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

Abstract

Exact solvability (typically, of harmonic oscillators) in quantum mechanics usually implies an elementary form of the spectrum while in all the "next-to-solvable" models, the energies E are only available in an implicit form (typically, as eigenvalues of an N-dimensional matrix). We demonstrate here that certain echoes of the unattainable harmonic-oscillator ideal may still survive in the latter (often called quasi-exact) cases exemplified here by the popular sextic anharmonic oscillator. In particular we show that whenever the spatial dimension D (or, equivalently, angular momentum L) happens to be "sufficiently" large, the surprisingly compact semi-explicit energies E remain available. In detail, using the Rayleigh-Schr\"{o}dinger perturbation theory in its appropriate "shifted-L" version we observe that: (1) all the k-th order approximants $E_k$ remain defined in integer arithmetics (i.e., without any errors); (2) an optimal auxiliary N-dependent shift $\beta(N)$ of L exists and is unique; (3) the resulting perturbative E degenerates to the series in powers of $1/[L+\beta(N)]^2$; (4) a certain optimal Pade re-summation formulae exist and possess a generic branched-continued-fraction structure.