QUARTIC, SEXTIC, AND OCTIC ANHARMONIC-OSCILLATORS - PRECISE ENERGIES OF GROUND-STATE AND EXCITED-STATES BY AN ITERATIVE METHOD BASED ON THEGENERALIZED BLOCH EQUATION

Recently, we proposed an iteration method for solving the eigenvalue p roblem of the time-independent Schrodinger equation [H. Meissner and E . O. Steinborn, Int. J. Quantum Chem. 61, 777 (1997)]. This method, wh ich is based on the generalized Bloch equation, calculates iteratively certain matrix elements of the wave operator which are the wave-funct ion expansion coefficients (WECs). It is valid for boson as well as fe rmion systems. In this article we show that the WEC-iteration method, together with a renormalization technique, allows us to calculate ener gy eigenvalues for the ground state and excited states of the quartic, sextic, and octic anharmonic oscillator with very high accuracy. In o rder to overcome slow convergence in the iteration scheme we use a ren ormalization technique introduced by F. Vinette and J. Cizek [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. Fit. 11, 77 (1947)] whic h is frequently used for the treatment of anharmonic oscillators in se cond quantization.