Partial sums and optimal shifts in shifted large−l perturbation expansions for quasi-exact potentials

For the N−plets of bound states in a quasi-exactly solvable (QES) toy model (sextic oscillator), the spectrum is known to be given as eigenvalues of an N by N matrix. Its determination becomes purely numerical for all the larger N > N0 = 9. We propose a new perturbative alternative to this construction. It is based on the fact that at any N , the problem turns solvable in the limit of very large angular momenta l → ∞. For all the finite l we are then able to define the QES spectrum by convergent perturbation series. These series admit a very specific rational resummation, having an analytic or branched continued-fraction form at the smallest N = 4 and 5 orN = 6 and 7, respectively. It is remarkable that among all the asymptotically equivalent small expansion parameters μ ∼ 1/(l + β), one must choose an optimal one, with unique shift β = β(N). AMS (MSC 2000): 81Q15