Quantum theory of coupled anharmonic oscillators

We study properties of the energy levels of systems of d coupled anharmonic oscillators, d = 1,2,, characterized by the Hamiltonian H = 1/2Σi = 1ᵈ (pᵢ² + ωᵢ²xᵢ²) + λP₂ₘ(x₁,,xd), where P₂ₘ(x₁,,xd) is some homogeneous polymomial in x₁,,xd of degree 2m, m = 1,2,, P₂ₘ(x₁,,xd)≥0, and λ > 0. On the basis of our exact numerical results for the cases $d = 1,2$ and $m = 2,3,4$, combined with Titchmarsh's rigorous analytical formulas, we suggest that the number of states $N(E)$ with energies not exceeding E can be approximated in the harmonic regime by a convergent series of the form N(E) ~ Eᵈ(a₀ + a₁θ + a₂θ² + ⋯), θ < c₁, where θ = λE^m-1 and c₁ is some constant which depends on the parameters, and in the anharmonic regime by a convergent series of the form $N(E) {~} {{λ}}^{{-}(1/2m)d}{E}[(1+m)/2m]d({b}₀ + {b}₁{{θ}}^{{{-}1}{m}} + {b}₂{{θ}}^{{{-}2}{m}} + {⋯})$, ${θ} > {c}₂$. Expressions for the first few $a's$ and $b's$ as well as the correction terms to $N(E)$ for some special cases of ${P}₂ₘ({x}₁,{},{x}d)$ are given.