Spectral analysis of singular Sturm-Liouville operators on time scales

Bilender P. Allahverdiev, Huseyin Tuna

Abstract


In this paper, we consider properties of the spectrum of a Sturm-Liouville
operator on time scales. We will prove that the regular symmetric
Sturm-Liouville operator is semi-bounded from below. We will also give some
conditions for the self-adjoint operator associated with the singular
Sturm-Liouville expression to have a discrete spectrum. Finally, we will
investigate the continuous spectrum of this operator.

Keywords


Sturm-Liouville operator; time scales; splitting method; discrete spectrum; continuous spectrum

Full Text:

PDF

References


Agarwal, R. P., Bohner, M., Li, W.-T., Nonoscillation and Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 2004.

Anderson, D. R., Guseinov, G. Sh., Hoffacker, J., Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math. 194 (2) (2006), 309-342.

Atici Merdivenci, F., Guseinov, G. Sh., On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (1-2) (2002), 75-99.

Berkowitz, J., On the discreteness of spectra of singular Sturm-Liouville problems, Comm. Pure Appl. Math. 12 (1959), 523-542.

Bohner, M., Peterson, A., Dynamic Equations on Time Scales, Birkhauser, Boston, 2001.

Bohner, M., Peterson, A. (Eds.), Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.

Dunford, N., Schwartz, J. T., Linear Operators, Part II: Spectral Theory, Interscience, New York, 1963.

Friedrics, K., Criteria for the discrete character of the spectra of ordinary differential equations, Courant Anniversary Volume, Interscience, New York, 1948.

Friedrics, K., Criteria for discrete spectra, Comm. Pure. Appl. Math. 3 (1950), 439-449.

Glazman, I. M., Direct methods of the qualitative spectral analysis of singular differential operators, Israel Program of Scientific Translations, Jerusalem, 1965.

Guseinov, G. Sh., Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math. 29 (4) (2005), 365-380.

Guseinov, G. Sh., An expansion theorem for a Sturm-Liouville operator on semiunbounded time scales, Adv. Dyn. Syst. Appl. 3 (1) (2008), 147-160.

Hilger, S., Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math. 18 (1-2) (1990), 18-56.

Hinton, D. B., Lewis, R. T., Discrete spectra criteria for singular differential operators with middle terms, Math. Proc. Cambridge Philos. Soc. 77 (1975), 337-347.

Huseynov, A., Weyl’s limit point and limit circle for a dynamic systems, in: Dynamical Systems and Methods, Springer, New York, 2012, 215-225.

Ismagilov, R. S., Conditions for semiboundedness and discreteness of the spectrum for one-dimensional differential equations, Dokl. Akad. Nauk SSSR 140 (1961), 33-36 (Russian).

Jones, M. A., Song, B., Thomas, D. M., Controlling wound healing through debridement, Math. Comput. Modelling 40 (9-10) (2004), 1057-1064.

Kreyszig, E., Introductory Functional Analysis with Applications, Wiley, New York, 1989.

Lakshmikantham, V., Sivasundaram, S., Kaymakcalan, B., Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.

Molchanov, A. M., Conditions for the discreteness of the spectrum of self-adjoint second-order differential equations, Trudy Moskov. Mat. Obs. 2 (1953), 169-200 (Russian).

Naimark, M. A., Linear Differential Operators, 2nd edition., Nauka, Moscow, 1969, English transl. of 1st edition, Frederick Ungar Publishing Co., New York, 1969.

Rollins, L. W., Criteria for discrete spectrum of singular self-adjoint differential operators, Proc. Amer. Math. Soc. 34 (1972), 195-200.

Rynne, B. P., (L^2) spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328 (2007), 1217-1236.

Spedding, V., Taming nature’s numbers, New Scientist 179 (2003), 28-31.

Thomas, D. M., Vandemuelebroeke, L., Yamaguchi, K., A mathematical evolution model for phytoremediation of metals, Discrete Contin. Dyn. Syst. Ser. B (2) (2005), 411-422.

Weyl, H., Uber gewohnliche Differentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen willkurlicher Funktionen, Math. Ann. 68 (2) (1910), 220-269.




DOI: http://dx.doi.org/10.17951/a.2018.72.1.1-11
Date of publication: 2018-06-25 09:04:02
Date of submission: 2018-06-24 16:20:27


Statistics


Total abstract view - 1175
Downloads (from 2020-06-17) - PDF - 613

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2018 Bilender P. Allahverdiev, Huseyin Tuna