Accurate Study from Adaptive Perturbation Method

The adaptive perturbation method decomposes a Hamiltonian by the diagonal elements and non-diagonal elements of the Fock state. The diagonal elements of the Fock state are solvable but can contain the information about coupling constants. We study the harmonic oscillator with the interacting potential, $lambda_1x^4/6+lambda_2x^6/120$, where $lambda_1$ and $lambda_2$ are coupling constants, and $x$ is the position operator. In this study, each perturbed term has an exact solution. We demonstrate the accurate study of the spectrum and $langle x^2rangle$ up to the next leading-order correction. In particular, we study a similar problem of Higgs field from the inverted mass term to demonstrate the possible non-trivial application of particle physics.