We introduce in a systematic way the properly defined arctangent of the mean value of the resolvent of a Hilbert space operator. We consider more precisely the case of semibounded self−adjoint operators H, in the region of the discrete spectrum. The arctangent of the Padé approximations to the mean value of the resolvent are constructed out of the moments. They are shown to provide converging monotonic increasing sequences of lower bounds to the arctangent of the resolvent. Consequently bounds with the same properties are derived for the discrete eigenvalues of H and, most remarkable, the ordering of the corresponding poles of the approximations reproduces the ordering of the exact poles. The Padé method is shown to provide a way to fully exploit the content of the Rayleigh−Ritz variational method, by providing a simple mechanical procedure to build up the variational subspaces: It defines a powerful both perturbative and variational approximation to semibounded operators. The difficulties of the Ritz method in the degenerate case are overcome by the fact that all bounds in the Padé method are strict bounds. In a second part, we consider the important case in which the moments are given by diverging algorithms. By properly regularizing them, we show that the Padé−Ritz variational principle generalizes to produce absolute maxima of the arctangent of the Padé approximations in the regulator and that these maxima form monotonic converging sequences of lower bounds. In the last part, we discuss an application to quantum mechanical systems for which the perturbative variational method is applied to the energy, allowing us to treat the case of strong coupling. As a consequence it appears possible to solve (approximately) the R−dimensional anharmonic oscillator in a purely algebraic way.
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Bessis et al. (1975) studied this question.