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The auxiliary field method, defined through introducing an auxiliary (alsocalled as the Hubbard-Stratonovich or the Mean-) field and utilizing aloop-expansion, gives an excellent result for a wide range of a couplingconstant. The analysis is made for Anharmonic-Oscillator and Double-Wellexamples in 0-(a simple integral) and 1-(quantum mechanics)dimension. It isshown that the result becomes more and more accurate by taking a higher loopinto account in a weak coupling region, however, it is not the case in a strongcoupling region. The 2-loop approximation is shown to be still insufficient forthe Double-Well case in quantum mechanics.Comment: 18 pages, 5 figure

Auxiliary field method as a powerful tool for nonperturbative study