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A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised in the literature to deal with $t$-dependent Schr\"odinger equations determined by a particular class of $t$-dependent Hamiltonian operators, the quantum Lie systems, and other differential equations through the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and quantum Lie systems to cope with $t$-dependent Schr\"odinger equations associated with the here called quantum quasi-Lie systems. To illustrate our methods, we propose and study a quantum analogue of the classical nonlinear oscillator searched by Perelomov and we analyse a quantum one-dimensional fluid in a trapping potential along with quantum $t$-dependent Smorodinsky--Winternitz oscillators.

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{\alpha=1}^rg_\alpha X_\alpha$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification $\mathcal{F}$ of $M$ while $g_1,\ldots,g_r:\mathbb{R}\times M\rightarrow \mathbb{R}$ are functions depending on $t$ that are constant along integral curves of $X_1,\ldots,X_r$ for each fixed $t$. We analyse the particular solutions of stratified Lie systems and how their properties can be obtained as generalisations of those of Lie systems. We illustrate our results by studying Lax pairs and a class of $t$-dependent Hamiltonian systems. We study stratified Lie systems with compatible geometric structures. In particular, a class of stratified Lie systems on Lie algebras are studied via Poisson structures induced by $r$-matrices., Comment: 34 pages. Much improved presentation: many more details in proofs, terminological changes in some definitions, and new applications added

A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski-Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skew-symmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products., Comment: 31 pages, 2 figures. Minor corrections

In this paper we consider a manifold with a dynamical vector field and inquire about the possible tangent bundle structures which would turn the starting vector field into a second order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second order and apply the procedure to vector fields conformally related with the harmonic oscillator (f-oscillators) . We select one which covers the vector field describing the Kepler problem., Comment: 17 pages, 2 figures

We prove that $t$-dependent Schr\"odinger equations on finite-dimensional Hilbert spaces determined by $t$-dependent Hermitian Hamiltonian operators can be described through Lie systems admitting a Vessiot--Guldberg Lie algebra of K\"ahler vector fields. This result is extended to other related Schr\"odinger equations, e.g. projective ones, and their properties are studied through Poisson, presymplectic and K\"ahler structures. This leads to derive nonlinear superposition rules for them depending in a lower (or equal) number of solutions than standard linear ones. Special attention is paid to applications in $n$-qubit systems., Comment: 36 pages

We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures., Comment: 14 pages. Published version

Constants of motion, Lagrangians and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained by Krasil'shchi, Vinogradov and coworkers. Finally the existence of non-local symmetries for such two nonlinear oscillators is proved., Comment: 20 pages

This work analyses types of group actions on families of $t$-dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to a deep geometrical understanding of quasi-Lie schemes and quasi-Lie systems giving rise to several new results. Our achievements are illustrated by studying Abel and Riccati equations. We retrieve the Liouville invariant and study other new quasi-Lie invariants of Abel equations. Several Abel equations with a superposition rule are described and we characterise Abel equations via quasi-Lie schemes., Comment: 23 pages

In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way. It turns out that the conditions can be expressed in a purely algebraic way. In a second step we generalize the construction to the case in which we substitute the Lie algebra of vector fields by a module (generalized distribution). We obtain much larger class of integrable systems replacing standard concepts of solvable (or nilpotent) Lie algebra with distributional solvability (nilpotency)., Comment: 18 pages

In a recent paper, Post and Winternitz studied the properties of two-dimensional Euclidean potentials that are linear in one of the two Cartesian variables. In particular, they proved the existence of a potential endowed with an integral of third-order and an integral of fourth-order. In this paper we show that these results can be obtained in a more simple and direct way by noting that this potential is directly related with the Holt potential. It is proved that the existence of a potential with higher order superintegrability is a direct consequence of the integrability of the family of Holt type potentials., Comment: to appear in J. of Phys. A (2013)

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the Dirac's description of constrained systems, we introduce and analyse a particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems, which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this `Dirac setting' and new applications of Dirac geometry in differential equations are presented. As an application, we analyze traveling wave solutions of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics., Comment: 41 pages

A Lie system is a nonautonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie-Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie-Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods to derive in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer-Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky-Winternitz systems with time-dependent frequency., Comment: 30 pages

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest., Comment: 20 pages

We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed., Comment: 50 pages, 24 figures

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations and analyse their properties. Several results concerning the existence of various types of superposition rules for higher-order systems are proved and illustrated with examples extracted from the physics and mathematics literature. In particular, two new superposition rules for second- and third-order Kummer--Schwarz equations are derived., Comment: (v2) 33 pages, some typos corrected, added some references and minor commentaries

This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre transforms. This result allows us to construct new superposition rules for studying second-order Riccati equations and to reduce their integration to solving (first-order) Riccati equations., Comment: 8 pages

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications., Comment: 161 pages, 2 figures

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently (Maruskin and Bloch, 2007). We also provide several examples to illustrate our construction., Comment: Revtex 4.1, 20 pages. To appear in Int. J. Geom. Meth. Modern Physics

The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply the theories of Lie and quasi-Lie systems to analyse second-order differential equations. In order to illustrate our results, several second-order differential equations appearing in the physics and mathematical literature are analysed and some superposition rules for these equations are derived by means of our methods., Comment: 6 pages. Corrected typos

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose several generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest., Comment: 24 pages. Presentation improved and several relevant remarks added

We describe a programme to quantize a particle in the field of a (three dimensional) magnetic monopole using a Weyl system. We propose using the mapping of position and momenta as operators on a quaternionic Hilbert module following the work of Emch and Jadczyk., Comment: Contribution to the volume: Mathematical Physics and Field Theory, Julio Abad, In Memoriam}, M. Asorey, J.V. Garcia Esteve, M.F. Ranada and J. Sesma Editors, Prensas Universitaria de Zaragoza, (2009)

We present a deformed star-product for a particle in the presence of a magnetic monopole. The product is obtained within a self-dual quantization-dequantization scheme, with the correspondence between classical observables and operators defined with the help of a quaternionic Hilbert space, following work by Emch and Jadczyk. The resulting product is well defined for a large class of complex functions and reproduces (at first order in hbar) the Poisson structure of the particle in the monopole field. The product is associative only for quantized monopole charges, thus incorporating Dirac's quantization requirement.

Time-dependent frequency harmonic oscillators (TDFHO's) are studied through the theory of Lie systems. We show that they are related to a certain kind of equations in the Lie group SL(2,R). Some integrability conditions appear as conditions to be able to transform such equations into simpler ones in a very specific way. As a particular application of our results we find t-dependent constants of the motion for certain one-dimensional TDFHO's. Our approach provides an unifying framework which allows us to apply our developments to all Lie systems associated with equations in SL(2,R) and to generalise our methods to study any Lie system.

The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the symplectic structure defined from the Lagrangian function and the constraints is studied. The concept of complete solutions and their relationship with constants of motion, are also studied in detail. Local expressions using quasivelocities are provided. As an example, the nonholonomic free particle is considered., Comment: 22 pp

The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper we use both developments to obtain a geometric theory of integrability in Quantum Mechanics and we use it to provide a series of non-trivial integrable quantum mechanical models and to recover some known results from our unifying point of view.

The recently developed theory of quasi-Lie schemes is studied and applied to investigate several equations of Emden type and a scheme to deal with them and some of their generalisations is given. As a first result we obtain t-dependent constants of the motion for particular instances of Emden equations by means of some of their particular solutions. Previously known results are recovered from this new perspective. Finally some t-dependent constants of the motion for equations of Emden type satisfying certain conditions are recovered.

We present a new setting of the geometric Hamilton-Jacobi theory by using the so-called time-evolution operator K. This new approach unifies both the Lagrangian and the Hamiltonian formulation of the problem developed in a previous paper [7], and can be applied to the case of singular Lagrangian dynamical systems., Comment: 10 pp

We develop a theory of gauge and dynamical equivalence for Lagrangian systems on Lie algebroids, also studying its relationship with Noether and non-Noether conserved quantities.

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne--Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants., Comment: To be published in the Int. J. Geom. Methods Mod. Phys

The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the literature will be analysed from this new perspective, and some applications in physics will be given., Comment: 10 pages

A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends of a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schr\"odinger solvable and then the wave functions $\Psi_n$ and the energies $E_n$ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials ${\cal P}_n(x)$ related with the Hermite polynomials and such that: (i) Every ${\cal P}_n$ is a linear combination of three Hermite polynomials, and (ii) They are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure., Comment: 11 pages, 11 figures

We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations., Comment: AMS-LaTeX, 35 pages. Expanded version of the Invited review talk delivered by G. Marmo at XXIst International Workshop On Differential Geometric Methods In Theoretical Mechanics, Madrid (Spain), 2006. To appear in Int. J. Geom. Methods in Modern Physics

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel's characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(h^2) because semiclassical corrections of energy levels of order O(h) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems., Comment: 29 pages, 8 eps figures

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level., Comment: Survey paper based on the lectures delivered at the XV International Workshop on Geometry and Physics Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11-16, 2006. To appear in Publ. de la RSME

We show that it is possible to represent various descriptions of Quantum Mechanics in geometrical terms. In particular we start with the space of observables and use the momentum map associated with the unitary group to provide an unified geometrical description for the different pictures of Quantum Mechanics. This construction provides an alternative to the usual GNS construction for pure states., Comment: 16 pages. To appear in Theor. Math. Phys. Some typos corrected. Definition 2 in page 5 rewritten

The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota-Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus we seek an algebraic theory of integration, with the Rota-Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson-Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota-Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew)differential equations., Comment: International Journal of Geometric Methods in Modern Physics, in press

The dynamical equations of an electromagnetic field coupled with a conducting material are studied. The properties of the interaction are described by a classical field theory with tensorial material laws in space-time geometry. We show that the main features of superconducting response emerge in a natural way within the covariance, gauge invariance and variational formulation requirements. In particular, the Ginzburg-Landau theory follows straightforward from the London equations when fundamental symmetry properties are considered. Unconventional properties, such as the interaction of superconductors with electrostatic fields are naturally introduced in the geometric theory, at a phenomenological level. The BCS background is also suggested by macroscopic fingerprints of the internal symmetries. It is also shown that dissipative conducting behavior may be approximately treated in a variational framework after breaking covariance for adiabatic processes. Thus, nonconservative laws of interaction are formulated by a purely spatial variational principle, in a quasi-stationary time discretized evolution. This theory justifies a class of nonfunctional phenomenological principles, introduced for dealing with exotic conduction properties of matter [Phys. Rev. Lett. 87, 127004 (2001)]., Comment: To be published in Journal of Physics A: Mathematical and General

The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron-monopole system., Comment: 40 pages

We reconsider the basic properties of ray-transfer matrices for first-order optical systems from a geometrical viewpoint. In the paraxial regime of scalar wave optics, there is a wide family of beams for which the action of a ray-transfer matrix can be fully represented as a bilinear transformation on the upper complex half-plane, which is the hyperbolic plane. Alternatively, this action can be also viewed in the unit disc. In both cases, we use a simple trace criterion that arranges all first-order systems in three classes with a clear geometrical meaning: they represent rotations, translations, or parallel displacements. We analyze in detail the relevant example of an optical resonator., Comment: 11 pages. 4 figures. Accepted for publication in Journal of Optics A: Pure and Applied Optics

In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with position-dependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the $\lambda\to 0$ limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.

We present a self-contained discussion of the use of the transfer-matrix formalism to study one-dimensional scattering. We elaborate on the geometrical interpretation of this transfer matrix as a conformal mapping on the unit disk. By generalizing to the unit disk the idea of turns, introduced by Hamilton to represent rotations on the sphere, we develop a method to represent transfer matrices by hyperbolic turns, which can be composed by a simple parallelogramlike rule., Comment: 8 pages, 5 figures. This paper is dedicated to Alberto Galindo the occasion of his seventieth birthday. Submitted for publication

We resort to the concept of turns to provide a geometrical representation of the action of any lossless multilayer, which can be considered as the analogous in the unit disk to the sliding vectors in Euclidean geometry. This construction clearly shows the peculiar effects arising in the composition of multilayers. A simple optical experiment revealing the appearance of the Wigner angle is analyzed in this framework., Comment: 6 pages, submitted for publication

We reelaborate on the basic properties of lossless multilayers by using bilinear transformations. We study some interesting properties of the multilayer transfer function in the unit disk, showing that hyperbolic geometry turns out to be an essential tool for understanding multilayer action. We use a simple trace criterion to classify multilayers into three classes that represent rotations, translations, or parallel displacements. Moreover, we show that these three actions can be decomposed as a product of two reflections in hyperbolic lines. Therefore, we conclude that hyperbolic reflections can be considered as the basic pieces for a deeper understanding of multilayer optics., Comment: 7 pages, 7 figures, accepted for publication in J. Opt. Soc. Am. A

We elaborate on the consequences of the factorization of the transfer matrix of any lossless multilayer in terms of three basic matrices of simple interpretation. By considering the bilinear transformation that this transfer matrix induces in the complex plane, we introduce the concept of multilayer transfer function and study its properties in the unit disk. In this geometrical setting, our factorization translates into three actions that can be viewed as the basic pieces for understanding the multilayer behavior. Additionally, we introduce a simple trace criterion that allows us to classify multilayers in three types with properties closely related to one (and only one) of these three basic matrices. We apply this approach to analyze some practical examples that are representative of these types of matrices., Comment: 8 pages, 5 figures. To be published in J. Opt. Soc. Am. A

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that classifies multilayers in three classes with properties closely related with one (and only one) of these three basic matrices., Comment: To be published in Optics Letters

We reelaborate on the basic properties of lossless multilayers. We show that the transfer matrices for these multilayers have essentially the same algebraic properties as the Lorentz group SO(2,1) in a (2+1)-dimensional spacetime, as well as the group SL(2,R) underlying the structure of the ABCD law in geometrical optics. By resorting to the Iwasawa decomposition, we represent the action of any multilayer as the product of three matrices of simple interpretation. This group-theoretical structure allows us to introduce bilinear transformations in the complex plane. The concept of multilayer transfer function naturally emerges and its corresponding properties in the unit disc are studied. We show that the Iwasawa decomposition reflects at this geometrical level in three simple actions that can be considered the basic pieces for a deeper undestanding of the multilayer behavior. We use the method to analyze in detail a simple practical example., Comment: 8 pages, 5 figures

Generalizations in many directions of the contraction procedure for Lie algebras introduced by E.J.Saletan are proposed. Products of arbitrary nature, not necessarily Lie brackets, are considered on sections of finite-dimensional vector bundles. Saletan contractions of such infinite-dimensional algebras are obtained via a generalization of the Nijenhuis tensor approach. In particular, this procedure is applied to Lie algebras, Lie algebroids, and Poisson structures. There are also results on contractions of n-ary products and coproducts., Comment: 25 pages, LateX, corrected typos

Previous $\lambda$-deformed {\it non-Hermitian} Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed $\lambda$-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view., Comment: 10 pages

The basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are greatly generalized in a simple way., Comment: Latex file, 35 pages

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei-Norman method is applied to obtain the associated differential equation in the group $SL(2,R)$. The superposition principle for first order differential equation systems and Lie-Scheffers theorem are also analysed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time-independent Schroedinger equation