Calculation of Expectation Values of Operators in the Complex Scaling Method

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 θ , with θ being the angle of rotation in the complex energy plane. Within the CSM, resonance and scattering solutions have fall-off 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 using the regularized backrotation technique 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.

Heisenberg uncertainty relations for the non-Hermitian resonance-state solutions to the Schrödinger equation

“…Notice that for the Hamiltonian to the system, the dispersion given by (40) yields, in view of (19), ∆H = 0, which is consistent with the fact that the particle is described by the eigenfunction u n (r). An issue of interest for future work is to consider the expectation value of an operator involving an arbitrary wave function Ψ(r) that may be expanded in terms of resonance states to address the issue of measurement from a non-Hermitian perspective.…”

“…This issue was addressed by a number of authors some decades ago [8,13,36,37]. We shall not be concerned here with the complex scaling method [38][39][40] mainly because it requires distinct mathematical considerations. In any case, as far as we know, the study of the Heisenberg uncertainty relations involving resonance states has not been addressed before.…”