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On discreteness of spectrum of a second order functional differential operator

Annotation

Necessary and sufficient conditions for discreteness of spectrum for the singular second order functional differential operator of the form 1/ρ(x) (-(p(x) u^' )^'+q(x)u-∫_a^b▒u(s)r(x,ds) ),x∈(a,b),- ∞≤a<b≤∞ are obtained.

Keywords

discreteness of spectrum; functional differential operator

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DOI

10.20310/1810-0198-2018-23-121-5-9

UDC

517.929.7

Pages

5-9

References

1. Molchanov A.M. Ob usloviyakh diskretnosti spektra dlya samosopryazhennykh uravneniy vtorogo poryadka [On conditions of discreteness of the spectrum for selfadjoint second order differential equations]. Trudy Moskovskogo matematicheskogo obshchestva [Proceedings of Moscow Mathematical Society], 1953, no. 2, pp. 169-199. Zbl 0052.10201. (In Russian). 2. Kats I., Kreyn M.G. Kriteriy diskretnosti spektra singulyarnoy struny [Criteria for the discreteness of the spectrum of a singular string]. Izvestiya vysshikh uchebnykh zavedeniy. Matematika [Bulletin of Higher Education Institutions. Mathematics], 1958, vol. 3, no. 2, pp. 136-153. Zbl 0272.34094, MR139804 34.30. (In Russian). 3. Friedrichs K. Spektraltheorie halbbeschr ankter operatoren und anwendung auf die spektralzerlegung von differentialoperatoren. Math. Ann., 1934, no. 109, pp. 465-487, 687-713. Zbl 0008.39203 & Zbl 0009.07205, MR1512905 & MR1512919. 4. Labovskiy S. On the Sturm-Liouville problem for a linear singular functional-differential equation. Russ. Math., 1996, vol. 40, no. 11, pp. 50-56. (English. Russian original. Translation from Iz Vyssh. Uchebn. Zaved., Mat., 1996, vol. 11, no. 414, pp. 48-53). Zbl 0909.34070, MR1442139 (97m:34120). 5. Labovskiy S. On spectral problem and positive solutions of a linear singular functional-differential equation. Functional Differential Equations, 2013, vol. 20, no. 3-4, pp. 179-200. 6. Birman M.Sh. O spektre singulyarnykh kraevykh zadach [On the spectrum of singular boundary-value problems]. Matematicheskiy sbornik [Sbornik: Mathematics], 1961, vol. 55, no. 2, pp. 125-174. Zbl 0104.32601, MR142896. (In Russian). 7. Birman M.S., Solomjak M.Z. Spectral Theory of Self-Adjoint Operators in Hilbert Space. Holland, D. Reidel Publishing Company, 1987. Zbl 0744.47017.

Received

2018-01-18

Section of issue

Scientific articles

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