Geodesic
Asymptotics of the Solutions of the Sturm–Liouville Equation with Singular Coefficients
Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 832-841 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We obtain asymptotic representations as $\lambda \to \infty$ in the upper and lower half-planes for the solutions of the Sturm–Liouville equation $$ -y''+p(x)y'+q(x)y= \lambda ^2 \rho(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ under the condition that $q$ is a distribution of first-order singularity, $\rho$ is a positive absolutely continuous function, and $p$ belongs to the space $L_2[a,b]$.
Mots-clés : Sturm–Liouville equation, singular coefficient
Keywords: asymptotic solution, Volterra integral operator, fundamental system of solutions, space of bounded functions. 
@article{MZM_2015_98_6_a3,
     author = {V. E. Vladikina and A. A. Shkalikov},
     title = {Asymptotics of the {Solutions} of the {Sturm{\textendash}Liouville} {Equation} with {Singular} {Coefficients}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {832--841},
     year = {2015},
     volume = {98},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a3/}
}
TY  - JOUR
AU  - V. E. Vladikina
AU  - A. A. Shkalikov
TI  - Asymptotics of the Solutions of the Sturm–Liouville Equation with Singular Coefficients
JO  - Matematičeskie zametki
PY  - 2015
SP  - 832
EP  - 841
VL  - 98
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a3/
LA  - ru
ID  - MZM_2015_98_6_a3
ER  - 
%0 Journal Article
%A V. E. Vladikina
%A A. A. Shkalikov
%T Asymptotics of the Solutions of the Sturm–Liouville Equation with Singular Coefficients
%J Matematičeskie zametki
%D 2015
%P 832-841
%V 98
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a3/
%G ru
%F MZM_2015_98_6_a3
V. E. Vladikina; A. A. Shkalikov. Asymptotics of the Solutions of the Sturm–Liouville Equation with Singular Coefficients. Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 832-841. http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a3/

[1] G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter”, Trans. Amer. Math Soc., 9 (1908), 219–231 | DOI | MR | Zbl

[2] G. D. Birkhoff, “Boundary value and expansion problem of ordinary linear differential equations”, Trans. Amer. Math Soc., 9 (1908), 373–395 | DOI | MR | Zbl

[3] Ya. D. Tamarkin, O nekotorykh obschikh zadachakh teorii obyknovennykh lineinykh differentsialnykh uravnenii, Tipografiya M. P. Frolovoi, Petrograd, 1917

[4] I. M. Rapoport, O nekotorykh asimptoticheskikh metodakh v teorii differentsialnykh uravnenii, Izd-vo AN USSR, Kiev, 1954 | MR

[5] M. A. Naimark, Lineinye differentsialnye operatory, Klassika i sovremennost, Fizmatlit, M., 2010 | MR | Zbl

[6] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Tr. MMO, 64, 2003, 159–212 | MR | Zbl

[7] A. M. Savchuk, A. A. Shkalikov, “Obratnye zadachi dlya operatora Shturma–Liuvillya s potentsialami iz prostranstv Soboleva. Ravnomernaya ustoichivost”, Funkts. analiz i ego pril., 44:4 (2010), 34–53 | DOI | MR | Zbl

[8] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 66:6 (1999), 897–912 | DOI | MR | Zbl