Second-Order Perturbation in Adaptive Perturbation Method

The perturbation method is an approximation scheme with a solvable leading-order. The standard way is to choose a non-interacting sector for the leading order. The adaptive perturbation method improves the solvable part in bosonic systems by using all diagonal elements of the Fock state. We consider the harmonic oscillator with the interacting term, $lambda_1x^4/6+lambda_2x^6/120$, where $lambda_1$ and $lambda_2$ are coupling constants, and $x$ is the position operator. The spectrum shows a quantitative result, less than 1 percent error, compared to a numerical solution when we use the adaptive perturbation method up to the second-order and turn off the $lambda_2$. When we turn on the $lambda_2$, the deviation becomes larger, but the error is still less than 2 percent error. Our qualitative results are demonstrated in different values of coupling constants, not only focused on a weakly coupled region.