The damped Pinney equation and its applications to dissipative quantum mechanics

TL;DR

This paper investigates the damped Pinney equation, deriving perturbative solutions for weak damping, and applies these results to analyze the evolution of Gaussian wave-functions in dissipative quantum mechanics.

Contribution

It introduces a perturbation method for the damped Pinney equation and applies it to dissipative quantum systems where traditional symmetries are absent.

Findings

Perturbative solutions agree well with numerical results.

The method effectively describes wave-function evolution in dissipative quantum mechanics.

The approach extends understanding of damped nonlinear oscillators in quantum contexts.

Abstract

The work considers the damped Pinney equation, defined as the model arising when a linear in velocity damping term is included in the Pinney equation. In the general case the resulting equation does not admit Lie point symmetries or is reducible to a simpler form by any obvious coordinate transformation. In this context the method of Kuzmak-Luke is applied to derive a perturbation solution, for weak damping and slow time-dependence of the frequency function. The perturbative and numerical solutions are shown to be in good agreement. The results are applied to examine the time-evolution of Gaussian shaped wave-functions in the Kostin formulation of dissipative quantum mechanics.