Two theorems on the variation perturbation method for the excited states of stationary quantum systems

Two theorems are proved which are of considerable use in understanding the situation which arises while applying the variation perturbation method for calculating the state vectors and the energies of stationary quantum systems. These theorems also enable us to ensure that the second-order energy calculated by the use of the method is a rigorous upper bound to the true value The earlier results of Sinanoulu and Miller follow as immediate corollaries of these theorems.