# Article

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Keywords:
singular differential operators; property BD; oscillation criteria; principal solution
Summary:
Necessary and sufficient conditions for discreteness and boundedness below of the spectrum of the singular differential operator $\ell(y)\equiv{1\over w(t)}\ddif{(r(t)\ddif{y})}$, $t\in[a,\infty)$ are established. These conditions are based on a recently proved relationship between spectral properties of $\ell$ and oscillation of a certain associated second order differential equation.
References:
[1] C. D. Ahlbrandt D. B. Hinton. R. T. Lewis: Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators. Canad. J. Math. 33 (1981), 229-246. DOI 10.4153/CJM-1981-019-1 | MR 0608867
[2] C. D. Ahlbrandt D. B. Hinton. R. T. Lewis: The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory. J. Math. Anal. Appl. 81 (1981), 234-277. DOI 10.1016/0022-247X(81)90060-3 | MR 0618771
[3] W. A. Coppel: Disconjugacy. Lectures Notes in Math., No. 220, Springer-Verlag, Berlin, 1971. MR 0460785 | Zbl 0224.34003
[4] O. Došlý: Spectral properties of a class of singular differential operators. To appear in Math. Nachr. MR 1484668
[5] O. Došlý J. Komenda: Conjugacy criteria and principal solutions of self-adjoint differential equations. Arch. Math. (Brno) 31 (1995), 217-238. MR 1368260
[6] F. Fiedler: Oscillation criteria for a special class of 2n-order ordinary differential equations. Math. Nachr. 131 (1987), 205-218. DOI 10.1002/mana.19871310119 | MR 0908812 | Zbl 0644.34024
[7] F. Fiedler: About certain fourth-order ordinary differential operators-oscillation and discreteness of their spectrum. Math. Nachr. 142 (1989), 235-250. DOI 10.1002/mana.19891420116 | MR 1017382 | Zbl 0685.34033
[8] I. M. Glazman: Direct Methods of Qualitative Analysis of Singular Differential Operators. Jerusalem, 1965.
[9] P. Hartman: Self-adjoint, non-oscillatory systems of ordinary, second order, linear differential equations. Duke J. Math. 24 (1956), 25-35. DOI 10.1215/S0012-7094-57-02405-5 | MR 0082591
[10] D. B. Hinton R. T. Lewis: Discrete spectra criteria for singular differential operators with middle terms. Math. Proc. Cambridge Philos. Soc. 77 (1975), 337-347. MR 0367358
[11] R. T. Lewis: The discreteness of the spectrum of self-adjoint, even order, differential operators. Proc. Amer. Mat. Soc. 42 (1974), 480-482. DOI 10.1090/S0002-9939-1974-0330608-8 | MR 0330608
[12] M. A. Naimark: Linear Differential Operators. Part II. Ungar, New York, 1968. Zbl 0227.34020
[13] W. T. Reid: Sturmian Theory for Ordinary Differential Equations. Springer-Verlag, New York, 1980. MR 0606199 | Zbl 0459.34001
[14] R. L. Sternberg: Variational methods and nonoscillatory theorems for systems of differential equations. Duke J. Math. 19 (1952), 311-322. MR 0048668
[15] C. A. Swanson: Comparison and Oscillation Theory of Linear Differential Equations. Acad. Press, New York, 1968. MR 0463570 | Zbl 0191.09904
[16] J. Weidmann: Linear Operators in Hilbert Spaces. Springer-Verlag, New York, 1980. MR 0566954 | Zbl 0434.47001

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