Exactly solvable quantum mechanical models with Stückelberg divergences

We consider an exactly solvable quantum mechanical model with an infinite number of degrees of freedom that is an analogue of the model of $N$ scalar fields $(\lambda/N)(\varphi^a\varphi^a)^2$ in the leading order in $1/N$. The model involves vacuum and $S$-matrix divergences and also the Stückelberg divergences, which are absent in other known renormalizable quantum mechanical models with divergences (such as the particle in a $\delta$-shape potential or the Lee model). To eliminate divergences, we renormalize the vacuum energy and charge and transform the Hamiltonian by a unitary transformation with a singular dependence on the regularization parameter. We construct the Hilbert space with a positive-definite metric, a self-adjoint Hamiltonian operator, and a representation for the operators of physical quantities. Neglecting the terms that lead to the vacuum divergences fails to improve and, on the contrary, worsens the renormalizability properties of the model.