The authors developed a numerical non-iterative method of nding of the value of eigenfunctions of perturbed self-adjoint operators, which was called the method of regularized traces. It allows to nd the value of eigenfunctions of perturbed discrete operators, using the spectral characteristics of the unperturbed operator and the eigenvalues of the perturbed operator. In contrast to the known methods, in the method of regularized traces the value of eigenfunctions are found by the linear equations. It signicantly increases the computational eciency. The diculty of the method is to nd sums of functional series of "suspended" corrections of perturbation theory, which can be found only numerically. The formulas, which are convenient to nd "suspended" corrections such that one can approximate the amount of these functional series by summing up of them, are presented in the paper. However, if a norm of the perturbing operator is large, then the summation of "suspended" corrections can be not eective. We obtain analytical formulas, which allow to nd the values of sums of functional series of "suspended" corrections of perturbation theory in the discrete nodes without direct summation of its terms. Computational experiments are performed. These experiments allowed to nd the values of the eigenfunctions of the perturbed one-dimensional Laplace operator. The experimental results showed the accuracy and computational eciency of the developed method.
The method of nding the eigenvalues and eigenfunctions of abstract discrete semibounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators dened on geometric graphs. The method for nding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to nd the eigenvalues and eigenfunctions of the Sturm Liouville operator dened on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high eciency of the developed methods.
 ðàáîòå ïðèâîäÿòñÿ îñíîâíûå ïîëîaeåíèÿ íîâîãî ìåòîäà íàõîaeE äåíèÿ ñîáñòâåííûõ ôóíêöèé âîçìóùåííûõ äèñêðåòíûõ ïîëóîãðàE íè÷åííûõ îïåðàòîðîâD çàäàííûõ íà êîìïàêòíûõ ãðàôàõF Êëþ÷åâûå ñëîâàX ñîáñòâåííûå ôóíêöèèD âîçìóùåííûå îïåðàòîðûD êîìïàêòíûå ãðàôûD áàçèñ ýíåðãåòè÷åñêîãî ïðîñòðàíñòâàF Development of a new method for finding eigenfunctions of perturbed discrete operators given on compact graphs The paper presents the main provisions of a new method for finding eigenfunctions of perturbed discrete semi-bounded operators given on compact graphs.
Àâòîðàìè ñòàòüè áûë ðàçðàáîòàí íåèòåðàöèîííûé ìåòîä âû÷èñëåíèÿ çíà÷åíèé ñîáñòâåííûõ ôóíêöèé âîçìóùåííûõ ñàìîñîïðÿaeåííûõ îïåðàòîðîâ, íàçâàííûé ìåòî-äîì ðåãóëÿðèçîâàííûõ ñëåäîâ (ÐÑ). Îí ïîçâîëÿåò íàéòè çíà÷åíèÿ ñîáñòâåííûõ ôóíê-öèé âîçìóùåííûõ îïåðàòîðîâ, çíàÿ ñïåêòðàëüíûå õàðàêòåðèñòèêè íåâîçìóùåííîãî îïåðàòîðà è ñîáñòâåííûå ÷èñëà âîçìóùåííîãî îïåðàòîðà.  îòëè÷èå îò èçâåñòíûõ ìå-òîäîâ íàõîaeäåíèÿ ñîáñòâåííûõ ôóíêöèé, ìåòîä ÐÑ íå èñïîëüçóåò ìàòðèöû è çíà÷åíèÿ ñîáñòâåííûõ ôóíêöèé íàõîäÿòñÿ ïî ëèíåéíûì ôîðìóëàì. Ýòî çíà÷èòåëüíî óâåëè÷èâà-åò åãî âû÷èñëèòåëüíóþ ýôôåêòèâíîñòü ïî ñðàâíåíèþ ñ êëàññè÷åñêèìè ìåòîäàìè. Äëÿ ïðèìåíåíèÿ ìåòîäà ÐÑ íà ïðàêòèêå íåîáõîäèìî óìåòü ñóììèðîâàòü ôóíêöèîíàëüíûå ðÿäû Ðåëåÿ Øðåäèíãåðà âîçìóùåííûõ äèñêðåòíûõ îïåðàòîðîâ. Ðàíåå áûëè ïîëó÷å-íû ôîðìóëû íàõîaeäåíèÿ ≪âçâåøåííûõ≫ ïîïðàâîê òåîðèè âîçìóùåíèé, ÷òî ïîçâîëÿëî ïðèáëèaeåííî íàõîäèòü ñóììû ôóíêöèîíàëüíûõ ðÿäîâ Ðåëåÿ Øðåäèíãåðà, çàìåíÿÿ èõ ÷àñòè÷íûìè ñóììàìè, ñîñòîÿùèìè èç ýòèõ ïîïðàâîê.  ñòàòüå âïåðâûå ïîëó÷åíû ôîðìóëû íàõîaeäåíèÿ çíà÷åíèé ñóìì ôóíêöèîíàëüíûõ ðÿäîâ Ðåëåÿ Øðåäèíãåðà âîçìóùåííûõ äèñêðåòíûõ îïåðàòîðîâ â óçëîâûõ òî÷êàõ. Ïðîâåäåíû âû÷èñëèòåëüíûå ýêñïåðèìåíòû ïî íàõîaeäåíèþ çíà÷åíèé ñîáñòâåííûõ ôóíêöèé âîçìóùåííîãî îäíîìåð-íîãî îïåðàòîðà Ëàïëàñà. Ðåçóëüòàòû ýêñïåðèìåíòà ïîêàçàëè âûñîêóþ âû÷èñëèòåëüíóþ ýôôåêòèâíîñòü ðàçðàáîòàííîãî ìåòîäà ñóììèðîâàíèÿ ðÿäîâ Ðåëåÿ Øðåäèíãåðà.Êëþ÷åâûå ñëîâà: âîçìóùåííûå îïåðàòîðû; ñîáñòâåííûå ÷èñëà; ñîáñòâåííûå ôóíêöèè; êðàòíûé ñïåêòð; ñóììû ôóíêöèîíàëüííûõ ðÿäîâ Ðåëåÿ Øðåäèíãåðà, ≪âçâåøåííûå≫ ïîïðàâêè òåîðèè âîçìóùåíèé. ÂâåäåíèåÐàññìîòðèì äèñêðåòíûé ïîëóîãðàíè÷åííûé ñíèçó îïåðàòîð T è îãðàíè÷åííûé îïåðàòîð P , çàäàííûå â ñåïàðàáåëüíîì ãèëüáåðòîâîì ïðîñòðàíñòâå H ñ îáëàñòüþ îïðåäåëåíèÿ â D. Ïðåäïîëîaeèì, ÷òî èçâåñòíû ñîáñòâåííûå ÷èñëà {λ n } ∞ n=1 îïåðà-òîðà T , çàíóìåðîâàííûå â ïîðÿäêå íåóáûâàíèÿ èõ âåëè÷èí, è îðòîíîðìèðîâàííûå ñîáñòâåííûå ôóíêöèè {v n (x)} ∞ n=1 (x ∈ D), îòâå÷àþùèå ýòèì ñîáñòâåííûì ÷èñëàì è îáðàçóþùèå áàçèñ â H. Îáîçíà÷èì ÷åðåç ν n êðàòíîñòü ñîáñòâåííîãî ÷èñëà λ n , à êîëè÷åñòâî âñåõ íåðàâíûõ äðóã äðóãó λ n , ëåaeàùèõ âíóòðè îêðóaeíîñòè T n 0 ðàäèó-ñîáñòâåííûå ÷èñëà îïåðàòîðà T + P , çàíóìåðîâàííûå â ïîðÿäêå íåóáûâàíèÿ èõ äåéñòâèòåëüíûõ ÷àñòåé, à {u n (x)} ∞ n=1 (x ∈ D) ñîîòâåòñòâóþùèå èì ñîáñòâåííûå ôóíêöèè. ðàáîòàõ [13] áûëà ïîëó÷åíà ñèñòåìà óðàâíåíèé:
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