The seven lower-lying energy levels E_g,0, k = (0,1,2) as a function of f, α, and detuning δ, respectively. (a) δ = 0, |α| = 1; (b) δ = 0, f = 0.01,0.1; (c) |α| = 1, f = 0.05. In panels (a) and (c), blue dot–dashed lines represent the results of our dressed perturbation theory, while red solid lines represent the results of numerical simulation. In panel (b), dot–dashed lines represent the results of dressed perturbation theory, solid lines represent the results of numerical simulation, red and blue lines represent the energy spectrum of f = 0.01, and black and green lines represent the energy spectrum of f = 0.1. For the parameter regimes with f < 0.1, |α| < 1, and a wide range of detuning values −1 < δ/ω_c < 1, our analytical energy spectrums are in good agreement with the direct numerical simulation.