Diagram summation method by means of the Legendre transformation in a scalar parameter and its application to anharmonic oscillator

The method of summing Feynman diagrams is suggested which permits to increase by one order the range of application of standard perturbation theory, at least, for one-dimensional theories. The peculiarities of the method lies in the fact that for summing diagrams it is nessary to calculate the final number of diagrams in each order in the number of lines. On the basis of results previously obtained the summation of diagrams up to the eighth order in the line number is given. The formulae obtained are applied to anharmonic oscillator. Expressions for the energy of the ground state E0(lambda) in the range of constants Q <= tambda < or approximately 1 are in satisfactory agreement with exact results. The best agreement (approximately 10%) with lambda < or approximately 0.3 is achieved when the standard perturbation theory is applied in the range lambda approximately 0.01