Arturo Ramos

INTEGRABILITY OF THE RICCATI EQUATION FROM A GROUP-THEORETICAL VIEWPOINT

In this paper we develop some group-theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation, and we discuss some of its integrability conditions from a group-theoretical perspective. The nonlinear superposition principle also arises in a simple way.

Second order SUSY transformations with 'complex energies'

Abstract Second order supersymmetry transformations which involve a pair of complex conjugate factorization energies and lead to real nonsingular potentials are analyzed. The generation of complex potentials with real spectra is also studied. The theory is applied to the free particle, one-soliton well and one-dimensional harmonic oscillator.

Reduction of Time-Dependent Systems Admitting a Superposition Principle

The problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective. We show how it is possible to reduce the problem of finding the general solution of such a differential equation system defined by a Lie group G to a pair of simpler problems, one in a subgroup H and the other on a homogeneous space. The theory is illustrated with several examples and applications.

Group Theoretical Approach to the Intertwined Hamiltonians

Abstract We show that the finite difference Backlund formula for the Schrodinger Hamiltonians is a particular element of the transformation group on the set of Riccati equations considered by two of us in a previous paper. Then, we give a group theoretical explanation to the problem of Hamiltonians related by a first order differential operator. A generalization of the finite difference algorithm relating eigenfunctions of three different Hamiltonians is found, and some illustrative examples of the theory are analyzed, finding new potentials for which one eigenfunction and its corresponding eigenvalue is exactly known.

RICCATI EQUATION, FACTORIZATION METHOD AND SHAPE INVARIANCE

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 generalized in a simple way.

Poisson structures for reduced non-holonomic systems

Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank 2 and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing two integrals of motion. We generalize the form of the Poisson structures and extend their domain of definition. We apply the theory to the rolling disc, the Rouths sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder.

A New Geometric Approach to Lie Systems and Physical Applications

The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei–Norman method and the reduction method, which are very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.

Shape invariant potentials depending on n parameters transformed by translation

Shape-invariant potentials in the sense of Gendenshtein (1983 JETP Lett. 38 356) which depend on more than two parameters are not known to date. Cooper et al (1987 Phys. Rev. D 36 2458) posed the problem of finding a class of shape-invariant potentials which depend on n parameters transformed by translation, but it was not solved. We analyse the problem using some properties of the Riccati equation and find the general solution.

On the new translational shape-invariant potentials

Recently, several authors have found new translational shape-invariant potentials not present in classic classifications like those of Infeld and Hull. For example, Quesne on the one hand and Bougie, Gangopadhyaya and Mallow on the other have provided examples of them, consisting on deformations of the classical ones. We analyze the basic properties of the new examples and observe a compatibility equation which has to be satisfied by them. We study particular cases of such an equation and give more examples of new translational shape-invariant potentials.

THE PARTNERSHIP OF POTENTIALS IN QUANTUM MECHANICS AND SHAPE INVARIANCE

The concept of partnership of potentials is studied in detail and in particular the non-uniqueness due to the ambiguity in the election of the factorization energy and in the choice of the solution of certain Riccati equation. We generate new factorizations from old ones using invariance under parameter transformations. The theory is illustrated with some examples.