Quartic, sextic, and octic anharmonic oscillators: Precise energies of ground state and excited states by an iterative method based on the generalized Bloch equation

Recently, we proposed an iteration method for solving the eigenvalue problem of the time-independent Schrödinger equation [H. Meiβner and E. O. Steinborn, Int. J. Quantum Chem. 61, 777 (1997)]. This method, which is based on the generalized Bloch equation, calculates iteratively certain matrix elements of the wave operator which are the wave-function expansion coefficients (WECs). It is valid for boson as well as fermion systems. In this article we show that the WEC-iteration method, together with a renormalization technique, allows us to calculate energy eigenvalues for the ground state and excited states of the quartic, sextic, and octic anharmonic oscillator with very high accuracy. In order to overcome slow convergence in the iteration scheme we use a renormalization technique introduced by F. Vinette and J. Čížek [J. Math. Phys. (N.Y.) 32, 3392 (1991)] and show that this method is equivalent to the renormalization scheme based on the Bogoliubov transformation [N. N. Bogoliubov, Izv. Akad. Nauk SSSR, Ser. Fiz. 11, 77 (1947)] which is frequently used for the treatment of anharmonic oscillators in second quantization.