General anharmonic oscillators

Abstract

The eigenvalue problem of the general anharmonic oscillator (Hamiltonian $H_{2\mu}(k,\lambda)=\text{-d}^{2}/\text{d}x^{2}+kx^{2}+\lambda x^{2\mu},(k,\lambda)>0$, $\mu =2,3,4,...)$ is investigated in this work. Very accurate eigenvalues are obtained in all regimes of the quantum number $n$ and the anharmonicity constant $\lambda $. The eigenvalues, as functions of $\lambda $, exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W.K.B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.