  The idea of adaptive perturbation theory is to divide a Hamiltonian into a
solvable part and a perturbation part. The solvable part contains the
non-interacting sector and the diagonal elements of Fock space from the
interacting terms. The perturbed term is the non-diagonal sector of Fock space.
Therefore, the perturbation parameter is not coupling constant. This is
different from the standard procedure of previous perturbation method. In this
letter, we use the adaptive perturbation theory to extract the solvable
elements in the strongly coupled double harmonic oscillator system and obtain
the energy spectrum of the solvable part. Then, we diagonalize the Hamiltonian
to obtain the numerical solution. In order to study the accuracy of adaptive
perturbation theory in the strongly coupled double harmonic oscillator system,
we demonstrate the analytical study from the leading order and second-order
perturbation. The deviations between the leading-order and the numerical
solution show that as the gap between the quasiparticle number n1 and n2
decreases, or the coupling constant $\lambda$ increases, the gap between the
exact solutions in the strongly coupling region of double harmonic oscillator
system and the numerical solutions becomes smaller. Overall, the numerical gap
is about 1% to 3%, which is not a bad result. The value of the second-order is
quite close to the numerical solution when n1 doesn't equal n2. In most cases,
the deviation is less than 1%, which means that the adaptive perturbation
theory is effective for the double harmonic oscillator system in strong
coupling field.
