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(theta) = E-R(theta) - i Gamma(theta)/2, as a function of the scaling parameter theta, we find that for potential barrier scattering, the condition d Gamma(theta(I))/d theta = 0 uniquely determines the scaling parameter theta. The condition dE(R) (theta(R))/d theta = 0 is merely a consequence of the Virial theorem and theta(I) = theta(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.