Abstract
We carry out spectral analysis of a nonself-adjoint fourth-order differential operator L defined on the Hilbert space L 2[0, 1] by Dirichlet type boundary conditions. We obtain results on the asymptotics of the spectrum of this operator and present equiconvergence estimates for spectral expansions. We show that the operator — L is the generator of an analytic operator semigroup.
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References
Bolotin, V.V., Nekonservativnye zadachi teorii uprugoi ustoichivosti (Nonconservative Problems in the Theory of Elastic Stability), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1961.
Badanin A.V. and Korotyaev E.L., Spectral Estimates for a Periodic Fourth-Order Operator, St. Petersburg Math. J., 2011 vol. 22, no. 5, pp. 703–736.
Agranovich, M.S., Spectral Properties of Diffraction Problems, in Obobshchennyi metod sobstvennykh kolebanii v teorii difraktsii (Generalized Method of Fundamental Oscillations in Diffraction Theory), Voitovich, N.N., Katselembaum, B.Z., and Sivov, A.N., Eds., Moscow, 1977, pp. 289–416.
Minkin A.M., Equiconvergence. A Generalization of the Tamarkin-Stone Theorem, Math. Notes, 1997, vol. 62, no. 6, pp. 789–791.
Sadovnichaya I.V. Equiconvergence of Expansions in Eigenfunctions of Sturm-Liouville Operators with Distributional Potentials in Holder Spaces, Differ. Equ., 2012, vol. 48, no. 5, pp. 681–692.
Djakov, P. and Mityagin, B., Convergence of Spectral Decompositions of Hill Operators with Trigonometric Polynomial Potentials, Math. Ann., 2011, vol. 351, no. 3, pp. 509–540.
Minkin, A.M., Equiconvergence Theorems for Differential Operators, J. Math. Sci., New York, 1999, vol. 96, no. 6, pp. 3631–3715.
Baskakov A.G., Methods of Abstract Harmonic Analysis in the Perturbation of Linear Operators, Siberian Math. J., 1983, vol. 24, no. 1, pp. 17–32.
Baskakov A.G., A Theorem on Splitting an Operator, and Some Related Questions in the Analytic Theory of Perturbations, Math. USSR Izv., 1987, vol. 28, no. 3, pp. 421–444.
Baskakov A.G., Spectral Analysis of Perturbed Nonquasianalytic and Spectral Operators, Izv. Math., 1995, vol. 45, no. 1, pp. 1–31.
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., 2011, vol. 75, no. 3, pp. 445–469.
Dunford, N. and Schwartz, J., Linear Operators. Spectral Operators, New York, 1958. Translated under the title Lineinye operatory. T. 3. Spektral’nye operatory, Moscow: Inostrannaya Literatura, 1974.
Polyakov, D.M., Spectral Properties of Fourth-Order Differential Operator, Vestn. VGU Ser. Fiz.-Mat., 2012, no. 1, pp. 179–181.
Engel, K.-J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Grad. Texts in Math. New York; Berlin; Heidelberg: Springer, 1999.
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Original Russian Text © D.M. Polyakov, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 3, pp. 417–420.
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Polyakov, D.M. Method of similar operators in spectral analysis of a fourth-order nonself-adjoint operator. Diff Equat 51, 421–425 (2015). https://doi.org/10.1134/S0012266115030131
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DOI: https://doi.org/10.1134/S0012266115030131