The rigged Hilbert space approach to the Gamow states

TL;DR

This paper employs the rigged Hilbert space framework to rigorously describe Gamow states, emphasizing the importance of test functions that decay faster than Gaussians for accurate representation.

Contribution

It introduces a rigorous mathematical approach to Gamow states using rigged Hilbert spaces with specific test function decay properties, advancing the theoretical understanding.

Findings

Gamow states are best described within a rigged Hilbert space framework.

Test functions must decay faster than Gaussians for proper Gamow state representation.

The approach clarifies the mathematical foundation of resonances in quantum mechanics.

Abstract

We use the resonances of the spherical shell potential to present a thorough description of the Gamow (quasinormal) states within the rigged Hilbert space. It will be concluded that the natural setting for the Gamow states is a rigged Hilbert space whose test functions fall off at infinity faster than Gaussians.