Structure of the essential spectrum and the discrete spectrum of the energy operator of six-electron systems in the Hubbard model. Fourth triplet state

In this paper, we analyze the energy operator of six-electron systems within the framework of the Hubbard model and examine the structure of the essential spectrum and the discrete spectrum of the system in the fourth triplet state. We prove that in the one- and two-dimensional cases, the essential spectrum of the six-electron fourth triplet state operator is the union of seven segments, whereas the discrete spectrum contains at most one eigenvalue. In the three-dimensional case, the following situations can occur: (a) the essential spectrum of the operator is the union of seven segments and the discrete spectrum contains at most one eigenvalue; (b) the essential spectrum is the union of four segments and the discrete spectrum is empty; (c) the essential spectrum is the union of two segments and the discrete spectrum is empty; (d) the essential spectrum consists of a single segment and the discrete spectrum is empty. We found conditions under which each of these situations occurs.