Study on effective Hamiltonian and decoupling equation

The effective Hamiltonian in a model space has been derived from the decoupling equation. We present a rigorous proof of the proposition that the decoupling equation is a necessary and sufficient condition to give an effective Hamiltonian, establishing a robust one-to-one correspondence between a solution to the decoupling equation and an effective Hamiltonian. We then present discussions based on this result, emphasizing (i) that the current situation of the theory is far from being satisfactory, and (ii) that the proposition gives a rigorous mathematical foundation to any effort to improve the theory of the effective Hamiltonian on the basis of the decoupling equation.