najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-2-135-141najo-350Research ArticleMATHEMATICSМАТЕМАТИКАNon-compact perturbations of the spectrum of multipliers given with functionsНекомпактные возмущения спектра заданных мультипликаторов с функциямиKucharovR. R.KucharovR. R.

100174, Tashkent

100174, Tashkent

ramz3364647@yahoo.comKhamraevaR. R.KhamraevaR. R.

100174, Tashkent,

100010, 12, Istiqbol str., Tashkent

100174, Tashkent,

100010, 12, Istiqbol str., Tashkent

r.khamraeva@wiut.uzNational University of UzbekistanNational University of UzbekistanNational University of Uzbekistan; Westminster International University in TashkentNational University of Uzbekistan; Westminster International University in Tashkent202128072025122135141Copyright © Kucharov R.R., Khamraeva R.R., 20252025Kucharov R.R., Khamraeva R.R.Kucharov R.R., Khamraeva R.R.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/350

The change in the spectrum of the multipliers H0 f (x, y) = xa + yβ f (x, y) and H0 f (x, y) = xayβ f (x, y) for perturbation with partial integral operators in the spaces L2 [0, 1]2 is studied. Precise description of the essential spectrum and the existence of simple eigenvalue is received. We prove that the number of eigenvalues located below the lower edge of the essential spectrum in the model is finite.

Изучается изменение спектра множителей H0 f(x;y) = xα + yβ f(x;y) и H0 f(x;y) = xαyβ f(x;y) при возмущении операторами в частных интегралах в пространствах L2[0; 1]2. Получено точное описание существенного спектра и существование простого собственного значения. Доказано, что число собственных значений, расположенных ниже нижнего края существенного спектра в модели, конечно.

существенный спектрдискретный спектрнижняя граница существенного спектраоператор частного интегралаessential spectrumdiscrete spectrumlower bound of the essential spectrumpartial integral operatorReferencesUchiyama J. Finiteness of the Number of Discrete Eigenvalues of the Schrodinger Operator for a Three Particle System. 1969, Publ. Res. Inst. Math. Sci., 5 (1), P. 51–63.Uchiyama J. Finiteness of the Number of Discrete Eigenvalues of the Schrodinger Operator for a Three Particle System. 1969, Publ. Res. Inst. Math. Sci., 5 (1), P. 51–63.Uchiyama J. Corrections to ”Finiteness of the Number of Discrete Eigenvalues of the Schrodinger Operator for a Three Particle System”. Publ. Res. Inst. Math. Sci., 1970, 6 (1), P. 189–192.Uchiyama J. Corrections to ”Finiteness of the Number of Discrete Eigenvalues of the Schrodinger Operator for a Three Particle System”. Publ. Res. Inst. Math. Sci., 1970, 6 (1), P. 189–192.Uchiyama J. Finiteness of the Number of Discrete Eigenvalues of the Schrodinger Operator for a Three Particle System. Publ. Res. Inst. Math. Sci., 1970, 6 (1), P. 193–200.Uchiyama J. Finiteness of the Number of Discrete Eigenvalues of the Schrodinger Operator for a Three Particle System. Publ. Res. Inst. Math. Sci., 1970, 6 (1), P. 193–200.Zhislin G.M. On the finiteness of the discrete spectrum of the energy operator of negative atomic and molecular ions. Theor. Math. Phys., 1971, 7, P. 571–578.Zhislin G.M. On the finiteness of the discrete spectrum of the energy operator of negative atomic and molecular ions. Theor. Math. Phys., 1971, 7, P. 571–578.Appell J., Frolova E.V., Kalitvin A.S., Zabrejko P.P. Partial integral operators on C(a, b. × c, d.). Integral Equ. Oper. theory, 1997, 27, P. 125–140.Appell J., Frolova E.V., Kalitvin A.S., Zabrejko P.P. Partial integral operators on C(a, b. × c, d.). Integral Equ. Oper. theory, 1997, 27, P. 125–140.Faddeev L.D. On a model of Friedrichs in the theory of perturbations of the continuous spectrum. Trudy Mat. Inst. Steklov, 1964, 73, 292 in Russian..Faddeev L.D. On a model of Friedrichs in the theory of perturbations of the continuous spectrum. Trudy Mat. Inst. Steklov, 1964, 73, 292 in Russian..Albeverio S., Lakaev S.N., Muminov Z.I. On the number of eigenvalues of a model operator associated to a system of three-particles on lattices. Russ. J. Math. Phys., 2007, 14 (4), P. 377—387.Albeverio S., Lakaev S.N., Muminov Z.I. On the number of eigenvalues of a model operator associated to a system of three-particles on lattices. Russ. J. Math. Phys., 2007, 14 (4), P. 377—387.Rasulov T.Kh. Asymptotics of the discrete spectrum of a model operator assotiated with a system of three particles on a lattice. Theor. and Math. Phys., 2010, 163 (1), P. 429–437.Rasulov T.Kh. Asymptotics of the discrete spectrum of a model operator assotiated with a system of three particles on a lattice. Theor. and Math. Phys., 2010, 163 (1), P. 429–437.Eshkabilov Yu.Kh., Kucharov R.R. Essential and discrete spectra of the three-particle Schrodinger operator on a lattice. Theor. Math. Phys., 2012, 170 (3), P. 341-–353.Eshkabilov Yu.Kh., Kucharov R.R. Essential and discrete spectra of the three-particle Schrodinger operator on a lattice. Theor. Math. Phys., 2012, 170 (3), P. 341-–353.Eshkabilov Yu.Kh., Kucharov R.R. Efimov’s effect for partial integral operators of Fredholm type. Nanosystems: Physics, Chemistry, Mathematics, 2013, 4 (4), P. 529–537.Eshkabilov Yu.Kh., Kucharov R.R. Efimov’s effect for partial integral operators of Fredholm type. Nanosystems: Physics, Chemistry, Mathematics, 2013, 4 (4), P. 529–537.Eshkabilov Yu.Kh. On infinity of the discrete spectrum of operators in the Friedrichs model. Siberian Adv. Math., 2012, 22 (1).Eshkabilov Yu.Kh. On infinity of the discrete spectrum of operators in the Friedrichs model. Siberian Adv. Math., 2012, 22 (1).Reed M., Simon B. Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Acad. Press, New York, 1982.Reed M., Simon B. Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Acad. Press, New York, 1982.Eshkabilov Yu.Kh. Efimov’s effect for a 3-particle model discrete Schrodinger operator. Theor. Math. Phys., 2010, 164 (1), P. 896-–904.Eshkabilov Yu.Kh. Efimov’s effect for a 3-particle model discrete Schrodinger operator. Theor. Math. Phys., 2010, 164 (1), P. 896-–904.Eshkabilov Yu.Kh. On a discrete “three-particle” Schrodinger operator in the Hubbard model. Theor. Math. Phys., 2006, 149 (2), P. 1497–1511.Eshkabilov Yu.Kh. On a discrete “three-particle” Schrodinger operator in the Hubbard model. Theor. Math. Phys., 2006, 149 (2), P. 1497–1511.

The authors declare that there are no conflicts of interest present.