Geodesic
The theory of singular perturbations in the case of spectral singularities of a limit operator
Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 541-555 
A. G. Eliseev; S. A. Lomov. The theory of singular perturbations in the case of spectral singularities of a limit operator. Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 541-555. http://geodesic.mathdoc.fr/item/SM_1988_59_2_a16/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A new method of asymptotic integration is developed – the method of regularization – in the case when the spectrum of the variable limit operator is zero at isolated points. To describe the singular dependence of a solution on the perturbation, additional independent variables are introduced; the space of resonance-free solutions is introduced, in which the coefficients of regularized series (the solution of the extended problem) are defined. Asymptotic convergence of the series thus obtained to the exact solution of the original singularly perturbed problem is proved. Bibliography: 14 titles.

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[5] Lomov S. A., “Asimptoticheskoe integrirovanie pri izmenenii kharaktera spektra”, Tr. MEI, 357 (1978), 56–62 | MR | Zbl

[6] Bobochko V. N., Lomov S. A., “Vnutrennii pogranichnyi sloi”, Tr. MEI, 499 (1980), 57–60 | MR | Zbl

[7] Rasschepkina N. A., Asimptoticheskoe integrirovanie zadachi Koshi v usloviyakh nestabilnosti spektra, Metody malogo parametra i ikh prilozhenie, Minsk, 1982

[8] Bobochko V. N., “Zadacha Valle–Pussena dlya sistemy singulyarno vozmuschennykh differentsialnykh uravnenii s nestabilnym spektrom predelnogo operatora”, Ukr. matem. zhurn., XVII:2 (1984), 34–45

[9] Dzyadyk V. K., “Nekotorye spetsialnye funktsii i ikh rol pri reshenii neodnorodnykh differentsialnykh uravnenii s tochkoi povorota”, Teoriya funktsii i ee prilozheniya, Naukova dumka, Kiev, 1979 | Zbl

[10] Dzyadyk S. Yu., “Asimptoticheskoe predstavlenie reshenii singulyarno vozmuschennykh differentsialnykh uravnenii s tochkoi povorota”, Teoriya funktsii i ee prilozheniya, Naukova dumka, Kiev, 1979

[11] Lomov S. A., “Asimptoticheskie resheniya singulyarno vozmuschennykh zadach”, DAN, 265:3 (1982), 529–532 | MR

[12] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966 | MR

[13] Sadovnichii V. A., Teoriya operatorov, Izd-vo MGU, M., 1979 | MR | Zbl

[14] Lioville J., “Second memoire sur le development des fonctions en series dont divers termes sont assujettis, a une meme equation”, J. Math. Pure Appl., 2 (1837), 16–35