General Riccati Solution Research Articles

Shifted one-parameter supersymmetric family of quartic asymmetric double-well potentials

Extending our previous work (Rosu, 2014), we define supersymmetric partner potentials through a particular Riccati solution of the form F(x)=(x−c)2−1, where c is a real shift parameter, and work out the quartic double-well family of one-parameter isospectral potentials obtained by using the corresponding general Riccati solution. For these parametric double well potentials, we study how the localization properties of the two wells depend on the parameter of the potentials for various values of the shifting parameter. We also consider the supersymmetric parametric family of the first double-well potential in the Razavy chain of double well potentials corresponding to F(x)=12sinh2x−2(1+2)sinh2x(1+2)cosh2x+1, both unshifted and shifted, to test and compare the localization properties.

One-parameter families of supersymmetric isospectral potentials from Riccati solutions in function composition form

In the context of supersymmetric quantum mechanics, we define a potential through a particular Riccati solution of the composition form (F∘f)(x)=F(f(x)) and obtain a generalized Mielnik construction of one-parameter isospectral potentials when we use the general Riccati solution. Some examples for special cases of F and f are given to illustrate the method. An interesting result is obtained in the case of a parametric double well potential generated by this method, for which it is shown that the parameter of the potential controls the heights of the localization probability in the two wells, and for certain values of the parameter the height of the localization probability can be higher in the smaller well.

INHOMOGENEOUS BAROTROPIC FRW COSMOLOGIES IN CONFORMAL TIME

It is known that the barotropic FRW system of differential equations for zero cosmological constant can be reduced to simple harmonic oscillator (HO) differential equations in the conformal time variable. This is due to the fact that the Hubble rate parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. In previous works, we have used this mathematical result to set the barotropic HO equations in the nonrelativistic supersymmetric approach by factorizing them. If a constant additive parameter, denoted by S, is added to the common Riccati solution of these supersymmetric partner cosmologies one obtains inhomogeneous barotropic cosmologies with periodic singularities in their spatial curvature indices that are counterparts of the non-shifted supersymmetric partners. The zero-mode solutions of these cyclic singular cosmologies are reviewed here as a function of real and imaginary shift parameter. We also notice the modulated zero modes obtained by using the general Riccati solution and comment on their cosmological application.

A note on Verhulst's logistic equation and related logistic maps

We consider the Verhulst logistic equation and a couple of forms of the corresponding logistic maps. For the case of the logistic equation we show that using the general Riccati solution only changes the initial conditions of the equation. Next, we consider two forms of corresponding logistic maps reporting the following results. For the map xn + 1 = rxn(1 − xn) we propose a new way to write the solution for r = −2 which allows better precision of the iterative terms, while for the map xn + 1 − xn = rxn(1 − xn + 1) we show that it behaves identically to the logistic equation from the standpoint of the general Riccati solution, which is also provided herein for any value of the parameter r.

Supersymmetric Free-Damped Oscillators: Adaptive Observer Estimation of the Riccati Parameter

A supersymmetric class of free damped oscillators with three parameters has been obtained in 1998 by Rosu and Reyes through the factorization of the Newton equation. The supplementary parameter is the integration constant of the general Riccati solution. The estimation of the latter parameter is performed here by employing the recent adaptive observer scheme of Besançon et al., but applied in a nonstandard form in which a time-varying quantity containing the unknown Riccati parameter is estimated first. Results of computer simulations are presented to illustrate the good feasibility of this approach for a case in which the estimation is not easily accomplished by other means.

Strictly Isospectral Potentials from Excited Quantum States

The strictly nonrelativistic isospectral scheme based on the general Riccati solution and Darboux transformation function corresponding to excited states is presented on the following examples: the harmonic oscillator, the square well, and the hydrogen atom.

Ermakov-Lewis angles for one-parameter supersymmetric families of Newtonian free damping modes.

We apply the Ermakov-Lewis procedure to the one-parameter damped modes y recently introduced by Rosu and Reyes, which are related to the common Newtonian free damping modes y by the general Riccati solution [H. C. Rosu and M. Reyes, Phys. Rev. E 57, 4850 (1998)]. In particular, we calculate and plot the angle quantities of this approach that can help to distinguish these modes from the common y modes.

Multiple Parameter Structure of Mielnik's Isospectrality in Unbroken SUSYQM

Within unbroken SUSYQM and for zero factorization energy, I present an iterativegeneralization of Mielnik's isospectral method by employing a Schrodinger truezero mode in the first-step general Riccati solution and imposing the physicalcondition of normalization at each iterative step. This procedure leads to awell-defined multiple-parameter structure within Mielnik's construction for both zeromodes and potentials.