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Journal of Applied Mathematics and Physics

Structure of Essential Spectra and Discrete Spectrum of the Energy Operator of Six-Electron Systems in the Hubbard Model. First Quintet and First Singlet States

Sadulla Mamarazhabovich Tashpulatov

Published Jan 1, 2022• 1 citation
Publisher

Abstract

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 first quintet and first singlet states in the v-dimensional lattice.
DOI
10.4236/jamp.2022.1011227

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We consider the energy operator of six-electron systems in the Hubbard model and study the structure of the essential spectrum and the discrete spectrum of the system for the second singlet state of the system. In the one- and two-dimensional cases, it is shown that the essential spectrum of the six-electron second singlet state operator is the union of seven closed intervals, and the discrete spectrum of the system consists of a single eigenvalue lying below (above) the domain of the lower (upper, respectively) edge of the essential spectrum of this operator. In the three-dimensional case, there are the following situations for the essential and discrete spectra of the six-electron second singlet state operator: (a) the essential spectrum is the union of seven closed intervals, and the discrete spectrum consists of a single eigenvalue; (b) the essential spectrum is the union of four closed intervals, and the discrete spectrum is empty; (c) the essential spectrum is the union of two closed intervals, and the discrete spectrum is empty; (d) the essential spectrum is a closed interval, and the discrete spectrum is empty. Conditions are found under which each of the situations takes place.
We consider a three-electron system in the Impurity Hubbard model with a coupling between nearest-neighbors.Our research aim consists of studying the structure of essential spectrum and discrete spectra of the energy operator of three-electron systems in the impurity Hubbard model in the quartet state of the system in a ν -dimensional lattice.We have reduced the study of the spectrum of the three-electron quartet state operator in the impurity Hubbard model to the study of the spectrum of a simpler operator.We proved the essential spectra of the three-electron systems in the Impurity Hubbard model in the quartet state is the union of no more than six segments, and the discrete spectrum of the system is consists of no more than four eigenvalues.
Ðàññìàòðèâàåòñÿ øåñòèýëåêòðîííàÿ ñèñòåìà â ìîäåëè ÕàááàðäàF Îïèñàíà ñòðóêòóðà ñóùåñòâåííîãî ñïåêòðà è äèñêðåòíûé ñïåêòð â ýòîé ìîäåëè â ñëó÷àå ïåðâîãî êâèíòåòíîãî ñîñòîÿíèÿF ÄîêàçûE âàåòñÿD ÷òîD â îäíîìåðíîì ñëó÷àåD ñóùåñòâåííûé ñïåêòð ñèñòåìû åñòü îáúåäèíåíèå ñåìüè îòðåçêîâD à äèñêðåòíûé ñïåêòð ñèñòåìû ñîñòîèò èç íå áîëåå îäíîãî ñîáñòâåííîãî çíà÷åíèÿF Êëþ÷åâûå ñëîâàX ìîäåëü
We consider the energy operator of a three-electron systems in the impurity Hubbard model and investigate the structure of essential spectra and discrete spectrum of the system in the first doublet state on the ν\nuν -dimensional lattice.