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

摘要

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_1)/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.