Geodesic
Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary-value problem on a segment with a summable potential
Izvestiya. Mathematics, Tome 64 (2000) no. 4, pp. 695-754 
V. A. Vinokurov; V. A. Sadovnichii. Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary-value problem on a segment with a summable potential. Izvestiya. Mathematics, Tome 64 (2000) no. 4, pp. 695-754. http://geodesic.mathdoc.fr/item/IM2_2000_64_4_a1/
@article{IM2_2000_64_4_a1,
     author = {V. A. Vinokurov and V. A. Sadovnichii},
     title = {Asymptotics of any order for the eigenvalues and eigenfunctions of the {Sturm{\textendash}Liouville} boundary-value problem on a~segment with a~summable potential},
     journal = {Izvestiya. Mathematics},
     pages = {695--754},
     year = {2000},
     volume = {64},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2000_64_4_a1/}
}
TY  - JOUR
AU  - V. A. Vinokurov
AU  - V. A. Sadovnichii
TI  - Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary-value problem on a segment with a summable potential
JO  - Izvestiya. Mathematics
PY  - 2000
SP  - 695
EP  - 754
VL  - 64
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/IM2_2000_64_4_a1/
LA  - en
ID  - IM2_2000_64_4_a1
ER  - 
%0 Journal Article
%A V. A. Vinokurov
%A V. A. Sadovnichii
%T Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary-value problem on a segment with a summable potential
%J Izvestiya. Mathematics
%D 2000
%P 695-754
%V 64
%N 4
%U http://geodesic.mathdoc.fr/item/IM2_2000_64_4_a1/
%G en
%F IM2_2000_64_4_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

For the Sturm–Liouville boundary-value problem on a segment we construct asymptotics for $s_n=\sqrt{\lambda_n}$, where $\lambda_n$ are the eigenvalues, and for the normalized eigenfunctions $y_n(x)$ of the form $$ s_n=s_{n,m}(q)+\psi_{n,m}, \qquad y_n(x)=y_{n,m}(q,x)+\Delta y_{n,m}(x) $$ for any $m=0,1,2,\dots$, where $s_{n,m}(q)$ and $y_{n,m}(q,x)$ are expressed explicitly in terms of the potential $q(x)$. Under the assumption that $q(x)$ is a real summable function, the terms $\psi_{n,m}$ and $\Delta y_{n,m}(x)$ are $O\biggl(\dfrac1{n^{m+1}}\biggr)$ as $n\to\infty$.

[1] Vinokurov V. A., “Logarifm resheniya lineinogo differentsialnogo uravneniya, formula Khausdorfa i zakony sokhraneniya”, DAN SSSR, 319:4 (1991), 792–797 | MR | Zbl

[2] Sadovnichii V. A., Teoriya operatorov, Izd-vo MGU, M., 1986 | MR

[3] Kollatts L., Zadachi na sobstvennye znacheniya, Nauka, M., 1968

[4] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971 | MR

[5] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970 | MR | Zbl

[6] Marchenko V. A., Operatory Shturma–Liuvillya i ikh prilozheniya, Nauk. dumka, Kiev, 1977 | MR

[7] Landau L. D., Lifshits E. M., Kvantovaya mekhanika, Nauka, M., 1974

[8] Kim G., Khushev A. V., “Metod Shturma–Liuvillya i problema kvazisvyazannykh sostoyanii”, Izv. RAN. Ser. fiz., 1996, no. 5, 123–125

[9] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967 | MR

[10] Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR

[11] Khorn R., Dzhonson I., Matrichnyi analiz, Mir, M., 1989 | MR

[12] Shvarts L., Analiz, T. 1, Mir, M., 1972

[13] Vinokurov V. A., “Yavnoe reshenie lineinogo obyknovennogo differentsialnogo uravneniya i osnovnoe svoistvo eksponenty”, Diff. uravn., 33:3 (1997), 302–308 | MR | Zbl

[14] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR | Zbl