Darboux Transformations for orthogonal differential systems and differential Galois Theory
Primitivo Acosta-Humanez, Moulay Barkatou, Raquel Sánchez-Cauce, Jacques-Arthur Weil
aa r X i v : . [ m a t h . C A ] J a n D AR B OUX T R ANSFORMATIONS FOR ORTHOGONALDIFFER ENTIAL SYSTEMS AND DIFFER ENTIAL G ALOIS T HEORY
Primitivo ACOSTA-HUMÁNEZ
Universidad Simón Bolivar, Barranquilla - ColombiaInstituto Superior de Formación Docente Salomé UreñaSantiago de los Caballeros - Dominican Republic [email protected]
Moulay BARKATOU
XLim - Université de Limoges & CNRSLimoges, France [email protected]
Raquel SÁNCHEZ-CAUCE
Department of Artificial IntelligenceUniversidad Nacional de Educación a Distancia (UNED)Madrid, Spain [email protected]
Jacques-Arthur WEIL
XLim - Université de Limoges & CNRSLimoges, France [email protected] A BSTRACT
Darboux developed an algebraic mechanism to construct an infinite chain of “integrable" secondorder differential equations as well as their solutions. After a surprisingly long time, Darboux’sresults had important features in the analytic context, for instance in quantum mechanics where itprovides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot ofpapers regarding the use of Darboux transformations in various contexts, not only in mathematicalphysics. In this paper, we develop a generalization of the Darboux transformations for tensor productconstructions on linear differential equations or systems. Moreover, we provide explicit Darbouxtransformations for s y m (SL(2 , C )) systems and, as a consequence, also for so (3 , C K ) systems, toconstruct an infinite chain of integrable (in Galois sense) linear differential systems. We introduceSUSY toy models for these tensor products, giving as an illustration the analysis of some shapeinvariant potentials. Keywords
Darboux transformations · differential Galois group · differential Galois Theory · Frenet-Serret formu-las · orthogonal differential systems · rigid solid problem · Schrödinger equation · shape invariant potentials · supersymmetric quantum mechanics · symmetric power · tensor product. MSC 2010.
Introduction
In this paper we study, from a Galoisian point of view, a combination of two results of Darboux: the first one isthe celebrated Darboux transformation [15], see also [17, 18], and the second one, from [18, 19], shows how toexpress solutions of a system with an orthogonal matrix using solutions of a second order differential equation. Thiscombination will allow us to generalize Darboux transformation to higher order systems.The first result, the Darboux transformation, has been studied from different points of view for ordinary differentialequations as well as for partial differential equations and as application in physics (supersymmetric quantum mechan-ics). The starting point was Darboux’s result given in [15], see also [17, 18], followed by plenty of papers and books:for example, we refer to the seminal works of Witten (supersymmetric quantum mechanics [33]) and Gendenshteïn(shape invariant potentials [23] ). A matrix formalism for Darboux transformations for Schrödinger equation wasdeveloped in the beginning of the twenty first century by Pecheritsin, Pupasov and Samsonov, see [29]. Recently, aGaloisian approach to Darboux transformations and shape invariant potentials has been proposed in [1, 2, 4], where itwas proved that the Darboux transformation preserves the galoisian structure of the differential equation (the DarbouxT
FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY transformation is isogaloisian ). A similar approach was presented in [25, 26, 27] in the context of integrable systems.There, the authors studied the behavior of the galoisian structure of some families of linear systems with respect to Dar-boux transformations. An extension of the Darboux transformation for polynomial vector fields has been developed bythe first author and Pantazi in [5]. They give a mechanism to construct an infinity chain of Darboux-integrable polyno-mial vector fields using the same philosophy of Darboux from [15, 18]. In the formalism of supersymmetric quantummechanics, the shape invariance property was also among these tools. An important feature of the Darboux trans-formation is that it preserves algebraic and analytic conditions. Thus, any generalization of Darboux transformationshould preserve these kind of conditions.The second result of Darboux used here concerns the transformation of three dimensional linear differential systems X ′ = − AX , where A is a skew-symmetric matrix ( A ∈ so (3 , C ) ) so that it can be reduced to solving a Riccatiequation, see [18, 19]. This reduction has been used for example to study the integrability of dynamical systems, see[3, 21]. We will show how to understand and systematize this little miracle by viewing so (3 , C ) as a tensor productof SL(2 , C ) . To achieve this, we introduce gauge transformations and the tensor constructions from representationtheory: see [6, 8, 7] for an introduction to these objects in the context of integrability for dynamical systems.In this mostly self-contained paper, we present a methodology to extend Darboux transformations to classes of higherorder systems. Our approach maintains the original "shape invariance" which is the core of Darboux transforma-tion. We use linear differential equations tools such as gauge transformations, differential Galois theory and tensorconstructions.The Darboux transformation may be viewed as a gauge transformation, an idea present in [4, 1, 2]; we give here(Proposition 1) a matrix factorization of this gauge transformation which in turn gives a simpler expression for thesolutions of a Darboux transformation.This allows us to extend Darboux transformations to systems whose Galois group is represented as a symmetricpower of SL(2 , C ) . We then extend Darboux transformation to so (3 , C ) -systems which allows us to construct infinitechains of integrable (in Galois sense) linear differential systems. As an application of this approach, we study shapeinvariant potentials in Supersymmetric Quantum Mechanics for Schrödinger equations and we apply this formalismto recover some results from other authors (Fedorov, Maciejewski, Przybylska and others) related to so (3 , C ) systems,in particular, the rigid solid problem and Frenet-Serret formulas in section 3.2. This section contains a brief summary that will be used along this paper. We present the original results of Darboux aswell as a preliminary material concerning invariants and symmetric powers.
Differential Galois theory, also known as Picard-Vessiot theory, is analogous to the classical Galois theory for poly-nomials; it describes algebraic relations that may exist between solutions of linear differential equations and theirderivatives, see [14, 30]. A differential field K , depending on a variable x , is a field equipped with a derivation ∂ x = ′ . We denote by C K the field of constants of K , the set of c ∈ K such that c ′ = 0 . Along this paper, weconsider differential equations or systems whose coefficients belong to a differential field K whose constant field C K is algebraically closed and of characteristic zero. For simplicity, we will explain this Galois theory on operators oforder two but it applies similarly to linear differential equations or systems of any order.Consider the differential operator L := ∂ x + p∂ x + q, p, q ∈ K. (1)Let { y , y } denote a basis of solutions of L y = 0 . We let F := K ( y , y , y ′ , y ′ ) be the smallest differentialextension of K containing these solutions of L y = 0 and such that C K = C L . The differential extension F is calleda Picard-Vessiot extension of K associated to L y = 0 . The differential automorphisms of F are the automorphismsthat commutes with the derivation. The differential Galois group of L y = 0 , denoted by DGal(
F/K ) , is the group ofdifferential automorphisms of L which leave invariant each element of K .Let σ ∈ DGal(
F/K ) . Then, { σ ( y ) , σ ( y ) } is another basis of solutions of L y = 0 . It follows that σ ( y , y ) =( y , y ) M σ . This M σ is the matrix of the automorphism σ . We see that DGal(
F/K ) is a group of matrices and it isactually a linear algebraic group. DGal(
F/K ) ⊆ GL(2 , C K ) . The Wronskian W of the solutions y and y satisfies the differential equation W ′ + pW = 0 . Thus, W =exp( R ( − p ) dx ) . We find that W ∈ K if and only if p = w ′ w = (ln w ) ′ for some w ∈ K . In this case, we have2T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY σ ( W ) = W (because W ∈ K ). As σ ( W ) = det( M σ ) W with W = 0 , we obtain det( M σ ) = 1 , that is: DGal(
F/K ) ⊆ SL(2 , C K ) ⇐⇒ p = w ′ w = (ln w ) ′ , w ∈ K. We say that an algebraic group G is virtually solvable when the connected identity component of G , denoted by G ◦ ,is a solvable group. In this paper, we say that L y = 0 is integrable whenever DGal(
F/K ) is virtually solvable. Thiscorresponds to cases when one can compute formulas for the solutions.Given a non-zero solution y and some non-zero function c , the standard change of variables (see e.g the book of Ince[24]) u = − c · y ′ y changes our second order linear differential equation to the first order (non linear) Riccati equation ( R ) : u ′ = a + bu + cu , where b = − p − c ′ c and a = 1 c q. Differential Galois theory shows that the equation L ( y ) = 0 is integrable if and only if the Riccati equation ( R ) has analgebraic solution. Similar statements (although more technical) can be obtained for higher order equations, see [30].A scalar differential equation such as L y = 0 is equivalent to its companion linear differential system [ A ] : [ A ] : X ′ = − AX, where A = (cid:18) − q p (cid:19) and X = (cid:18) yy ′ (cid:19) . (2)Differential Galois theory applies naturally to linear differential systems as well (solution spaces are vector spaces, thegroups act on these vector spaces).Given a linear differential system [ A ] : X ′ = − AX , a gauge transformation is a linear change of variables X = P Y with P ∈ GL ( n, K ) . It transforms the system [ A ] into a linear differential system Y ′ = − P [ A ] Y with P [ A ] := P − AP + P − P ′ . We say that two linear differential systems [ A ] and [ B ] are equivalent over K when there exists a gauge transformation P ∈ GL ( n, K ) such that B = P [ A ] . Equivalent differential systems share the same differential Galois group.Given any linear differential system [ A ] : X ′ = − AX , the cyclic vector method (see [9] for a simple constructiveprocess) allows to construct an equivalent differential system in companion form. So we may go from operator tosystem and vice-versa without altering the theory. In what follows, we will start from a Darboux transformation onoperators, recast it as a transformation on systems and this will allow us to use the machinery on systems to buildhigher order Darboux transformations. In [15], Darboux proposes a transformation which, given a family of linear differential equations produces a newfamily of differential equations with a similar shape and similar properties. This transformation has proved to bepowerful, for example, in the study of Shrödinger equations. We recast it here in modern language. This propositionappears in Darboux’s note [16] and then in his book [15]. Ince mentions it in [24] page 132.
Theorem 1 (Darboux, [16])
Consider the family of differential equations L ( y ) = m ry : y ′′ + py ′ + ( q − mr ) y = 0 , (3) where p, q, r are functions ( r = 0 by hypothesis) and m is a constant parameter. Given a non-zero value for m , we let y m denote a general solution of (3). Suppose that we know a non-zero solution y of equation (3) for m = 0 . Let ˜ y m be a function defined by ˜ y m = 1 √ r ( y ′ m − θ y m ) with θ := y ′ y . (4) Then, for m = 0 , ˜ y m is a general solution of the new differential equation ˜ y ′′ + p ˜ y ′ + (˜ q − mr )˜ y = 0 , with ˜ q = q + q (5) where, letting ˆ r := r ′ r , the new part q is given by q := 2 θ ′ + ˆ r ′ + p ′ − ˆ r (ˆ r + 2 p − θ ) (6)3T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
The new ˜ q is given by Darboux [16] in the compact expression ˜ q = y √ r py √ r − (cid:18) y √ r (cid:19) ′ ! ′ . This transformation has been made famous by its applications to Schrödinger equations where p = 0 and r = 1 . Inthis case, the formula is much simpler: ˜ q = q + 2 θ ′ . Definition 1
The map (4) transforming the family of equations (3) to the family (5) is called a
Darboux transformation . Remark 1
A starting point in the philosophy of the Darboux transformation is that there is a sort of covariance ,which can be seen in Theorem 1. Both equations have the same structure. Their only difference is that q is changedinto ˜ q . Darboux presented in [15, 18] the particular case for r = 1 and p = 0 , which today is known as Darbouxtransformation, but it is really a corollary of the general Darboux transformation given in Theorem 1 . Consider the differential field K and the family of differential operators L m := ∂ x + p∂ x + ( q − mr ) , p, q, r ∈ K with r = 0 . (7)Here, m is a constant parameter. When m is given any value, we let F m be a Picard-Vessiot extension of K corre-sponding to the equation L m y = 0 . Without loss of generality, we assume from now on that there exists w ∈ K suchthat p = w ′ /w = (ln w ) ′ . Therefore, the differential Galois groups DGal( F m /K ) of L m y = 0 are subgroups of SL(2 , C K ) .Following Acosta-Humánez and co-authors in [4], see also [1, 2], we denote by Λ the set of values of m for which L m y = 0 is integrable (over K ), the so-called algebraic spectrum . If we have a family of parameters for which theequations L m are integrable (over K ), Darboux transformation will construct a new family with the same shape whilepreserving the integrability properties.After performing a Darboux transformation, our new family e L m of differential equations has a new coefficients field e K := K ( θ ) (with notations of Theorem 1). Note that the Darboux transformation itself is defined over the biggerfield e K ( √ r ) .We say that the Darboux transformation is isogaloisian when the differential Galois group is preserved (see [4, 1, 2]),i.e DGal( e F m / e K ) = DGal( F m /K ) ; it is strong isogaloisian when e K = K (i.e. when θ ∈ K ). Results on theisogaloisian character of the Darboux transformation appear in [4], see also [1, 2] ; they will reappear in the nextsection as a consequence of the view of the Darboux transformation as a gauge transformation. When θ is algebraicover K , then Darboux transformation is virtually isogaloisian; in this case, for any value m such that L m is integrable,its Darboux tranformation e L m is also integrable (see [4]). We see that the Darboux transformation transforms a familyof integrable equations into another integrable family with the same shape.We now turn to a rather different result, also due to Darboux, which allows one to solve third order orthogonal systemsusing only solutions of a first order Riccati equation, see [19, part I, chap. II] for details and proofs. A third-orderorthogonal system is one of the form αβγ ! ′ = h − g − h fg − f ! · αβγ ! . A simple calculation shows that α + β + γ is always constant for such a system. Theorem 2 (Darboux, [19], chap. II, pages 30-31)
Consider a differential system αβγ ! ′ = h − g − h fg − f ! · αβγ ! . (8) A solution ( α, β, γ ) such that α + β + γ = 1 can be parametrized by α = 1 − uvu − v , β = i uvu − v and γ = u + vu − v , (9)4T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY where u and v are distinct solutions of the same Riccati equation θ ′ = ω + µθ + ω θ , with ω = g − if , ω = g + if , µ = − ih. (10) Furthermore, u = α + iβ − γ = 1 + γα − iβ and v = − − γα − iβ = − α + iβ γ . (11) Remark 2
As we recalled, solutions of a first order Riccati equation are logarithmic derivatives of solutions of asecond order linear differential equation. So, this result shows that one can solve third order orthogonal systems usingsolutions of second order linear differential equations. Namely, performing the change of variable u = − ω y ′ y , wesee that y is a solution of y ′′ + (cid:18) µω + ω ′ ω (cid:19) y ′ + ω ω y = 0 (12) So, the parametrization given by (9) can be restated as αβγ ! = 1 w − ω ω − iω − iω y y y ′ y + y y ′ y ′ y ′ ! (13) where y , y are linearly independent solutions of the second order linear differential equation (12) and w := y ′ y − y y ′ is their wronskian. This gives an explicit solution for an orthogonal system in terms of solutions of second orderequations. This will be further explored in the next section. In the study of the rigid solid, see for example Fedorov et. al. [21, 22], orthogonal systems such as (8) are traditionallywritten in a more compact way, using the cross-product × : Z ′ = Z × Ω , Z = ( α, β, γ ) T , Ω = ( f, g, h ) T , f, g, h ∈ K. (14)From now on, we work with either equation (14) or equation (8): they are the same equation, although presenteddifferently. In what follows, the terminology orthogonal systems will refer to systems of the form (14) or (8). In differential Galois theory, one classically translates linear algebra constructions on the solution space (tensor prod-uct, symmetric powers, etc.) into constructions on differential systems. This allows to measure properties on solutionsby looking for rational function solutions in these tensor constructions (see chapters 3 and 4 in [30]). We review herethe construction of symmetric powers for later use.Let V denote a vector space over C K of dimension n . We fix a basis B of V and consider g ∈ End( V ) . Let M = ( m i,j ) ∈ M n ( C K ) denote the n × n matrix of the endomorphism g in that basis B . We define a linear action of g on the variables X j by g ( X j ) := P ni =1 m i,j X i so g acts on the indeterminates X i as if they were the basis B .Consider a homogeneous polynomial P ∈ K [ X , . . . , X n ] m of degree m . We may identify the m -th symmetric powerof V with the linear span of all monomials { X m , X m − X , . . . , X mn } of degree m . This way, our polynomial P maybe identified with its vector v P of coefficients on the monomial basis { X m , X m − X , . . . , X mn } . Using the relations g ( X j ) := P ni =1 m i,j X i , we can define an action of g on P by linear substitution g ( P ) := P ( g ( X ) , . . . , g ( X n )) .This action by substitution translates into a natural action of g on the coefficient vector v P by g ( v P ) := v g ( P ) . Usingthis action, we define the m -th symmetric power in the sense of groups Sym m ( M ) of M as the matrix of the linearmap v P v g ( P ) : it is defined by the relation v g ( P ) = Sym m ( M ) · v P . The map M Sym m ( M ) is a group morphism. Given a group G ⊂ GL ( V ) , an invariant of G (in Sym( V ) ) is apolynomial P such that ∀ g ∈ G , g ( P ) = P .Similarly, one can associate to the matrix M the derivation D M = n X j =1 n X i =1 m i,j X i ! ∂∂X j . FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Then, we define the m -th symmetric power in the sense of Lie algebras s y m m ( M ) of M as the matrix of the linearmap v P v D M ( P ) : it is defined by the relation v D M ( P ) = s y m m ( M ) · v P . The map M s y m m ( M ) is a Lie algebra morphism. Given a Lie algebra g , an invariant of g is a polynomial P suchthat, ∀ D ∈ g , D ( P ) = 0 . We have the equivalence: P is an invariant of G ◦ if and only if P is an invariant of its Liealgebra Lie ( G ) .Take a linear differential system [ A ] : X ′ = − AX . Its m -th symmetric power system is [ s y m m ( A )] . If X is a solutionmatrix of [ A ] then Sym m ( X ) is a solution matrix of [ s y m m ( A )] : Sym m ( X ) ′ = − s y m m ( A )Sym m ( X ) , see e. g.[6, 30]. Example 1
We will make this explicit on a construction used later on in the paper. Consider a system X ′ = − AX with A = (cid:18) − q p (cid:19) where p = w ′ w with w ∈ K. It is in companion form so it admits a fundamental solution matrix of the form X = (cid:18) y y y ′ y ′ (cid:19) . (15) The second symmetric power system is [ s y m ( A )] : Y ′ = − s y m ( A ) Y with s y m ( A ) = − q p − q p ! . It admits the fundamental solution matrix Y = Sym ( X ) = y y y y y y ′ y ′ y + y y ′ y y ′ ( y ′ ) y ′ y ′ ( y ′ ) . (16) Now, let σ ∈ DGal( F /K ) be an automorphism of the differential Galois group of [ A ] with matrix representation M σ = (cid:18) λ λ λ λ (cid:19) ∈ DGal( F /K ) , λ λ − λ λ = 1 . This means that σ ( X ) = X · M σ . As Sym( • ) is a group morphism, we have σ ( Y ) = σ (Sym ( X )) = Sym ( σ ( X )) = Sym ( X · M σ ) = Y · Sym ( M σ ) . So the matrix of σ acting on the solution Y of [ s y m ( A )] is Sym ( M σ ) (computed as in formula (16) ). By a slightabuse of notation, we will denote the set of such matrices by Sym (DGal( F /K )) ⊆ SL (3 , C K ) . Note that, using the cyclic vector (1 , , T , this symmetric power system can be written as the traditional third orderdifferential operator known as the second symmetric power of L :sym ( L ) := ∂ x + 3 p∂ x + (4 q + p ′ + 2 p ) ∂ x + 2( q ′ + 2 pq ) . (17)Last, we look at gauge transformations P . The change X = P Y transforms the system [ A ] into P [ A ] . Then,as Sym m ( • ) is a group morphism, Sym m ( X ) = Sym m ( P ) Sym m ( Y ) which shows that s y m m ( P [ A ]) = Sym m ( P )[ s y m m ( A )] . We write the differential equation L m y = 0 , equation (7), in its companion system form: [ A m ] : X ′ = − ( A + mN ) X, X = (cid:18) yy ′ (cid:19) , A = (cid:18) − q p (cid:19) , N = (cid:18) − r (cid:19) . (18)6T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Note that N is the null matrix. The Darboux transformation transforms the family L m y = 0 into a family e L m y = 0 whose companion form is now [ e A m ] : e X ′ = − ( e A + mN ) e X, e X = (cid:18) ˜ y ˜ y ′ (cid:19) , e A = (cid:18) − q p (cid:19) , (19)where ˜ y and ˜ q have the explicit form expressed in Theorem 1. We denote by e K = K ( θ ) the field of coefficients ofsystem (19). Proposition 1
The Darboux transformation given in Theorem 1 is equivalent to a gauge transformation between theabove families of systems [ ˜ A m ] and [ A m ] , whose matrix P m is given by P m := 1 √ r (cid:18) − θ mr − θ ρ ρ (cid:19) = L m .R where L m := (cid:18) mr ρ (cid:19) , R := 1 √ r (cid:18) − θ (cid:19) , (20) with θ = y ′ y , p = w ′ w and ρ := − θ − p − r ′ r . When θ is algebraic over K , this gauge transformation preservesthe identity component of the differential Galois group. Moreover, whenever θ ∈ K , this Darboux transformationpreserves the differential Galois group. Remark 3
Note that, in this factorization, the first matrix L m contains the dependence on m and the second matrix R only depends on the known solution y . Proof.
Let Y := P m X with Y = ( z , z ) T and X given in (18). The first line is z = √ r ( − θ y + y ′ ) so we recognizethe Darboux transformation from Theorem 1 and we have z = ˜ y . Now we would like z to be ˜ y ′ . So we differentiate √ r ( − θ y + y ′ ) modulo the relation L m y = 0 : this gives us ˜ y ′ as a linear combination of y and y ′ and we find theexpression of P m giving ˜ X = P m X .The new coefficient field is e K := K ( θ ) . When θ is an algebraic function over K , e K is an algebraic extension ofthe differential field K and the Picard-Vessiot extension e F m of equation e L m u = 0 is an algebraic extension of thePicard-Vessiot extension F m of L m y = 0 . Thus, (DGal( F m /K )) ◦ = (DGal( e F m /K )) ◦ . Finally, if θ belongs to K ,then e K = K and e F m = F m , which implies that the Darboux transformation preserves the Galois groups. (cid:4) Remark 4
This matrix factorization has an interesting interpretation. The gauge transformation is e X = P m X . Now,it is easily seen, from Theorem 1, that RX = (cid:18) y ˜ y (cid:19) : indeed, the second row is the Darboux transformation. The matrix factorization P m = L m R , combined with thisrelation, thus shows that (cid:18) ˜ y ˜ y ′ (cid:19) = L m (cid:18) y ˜ y (cid:19) . The first row is an obvious identity. The second row provides us a notably simple first order link between solutions y of L m and solutions ˜ y of its Darboux transformation e L m : ˜ y ′ − ρ ˜ y = mry. (21) Note that ρ does not depend on the parameter: m appears only in the right hand side. s y m (SL(2 , C K )) and so (3 , C K ) Now we use the method of section 1.3 to build linear differential systems in s y m (SL(2 , C K )) and give their relationsto systems in so (3 , C K ) .Consider the linear differential system [ A ] as in equation (18) for m = 0 , where p = w ′ /w and w, q ∈ K . We recallthat its second symmetric power system is given by the linear differential system [ s y m ( A )] : Y ′ = − S Y, (22)for Y := Sym ( X ) = y yy ′ ( y ′ ) and S := s y m ( A ) = − q p − q p ! . FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
We recall that we showed, in Example 1, how to build a solution matrix and a representation of the Galois group inthis construction. We record that for further use in this classical Lemma.
Lemma 1
Let X be a fundamental matrix of system (18) and DGal( F m /K ) be its differential Galois group. Then, Y = Sym ( X ) is a fundamental matrix for system (22) and Sym (DGal( F m /K )) is a representation of its differentialGalois group. This situation can be illustrated by the following diagram: [ A ] : X ′ = − AX / / /o/o/o/o/o/o/o/o/o/o/o/o/o s y m (cid:15) (cid:15) DGal( F /K ) Sym (cid:15) (cid:15) [ s y m ( A )] : Y ′ = − s y m ( A ) Y / / /o/o/o/o/o/o/o Sym (DGal( F /K )) (23)The next result provides us with a gauge transformation to go from an so (3 , C K ) system to a second symmetric powersystem of the form (22). This enables us to express equation L y = 0 as a linear differential system in so (3 , C K ) written in the form Z ′ = Z × Ω (equation (14)). As a consequence of this result, we can extend previous reasoningfor differential Galois groups to so (3 , C K ) systems, as we will show next. Lemma 2
Let Q be the gauge matrix given by Q = − i i − . (24) The gauge transformation Z = QY transforms the so (3 , C K ) system Z ′ = Z × Ω , where Ω = ( f, g, h ) T into thesystem Y ′ = s y m ( C ) Y with C := 12 ih ( g + if ) − ( g − if ) − ih ! Proof.
We have s y m ( C ) = ih ( g + if ) 0 − ( g − if ) 0 g + if − ( g − if ) − ih As Q is constant, the effect of the gauge transformation is just conjugation by Q . We have Q. s y m ( C ) = ih g ih − h − f hg − if − g − if and Q. s y m ( C ) .Q − = h − g − h fg − f . (cid:4) This shows that, given an orthogonal system, we have an explicit gauge transformation formula to view it as a sym-metric square of a second order system.
Remark 5
Using an additional gauge transformation, one can transform the matrix C of the above Lemma 2 into acompanion form. This allows one to recover the formulas from Remark 2 and reprove Theorem 2.Conversely, if we start from L y := y ′′ + py ′ + qy = 0 , reversing the above process will produce an equivalentorthogonal system Z ′ = Z × Ω with Ω = ( i ( q − , q + 1 , i p ) T =: ( f, g, h ) T D. Blázquez-Sanz and J.J. Morales-Ruiz in [13], see also [12], have also given such a classical isomorphism between so (3 , C K ) and sl (2 , C K ) , see Proposition 6.7 of [13]: − ! (cid:18) i − i (cid:19) , − ! (cid:18) − (cid:19) , − ! (cid:18) − i − i (cid:19) . FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Corollary 1
A fundamental matrix for system Z ′ = Z × Ω is Z := Q Y = w y − ( y ′ ) y y − y ′ y ′ y − ( y ′ ) i ( y + ( y ′ ) ) i ( y y + y ′ y ′ ) i ( y + ( y ′ ) ) − y y ′ − y y ′ − y ′ y − y y ′ , (25) for Q and Y defined by (24) and (16) respectively. Finally, we can compute the differential Galois group of system (14).
Corollary 2
Using the fundamental matrix (25) , the matrices in the differential Galois group of the so (3) system Z ′ = Z × Ω are the matrices Sym ( M σ ) of Lemma 1. Proof.
Let σ be in the Galois group. It acts on Y via σ ( Y ) = Y · Sym ( M σ ) (Lemma 1). Now, σ ( Q ) = Q because w ∈ K . So, we have σ ( Z ) = σ ( Q Y ) = Q · Y · Sym ( M σ ) = Z · Sym ( M σ ) . (cid:4) Now, we can complete diagram (23) by adding the action on the so (3 , C K ) system: [ A ] : X ′ = − A X / / /o/o/o/o/o/o/o/o/o/o/o s y m (cid:15) (cid:15) DGal( L /K ) Sym (cid:15) (cid:15) [ s y m ( A )] : Y ′ = − S Y / / /o/o/o/o/o/o/o Q (cid:15) (cid:15) O O Sym (DGal( L /K )) O O [Ω] : Z ′ = − Ω Z / / /o/o/o/o/o/o/o/o/o/o O O Sym (DGal( L /K )) (26)Finally we record an easy Corollary. Corollary 3
Consider the orthogonal system [Ω] : Z ′ = Ω × Z with Z = ( α, β, γ ) T and the equivalent secondsymmetric power [ s y m ( A )] : Y ′ = s y m ( A ) Y with Y = ( z , z , z ) T . The system [Ω] admits the first integral α + β + γ and [ s y m ( A )] admits the first integral w (4 z z − z ) . Proof.
The first part is well known and is proved by Darboux in [19], see part I, chap. I, page 8 and chap. II, page 28.The second part follows from the application of the gauge transformation of Lemma 2: it transforms the first integral α + β + γ of [Ω] into w (4 z z − z ) : this is still a constant of motion and hence a first integral of [ s y m ( A )] . (cid:4) Sym (SL(2 , C K )) and so (3 , C K ) The aim of this subsection is to construct Darboux transformations for third order orthogonal systems using diagram(26). Our construction will ensure that these Darboux transformations will preserve the identity component of thedifferential Galois group of each equation.In order to construct the Darboux transformations for the second symmetric power system coming from the generalsecond order linear differential equation L m y = 0 we extend Proposition 1. We present two ways to extend it. Like inprevious subsection, this will allow us to obtain Darboux transformations for the so (3 , C K ) system.For the first one, consider the linear differential system [ A m ] : X ′ = − ( A + mN ) X given by equation (18). Itssecond symmetric power system is given by the linear differential system [ s y m ( A m )] : Y ′ = − ( S + mN ) Y, (27)where Y = Sym ( X ) = ( y , yy ′ , ( y ′ ) ) T and S + mN = s y m ( A + mN ) are given by S := − q p − q p ! and N := − r − r ! . A fundamental matrix for this system is given by matrix (16), where { y , y } is a basis of solutions of equation (7).9T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Recall that, after applying the gauge transformation (20), system (18) is transformed into the linear differential system [ e A m ] : e X ′ = − ( e A + mN ) e X , defined by equation (19), whose second symmetric power system is given by the lineardifferential system [ s y m ( e A m )] : e Y ′ = − ( e S + mN ) e Y , (28)where e Y = Sym ( e X ) = ( u , uu ′ , ( u ′ ) ) T and e S + mN = s y m ( e A + mN ) , for e S := − q p −
20 ˜ q p ! , where u and ˜ q have the explicit form expressed in Theorem 1. Thus, since e Y = Sym ( e X ) = Sym ( P m ) · Sym ( X ) =Sym ( P m ) · Y , the gauge transformation (20) also induces a transformation in the second symmetric power systemswhich sends system (27) to system (28). The following result formalizes this idea. Proposition 2 (First Darboux Transformation for
Sym (SL(2 , C K )) ) Let P ,m := Sym ( P m ) . We have P ,m = 1 r θ − θ θ ν ν − θ ρ ρν − ρ ν ρ = 1 r mr ρm r ρ mr ρ · − θ θ − θ , (29) where P m is defined by expression (20) , θ = y ′ y , ρ = − θ − p − r ′ r and ν = mr − θ ρ .Then, P ,m is a gauge transformation which sends system s y m ([ A m ]) to system s y m ([ e A m ]) . Proof.
Since matrix P ,m is the second symmetric power matrix of matrix P m , defined by (20), given Y and e Y solutions of equations (27) and (28) respectively, it satisfies e Y = Sym ( P m ) Y . Proposition 1 shows that P m = L m .R .As Sym( • ) is a group morphism, we have Sym ( P m ) = Sym ( L m )Sym ( R ) , which gives the matrix factorizationand the result. (cid:4) As systems s y m ([ A m ]) and s y m ([ e A m ]) have the same shape, we may say that P ,m is a Darboux transformation from s y m ([ A m ]) to s y m ([ e A m ]) . This new Darboux transformation is induced by the original Darboux transforma-tion for second order systems. We recover the matrix factorization from our expression of the Darboux transformationas a gauge transformation in Proposition 1.There is another way to build a Darboux transformation for second symmetric power systems. That is, transform thefirst system [ A m ] : X ′ = − ( A + mN ) X , given by equation (18), into a system in sl (2 , K ) and perform all theprevious process with the resulting system. By doing that, we ensure that the differential Galois group of equation (18)lays in SL (2 , C K ) . In order to obtain an sl (2 , K ) system, we consider the gauge change: X := ∆ X, ∆ := (cid:18) w (cid:19) . (30)Thus, X = ( y, wy ′ ) T and the resulting system is the sl (2 , K ) system [ B m ] : X ′ = − ( B + mN ) X , (31)with B + mN ∈ sl (2 , K ) , given by B := − w wq ! and N := − wr ! . By an elimination process (cyclic vector), we can show that this systems is still equivalent to the second order lineardifferential equation (7). And a fundamental matrix for this system is given by: X = (cid:18) y y wy ′ wy ′ (cid:19) , (32)where { y , y } is a basis of solutions of equation (7).The Darboux transformation P m , given by (20), for system (18) induces the Darboux transformation ∆ P m ∆ − forsystem (31). 10T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Now, we consider the second symmetric power system of system (31). This system is given by [ s y m ( B m )] : Y ′ = − ( b S + m b N ) Y , (33)where Y := Sym ( X ) = ( y , y y , y ) T and b S + m b N = s y m ( B + mN ) ∈ sl (3 , K ) , for b S := − w wq − w wq and b N := − wr − wr . From the above, we can easily compute the expression of a fundamental matrix for system (33): Y = Sym ( X ) = y y y y wy y ′ w ( y y ′ + y ′ y ) 2 wy y ′ w ( y ′ ) w y ′ y ′ w ( y ′ ) , (34)where X is given by (32).Next, we find the second expression for the Darboux transformation for systems that can be written as a secondsymmetric power. Corollary 4 (Second Darboux Transformation for
Sym (SL(2 , C K )) ) Consider the system [ s y m ( B m )] : Y ′ = − ( b S + m b N ) Y given by (33) .Let P ,m := Sym (∆ P m ∆ − ) . We have P ,m = 1 r θ − w θ w − θ νw ν − ρ θ w ρ ν w ρ νw ρ = 1 r wmr wρw m r w ρ mr w ρ · − θ w θ − θ w w (35) where matrices P m and ∆ are defined by (20) and (30) respectively, θ = y ′ y , ρ = − θ − p − r ′ r and ν = mr − ρθ .Then, P ,m is a Darboux transformation for system [ s y m ( B m )] . Proof.
As we have seen, a Darboux transformation for the sl (2 , K ) system (31) is given by e X = ∆ P m ∆ − X ,where ∆ and P m are defined by (30) and (20) respectively. The transformed system is the sl (2 , K ) system [ e B m ] := e X ′ = − ( e B + mN ) e X , (36)with e B + mN ∈ sl (2 , K ) , for e X = (cid:18) u u (cid:19) and e B := (cid:18) w − ˜ qw (cid:19) . Its corresponding second symmetric power system is [ s y m ( e B m )] : e Y ′ = − ( eb S + m b N ) e Y , (37)where e Y := Sym ( e X ) = ( u , u u , u ) T and eb S + m b N = s y m ( e B + mN ) ∈ sl (3 , K ) , for eb S := − w w ˜ q − w w ˜ q , where u and ˜ q have the explicit form expressed in Theorem 1.Now, consider the second symmetric power system (33). Since e Y = Sym ( e X ) = Sym (∆ P m ∆ − ) · Sym ( X ) = Sym (∆ P m ∆ − ) · Y , FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY we get that matrix P ,m := Sym (∆ P m ∆ − ) is a Darboux transformation which sends system (33) into system (37).We had the matrix factorization P m = L m R (Proposition 1). So, as Sym( • ) is a group morphism, Sym (∆ P m ∆ − ) = Sym (∆ L m ) · Sym (cid:0) R ∆ − (cid:1) and this gives us the desired matrix factorization above. (cid:4) We recall that if w = 1 , Darboux transformations P ,m and P ,m are the same. This is useful in the applications tonon-relativistic and one dimensional Quantum Mechanics where also r = 1 , see Section 3.1.Once we have defined the Darboux transformations for second symmetric power systems, we can state the Darbouxtransformation for so (3 , C K ) systems as follows. As we have found two Darboux transformations for second symmet-ric power systems, we will have two Darboux transformations for so (3 , C K ) systems as well: one using Proposition 2and another one using Corollary 4.Applying Lemma 2, we can transform the second symmetric power system (27) into the so (3 , C K ) system [Ω m ] : Z ′ = − (Ω + mN ) Z, (38)where Z = QY and − (Ω + mN ) = Q ′ Q − − Q ( S + mN ) Q − are given by Z = αβγ ! , Ω = − ip − ( q + 1) ip i ( q − q + 1 − i ( q −
1) 0 and N = r −
10 0 i − i ! . A fundamental matrix for this system is given by matrix (25), where { y , y } is a basis of solutions of equation (7).After performing the Darboux transformation (29), system (27) is transformed into system (28), which, again byLemma 2, can be transformed into the so (3 , C K ) system [ e Ω m ] : e Z ′ = − ( e Ω + mN ) e Z, (39)where e Z = Q e Y and − ( e Ω + mN ) = Q ′ Q − − Q ( e S + mN ) Q − for e Z = e α e β e γ and e Ω = − ip − (˜ q + 1) ip i (˜ q − q + 1 − i (˜ q −
1) 0 , for ˜ q as in Theorem 1. Thus, the Darboux transformation (29) also induces a transformation in the corresponding so (3 , C K ) systems which sends system (38) to system (39).The following result shows that this induced transformation is indeed a Darboux transformation for so (3 , C K ) systems. Proposition 3 (First Darboux Transformation for so (3 , C K ) ) Consider the systems [Ω m ] : Z ′ = − (Ω + mN ) Z and [ e Ω m ] : e Z ′ = − ( e Ω + mN ) e Z given by (38) and (39) respectively. Let T ,m be the matrix defined by T ,m = QP ,m Q − = 12 r − ν + ρ + θ − i (cid:0) ν + ρ − θ − (cid:1) ν ρ + θ ) i (cid:0) ν − ρ + θ − (cid:1) ν + ρ + θ + 1 2 i ( θ − ν ρ )2( ν θ + ρ ) − i ( ν θ − ρ ) 2( ν − θ ρ ) , (40) where matrix Q is defined by (24) , matrix P ,m is defined by expression (29) and ν = mr − θ ρ .Then, T ,m is a Darboux transformation, i.e., a virtually strong isogaloisian gauge transformation, which sends system [Ω m ] : Z ′ = − (Ω + mN ) Z to a system [ e Ω m ] : e Z ′ = − ( e Ω + mN ) e Z of the same shape. Remark 6
As in the previous results, the matrix can be factored into an m -dependent part and an independent part : T ,m = − m r − ρ mr − ρ + 1 im r iρ mr i + iρ − mr − ρ . − i − θ i θ − ( θ − − i ( θ + 1) θ where the m -dependent matrix is Q Sym (∆ L m ) and the matrix on the right is Sym (cid:0) R ∆ − (cid:1) Q − . FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Proof.
The proof follows from the application of Lemma 2 and Theorem 2. Given Y and e Y solutions of equations(27) and (28) respectively, and Z and e Z solutions of (38) and (39) respectively, by Lemma 2, we have that Z = QY and e Z = Q e Y . On the other hand, by Theorem 2, we know that e Y = P ,m Y . Thus, we get the gauge transformation e Z = ( QP ,m Q − ) Z = T ,m Z . From this, we immediately obtain the gauge transformation for the coefficient matrix: − ( e Ω + mN ) = T ′ ,m T − ,m − T ,m (Ω + mN ) T − ,m . The rest of the corollary is proved following the same argument as in Proposition 1. (cid:4)
Propositions 2 and 3 can be summarized in the following commutative diagram: [ A m ] : X ′ = − ( A + mN ) X P m / / s y m (cid:15) (cid:15) [ e A m ] : e X ′ = − ( e A + mN ) e X s y m (cid:15) (cid:15) [ s y m ( A m )] := Y ′ = − ( S + mN ) Y P ,m / / Q (cid:15) (cid:15) O O [ s y m ( e A m )] := e Y ′ = − ( e S + mN ) e Y Q (cid:15) (cid:15) O O [Ω m ] : Z ′ = − (Ω + mN ) Z T ,m / / O O [ e Ω m ] : e Z ′ = − ( e Ω + mN ) e Z O O (41)We end this section by establishing a second Darboux transformation for so (3 , C K ) systems. For that, we transformthe Sym ( SL (2 , K )) system (33) into an SO (3 , C K ) system. Consider the matrix S = i i − i and the gauge change Z = S · Y . Then, the system [ b Ω m ] : Z ′ = − ( b Ω + m b N ) Z , (42)where b Ω = i (cid:0) w − wq (cid:1) − i (cid:0) w − wq (cid:1) w + wq − (cid:0) w + wq (cid:1) and b N = wr i − i −
10 1 0 ! , (43)is an so (3 , C K ) system corresponding to the linear differential equation (18). A fundamental matrix for this system is: Z = S · Y = y + w ( y ′ ) y y + w y ′ y ′ y + w ( y ′ ) iwy y ′ iw ( y y ′ + y ′ y ) 2 iwy y ′ i ( y − w ( y ′ ) ) i ( y y − w y ′ y ′ ) i ( y − w ( y ′ ) ) , (44)where Y is given by (34) and { y , y } is a basis of solutions of equation (7).Then, the second expression for the Darboux transformation for so (3 , C K ) systems is given by: Corollary 5 (Second Darboux Transformation for so (3 , C K ) ) Consider the system [ b Ω m ] : Z ′ = − ( b Ω + m b N ) Z given by (42) . Let T ,m be the matrix T ,m = SP ,m S − = 12 r w + ρ + θ + ν w i (cid:0) w θ − ν ρw (cid:1) i (cid:16) w + ρ − θ − ν w (cid:17) i (cid:0) w ρ − ν θ w (cid:1) ν − ρ θ ) − (cid:0) w ρ + ν θ w (cid:1) i (cid:16) w − ρ + θ − ν w (cid:17) − (cid:0) w θ + ν ρw (cid:1) − w + ρ + θ − ν w , (45) where matrix P ,m is defined by expression (35) and ν = mr − θ ρ . Then, the gauge transformation T ,m is a Darbouxtransformation for system [ b Ω m ] . FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Proof.
Since e Y = P ,m · Y by Corollary 4, it follows that: T ,m = S · P ,m · S − . By construction, the image of b Ω + m b N by this gauge transformation is eb Ω + m b N , where eb Ω is obtained from b Ω by changing q by the function q obtained in (5) by the Darboux transformation: eb Ω = i (cid:0) w − w ˜ q (cid:1) − i (cid:0) w − w ˜ q (cid:1) w + w ˜ q − (cid:0) w + w ˜ q (cid:1) . The rest of the corollary is proved following the same argument as in Proposition 1. (cid:4)
As in all the above results, thetransformation can be factored into an m -dependent part and an independent part. The formula is not as compact as theprevious ones but is easily found with a computer algebra system once one is equipped with this paper’s methodologyand results.We note that Darboux transformations T ,m and T ,m are not equivalent when w = 1 because matrices Q and S arethen different.Corollaries 4 and 5 can be summarized in the following commutative diagram: [ A m ] : X ′ = − ( A + mN ) X P m / / ∆ (cid:15) (cid:15) [ e A m ] : e X ′ = − ( e A + mN ) e X ∆ (cid:15) (cid:15) [ B m ] : X ′ = − ( B + mN ) X P m ∆ − / / s y m (cid:15) (cid:15) O O [ e B m ] : e X ′ = − ( e B + mN ) e X s y m (cid:15) (cid:15) O O [ s y m ( B m )] : Y ′ = − ( b S + m b N ) Y P ,m / / S (cid:15) (cid:15) O O [ s y m ( e B m )] : e Y ′ = − ( eb S + m b N ) e Y S (cid:15) (cid:15) O O [ b Ω m ] : Z ′ = − ( b Ω + m b N ) Z T ,m / / O O [ eb Ω m ] : e Z ′ = − ( eb Ω + m b N ) e Z O O (46) The results can be extended to the general case of a linear differential system Y ′ = − AY whose differential Galoisgroup is in the special orthogonal group SO (3 , C K ) . This can be tested the following way: the system should beirreducible (it has no hyperexponential solution), the trace of A should be a logarithmic derivative (i.e the equation y ′ = − T r ( A ) y has a rational solution) and the second symmetric power system Z ′ = − s y m ( A ) Z should have arational solution, corresponding to the quadratic invariant of the special orthogonal group.In general, the matrix of such a differential systems is not in so (3 , C K ) ; then, the results of the previous sectionwould not apply directly. However, using the constructive theory of reduced forms from [10, 11], we may find agauge transformation matrix P such that, letting Y = P Z , the new unknown Z satisfies a system Z ′ = − BZ with B ∈ so (3 , K ) . Then the results of the previous sections apply: we may solve using solutions of second order equationsand construct families of equations of similar shapes via Darboux transformation.Another version of such a process also appears in Singer’s work [31] and several subsequent works on solving lineardifferential equations in terms of lower order equations.Note that this observation already appears in the book of Darboux [19], pages 28-29: he shows (in old language) howto identify third order linear differential systems with an orthogonal Galois group using a first integral ; he then showshow to transform such a system into the orthogonal form treated in this paper and calls it the type or la forme réduite of the class of third order systems admitting a quadratic first integral.14T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
In this section, to motivate the results of this paper, we present some examples coming from supersymmetric quantummechanics and differential geometry.
The Schrödinger equation for the stationary and non-relativistic case is given by Hψ = λψ, H = − ∂ x + V ( x ) , where λ is called the energy , V is called the potential and ψ is called the wave function . Supersymmetric quantummechanics in the Witten’s formalism was introduced by himself in [33, §6] as a toy model. Witten introduced the supersymmetric partner hamiltonians H ± as follows H ± = − ∂ x + V ± ( x ) , V ± = W ± W ′ , where V ± are called the supersymmetric partner potentials and W is called the superpotential which satisfies W = − ψ ′ ψ , H − ψ = λ ψ , where ψ is called the ground state and λ is an specific value of the energy λ .We can go from H − ψ = λψ to H + e ψ = λ e ψ through a Darboux transformation, where θ = ψ , m = − λ , y = ψ , u = e ψ , p = 0 , q = − V , and r = 1 .Gendenshtein in [23] introduced what today is called shape invariant potentials, that is, potentials with the shapeinvariance property : the potential V = V − = V − ( x ; a ) has the shape invariance property if and only if its super-symmetric partner potential V + = V + ( x ; a ) can be written as V + ( x ; a ) = V − ( x ; a ) + R ( a ) , where a is a set ofparameters and a = f ( a ) . In other words, the supersymmetric partner potentials differs only in parameters. Apply-ing systematically this procedure, one can obtain the spectrum as the values of energy λ such that λ = n X k =1 R ( a k ) . Moreover, H − = A † A and H + = AA † , where A † = − ∂ x + W and A = ∂ x + W are called the ladder (raising andlowering) operators , see [20]. We can rewrite the starting potential V as V − − λ to apply Darboux transformations.Thus, we can obtain the rest of wave functions applying it as follows: ψ = A ( x ; a ) † ψ , and in general as ψ k = A ( x ; a k − ) † ψ k − , where a = a , a = f ( a ) . First examples of rational shape invariant potentials correspond toharmonic oscillator and Coulomb potentials, for one dimensional and three dimensional cases.In the following, we combine this theoretical background regarding the Schrödinger equation and SupersymmetricQuantum Mechanics with our results from the previous section. We apply our previous results to a matrix form of the Schrödinger equation. We start by introducing the following × matrix Schrödinger operators, with supersymmetric partner potentials, related to the systems (18) and (19) asfollows. H ± = − ∂ x + V ± , V ± = (cid:18) V ± (cid:19) , V − = − A , V + = − e A , where H − Ψ = E λ Ψ , H + e Ψ = E λ e Ψ , Ψ = (cid:18) ψψ ′ (cid:19) , e Ψ = (cid:18) e ψ e ψ ′ (cid:19) , E λ = λ ( − N ) , − N = (cid:18) (cid:19) . According to Proposition 1, the relevant Darboux transformation in this × matrix formalism is given by e Ψ = P λ Ψ , P λ = (cid:18) W W − λ W (cid:19) = (cid:18) − λ W (cid:19) · (cid:18) W (cid:19) . and the supersymmetric partner potentials V ± will depend on the supersymmetric partner potentials V ± according tothe original Witten’s formalism, i.e., V + = V − + 2 W ′ , which lead us to V + = V − + 2 W ′ ( − N ) .15T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
Now we present the shape invariance property for this × matrix formalism of Schrödinger equation as follows.Consider the parametric supersymmetric partner potentials and matrix R ( a ) as follows V ± ( x ; a ) = (cid:18) V ± ( x ; a ) 0 (cid:19) , R ( a ) = R ( a )( − N ) , where, as in the classical case, a is a set of parameters and a = f ( a ) . The potential V = V − = V − ( x ; a ) has theshape invariance property if and only if its supersymmetric partner potential satisfies V + = V + ( x ; a ) = V − ( x ; a ) + R ( a ) . We can see again that supersymmetric partner potentials differ only in parameters. Applying systematically thisprocedure, we can obtain the spectrum as the values of energy E λ ( a ) , where E λ (1) = E λ , such that E λ ( a ) = n X k =1 R ( a k ) . Also we present the ladder operators for this × matrix formalism of Schrödinger equation: A = (cid:18) A W A (cid:19) , A † = (cid:18) A † W ′ − W A † (cid:19) . We illustrate this formalism with the 1D-harmonic oscillator, which is a classical rational shape invariant potential.The superpotential for harmonic oscillator is W = x , thus the supersymmetric partner potentials are given by V − = (cid:18) x − (cid:19) , V + = (cid:18) x + 1 0 (cid:19) = V − + (cid:18) (cid:19) . Therefore, introducing a multiplicative parameter a in V − , such that V − ( x ; a ) = V − ( x ) for a = 1 , we obtain f ( a ) = 2 a, R ( a ) = (cid:18) a (cid:19) , E λ ( a ) = n X k =1 R ( a k ) = (cid:18) na (cid:19) . Thus, for a = 1 the spectrum of H − is Spec( H − ) = { E λ : λ ∈ Z + } . For instance, we have e Ψ = P λ Ψ = (cid:18) x x − λ x (cid:19) H λ ( x ) H ′ λ ( x ) − xH λ ( x ) ! exp (cid:18) − x (cid:19) , where H λ denotes the Hermite polynomial of degree λ . The ladder operators are given respectively by A = (cid:18) A xA (cid:19) , A † = (cid:18) A † − xA † (cid:19) , where A = ∂ x + x and A † = − ∂ x + x . Using these ladder operators we can obtain Ψ n = A † Ψ n − ( x ) , where λ = 2 n and Ψ = exp( − x ) − x exp( − x ) ! . The following results correspond to the second symmetric power of Schrödinger equation in the previous matrixformalism. Thus, we obtain the following × matrix Schrödinger operators, with supersymmetric partner potentialsaccording to equations (27), (28), (33) and (37) as follows. H ± = − ∂ x + V ± , V − = − S = − b S, V + = − e S = − eb S, V ± = V ± V ± ! , FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY where H − Ψ = E λ Ψ , H + e Ψ = E λ e Ψ , Ψ = ψ ψψ ′ ( ψ ′ ) , e Ψ = e ψ e ψ e ψ ′ ( e ψ ′ ) , E λ = λ ( − N ) , and − N = − N = ! . Using (29), our generalized Darboux transformation in this × matrix formalism is given by e Ψ = P λ Ψ , P λ = P ,λ = P ,λ = W W W ( W − λ ) 2 W − λ W ( W − λ ) W ( W − λ ) W with the factorization into a λ -dependent part and a part with only W : P λ = − λ Wλ − λW W · W W W . As in the previous case, the supersymmetric partner potentials V ± will depend on the supersymmetric partner poten-tials V ± according to the original Witten’s formalism, i.e., V + = V − + 2 W ′ , which lead us to V + = V − + 2 W ′ ( − N ) , Now we present the shape invariance property for this × matrix formalism of Schrödinger equation as follows.Consider the parametric supersymmetric partner potentials and matrix R ( a ) as follows V ± ( x ; a ) = V ± ( x ; a ) 0 20 V ± ( x ; a ) 0 ! , R ( a ) = R ( a )( − N ) , where, as in the previous case, a is a set of parameters and a = f ( a ) . The potential V = V − = V − ( x ; a ) has theshape invariance property if and only if its supersymmetric partner potential can be written as V + = V + ( x ; a ) = V − ( x ; a ) + R ( a ) . We can see again that supersymmetric partner potentials differ only in parameters. Applying systematically thisprocedure, we can obtain the spectrum as the values of energy E λ ( a ) given by E λ (1) = E λ and E λ ( a ) = n X k =1 R ( a k ) . The ladder operators for this × matrix formalism of Schrödinger equation are: A = W A W A WW A W , A † = W A † − W A † − WW A † W . We illustrate this formalism with the 1D-harmonic oscillator, which is a classical rational shape invariant potential.The superpotential for harmonic oscillator is W = x , thus the supersymmetric partner potentials are given by V − = x − x − , V + = x + 2 0 20 x + 1 0 = V − + ! . Therefore, introducing a multiplicative parameter a in V − , such that V − ( x ; a ) = V − ( x ) for a = 1 , we obtain f ( a ) = 2 a, R ( a ) = a a ! , E λ ( a ) = n X k =1 R ( a k ) = na na ! . Thus, for a = 1 the spectrum of H − is Spec( H − ) = { E λ : λ ∈ Z + } . FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY
For instance we have e Ψ = P λ Ψ = x x x − λx x − λ xx − λx + λ x − λx x H λ ( x ) (cid:0) H (cid:1) ′ λ ( x ) − xH λ ( x ) (cid:16) H ′ λ ( x ) − xH λ ( x ) (cid:17) exp (cid:0) − x (cid:1) , where H λ denotes the Hermite polynomial of degree λ . The ladder operators are given respectively by A = xA x A xx A x , A † = xA † − x A † − xx A † x . so (3 , C K ) Systems
In this section, we revisit from the point of view developed in this article two well-known problems which ariseexpressed as so (3 , C K ) systems. Given a non degenerate curve in the space, denote by T the tangent unit vector to the curve, by N the normal unitvector and by B = T × N the binormal unit vector. Then, the Frenet-Serret formulas can be formulated as thefollowing so (3 , C K ) system: TNB ! ′ = − − κ κ − τ τ ! · TNB ! = − Ω · TNB ! , (47)where ′ denotes the derivative with respect to arclength, κ is the curvature of the curve and τ is its torsion, see [32,Chap. 1] for more details.In order to apply our previous formalism to this so (3 , C K ) system, we have two possibilities: we can use system (38)or system (42).In the first case, the identification of matrix Ω with matrix Ω given by Lemma 2 yields the degenerate situation: κ = − ip , τ = i ( q − and q + 1 , hence, q = − and τ = − i . The second order linear differential equationassociated to this system is: L y = y ′′ + iκy ′ − y = 0 . (48)Since p = w ′ w = iκ , we find that w = e ( i R κdx ) . As immediate application of so (3 , C K ) systems to Frenet-Serretformulas, we obtain directly a fundamental matrix of solutions through Corollary 1: Z = e ( i R κdx ) · y − ( y ′ ) y y − y ′ y ′ y − ( y ′ ) i ( y + ( y ′ ) ) i ( y y + y ′ y ′ ) i ( y + ( y ′ ) ) − y y ′ − y y ′ − y ′ y − y y ′ , where { y , y } is a basis of solutions of equation (48).Notice that this framework is only valid for curves with torsion τ = − i . However, we can avoid this restrictionby using the second approach, given by equation (42). In this case, the gauge change given by S transform thematrix Ω into a matrix with the same structure as Ω , namely b Ω . Identifying the entries of both matrices we obtain: κ = − i (1 /w − wq ) and τ = − (1 /w + wq ) . Thus, w = iκ − τ and q = κ + τ and the second order linear differentialequation associated to the system in this case is: L y = y ′′ − ( iκ − τ ) ′ iκ − τ y ′ + κ + τ y = 0 . (49)Under this assumptions, a fundamental matrix for this system, given by (44), becomes: Z = y + 4( y ′ ) ( iκ − τ ) y y + 4 y ′ y ′ ( iκ − τ ) y + 4( y ′ ) ( iκ − τ ) iy y ′ iκ − τ i ( y y ′ + y ′ y ) iκ − τ iy y ′ iκ − τi (cid:18) y − y ′ ) ( iκ − τ ) (cid:19) i (cid:18) y y − y ′ y ′ ( iκ − τ ) (cid:19) i (cid:18) y − y ′ ) ( iκ − τ ) (cid:19) , FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY for { y , y } a basis of solutions of equation (49).Finally, we consider the following perturbation for the Frenet-Serret system, according to the second approach for so (3 , C K ) systems (see equation (42)): TNB ! ′ = − − κ κ − τ τ ! + 2 miκ − τ i − i −
10 1 0 !! · TNB ! . Following the philosophy of Darboux we can construct an infinite chain of such perturbed Frenet-Serret systemsapplying Corollary 5 (we restrict ourselves to the case r = 1 since it is the usual situation in the applications, see, forinstance, the classical book of Matveev and Salle [28]). The Darboux transformation is given by: T ,m = 12 η + ρ + θ + ν η i (cid:18) θ η − νρη (cid:19) i (cid:18) η + ρ − θ − ν η (cid:19) i (cid:18) ρη − νθ η (cid:19) ν − ρθ ) − ρη − νθ ηi (cid:18) η − ρ + θ − ν η (cid:19) − θ η − νρη ρ + θ − η − ν η , for θ = y ′ y , ρ = − θ + η ′ η , ν = m − θ ρ and η = iκ − τ , where y is a solution of equation (49). As before, thismatrix could be factored into an m -dependent and an independant part. A rigid solid consists of a set of points in the space preserving the distance among them under the action of someapplied forces. The transformations allowed for a rigid solid are translations and rotations.The Poisson equation describes the motion of the rigid body in space: γ ′ = γ × ω, where γ = ( γ , γ , γ ) T is a unit vector fixed in space, ω = ( ω , ω , ω ) T the angular velocity vector and ′ = ∂ t . See[3, 21, 22] for more details.We follow the constraints and notation of [21]. Hence, we take ω = 0 and restrict ourselves to rigid transformationsin the plane. In this case, the Poisson equation can be rewritten as the so (3 , C K ) system: γ γ γ ! ′ = − ω − ω − ω ω ! · γ γ γ ! = − A · γ γ γ ! . (50)Fedorov et al. studied in [21] this system for a general case of matrix A . Acosta-Humánez et al. considered in [3] aparticular case with ω = e x − e − x e x + e − x and ω = √ e x + e − x . In this work, we are going to restrict the rigid transformationsallowed to the coupled case iω + ω = 2 . For that, we consider the so (3 , C K ) system given in Lemma 2 and applythe formalism developed for the so (3 , C K ) system (38). The identification of matrix A with matrix Ω leads to: p = 0 , ω = i ( q − , ω = q + 1 , hence, q = 1 − iω = ω − . This yields the announced coupled situation iω + ω = 2 .The second order linear differential equation associated to this problem is: L y = y ′′ + (1 − iω ) y = y ′′ + ( ω − y = 0 . (51)Since p = w ′ w = 0 , we find that w ∈ C K . Without lost of generality, we can assume that w = 1 . Applying Corollary 1we obtain a fundamental matrix of solutions for the rigid solid problem: Z = y − ( y ′ ) y y − y ′ y ′ y − ( y ′ ) i ( y + ( y ′ ) ) i ( y y + y ′ y ′ ) i ( y + ( y ′ ) ) − y y ′ − y y ′ − y ′ y − y y ′ , where { y , y } is a basis of solutions of equation (51).Next, we consider the second approach for so (3 , C K ) systems, given by equation (42). The identification of matrices A and b Ω leads to the degenerate situation: ω = − (1 /w + wq ) , ω = 0 and /w − wq = 0 . Thus, w = − /ω and19T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY q = ω / . This case also produces a coupled situation, this time between w and q : q = 1 /w . The fact that ω = 0 means that we are only considering rigid transformations in the line, which is a simplification of the problem. Thesecond order linear differential equation associated to the system in this case is: L y = y ′′ − ω ′ ω y ′ + ω y = 0 . (52)Under this assumptions, a fundamental matrix for this system, given by (44), becomes: Z = y + 4( y ′ ) ω y y + 4 y ′ y ′ ω y + 4( y ′ ) ω − iy y ′ ω − i ( y y ′ + y ′ y ) ω − iy y ′ ω i (cid:18) y − y ′ ) ω (cid:19) i (cid:18) y y − y ′ y ′ ω (cid:19) i (cid:18) y − y ′ ) ω (cid:19) , for { y , y } a basis of solutions of equation (52).Finally, we consider the following perturbation for the rigid solid system, according to the first approach for so (3 , C K ) systems (see equation (38)): γ γ γ ! ′ = − ω − ω − ω ω ! + m −
10 0 i − i !! · γ γ γ ! . Following the philosophy of Darboux in the same vein as we did for the Frenet-Serret system, we can construct aninfinite chain of such perturbed Poisson equations for the rigid body problem by applying Proposition 3 (again, werestrict ourselves to the case r = 1 ). The Darboux transformation is given by: T ,m = 12 − ν + 2 θ − i (cid:0) ν − (cid:1) θ (1 − ν ) i (cid:0) ν − (cid:1) ν + 2 θ + 1 2 iθ (1 + ν )2 θ ( ν − − iθ ( ν + 1) 2 (cid:0) ν + θ (cid:1) , for θ = y ′ y and ν = m + θ , where y is a solution of equation (51). This transformation factors as T ,m = − m θ m − θ + 1 im − iθ m i + iθ − m θ · − i − θ iθ − ( θ − − i ( θ + 1) θ . Final Remarks
In this paper, we have shown how, using tensor construction on SL (2 , C K ) , we can define Darboux transformationsfor higher order linear differential systems such as Sym ( SL (2 , C K )) -systems or SO (3 , C K ) systems, as summarizedin the diagrams (41) and (46).Our tool to achieve this is the observation that Darboux transformations can be viewed as gauge transformations andhence may be extended using the tools of Tannakian constructions.We present an approach to solve Sym (SL(2 , C K )) systems together with two Darboux transformations for these kindof systems. From that, in a natural way, we fall into so (3 , C K ) systems. For that, we make explicit a direct methodto transform an so (3 , C K ) system into a construction on a second order linear differential equation by means of theisomorphism between the Lie algebras so (3 , C K ) and sl (2 , C K ) . This allows us to explicitly compute fundamentalmatrices and “generalized” Darboux transformations for Sym (SL(2 , C K )) systems as well as for so (3 , C K ) systemsin terms of the solutions of the initial second order linear differential equation. The Darboux transformations obtainedin this form come up in a natural way.These constructions are applied to toy formalisms for Supersymmetric Quantum Mechanics in the non-relativistic case.We construct systems-like Schrödinger equations following these approaches. Some well-known so (3 , C K ) systemssuch as Frenet-Serret formulas and the rigid solid problem are also included in these constructions.Finally, we notice that both approaches for so (3 , C K ) systems developed in this article are not strictly equivalent,since they produce different results. This can be seen in Section 3.2, where two applications to so (3 , C K ) systems are20T FOR ORTHOGONAL DIFFERENTIAL SYSTEMS & DIFFERENTIAL G ALOIS T HEORY showed. In the first one, the Frenet-Serret formulas, the first approach leads to a degenerate situation for curves withtorsion τ = − i , whilst the second approach allows us to deal with any curve. However, in the rigid solid application,it is the other way around: the first approach produces a situation that, despite being a coupled case, is more generalthan the one given by the second approach, which restricts to rigid transformations in the line.The philosophy developed in this work can allow one to construct Darboux transformations to differential systems oforder higher than two, namely those which can be obtained from a tensor construction on SL(2 , C K ) ; for any suchsystem, the methodology exposed here allows to solve via solutions of second order equations and extends Darbouxtransformations to these families. Acknowledgements
The first author thanks the hospitality of XLim and suggestions of J.J. Morales-Ruiz during the initial stage of thiswork. The third author thanks Autonomous University of Madrid for the financial support for a research stay at XLim,where she started to work in this article. She also thanks the hospitality of XLim and the support of J.J. Morales-Ruizto participate in this work.This work was partially supported by the grant TIN2016-77206-R from the Spanish Government, co-financed by theEuropean Regional Development Fund. The third author received a postdoctoral grant (PEJD-2018-POST/TIC-9490)from Universidad Nacional de Educación a Distancia (UNED), co-financed by the Regional Government of Madridand the Youth Employment Initiative (YEI) of the European Social Fund.
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