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January 2019, 18(1): 397-424. doi: 10.3934/cpaa.2019020

## Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators

Received  January 2018 Revised  April 2018 Published  August 2018

In this paper we construct the spectral expansion for the differential operator generated in $L_{2}(-∞, ∞)$ by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.

Citation: Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020
##### References:
 [1] M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential oper ators, Fankts. Anal. Prilozhen, 14 (1980), 14-19. [2] I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120. [3] F. Gesztesy and V. Tkachenko, A criterion for Hill's operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287-353. doi: 10.1007/s11854-009-0012-5. [4] D. C. McGarvey, Differential operators with periodic coefficients in Lp(-∞, ∞), Journal of Mathematical Analysis and Applications, 11 (1965), 564-596. doi: 10.1016/0022-247X(65)90105-8. [5] V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984. [6] M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967. [7] F. S. Rofe-Beketov, The spectrum of nonself-adjoint differential operators with periodic coef ficients, Sov. Math. Dokl., 4 (1963), 1563-1564. [8] E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. [9] V. A. Tkachenko, Eigenfunction expansions associated with one-dimensional periodic differ ential operators of order 2n, Funktsional. Anal. i Prilozhen, 41 (2007), 66-89. doi: 10.1007/s10688-007-0005-z. [10] O. A. Veliev, The spectrum and spectral singularities of differential operators with complex valued periodic coefficients, Differential Cprime Nye Uravneniya, 19 (1983), 1316-1324. [11] O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059. [12] O. A. Veliev, Spectral expansion for a non-self-adjoint periodic differential operator, Russian Journal of Mathematical Physics, 13 (2006), 101-110. doi: 10.1134/S1061920806010109. [13] O. A. Veliev, Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Boundary Value Problems, Volume 2008, Article ID 628973, 22 pp. (2008). [14] O. A. Veliev, Asymptotic analysis of non-self-adjoint Hill's operators, Central European Jour nal of Mathematics, 11 (2013), 2234-2256. doi: 10.2478/s11533-013-0305-x. [15] O. A. Veliev, On the spectral singularities and spectrality of the Hill operator, Operators and Matrices, 10 (2016), 57-71. doi: 10.7153/oam-10-05. [16] O. A. Veliev, Essential spectral singularities and the spectral expansion for the Hill operator, Communication on Pure and Applied Analysis, 16 (2017), 2227-2251. doi: 10.3934/cpaa.2017110.

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##### References:
 [1] M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential oper ators, Fankts. Anal. Prilozhen, 14 (1980), 14-19. [2] I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120. [3] F. Gesztesy and V. Tkachenko, A criterion for Hill's operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287-353. doi: 10.1007/s11854-009-0012-5. [4] D. C. McGarvey, Differential operators with periodic coefficients in Lp(-∞, ∞), Journal of Mathematical Analysis and Applications, 11 (1965), 564-596. doi: 10.1016/0022-247X(65)90105-8. [5] V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984. [6] M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967. [7] F. S. Rofe-Beketov, The spectrum of nonself-adjoint differential operators with periodic coef ficients, Sov. Math. Dokl., 4 (1963), 1563-1564. [8] E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. [9] V. A. Tkachenko, Eigenfunction expansions associated with one-dimensional periodic differ ential operators of order 2n, Funktsional. Anal. i Prilozhen, 41 (2007), 66-89. doi: 10.1007/s10688-007-0005-z. [10] O. A. Veliev, The spectrum and spectral singularities of differential operators with complex valued periodic coefficients, Differential Cprime Nye Uravneniya, 19 (1983), 1316-1324. [11] O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059. [12] O. A. Veliev, Spectral expansion for a non-self-adjoint periodic differential operator, Russian Journal of Mathematical Physics, 13 (2006), 101-110. doi: 10.1134/S1061920806010109. [13] O. A. Veliev, Uniform convergence of the spectral expansion for a differential operator with periodic matrix coefficients, Boundary Value Problems, Volume 2008, Article ID 628973, 22 pp. (2008). [14] O. A. Veliev, Asymptotic analysis of non-self-adjoint Hill's operators, Central European Jour nal of Mathematics, 11 (2013), 2234-2256. doi: 10.2478/s11533-013-0305-x. [15] O. A. Veliev, On the spectral singularities and spectrality of the Hill operator, Operators and Matrices, 10 (2016), 57-71. doi: 10.7153/oam-10-05. [16] O. A. Veliev, Essential spectral singularities and the spectral expansion for the Hill operator, Communication on Pure and Applied Analysis, 16 (2017), 2227-2251. doi: 10.3934/cpaa.2017110.
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