Complex Scaling Method applied to the study of the Swanson Hamiltonian in the broken PT-symmetry phase

TL;DR

This paper applies the Complex Scaling Method to analyze the non-PT symmetric phase of the Swanson Hamiltonian, focusing on time evolution, exceptional points, and comparing with Rigged Hilbert Space approaches.

Contribution

It introduces a bi-orthogonality framework and response function formalism to study the dynamics of the Swanson Hamiltonian in the broken PT-symmetry phase.

Findings

Time evolution analyzed near Exceptional Points

Continuity equation derived for the system

Comparison with Rigged Hilbert Space results

Abstract

In this work, we study the non-PT symmetry phase of the Swanson Hamiltonian in the framework of the Complex Scaling Method. By constructing a bi-orthogonality relation, we apply the formalism of the response function to analyse the time evolution of different initial wave packages. The Wigner Functions and mean value of operators are evaluated as a function of time. We analyse in detail the time evolution in the neighbourhood of Exceptional Points. We derive a continuity equation for the system. We compare the results obtained using the Complex Scaling Method to the ones obtained by working in a Rigged Hilbert Space.