Improved Calculations for Anharmonic Oscillators Using the Gradient Method

We report results of energy calculations for the ground state and first excited state of anharmonic oscillators with HamiltonianH=p 2+?x 2+?x 2q . The calculations have been performed for ?=0, ± 1,q=2, 3,..., 6 and some sets of ? in the range 10-6<=?<=106. We have used as a method of direct energy minimization a simple form of the gradient method. The energy has been computed with at least 15 exact figures (in the cases ?=-1, ?=10-3,q=2 and ?=0 or 1, ?=104,q=6) in virtue of a new, general criterion for the optimal choice of both the dimension and the size parameter of the basis.