Quantum Mechanics of Damped Systems II. Damping and Parabolic Potential Barrier

TL;DR

This paper explores the quantum resonant states of an inverted harmonic oscillator, revealing a connection between special functions and distribution theory, advancing understanding of quantum damping phenomena.

Contribution

It introduces a novel analysis of resonant states for the parabolic potential barrier and links special functions to Gel'fand distributions in this context.

Findings

Resonant states correspond to poles of the resolvent in the complex energy plane.

Established a relation between parabolic cylinder functions and Gel'fand distributions.

Provided insights into quantum damping mechanisms for inverted oscillators.

Abstract

We investigate the resonant states for the parabolic potential barrier known also as inverted or reversed oscillator. They correspond to the poles of meromorphic continuation of the resolvent operator to the complex energy plane. As a byproduct we establish an interesting relation between parabolic cylinder functions (representing energy eigenfunctions of our system) and a class of Gel'fand distributions used in our recent paper.