The peak model for the triplet extensions and their transformations to the reference Hilbert space in the case of finite defect numbers

We develop the so-called peak model for the triplet extensions of supersingular perturbations in the case of a not necessarily semibounded symmetric operator with finite defect numbers. The triplet extensions in scales of Hilbert spaces are described by means of abstract boundary conditions. The resolvent formulas of Krein-Naimark type are presented in terms of the $\gamma$-field and the abstract Weyl function. By applying certain scaling transformations to the triplet extensions in an intermediate Hilbert space we investigate the obtained operators acting in the reference Hilbert space and we show the connection with the classical extensions

Irreducible tensor form of three-particle operator for open-shell atoms

The three-particle operator in a second quantized form is studied. The operator is transformed into irreducible tensor form. Possible coupling schemes, distinguished by the classes of symmetric group \mathrm{S_{6}}, are presented. Recoupling coefficients, which allow one to transform given scheme into another, are produced by using the angular momentum theory, combined with quasispin formalism. The classification of three-particle operator, which acts on n=1,2,...,6 open shells of equivalent electrons of atom, is considered. The procedure to construct three-particle matrix elements are examined