Gamow Vectors in Exactly Solvable Models

Through two examples: the Friedrichs model and a particular case of central potential scattering, we illustrate the way of constructing Gamow vectors.     

One Dimensional Models with a Singular Potential of the Type −<Emphasis Type="Italic">αδ</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)+<Emphasis Type="Italic">βδ</Emphasis>′(<Emphasis Type="Italic">x</Emphasis>)

We study a one-dimensional singular potential plus two types of regular interactions: constant electric field and harmonic oscillator. In order to search for the bound state energies, we shall use the Lippman-Schwinger Green function technique. Another direct method will be mentioned for the harmonic oscillator. In the electric field case the unique bound state coincides with that found in an earlier study as the field is switched off. For non-zero field the ground state is shifted and positive energy “quasibound states” appear. The harmonic oscillator demonstrates the general result that for a symmetric potential the odd states are not altered whereas the even states energies are lowered or raised accordingly as the delta perturbation is attractive or repulsive. No states are created or annihilated.     

Iterative solution of some nonlinear differential equations

We propose an iterative method to solve some non-linear ordinary differential equations. Comparing on the Mathieu, van der Pol and Hill equation of fourth order, we see that this method is much more efficient than the well known methods by Lyapunov or Picard.     

On the Concept of Entropy for Quantum Decaying Systems

The classical concept of entropy was successfully extended to quantum mechanics by the introduction of the density operator formalism. However, further extensions to quantum decaying states have been hampered by conceptual difficulties associated to the particular nature of these states. In this work we address this problem, by (i) pointing out the difficulties that appear when one tries a consistent definition for this entropy, and (ii) building up a plausible formalism for it, which is based on the use of coherent complex states in the context of a path integration.     

Resonant branch cuts in a generalized Friedrichs model

We study an example of a generalized Friedrichs model, in which a continuous-continuous coupling produces a pair of resonances as branch cuts of the analytic continuation of the reduced resolvent of the perturbed Hamiltonian to the second sheet of the Riemann surface associated to a transformation of the type w = z + α √z. To define the perturbation, we use the theory of self-adjoint extensions of symmetric operators proposed by the group of St. Petersburg. © 1996 John Wiley & Sons, Inc.     

Action–angle variables,ladder operators and coherent states

This Letter is devoted to the building of coherent states from arguments based on classical action–angle variables. First, we show how these classical variables are associated to an algebraic structure in terms of Poisson brackets. In the quantum context these considerations are implemented by ladder type operators and a structure known as spectrum generating algebra. All this allows to generate coherent states and thereby the correspondence of classical–quantum properties by means of the aforementioned underlying structure. This approach is illustrated with the example of the one-dimensional Pöschl–Teller potential system.     

On the determination of approximate periodic solutions of some non-linear ODE

We propose a numerical-symbolic method for the approximation of periodic solutions of a type of non-linear ODE. The efficiency of our method is contrasted with the harmonic balance method and with another one which combines the differential transformation method with Padé approximants on a non trivial example: the relativistic oscillator. It is shown that our method is computationally more reliable.