After cca 30 years the group of mathematical physicists from Bologna
and Modena returns to their beloved problem of the Pad\'e
(Stieltjes) summability of the (cubic) anharmonic oscillator energy
levels and proves it. They add a simple proof of the absence of
non-perturbative eigenvalues and of the related Bessis - Zinn-Justin
conjecture (concerning the reality and also the simplicity and
positivity of energy levels). Relations are finally found between
zeros of wave functions and the semiclassical quantization rules.

It is worth adding that during the last ten years the model in
question acquired a lot of popularity in quantum physics (cf. also
the fairly incomplete list of references), especially because its
apparently non-Hermitian Hamiltonian $H_\alpha(\beta)$ (cf. eq. (1))
proves strictly self-adjoint in an {\em ad hoc} Hilbert space ${\cal
H}^{(S)}$ where, in the notation of our recent more detailed review
paper (M. Znojil, Three-Hilbert-space formulation of Quantum
Mechanics, SIGMA 5 (2009), 001, doi: 10.3842/SIGMA.2009.001), the
superscript $^{(S)}$ may be read as standing for either
``standardized" or ``sophisticated". This means that in ${\cal
H}^{(S)}$ the ``correct physical" Hermitian conjugation operation
${\cal T}^{(S)}: |\psi \rangle \to \langle\!\langle \psi|$ must be
introduced as defined in terms of a nontrivial (often called
``non-Dirac") metric operator $\Theta \neq I$. This yields
$\langle\!\langle \psi|:= \langle \psi|\Theta\ $. As a consequence
the apparently non-Hermitian Hamiltonian $H_\alpha(\beta)\neq \left
[H_\alpha(\beta) \right ]^\dagger$ may be perceived as an
isospectral transform [sometimes called a Dyson's non-unitary map --
cf. F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Quasi-Hermitian
Operators in Quantum Mechanics and the Variational Principle, Ann.
Phys. (NY) 213  (1992) 74-101] of a ``true" or ``physical"
self-adjoint Hamiltonian operator $\mathbb{h}=\mathbb{h}^\dagger$.
The readers are recommended to find [e.g., in A. Mostafazadeh,
PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model, J.
Phys. A: Math. Gen. 38 (2005) 6557–6570], the explicit perturbative
approximate representation of the latter, drastically more
complicated ``standard" self-adjoint Hamiltonian in question.

MR2546046 Grecchi, Vincenzo; Maioli, Marco; Martinez, André Padé
summability of the cubic oscillator. J. Phys. A 42 (2009), no. 42,
425208, 17 pp. 81Qxx (34L40 41A21)