Anharmonic-oscillator energies with operator recursion mechanics

The procedure of the new Tamm-Dancoff method of Heisenberg et al. is considered in the context of the nonlinear oscillator. Recurrence equations which result from the operator algebra and equation of motion of the oscillator's displacement operator are converted into a meaningful characteristic energy eigenvalue problem via an infinite-dimensional basis transformation of the form $\mathrm{SAT}$, where $A$ is a matrix representing the recursion equation and where $S$ is not the inverse of $T$. This procedure, appropriate to the situation where the recursive matrix $A$ is not orthogonal or Hermitian, is numerically seen to lead to convergent approximation sequences for energy eigenvalues for several nonlinear oscillators. The matrix elements of $\mathrm{SAT}$ are defined by summable but nonconvergent infinite series. In each order of approximation eigenvalues exist which are locally independent of the single parameter upon which $S$ and $T$ depend, a fact which implies that this recursive method belongs to an as yet not defined variational principle. As the approximation order is increased the eigenvalues are numerically seen to converge to the proper limits for several different oscillators, the convergence being independent of the parameter.