Determination of Scaling Parameter and Dynamical Resonances in Complex-Rotated Hamiltonian I: Theory

Abstract

In this paper we present a theoretical analysis on the determination of the scaling parameter in the complex-rotated Hamiltonian, which has served as a basis for successful applications of the rigged Hilbert space theory for resonances. Based on the complex energy eigenvalue, E(θ) = ER(θ) - iΓ(θ)/2, as a function of the scaling parameter θ, we find that for potential barrier scattering, the condition dΓ(θI)/dθ = 0 uniquely determines the scaling parameter θ. The condition dER(θR)/dθ = 0 is merely a consequence of the Virial theorem and θI = θR is not a necessary condition for a resonance state. We also provide a harmonic approximation formalism for resonances in scattering over a potential barrier.