Startseite Spectral analysis of dissipative fractional Sturm–Liouville operators
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Spectral analysis of dissipative fractional Sturm–Liouville operators

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Veröffentlicht/Copyright: 15. Juli 2016
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Abstract

We study fractional Sturm–Liouville operators. We give some basic definitions and properties of fractional calculus. Using the method of Pavlov [31, 30, 32], we prove a theorem on the completeness of the system of eigenvectors and associated vectors of dissipative fractional Sturm–Liouville operators.

References

[1] V. M. Adamjan and D. Z. Arov, Unitary couplings of semi-unitary operators (in Russian), Mat. Issled. 1 (1966), no. 2, 3–64; translation in Amer. Math. Soc. Transl. Ser. 2 95 (1970), 75–129. Suche in Google Scholar

[2] B. P. Allahverdiev, On the theory of dilatation and on the spectral analysis of dissipative Schrödinger operators in the case of the Weyl limit circle (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 242–257; translation in Math. USSR-Izv. 36 (1991), no. 2, 247–262. Suche in Google Scholar

[3] B. P. Allahverdiev, Spectral analysis of dissipative Dirac operators with general boundary conditions, J. Math. Anal. Appl. 283 (2003), no. 1, 287–303. 10.1016/S0022-247X(03)00293-2Suche in Google Scholar

[4] B. P. Allahverdiev, Dissipative Schrödinger operators with matrix potentials, Potential Anal. 20 (2004), no. 4, 303–315. 10.1023/B:POTA.0000009815.97987.26Suche in Google Scholar

[5] B. P. Allahverdiev, Dissipative second-order difference operators with general boundary conditions, J. Difference Equ. Appl. 10 (2004), no. 1, 1–16. 10.1080/1023619031000110912Suche in Google Scholar

[6] B. P. Allahverdiev, Dissipative discrete Hamiltonian systems, Comput. Math. Appl. 49 (2005), no. 7–8, 1139–1155. 10.1016/j.camwa.2004.07.024Suche in Google Scholar

[7] Q. M. Al-Mdallal, An efficient method for solving fractional Sturm–Liouville problems, Chaos Solitons Fractals 40 (2009), no. 1, 183–189. 10.1016/j.chaos.2007.07.041Suche in Google Scholar

[8] E. Bas, Fundamental spectral theory of fractional singular Sturm–Liouville operator, J. Funct. Spaces Appl. 2013 (2013), Article ID 915830. 10.1155/2013/915830Suche in Google Scholar

[9] E. Bas and F. Metin, Spectral properties of fractional Sturm–Liouville problem for diffusion operator, preprint (2012), http://arxiv.org/abs/1212.4761. Suche in Google Scholar

[10] E. Bas and F. Metin, Fractional singular Sturm–Liouville operator for Coulomb potential, Adv. Difference Equ. 2013 (2013), Paper No. 300. 10.1186/1687-1847-2013-300Suche in Google Scholar

[11] C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 3–4, 293–308. 10.1017/S030821050002521XSuche in Google Scholar

[12] C. T. Fulton, Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980/81), no. 1–2, 1–34. 10.1017/S0308210500012312Suche in Google Scholar

[13] Y. P. Ginzburg and N. A. Talyush, Exceptional sets of analytic matrix-functions, contracting and dissipative operators (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1984 (1984), no. 8(267), 9–14; translation in Sov. Math. (Iz. VUZ) 28 (1984), no. 8, 10–17. Suche in Google Scholar

[14] I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, American Mathematical Society, Providence, 1969. Suche in Google Scholar

[15] M. L. Gorbachuk and V. I. Gorbachuk, Boundary Value Problems for Operator-Differential Equations (in Russian), “Naukova Dumka”, Kiev, 1984; translation: Math. Appl. (Soviet Series) 48, Kluwer Academic Publishers, Dordrecht, 1991. 10.1007/978-94-011-3714-0Suche in Google Scholar

[16] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000. 10.1142/3779Suche in Google Scholar

[17] R. S. Johnson, An Introduction to Sturm–Liouville Theory, University of Newcastle, Newcastle, 2006. Suche in Google Scholar

[18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Suche in Google Scholar

[19] M. Klimek and O. P. Argawal, On a regular fractional Sturm–Liouville problem with derivatives of order in (0,1), Proceedings of the 13th International Carpathian Control Conference (High Tatras 2012), IEEE Press, Piscataway (2012), 284–289. 10.1109/CarpathianCC.2012.6228655Suche in Google Scholar

[20] M. Klimek and O. P. Argawal, Fractional Sturm–Liouville problem, Comput. Math. Appl. 66 (2013), no. 5, 795–812. 10.1016/j.camwa.2012.12.011Suche in Google Scholar

[21] A. Kuzhel, Characteristic Functions and Models of Nonselfadjoint Operators, Math. Appl. 349, Kluwer Academic Publishers, Dordrecht, 1996. 10.1007/978-94-009-0183-4Suche in Google Scholar

[22] V. Lakshmikantham and A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. 11 (2007), no. 3–4, 395–402. Suche in Google Scholar

[23] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), no. 8, 2677–2682. 10.1016/j.na.2007.08.042Suche in Google Scholar

[24] P. D. Lax and R. S. Phillips, Scattering Theory, Pure Appl. Math. 26, Academic Press, New York, 1967. Suche in Google Scholar

[25] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Transl. Math. Monogr. 39, American Mathematical Society, Providence, 1975. 10.1090/mmono/039Suche in Google Scholar

[26] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Suche in Google Scholar

[27] M. A. Naĭmark, Linear Differential Operators (in Russian), 2nd ed., Izdat. “Nauka”, Moscow, 1969; translation: Parts I, II, Frederick Ungar Publishing, New York, 1967/1968. Suche in Google Scholar

[28] N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Math. Surveys Monogr. 93, American Mathematical Society, Providence, 2002. Suche in Google Scholar

[29] Z. Odibat and S. Momani, Numerical methods for nonlinear partial differential equations of fractional order, Appl. Math. Model. 32 (2008), no. 1, 28–39. 10.1016/j.apm.2006.10.025Suche in Google Scholar

[30] B. S. Pavlov, Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in eigenfunctions (in Russian), Funkcional. Anal. i Priložen. 9 (1975), no. 2, 87–88; translation in Funct. Anal. Appl. 98 (1975), 172–173. Suche in Google Scholar

[31] B. S. Pavlov, Dilation theory and spectral analysis of nonselfadjoint differential operators (in Russian), Mathematical Programming and Related Questions (Drogobych 1974), Central. Èkonom. Mat. Inst. Akad. Nauk SSSR, Moscow (1976), 3–69; translation in Amer. Math. Soc. Transl. Ser. 2 115 (1981), 103–142. Suche in Google Scholar

[32] B. S. Pavlov, Selfadjoint dilation of a dissipative Schrödinger operator, and expansion in its eigenfunction (in Russian), Mat. Sb. (N.S.) 102(144) (1977), no. 4, 511–536; translation in Math. USSR-Sbornik 31 (1977), no. 4, 457–478. Suche in Google Scholar

[33] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Suche in Google Scholar

[34] M. Rivero, J. J. Trujillo and M. P. Velasco, A fractional approach to the Sturm–Liouville problem, Cent. Eur. J. Phys. 11 (2013), no. 10, 1246–1254. 10.2478/s11534-013-0216-2Suche in Google Scholar

[35] L. I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables (in Russian), Izdat. “Nauka”, Moscow, 1971; translation in Transl. Math. Monogr. 44, American Mathematical Society, Providence, 1974. 10.1090/mmono/044Suche in Google Scholar

[36] E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren Math. Wiss. 316, Springer, Berlin, 1997. 10.1007/978-3-662-03329-6Suche in Google Scholar

[37] S. Saltan and B. P. Allahverdiev, Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential, J. Math. Anal. Appl. 303 (2005), no. 1, 208–219. 10.1016/j.jmaa.2004.08.031Suche in Google Scholar

[38] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. Suche in Google Scholar

[39] A. A. Shkalikov, Boundary value problems with the spectral parameter in the boundary conditions, ZAMM Z. Angew. Math. Mech. 76 (1996), 233–235. 10.1002/zamm.19960761416Suche in Google Scholar

[40] B. Sz. -Nagy and C. Foiaş, Analyse Harmonique des Opérateurs de L’espace de Hilbert, Masson et Cie, Paris, 1967; translation in Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing, Amsterdam, 1970. Suche in Google Scholar

[41] A. Zettl, Sturm–Liouville Theory, Math. Surveys Monogr. 121, American Mathematical Society, Providence, 2005. Suche in Google Scholar

Received: 2014-5-26
Revised: 2014-12-4
Accepted: 2015-2-2
Published Online: 2016-7-15
Published in Print: 2017-9-1

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Heruntergeladen am 8.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/gmj-2016-0026/pdf
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