Geodesic
Trace of Order~$(-1)$ for a String with Singular Weight
Matematičeskie zametki, Tome 102 (2017) no. 2, pp. 197-215.

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

The Sturm–Liouville problem on a finite closed interval with potential and weight of first order of singularity is studied. Estimates for the $s$-numbers and eigenvalues of the corresponding integral operator are obtained. The spectral trace of first negative order is evaluated in terms of the integral kernel. The obtained theoretical results are illustrated by examples.
Keywords: spectral asymptotics, spectral trace.
Mots-clés : Sturm–Liouville problem 
@article{MZM_2017_102_2_a3,
     author = {A. S. Ivanov and A. M. Savchuk},
     title = {Trace of {Order~}$(-1)$ for a {String} with {Singular} {Weight}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {197--215},
     publisher = {mathdoc},
     volume = {102},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_2_a3/}
}
TY  - JOUR
AU  - A. S. Ivanov
AU  - A. M. Savchuk
TI  - Trace of Order~$(-1)$ for a String with Singular Weight
JO  - Matematičeskie zametki
PY  - 2017
SP  - 197
EP  - 215
VL  - 102
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2017_102_2_a3/
LA  - ru
ID  - MZM_2017_102_2_a3
ER  - 
%0 Journal Article
%A A. S. Ivanov
%A A. M. Savchuk
%T Trace of Order~$(-1)$ for a String with Singular Weight
%J Matematičeskie zametki
%D 2017
%P 197-215
%V 102
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2017_102_2_a3/
%G ru
%F MZM_2017_102_2_a3
A. S. Ivanov; A. M. Savchuk. Trace of Order~$(-1)$ for a String with Singular Weight. Matematičeskie zametki, Tome 102 (2017) no. 2, pp. 197-215. http://geodesic.mathdoc.fr/item/MZM_2017_102_2_a3/

[1] M. G. Krein, “Ob odnom obobschenii issledovanii Stiltesa”, DAN SSSR, 87:6 (1952), 881–884 | MR | Zbl

[2] M. G. Krein, I. S. Kats, “On the spectral functions of the string”, Amer. Math. Soc. Transl., 103:2 (1974), 19–102 | Zbl

[3] W. Feller, “Generalized second order differential operators and their lateral conditions”, Illinois J. Math., 1 (1957), 459–504 | MR | Zbl

[4] Y. Kasahara, “Spectral theory of generalized second order differential operators and its applications to Markov processes”, Japan J. Math. (N.S.), 1:1 (1975), 67–84 | MR | Zbl

[5] H. Dym, H. P. McKean, Gaussian Processes, Function Theory and the Inverse Spectral Problem, Probab. and Math. Stat., 31, Academic Press, New York, 1976 | MR | Zbl

[6] I. S. Kats, “The spectral theory of a string”, Ukrainian Math. J., 46:3 (1994), 159–182 | DOI | MR | Zbl

[7] S. Kotani, S. Watanabe, “Krein's spectral theory of strings and generalized diffusion processes”, Functional Analysis in Markov Processes, Lecture Notes in Math., 923, Springer-Verlag, Berlin, 1982, 235–259 | DOI | MR | Zbl

[8] A. Fleige, Spectral Theory of Indefinite Krein–Feller Differential Operators, Math. Res., 98, Akademie Verlag, Berlin, 1996 | MR | Zbl

[9] A. Zettl, Sturm–Liouville Theory, Math. Surveys Monogr., 121, Amer. Math. Soc., Providence, RI, 2005 | MR | Zbl

[10] J. Eckhardt, A. Kostenko, “The inverse spectral problem for indefinite strings”, Invent. Math., 204:3 (2016), 939–977 | DOI | MR | Zbl

[11] A. Constantin, V. S. Gerdjikov, R. I. Ivanov, “Inverse scattering transform for the Camassa–Holm equation”, Inverse Problems, 22:6 (2006), 2197–2207 | DOI | MR | Zbl

[12] C. Bennewitz, B. M. Brown, R. Weikard, “Scattering and inverse scattering for a left-definite Sturm–Liouville problem”, J. Differential Equations, 253:8 (2012), 2380–2419 | DOI | MR | Zbl

[13] J. Eckhardt, A. Kostenko, “An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation”, Comm. Math. Phys., 329:3 (2014), 893–918 | DOI | MR | Zbl

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

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

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

[17] A. A. Vladimirov, I. A. Sheipak, “Indefinitnaya zadacha Shturma–Liuvillya dlya nekotorykh klassov samopodobnykh singulyarnykh vesov”, Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz, Tr. MIAN, 255, Nauka, M., 2006, 88–98 | MR | Zbl

[18] I. A. Sheipak, “O konstruktsii i nekotorykh svoistvakh samopodobnykh funktsii v prostranstvakh $L_p[0,1]$”, Matem. zametki, 81:6 (2007), 924–938 | DOI | MR | Zbl

[19] A. A. Vladimirov, I. A. Sheipak, “Asimptotika sobstvennykh znachenii zadachi Shturma–Liuvillya s diskretnym samopodobnym vesom”, Matem. zametki, 88:5 (2010), 662–672 | DOI | MR | Zbl

[20] A. A. Vladimirov, I. A. Sheipak, “O zadache Neimana dlya uravneniya Shturma–Liuvillya s samopodobnym vesom kantorovskogo tipa”, Funkts. analiz i ego pril., 47:4 (2013), 18–29 | DOI | MR | Zbl

[21] M. Solomyak, E. Verbitsky, “On a spectral problem related to self-similar measures”, Bull. London Math. Soc., 27:3 (1995), 242–248 | DOI | MR | Zbl

[22] A. I. Nazarov, “Logarifmicheskaya asimptotika malykh uklonenii dlya nekotorykh gaussovskikh sluchainykh protsessov v $L_2$-norme otnositelno samopodobnoi mery”, Veroyatnost i statistika. 7, Zap. nauchn. sem. POMI, 311, POMI, SPb., 2004, 190–213 | MR | Zbl

[23] A. A. Vladimirov, “Nekotorye zamechaniya ob integralnykh kharakteristikakh vinerovskogo protsessa”, Dalnevost. matem. zhurn., 15:2 (2015), 156–165 | Zbl

[24] V. A. Sadovnichii, V. E. Podolskii, “Sledy operatorov”, UMN, 61:5 (371) (2006), 89–156 | DOI | MR | Zbl

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

[26] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR | Zbl

[27] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR | Zbl

[28] F. Khartman, Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[29] A. A. Vladimirov, “K ostsillyatsionnoi teorii zadachi Shturma–Liuvillya s singulyarnymi koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 49:9 (2009), 1609–1621 | MR | Zbl

[30] Kh. Tribel, Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR | Zbl

[31] R. Oloff, “Interpolation zwischen den Klassen $\mathfrak S_p$ von Operatoren in Hilberträumen”, Math. Nachr., 46 (1970), 209–218 | DOI | MR | Zbl