Determination of scaling parameter and dynamical resonances in complex-rotated Hamiltonian I: Theory

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) = E _R(θ) - iΓ(θ)/2, as a function of the scaling parameter θ, we find that for potential barrier scattering, the condition dΓ(θ_I)/dθ = 0 uniquely determines the scaling parameter θ. The condition dE_R(θ_R)/dθ = 0 is merely a consequence of the Virial theorem and θ_I = θ_R is not a necessary condition for a resonance state. We also provide a harmonic approximation formalism for resonances in scattering over a potential barrier.