A new iterative method for solving the time-independent Schr�dinger equation based on the generalized Bloch equation. I. Boson systems: The quartic anharmonic oscillator

The eigenvalue problem of the time-independent Schrödinger equation is solved as usual by expanding the eigenfunctions in terms of a basis set. However, the wave-function expansion coefficient (WECs), which are certain matrix elements of the wave operator, are determined by an iterative method. For these WECs, we deduced new nonlinear equations utilizing the concept of a reference space and the generalized Bloch equation for the wave operator. To test these equations, the method is used to calculate with great accuracy the energy eigenvalues of the ground state and first few excited states of the quartic anharmonic oscillator, whose Rayleigh-Schrödinger perturbation series diverges strongly for every nonzero coupling constant.