acosta-Humánez, J.-J. Morales-Ruiz, J.-A. Weil, Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys. 67 (2011) 305–374. [3] P.B. Acosta-Humánez, J.T. Lázaro-Ochoa, J.J. Morales-Ruiz, Ch. Pantazi, On the integrability of polynomial fields in the plane by means of Picard–Vessiot theory, arXiv:1012.4796. [4] P.B. Acosta-Humánez, C. Pantazi, Darboux integrals for Schrödinger planar vector fields via Darboux transformations, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012) 043. arXiv:1111.0120. [5] J.-A. Weil, Méthodes effectives en théorie de Galois différentielle et applications à l’intégrabilité de systèmes dynamiques (Habilitation thesis), Université de Limoges, 2013. [6] J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1981. [7] A. Borel, Linear Algebraic Groups, Springer-Verlag, New York, 1991. [8] E. Picard, Sur les groupes de transformation des équations différentielles linéaires, C. R. Acad. Sci., Paris 96 (1883) 1131–1134. [9] E. Picard, Sur équations différentielles et les groupes algébriques des transformation, Ann. Fac. Sci. Univ. Toulouse (1) 1 (1887) A1–A15. [10] E. Picard, Traité d’Analyse, Tome III, Gauthiers-Villars, Paris, 1928. [11] M.E. Vessiot, Sur l’intégration des équations différentielles linéaires, Ann. Sci. Éc. Norm. Supér. (3) 9 (1892) 197–280. [12] M. van der Put, M. Singer, Galois Theory of Linear Differential Equations, Springer, Berlin, 2003. [13] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, 1973. [14] I. Kaplansky, An Introduction to Differential Algebra, Hermann, 1957. [15] M. Singer, in: M.A.H. MacCallum, A.V. Mikhalov (Eds.), Introduction to the Galois Theory of Linear Differential Equations Algebraic Theory of Differential Equations, in: London Mathematical Society Lecture Note Series, vol. 357, Cambridge University Press, 2009, pp. 1–82. [16] J. Martinet, J.P. Ramis, Théorie de Galois differentielle et resommation, in: E. Tournier (Ed.), Computer Algebra and Differential Equations, Academic Press, London, 1989, pp. 117–214.

J.J. Morales-Ruiz / Journal of Geometry and Physics (

)

–

29

[17] J. Liouville, Mémoire sur l’intégration d’une classe d’équations différentielles du second ordre en quantités finies explicites, J. Math. Pures Appl. (1) 4 (1839) 423–456. [18] D. Blázquez-Sanz, J.J. Morales-Ruiz, Differential Galois theory of algebraic Lie–Vessiot systems, in: Differential Algebra, Complex Analysis and Orthogonal Polynomials, in: Contemp. Math., vol. 509, Amer. Math. Soc., Providence, RI, 2010, pp. 1–58. [19] D. Blázquez-Sanz, Differential Galois Theory and Lie–Vessiot Sytems (Ph.D. thesis), VDM Verlag, 2008. [20] J.J. Morales-Ruiz, J.P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal. 8 (2001) 33–96. [21] J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, in: Progress in Mathematics, vol. 179, Birkhäuser, Basel, 1999. [22] P. Acosta-Humánez, D. Blázquez-Sanz, Non-integrability of some hamiltonian systems with rational potential, Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 265–293. [23] H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996) 59–135. [24] B. Malgrange, Le groupoide de Galois d’un feuilletage, in: Essays on Geometry and Related Topics, Vols. 1, 2, Monogr. Enseign. Math. 38 (2) (2001) 465–501. [25] B. Malgrange, On the non linear Galois differential theory, Chin. Ann. Math. Ser. B 23 (2002) 219–226. [26] G. Casale, Sur le groupoïde de Galois d’un feuilletage (Ph.D. thesis), Univ. Toulouse, 2004. [27] G. Casale, Feuilletages singuliers de codimension un, groupoide de Galois et intègrales premieres, Ann. Inst. Fourier 56 (2006) 735–779. [28] G. Casale, The Galois groupoid of Picard Painlevé VI equation, in: Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and Their Deformations, Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu B2 (2007) 15–20. [29] G. Casale, Le groupe de Galois de P 1 et son irreductibilité, Comment. Math. Helv. 83 (2008) 471–519. [30] P. Cassidy, M. Singer, Galois theory of parameterized differential equations and linear differential algebraic groups, in: D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah, R. Schaefke (Eds.), Differential Equations and Quantum Groups, in: IRMA Lectures in Mathematics and Theoretical Physics, vol. 9, EMS Publishing house, 2006, pp. 113–157. [31] J.J. Morales-Ruiz, J.P. Ramis, C. Simó, Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations, Ann. Sci. Éc. Norm. Supér. 40 (2007) 845–884. [32] J.J. Morales-Ruiz, J.P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal. 8 (2001) 113–120. [33] H. Yoshida, A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D 29 (1987) 128–142. [34] A. Aparicio-Monforte, Méthodes effectives pour l’intégrabilité des systèmes dynamiques (Ph.D. Thesis), Université de Limoges, December 2010. [35] A.V. Tsygvintsev, Non-existence of new meromorphic first integrals in the planar three-body problem, Celestial Mech. Dynam. Astronom. 86 (2003) 237–247. [36] A. Aparicio-Monforte, J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems. Symmetries and related topics in differential and difference equations, in: Contemp. Math., vol. 549, Amer. Math. Soc., Providence, RI, 2011, pp. 1–15. [37] A. Aparicio-Monforte, M. Barkatou, S. Simon, J.-A. Weil, Formal first integrals along solutions of differential systems I. ISSAC 2011, in: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2011, pp. 19–26. [38] A. Aparicio-Monforte, J.-A. Weil, A reduced form for linear differential systems and its application to integrability of Hamiltonian systems, J. Symbolic Comput. 47 (2012) 192–213. [39] A. Aparicio-Monforte, E. Compoint, J.-A. Weil, A characterization of reduced forms of linear differential systems, J. Pure Appl. Algebra 217 (2013) 1504–1516. [40] S. Simon, Linearised higher variational equations, Preprint, 2013. arXiv:1304.0130. [41] S. Simon, Evidence and conditions for non-integrability in the Friedmann–Robertson–Walker Hamiltonian, Preprint, 2013. arXiv:1309.2754v1. [42] M. Ayoul, N.T. Zung, Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris 348 (2010) 1323–1326. [43] T. Combot, A note on algebraic potentials and Morales–Ramis theory, Celestial Mech. Dynam. Astronom. 115 (2013) 397–404. [44] T. Combot, Non-integrability of a self-gravitating Riemann liquid ellipsoid, Regul. Chaotic Dyn. 18 (2013) 497–507. [45] S. Rosemann, K. Schöbel, Open problems in the theory of finite-dimensional integrable systems and related fields, J. Geom. Phys. (2014) submitted for publication. [46] T. Combot, Non-intégrabilité algébrique et méromorphe de problèmes de n corps et de potentiels homogènes de degré −1 (Ph.D. thesis), Univ. Paris 7, 2013. [47] T. Combot, Integrability conditions at order 2 for homogeneous potentials of degree −1, Nonlinearity 26 (2013) 95–120. [48] G. Duval, A.J. Maciejewski, Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials, Ann. Inst. Fourier 59 (2009) 2839–2890. [49] T. Combot, Generic classification of homogeneous potentials of degree −1 in the plane, Preprint, 2011. arXiv:1110.6130. [50] T. Combot, C. Koutschan, Third order integrability conditions for homogeneous potentials of degree −1, J. Math. Phys. 53 (2012) 082704–26. [51] T. Combot, Non-integrability of the equal mass n-body problem with non-zero angular momentum, Celestial Mech. Dynam. Astronom. 114 (2012) 319–340. [52] G. Casale, G. Duval, A.J. Maciejewski, M. Przybylska, Integrability of Hamiltonian systems with homogeneous potentials of degree zero, Phys. Lett. A 374 (2010) 448–452. [53] A.J. Maciejewski, M. Prybylska, Integrable deformations of integrable Hamiltonian systems, Phys. Lett. A 376 (2011) 80–93. [54] A.J. Maciejewski, M. Prybylska, Non-integrability of the dumbbell and point mass problem, Preprint, 2013. arXiv:1304.6369v1. [55] J. Kovacic, An algorithm for solving second order linear homogeneus differential equations, J. Symbolic. Comput. 2 (1986) 3–43. [56] M. Studzinski, M. Prybylska, Darboux points and integrability analysis of Hamiltonian systems with homogeneous rational, Preprint, 2012. arxiv:1205.4395v1. [57] G. Duval, A.J. Maciejewski, Integrability of Homogeneous potential of degree k = ±2. An application of higher variational equations, Preprint, 2013. arXiv:1303.5550v1. [58] G. Duval, A.J. Maciejewski, Integrability of potentials of degree k ̸= ±2. Second order variational equations between Kolchin solvability and Abelianity, Preprint, 2013. arXiv:1303.5829v1. [59] M. Shibayama, Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities, Preprint, 2013. arXiv:1307.4509v1. [60] T. Waters, Regular and irregular geodesics on spherical harmonic surfaces, Physica D 241 (2012) 543–552. [61] T. Waters, Non-integrability of geodesic flow on certain algebraic surfaces, Phys. Lett. A 376 (2012) 1442–1445. [62] T. Combot, T. Waters, Integrability conditions of geodesic flow on homogeneous Monge manifolds, Ergodic Theory Dynam. Systems (2014) in press. [63] A.J. Maciejewski, M. Pryzbylska, T. Stachowiak, Non-existence of the final first integral in the Zipoy-Voorhees space–time, Perprint, 2013. arxiv:1302.4234v1. [64] P. Basu, L.A. Pando Zayas, Analytic non-integrability in string theory, Phys. Rev. D 48 (2011) 046006. arXiv:1105.2540v1. [65] A. Stepanchuk, A.A. Tseytlin, On (non) integrability of classical strings in p-brane backgrounts, J. Phys. A: Math. Ther. 46 (2013) 125401–125421. arXiv:1211.3727. [66] A. Ghosh, Time-dependent systems and chaos in string theory (Ph.D. thesis), Univ. Kentucky, 2012. [67] D. Giataganas, L.A. Pando Zayas, K. Zoubos, On marginal deformations and non-integrability, arXiv:1311.3241v1. [68] P. Basu, D. Das, A. Ghosh, L.A. Panda Zayas, Chaos around holographic Regge trajectories, J. High Energy Phys. 2012 (5) (2012) 077. arXiv:1201.5634v3. [69] Y. Chervonyi, O. Lunin, (Non)-integrability of Geodesic in D-brane backgrounds, Preprint, 2013. arXiv:1311.1521v2. [70] D. Giataganas, K. Sfetsos, Non-integrability in non-relativistic theories, arXiv:1403.2703v1.

30

J.J. Morales-Ruiz / Journal of Geometry and Physics (

)

–

[71] P. Basu, D. Das, A. Ghosh, Integrability lost: chaotic dynamics of classical strings on a confining holographic background, Phys. Lett. B 699 (2011) 388–393. arXiv:1103.4101. [72] L.J. Mason, N.M.J. Woodhouse, Integrability, Self-Duality and Twistor Theory, Clarendon Press, Oxford, 1996. [73] J.J. Morales-Ruiz, A Remark about the Painlevé transcendents, in: Théories asymptotiques et équations de Painlevé, S.M.F, Sémin. Congr. 14 (2005) 229–235. [74] T. Stoyanova, O. Christov, Non-integrability of the second Painlevé equation as a Hamiltonian system, C. R. Acad. Bulg. Sci. 60 (2007) 1. [75] E. Horozov, T. Stoyanova, Non-integrability of some Painlevé VI-equations and dilogarithms, Regul. Chaotic Dyn. 12 (2007) 622–629. [76] N. Beisert, et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (3) (2012). arXiv:1012.3982. [77] R. Martínez, C. Simó, Non-integrability of Hamiltonian systems through high order variational equations: summary of results and examples, Regul. Chaotic Dyn. 14 (2009) 323–348. [78] V. Salnikov, Effective algorithm of analysis of integrability via the Ziglin’s method, arXiv:1208.6252. [79] V. Salnikov, Integrability of the double pendulum–the Ramis question, Preprint, 2013. arXiv:1303.4904v1. [80] O. Christov, Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom, Celestial Mech. Dynam. Astronom. 112 (2012) 149–167. [81] J.J. Morales-Ruiz, C. Simó, S. Simon, Algebraic proof of the non-integrability of Hill’s problem, Ergodic Theory Dynam. Systems 25 (2005) 1237–1256. [82] L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, Pergamon Press, Oxford, 1977. [83] G. Darboux, Sur une proposition relative aux équations linéaires, C. R. Acad. Sci. 94 (1882) 1456–1459. [84] G. Darboux, Théorie des Surfaces, II, Gauthier-Villars, 1889. [85] V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, 1991. [86] J.J. Morales-Ruiz, J.M. Peris, On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse VIII (1999) 125–141. [87] C. Grotta-Ragazzo, Nonintegrability of some Hamiltonian systems, scattering and analytic continuation, Comm. Math. Phys. 166 (1994) 255–277. [88] N. Rosen, P.M. Morse, On the vibrations of polyatomic molecules, Phys. Rev. 42 (1932) 210–217. [89] P. Deift, E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979) 121–251. [90] S. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons, Consultants Bureau, New York, 1984. [91] F. Fauvet, J.-P. Ramis, F. Richard-Jung, J. Thomann, Stokes phenomenon for the prolate spheroidal wave equation, Appl. Numer. Math. 60 (2010) 1309–1319. [92] T. Stachowiak, M. Przybylska, On integrable rational potentials of the Dirac equation, Phys. Lett. A 377 (2013) 833–841. [93] A.J. Maciejewski, M. Przybylska, T. Stachowiak, Full spectrum of the Rabi model, Phys. Lett. A 378 (2014) 16–20. [94] D. Blázquez-Sanz, K. Yagasaki, Galoisian approach for a Sturm–Liouville problem on the infinite interval, Methods Appl. Anal. 19 (2012) 267–288. [95] P. Etingof, E. Rains, On algebraically integrable differential operators on an elliptic curve, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), 19 pp, Paper 062. [96] N.V. Krigorenko, Algebraic-geometric operators and differential Galois theory, Ukrainian Math. J. 61 (2009) 14–29. [97] Yu.V. Brezhnev, What does integrability of finite-gap or soliton potentials mean? Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (1867) (2008) 923–945. [98] Yu.V. Brezhnev, Spectral/quadrature duality: Picard–Vessiot theory and finite-gap potentials, in: Algebraic Aspects of Darboux Transformations, Quantum Integrable Systems and Supersymmetric Quantum Mechanics, in: Contemp. Math., vol. 563, Amer. Math. Soc., Providence, RI, 2012, pp. 1–31. [99] Yu.V. Brezhnev, Elliptic solitons, Fuchsian equations, and algorithms, St. Petersburg Math. J. 24 (2013) 555–574. [100] A. Duval, M. Loday-Richaud, Kovacic’s algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput. 3 (3) (1992) 211–246. [101] F. Ulmer, J.A. Weil, Note on Kovacic’s algorithm, J. Symbolic Comput. 22 (1996) 179–200. [102] T. Kimura, On Riemann’s equations which are solvable by quadratures, Funkcial. Ekvac. 12 (1969) 269–281. [103] E.G.C. Poole, Introduction to the Theory of Linear Differential Equations, Oxford Univ. Press, London, 1936. [104] T. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978. [105] E.T. Whittaker, E.T. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, UK, 1969. [106] G.H. Halphen, Traité des fonctions elliptiques, Vols. I, II, Gauthier-Villars, Paris, 1888. [107] F. Baldassarri, On algebraic solutions of Lamé’s differential equation, J. Differential Equations 41 (1981) 44–58.