Abstract. We study the perturbation theory for H=pZ+x2+flx 2~+1, n= 1,2 ..... It is proved that when Imfl~:0, H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schr6dinger perturbation expansion, and admits an analytic continuation to Im/~=0 where it can be interpreted as a resonance of the problem.

This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

It is proved that the divergent Rayleigh-Schrödinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are Borel summable in the distributional sense to the resonances naturally associated with the system.

Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PTsymmetric Schrödinger operators are discussed.

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