Geodesic
On the oscillation theory of the Sturm–Liouville problem with singular coefficients
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1609-1621 
A. A. Vladimirov. On the oscillation theory of the Sturm–Liouville problem with singular coefficients. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1609-1621. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a7/
@article{ZVMMF_2009_49_9_a7,
     author = {A. A. Vladimirov},
     title = {On the oscillation theory of the {Sturm{\textendash}Liouville} problem with singular coefficients},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1609--1621},
     year = {2009},
     volume = {49},
     number = {9},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a7/}
}
TY  - JOUR
AU  - A. A. Vladimirov
TI  - On the oscillation theory of the Sturm–Liouville problem with singular coefficients
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2009
SP  - 1609
EP  - 1621
VL  - 49
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a7/
LA  - ru
ID  - ZVMMF_2009_49_9_a7
ER  - 
%0 Journal Article
%A A. A. Vladimirov
%T On the oscillation theory of the Sturm–Liouville problem with singular coefficients
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2009
%P 1609-1621
%V 49
%N 9
%U http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a7/
%G ru
%F ZVMMF_2009_49_9_a7

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

The spectral Sturm–Liouville problem with distribution coefficients is examined. It is shown that the basic results concerning the number and the location of the zeros of eigenfunctions that are known in the smooth case remain valid in the general situation. The Chebyshev properties of systems of eigenfunctions are also investigated in the case where the weight function is positive.

[1] Rofe-Beketov F. S., Kholkin A. M., Spektralnyi analiz differentsialnykh operatorov, Mariupol, 2001

[2] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[3] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Tr. Mosk. matem. ob-va, 64, 2003, 159–212 | MR | Zbl

[4] Vladimirov A. A., “O skhodimosti posledovatelnostei obyknovennykh differentsialnykh operatorov”, Matem. zametki, 75:6 (2004), 941–943 | Zbl

[5] Binding P. A., Volmer H., “Oscillation theory for Sturm-Liouville problems with indefinite coefficients”, Proc. Roy. Soc. Edinburg., 131 (2001), 989–1002 | DOI | MR | Zbl

[6] Gantmakher F. R., Krein M. G., Ostsillyatsionnye matritsy i yadra i malye kolebaniya mekhanicheskikh sistem, Gostekhteorizdat, M.-L., 1950

[7] Borovskikh A. B., Pokornyi Yu. V., “Sistemy Chebysheva–Khaara v teorii razryvnykh yader Kelloga”, Uspekhi matem. nauk, 49:3 (1994), 3–42 | MR | Zbl

[8] Pokornyi Yu. V., Zvereva M. B., Ischenko A. C., Shabrov C. A., “O neregulyarnom rasshirenii ostsillyatsionnoi teorii spektralnoi zadachi Shturma–Liuvillya”, Matem. zametki, 82:4 (2007), 578–582 | MR | Zbl

[9] Riss F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR

[10] Kats I. S., Krein M. G., “O spektralnykh funktsiyakh struny”, F. Atkinson. Diskretnye i nepreryvnye granichnye zadachi, Mir, M., 1968, 648–733 | MR

[11] Levin A. Yu., Stepanov G. D., “Odnomernye kraevye zadachi s operatorami, ne ponizhayuschimi chisla peremen znaka”, Sibirskii matem. zhurnal, 17:3 (1976), 606–605 ; 17:4, 813–830 | MR | MR | Zbl

[12] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[13] Vladimirov A. A., Sheipak I. A., “Samopodobnye funktsii v prostranstve $L_2[0,1]$ i zadacha Shturma–Liuvillya s singulyarnym indefinitnym vesom”, Matem. sb., 197:11 (2006), 13–30 | MR | Zbl

[14] Vladimirov A. A., “O vychislenii sobstvennykh znachenii zadachi Shturma–Liuvillya s fraktalnym indefinitnym vesom”, Zh. vychisl. matem. i matem. fiz., 47:8 (2007), 1350–1355 | MR