Intrinsic relationship between dynamical resonance energy and decay rate

In the framework of dilatation transformation based on complex scaling, we show that the dilated non-Hermitian Hamiltonian may be decoupled into two separated Hermitian operators, namely, the resonance energy operator and the decay rate operator, i.e., the FWHM. We present an intrinsic relationship between these operators. Specifically, we show that the effective potential for resonance decay rate is a virial function of the effective potential for resonance energy. As examples of their application, we invoke the decoupled Hamiltonian operators for resonance states of a single particle in a central force field and to problems in potential scattering to identify the resonance states and determine resonance energy and decay rate.