Abstract
Using the method of similar operators, we study the spectral properties of a fourth-order differential operator with two types of classical boundary conditions. We obtain asymptotics for the spectrum and estimates for the spectral decompositions of this operator. We construct the semigroup generated by the opposite fourth-order differential operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kondrat’ev S. K. and Turbin M. V., “Vizualization of an attractor for a mathematical model of the motion of weakly concentrated water polymer emulsions,” Vestnik Voronezh Gos. Univ. Ser. Fiz. Mat., No. 2, 142–163 (2010).
Bolotin V. V., Nonconservative Problems of the Theory of Elastic Stability [in Russian], Fizmatlit, Moscow (1961).
Bardin B. S. and Furta S. D., “The local theory of existence of periodic wave motions of an infinite beam on nonlinear elastic base,” in: Current Problems of Classical and Celestial Mechanics [in Russian], Èl’f, Moscow, 1998, pp. 13–22.
Sidorkin A. S., Domain Structure in Ferroelectrics and Related Materials, Cambridge, Cambridge Intern. Sci. Publ. (2006).
Badanin A. V. and Belinskiĭ B. P., “Oscillations of a liquid in a bounded cavity with a plate on the boundary,” Comp. Math. Math. Phys., 33, No. 6, 829–835 (1993).
Badanin A. V. and Pokrovskiĭ A. A., “On a model supported by an edge of a plate,” Vestnik SPbGU Ser. 4, 3, No. 18, 94–97 (1994).
Badanin A. V. and Korotyaev E. L., “Spectral estimates for a periodic fourth-order operator,” St. Petersburg Math. J., 22, No. 5, 703–736 (2011).
Djakov P. B. and Mityagin B. S., “Instability zones of periodic 1-dimensional Schrödinger and Dirac operators,” Russian Math. Surveys, 61, No. 4, 663–766 (2006).
Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin etc. (1995).
Dunford N. and Schwartz J. T., Linear Operators. Vol. 3: Spectral Operators, Wiley-Interscience, New York and London (1971).
Naĭmark M. A., Linear Differential Operators. Pts. 1 and 2, Frederick Ungar Publishing Co., New York (1968, 1969).
Agranovich M. S., “Spectral properties of diffraction problems,” in: Generalized Method of Eigenoscillations in Diffraction Theory, Wiley, Berlin, 1999.
Baskakov A. G., “Methods of abstract harmonic analysis in the perturbation of linear operators,” Siberian Math. J., 24, No. 1, 17–32 (1983).
Baskakov A. G., “The averaging method in the perturbation of linear differential operators,” Differential Equations, 21, No. 4, 357–362 (1985).
Baskakov A. G., “A theorem on splitting an operator, and some related questions in the analytic theory of perturbations,” Math. USSR-Izv., 28, No. 3, 421–444 (1987).
Baskakov A. G., Harmonic Analysis of Linear Operators [in Russian], Voronezh Univ., Voronezh (1987).
Baskakov A. G., “Spectral analysis of perturbed nonquasianalytic and spectral operators,” Izv. Math., 45, No. 1, 1–31 (1995).
Baskakov A. G., Derbushev A. V., and Shcherbakov A. O., “The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials,” Izv. Math., 75, No. 3, 445–469 (2011).
Polyakov D. M., “Spectral properties of a differential operator of fourth order,” Vestnik Voronezh Gos. Univ. Ser. Fiz. Mat., No. 1, 179–181 (2012).
Gohberg I. Ts. and Krein M. G., An Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc., Providence, RI (1969).
Henry D., Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, Heidelberg, and New York (1981).
Engel K.-J. and Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, Berlin, and Heidelberg (1999) (Grad. Texts Math.; V. 194).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2015 Polyakov D.M.
The author was supported by the Russian foundation for Basic Research (Grants 13-01-00378 and 14-01-31196), and Grant 14-21-00066 of the Russian Scientific Foundation implemented at Voronezh State University (Section 3).
__________
Voronezh. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 1, pp. 165–184, January–February, 2015.
Rights and permissions
About this article
Cite this article
Polyakov, D.M. Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients. Sib Math J 56, 138–154 (2015). https://doi.org/10.1134/S0037446615010140
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446615010140