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Exactly solvable quantum mechanical models with infinite renormalization of the wavefunction

The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate renormalization beyond perturbation theory. However, known models of constructive field theory do not contain such difficulties as infinite renormalization of the wavefunction. In this paper an exactly solvable quantum mechanical model with such a difficulty is constructed. This model is a simplified analogue of the large-N approximation to the Φaa-model in six-dimensional spacetime. It is necessary to introduce an indefinite inner product to renormalize the theory. The mathematical results of the theory of Pontriagin spaces are essentially used. It is remarkable that not only the field but also the canonically conjugated momentum become well defined operators after adding counterterms.