Second-Order Perturbation in Adaptive Perturbation Method

The perturbation method is an approximation scheme with a solvableleading order. the standard way is to choose a non-interacting sector for the leadingorder. the adaptive perturbation method improves the solvable part by using alldiagonal elements for a fock state. we consider the harmonic oscillator with theinteracting term, 1x4=6 + 2x6=120, where 1 and 2 are coupling constants, and xis 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 o the2. when we turn on the 2, more deviation occurs, but the error is still less than2 percent. we show a quantitative result beyond a weak-coupling region. our studyshould provide interest in the holographic principle and strongly coupled boundarytheory.