  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 by using all
diagonal elements for a 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 from the second-order, less than 1 percent
error, compared to a numerical solution when turning off the $\lambda_2$. When
we turn on the $\lambda_2$, more deviation occurs, but the error is still less
than 2 percent. We show a quantitative result beyond a weak-coupling region.
Our study should provide interest in the holographic principle and strongly
coupled boundary theory.
