Аналитическая природа функции Грина в окрестности простого полюса
DOI:
https://doi.org/10.26577/JMMCS-2019-4-m1Кілттік сөздер:
Оценка, полюс, собственные значения, интегро-дифференциальные условия, единственное решение, ряд Лорана, сопряженной оператор, собственная функция, возмущенная краевая задача, граничные условия, резольвента, базис РиссаАннотация
Известно, что функция Грина краевой задачи представляет мероморфную функцию от
спектрального параметра. Когда краевые условия содержат интегро-диференциальные
члены, то мероморфность функции Грина такой задачи также можно доказать. При
этом удается выписать структуру вычета в особых точках функции Грина краевой
задачи с интегродифференциальными возмущениями. Анализ структуры вычета позволяет
утверждать, что собственные функций исходного оператора достаточно гладкие функции.
Удивительно, что сопряженный оператор может иметь негладкие собственные функций.
В работе выяснена степень негладкости собственной функции сопряженного оператора
к оператору с интегро-диференциальными краевыми условиями. Указывается, что даже
сопряженные к многоточечным граничным задачам обладают негладкими собственными
функциями.
Библиографиялық сілтемелер
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[17] Stankevich M. "Ob odnoy obratnoy zadache spektralnogo analiza dlya obyiknove inogo differentsialnogo operatora chetnogo poryadka / M. Stankevich [On an inverse problem of spectral analysis for an ordinary other differential operator of even order]" , Bestnik MGU. Ser. Mat. Meh. No 4 (1981): 24-28.
[18] Ahtyamov A.M. "Obobscheniya teoremyi edinstvennosti Borga na sluchay nerazdelennyih granichnyih usloviy / A.M. Ahtyamov V.A. Sadovnichiy, Ya.T. Sultanaev [Generalizations of Borg‘s uniqueness theorem to the case of nonseparated boundary conditions]" , Evraziyskaya matematika Vol. 3, No 4 (2012): 10-22.
[19] Ahtyamov A.M. "Obratnaya zadacha dlya puchka operatorov s nerazdelennyimi granichnyimi usloviyami / A.M. Ahtyamov V.A. Sadovnichiy, Ya.T. Sultanaev [Inverse problem for an operator pencil with nonseparated boundary conditions]" , Evraziyskiy matem. Vol. 1, No 2 (2010): 5-16.
[20] Sadovnichiy V.A. "Teorema edinstvennosti resheniya obratnoy zadachi spektralnogo analiza v sluchae differentsialnogo uravneniya s periodicheskimi granichnyimi usloviyami [Uniqueness theorem for the inverse problem of spectral analysis in the case of differential equations with periodic boundary conditions]" , Differents. uravneniya Vol. 9, No 2 (1973): 271–277.
[2] Keselman G.M. "Obezuslovnoy shodimosti razlozheniy po sobstvennyim funktsiyam nekotoryih differentsialnyih operatorov [Unconditional convergence of expansions in eigenfunctions of certain differential operators]" , Izv. Vuzov SSSR, Matematika No 2 (1964): 82-93.
[3] Naymark M.A. Lineynyie differentsialnyie operatoryi [Linear differential operators] (M.: 1969): 528.
[4] Shkalikov A.A. "O bazisnosti sobstvennyih funktsiy obyiknovennyih differentsialnyih operatorov s integralnyimi kraevyimi usloviyami [On the basis property of the Eigen functions of ordinary differential operators with integral boundary conditions]" , Vestn. MGU. Ser. Mat. Meh. No 6 (1982): 12-21.
[5] Kokebaev B.K., Otelbaev M., Shyinyibekov A.N. "K voprosam rasshireniy i suzheniy operatorov [To questions of extensions and restrictions of operators]" , Dokl. AN SSSR No 6 (271) (1983): 1307-1313.
[6] Kanguzhin B.E., Dairbaeva G., Madibayulyi Zh. "Identifikatsiya granichnyih usloviy differentsialnogo operatora [Identification of the boundary conditions of a differential operator]" , Vestnik KazNU. Seriya matematika, mehanika, informatika. No 3 (103) (2019): 13-18
[7] Dezin A.A. "Differentsialno-operatornyie uravneniya. Metod modelnyih operatorov v teorii granichnyih zadach [Differential operator equations. The method of model operators in the theory of boundary value problems]" , Tr. MIAN. M., Nauka. MAIK «Nauka/Interperiodika» No 229 (2000): 3-175
[8] Levitan B.M. "Obratnaya zadacha dlya operatora Shturma-Liuvillya v sluchae konechno-zonnyih i beskonechno-zonnyih potentsialov [The inverse problem for the Sturm-Liouville operator in the case of finite-band and infinitely-band potentials]" , Trudyi Mosk. Matem.ob-vo. MGU. M. (1982): 3-36.
[9] Berezanskiy Yu.M. Razlozhenie po sobstvennyim funktsiyam [Expansion in eigenfunctions] (M.-L.: 1950).
[10] Berezanskiy Yu.M. Razlozhenie po sobstvennyim funktsiyam samosopryazhennyih operatorov [Expansion in eigenfunctions of self-adjoint operators] (Kiev: Naukova Dumka, 1965): 798.
[11] Marchenko V.A. Operatoryi Shturma-Luivillya i ih prilozheniya [Sturm-Louisville Operators and their Applications] (Kiev: Naukova Dumka, 1977): 329.
[12] Kato T. Teoriya vozmuschenniy lineynyih operatorov [Perturbation theory of linear operators] (M.: Mir, 1972): 740. [13] Leybenzon Z.L. "Obratnaya zadacha spektralnogo analiza obyiknovennyih differentsialnyih operatorov vyisshih poryadkov / Z.L. Leybenzon [The inverse problem of spectral analysis of ordinary differential operators of higher orders]" , Trudyi Moskov. mat. ob-va Vol. 15 (1966): 70-144.
[14] Yurko V.A. "Obratnaya zadacha dlya differentsialnyih operatorov vtorogo poryadka s regulyarnyimi kraevyimi usloviyami / V.A. Yurko [The inverse problem for second-order differential operators with regular boundary conditions]" , Mat. zametki Vol. 18, No 4 (1975): 569-576.
[15] Sadovnichiy V.A. "O svyazi mezhdu spektrom differentsialnogo operatora s simmetrichnyimi koeffitsentami i kraevyimi usloviyami / V.A. Sadovnichiy, B.E. Kanguzhin [On the relationship between the spectrum of a differential operator with symmetric coefficients and boundary conditions]" , DAN SSSR Vol. 267, No 2 (1982): 310-313.
[16] Shkalikov A.A. "O bazisnosti sobstvennyih funktsiy obyiknovennyih differentsialnyih operatorov s integralnyimi kraevyimi usloviyami / A.A. Shkalikov [On the basis property of the eigenfunctions of ordinary differential operators with integral boundary conditions]" , Vestnik MGU. Ser. Mat. Meh. No 6 (1982): 12-21
[17] Stankevich M. "Ob odnoy obratnoy zadache spektralnogo analiza dlya obyiknove inogo differentsialnogo operatora chetnogo poryadka / M. Stankevich [On an inverse problem of spectral analysis for an ordinary other differential operator of even order]" , Bestnik MGU. Ser. Mat. Meh. No 4 (1981): 24-28.
[18] Ahtyamov A.M. "Obobscheniya teoremyi edinstvennosti Borga na sluchay nerazdelennyih granichnyih usloviy / A.M. Ahtyamov V.A. Sadovnichiy, Ya.T. Sultanaev [Generalizations of Borg‘s uniqueness theorem to the case of nonseparated boundary conditions]" , Evraziyskaya matematika Vol. 3, No 4 (2012): 10-22.
[19] Ahtyamov A.M. "Obratnaya zadacha dlya puchka operatorov s nerazdelennyimi granichnyimi usloviyami / A.M. Ahtyamov V.A. Sadovnichiy, Ya.T. Sultanaev [Inverse problem for an operator pencil with nonseparated boundary conditions]" , Evraziyskiy matem. Vol. 1, No 2 (2010): 5-16.
[20] Sadovnichiy V.A. "Teorema edinstvennosti resheniya obratnoy zadachi spektralnogo analiza v sluchae differentsialnogo uravneniya s periodicheskimi granichnyimi usloviyami [Uniqueness theorem for the inverse problem of spectral analysis in the case of differential equations with periodic boundary conditions]" , Differents. uravneniya Vol. 9, No 2 (1973): 271–277.
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Как цитировать
Ghulam Hazrat, A. R., Auzerkhan, G. S., & Beisenbay, A. A. (2019). Аналитическая природа функции Грина в окрестности простого полюса. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 104(4), 3–11. https://doi.org/10.26577/JMMCS-2019-4-m1
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