IJPAM: Volume 12, No. 1 (2004)

PARTIAL SUMS AND OPTIMAL SHIFTS IN
SHIFTED LARGE $-\ell$ PERTURBATION
EXPANSIONS

Miloslav Znojil
Nuclear Physics Institute of Czech Academy of Sciences
250 68 Rez near Prague, CZECH REPUBLIC
e-mail: znojil@ujf.cas.cz

Abstract.For the 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 by matrix. Its determination becomes purely numerical for all the larger . We propose a new perturbative alternative to this construction. It is based on the fact that at any , the problem turns solvable in the limit of very large angular momenta $\ell\to\infty$ . For all the finite $\ell$ 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 and or and , respectively. It is remarkable that among all the asymptotically equivalent small expansion parameters $\mu \sim 1/(\ell + \beta)$ , one must choose an optimal one, with unique shift $\beta=\beta(N)$ .