Quantum theory of coupled anharmonic oscillators

Abstract

We study properties of the energy levels of systems of $d$ coupled anharmonic oscillators, $d = 1,2,\dots$, characterized by the Hamiltonian $H = \frac{1}{2}{\Sigma }_{i = 1}^{ d} ({p_i}^2 + {{\omega }_i}^2{x_i}^2) + \lambda P_{2m}(x_1,\dots ,x_d)$, where $P_{2m}(x_1,\dots ,x_d)$ is some homogeneous polymomial in $x_1,\dots ,x_d$ of degree $2m, m = 1,2,\dots$, $P_{2m}(x_1,\dots ,x_d)\ge 0$, and $\lambda > 0$. 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 $N\mathrm{}\left(E\right)\mathrm{}$ with energies not exceeding $E$ can be approximated in the harmonic regime by a convergent series of the form $N\mathrm{}\left(E\right)\mathrm{} \sim E^d(a_0 + a_1\theta + a_2{\theta }^2 + \cdots )$, $\theta < c_1$, where $\theta = \lambda E^{m-1\mathrm{}}$ and $c_1$ is some constant which depends on the parameters, and in the anharmonic regime by a convergent series of the form $N\mathrm{}\left(E\right)\mathrm{} \sim {\lambda }^{-\left(\frac{1}{2m}\right)d}{E}^{\left[\frac{\mathrm{}(1+m)\mathrm{}}{2m}\right]d}(b_0 + b_1{\theta }^{\frac{-1\mathrm{}}{m}} + b_2{\theta }^{\frac{-2\mathrm{}}{m}} + \cdots )$, $\theta > c_2$. Expressions for the first few $a\text{'}\mathrm{s}$ and $b\text{'}\mathrm{s}$ as well as the correction terms to $N\mathrm{}\left(E\right)\mathrm{}$ for some special cases of $P_{2m}(x_1,\dots ,x_d)$ are given.