Hull-range potentials and M. G. Krein's formula for generalized resolvents

For non-additive finite dimensional perturbations of an Hermitian operator Krein's formula is used to obtain arepresentation of scattering matrices with Hermitian $(n\times n)$-matrix as a parameter. Relying on this representation, we obtain explicit formulae for a number of new models of quantum mechanics with null-range potential. We also establish a connection of the parametrization obtained with phenomenological $S$-matrices and relevant Wigner's $R$-functions.