The analytical nature of the Green’s function in the vicinity of a simple pole
DOI:
https://doi.org/10.26577/JMMCS-2019-4-m1Keywords:
Estimate, pole, eigenvalues, integro-differential conditions, unique solution, Laurent series, adjoint operator, eigenfunction, perturbed boundary, value problem, boundary conditions, Green’s function, resolution, Riesz basis, simple zeroAbstract
It is known that the Green function of a boundary value problem is a meromorphic function
of a spectral parameter. When the boundary conditions contain integro-differential terms,
then the meromorphism of the Green’s function of such a problem can also be proved. In
this case, it is possible to write out the structure of the residue at the singular points of the
Green’s function of the boundary value problem with integro-differential perturbations. An
analysis of the structure of the residue allows us to state that the eigenfunction functions of the
original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have
non-smooth eigenfunctions. The degree of non-smoothness of the eigenfunction of the adjoint
operator to an operator with integro-differential boundary conditions is clarified. It is indicated
that even those conjugate to multipoint boundary value problems have non-smooth eigenfunctions.
References
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[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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How to Cite
Ghulam Hazrat, A. R., Auzerkhan, G. S., & Beisenbay, A. A. (2019). The analytical nature of the Green’s function in the vicinity of a simple pole. Journal of Mathematics, Mechanics and Computer Science, 104(4), 3–11. https://doi.org/10.26577/JMMCS-2019-4-m1
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