Generalized pointlike interactions in $R_3$ and related models with rational $S$ matrix II. $l=1$

A description is given of “inequivalen” Hamiltonians on a Hilbert space $\mathfrak{H}^N$ which is obtained by restricting the Pontryagin space of the form
$$\Pi_1^N=\mathscr{H}_{+}^N[+]\mathscr{H}_{-},\quad\mathscr{H}_{+}^N=L_2(R_3)\oplus C_{N+1}\quad\mathscr{H}_{-}=C_1$$
to a hyperplane of unit codimensionality, the Hamiltonians leading to a rational $S$ matrix in the sense of scattering theory in the pair of spaces $L_2$ and $\mathfrak{H}^N$. The use in intermediate considerations of spaces with indefinite metric is an essential and distinctive feature of the ease considered. Hamiltonians on $\mathfrak{H}^1$ are characterized as models of generalized pointlike interactions.