Calculation of expectation values of operators in the Complex Scaling method

TL;DR

This paper explores calculating expectation values in the complex scaling method by retrieving Gamow asymptotics, enabling the use of unrotated operators for resonance states, demonstrated on model and realistic systems.

Contribution

It introduces a method to compute expectation values in CSM by restoring Gamow asymptotics, improving the analysis of resonance states.

Findings

Successful calculation of expectation values using unrotated operators.

Application to both schematic and realistic quantum systems.

Validation of the method with resonance and scattering solutions.

Abstract

The complex scaling method (CSM) provides with a way to obtain resonance parameters of particle unstable states by rotating the coordinates and momenta of the original Hamiltonian. It is convenient to use an L$^2$ integrable basis to resolve the complex rotated or complex scaled Hamiltonian H$_{\theta}$, with $\theta$ being the angle of rotation in the complex energy plane. Within the CSM, resonance and scattering solutions do not exhibit an outgoing or scattering wave asymptotic behavior, but rather have decaying asymptotics. One of the consequences is that, expectation values of operators in a resonance or scattering complex scaled solution are calculated by complex rotating the operators. In this work we are exploring applications of the CSM on calculations of expectation values of quantum mechanical operators by retrieving the Gamow asymptotic character of the decaying state and calculating hence the expectation value using the unrotated operator. The test cases involve a schematic two-body Gaussian model and also applications using realistic interactions.