Structure of essential spectra and discrete spectrum of the energy operator of six-electron systems in the Hubbard model. Second singlet state

We consider the energy operator of six electron systems in the Hubbard model and investigate the structure of essential spectra and discrete spectrum of the system in the second singlet state of the system. It is shown that in one-dimensional and two-dimensional cases, the essential spectrum of the operator of the six-electron second singlet state consists of the union of seven segments, and the discrete spectrum of the six-electron second singlet state consists of a single eigenvalue, lying below (above) the domain of the lower (upper) edge of the essential spectrum of this operator. And, in the three-dimensional case, the following situations occur: a) the essential spectrum of the six-electron second singlet state operator consists of the union of seven segments, and the discrete spectrum of the six-electron second singlet state operator consists of a single eigenvalue; b) the essential spectrum of the six-electron second singlet state operator consists of the union of four segments, and the discrete spectrum of the six-electron second singlet state operator consists of an empty set; c) the essential spectrum of the six-electron second singlet state operator consists of the union of two segments, and the discrete spectrum of the six-electron second singlet state operator consists of an empty set; d) the essential spectrum of the six-electron second singlet state operator consists of single segment, and the discrete spectrum of the six-electron second singlet state operator is empty set; provided that every situation occurs. Found the conditions when every situation to take place.