Moment recursions and the Schrödinger problem

We present new techniques for attacking the Schrödinger eigenvalue problem. They are based on asymptotic solutions to an exact set of recursion relations satisfied by moments of the coordinate operator. We apply these techniques to the generalized anharmonic oscillator $H=P^2+X^{2M}$ and show how to compute the energy levels, all of the moments $〈X^N〉$, and the value of the wave function and its derivatives at the origin. We specialize to the case $M=2\mathrm{}$ to obtain accurate numerical results for the low-lying energy levels as well as (all) the moments. We also discuss the case $V\left(x\right)=d{x}^2+x^4$. Transition moments are then treated in the same manner.