Geodesic
A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues
Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 129-145 
V. A. Sadovnichii; V. E. Podolskii. A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues. Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 129-145. http://geodesic.mathdoc.fr/item/SM_1998_189_1_a6/
@article{SM_1998_189_1_a6,
     author = {V. A. Sadovnichii and V. E. Podolskii},
     title = {A~class of {Sturm{\textendash}Liouville} operators and approximate calculation of the first eigenvalues},
     journal = {Sbornik. Mathematics},
     pages = {129--145},
     year = {1998},
     volume = {189},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_1_a6/}
}
TY  - JOUR
AU  - V. A. Sadovnichii
AU  - V. E. Podolskii
TI  - A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues
JO  - Sbornik. Mathematics
PY  - 1998
SP  - 129
EP  - 145
VL  - 189
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1998_189_1_a6/
LA  - en
ID  - SM_1998_189_1_a6
ER  - 
%0 Journal Article
%A V. A. Sadovnichii
%A V. E. Podolskii
%T A class of Sturm–Liouville operators and approximate calculation of the first eigenvalues
%J Sbornik. Mathematics
%D 1998
%P 129-145
%V 189
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1998_189_1_a6/
%G en
%F SM_1998_189_1_a6

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

A special class $S$ of Sturm–Liouville operators with simple asymptotic properties of eigenfunctions is investigated. The analytic properties of the potentials are analyzed and the operators in this class are described in terms of the transition function of the inverse problem. The following result is established: the class $S$ is dense in the set of Sturm–Liouville operators with potentials in $L_2$. A subset of $S$ that also has the density property is effectively distinguished. Based on the properties of the operators in this subset, a method of the approximate evaluation of the first eigenvalues of a Sturm–Liouville operator through its regularized traces is proposed and substantiated.

[1] Dorodnitsyn A. A., “Asimptoticheskie zakony raspredeleniya sobstvennykh znachenii dlya nekotorykh osobykh vidov differentsialnykh uravnenii vtorogo poryadka”, UMN, 7:6 (1952), 3–96 | MR | Zbl

[2] Gelfand I. M., Levitan B. M., “Ob odnom prostom tozhdestve dlya sobstvennykh znachenii differentsialnogo operatora vtorogo poryadka”, Dokl. AN SSSR, 88 (1953), 593–596 | MR | Zbl

[3] Gelfand I. M., “O tozhdestvakh dlya sobstvennykh znachenii differentsialnogo operatora vtorogo poryadka”, UMN, 11:1 (1956), 191–198 | MR | Zbl

[4] Dikii L. A., “Dzeta-funktsiya obyknovennogo differentsialnogo uravneniya na konechnom otrezke”, Izv. AN SSSR. Ser. matem., 19:4 (1955), 187–200 | MR | Zbl

[5] Lidskii V. B., Sadovnichii V. A., “Regulyarizovannye summy kornei odnogo klassa tselykh funktsii”, Funkts. analiz i ego prilozh., 1:2 (1967), 52–59 | MR | Zbl

[6] Lidskii V. B., Sadovnichii V. A., “Formuly sledov v sluchae uravneniya Orra–Zommerfelda”, Izv. AN SSSR. Ser. matem., 32:3 (1968), 633–648 | MR | Zbl

[7] Dikii L. A., “Novyi sposob priblizhennogo vychisleniya sobstvennykh chisel zadachi Shturma–Liuvillya”, Dokl. AN SSSR, 116:1 (1957), 12–14 | MR | Zbl

[8] Shkarin S. A., “O sposobe Gelfanda–Dikogo vychisleniya pervykh sobstvennykh znachenii operatora Shturma–Liuvillya”, Vestnik MGU. Ser. 1. Matem., mekh., 1996, no. 1, 39–44 | MR

[9] Sadovnichii V. A., Podolskii V. E., “O vychislenii pervykh sobstvennykh znachenii operatora Shturma–Liuvillya”, Dokl. AN, 346:2 (1996), 162–164 | MR

[10] Marchenko V. A., “Nekotorye voprosy teorii odnomernykh lineinykh differentsialnykh operatorov vtorogo poryadka, I”, Tr. MMO, 1, URSS, M., 1952, 327–420 | MR

[11] Vittikh G., Noveishie issledovaniya po odnoznachnym analiticheskim funktsiyam, Fizmatgiz, M., 1960 | MR

[12] Levitan B. M., Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR

[13] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR