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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

Oktay Veliev

Department of Mathematics, Dogus University, Acıbadem, 34722, Kadiköy, Istanbul, Turkey

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.

Keywords: Periodic operators, spectral singularity, spectral expansion, singular quasimomenta, nonself-adjoint operators.
Mathematics Subject Classification: Primary: 34L05; Secondary: 34L20.
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.   Google Scholar

[2]

I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984.   Google Scholar

[6]

M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967.  Google Scholar

[7]

F. S. Rofe-Beketov, The spectrum of nonself-adjoint differential operators with periodic coef ficients, Sov. Math. Dokl., 4 (1963), 1563-1564.   Google Scholar

[8]

E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059.   Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[2]

I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

V. P. Mikhailov, On the Riesz bases in L2(0, 1), Sov. Math. Dokl., 25 (1962), 981-984.   Google Scholar

[6]

M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967.  Google Scholar

[7]

F. S. Rofe-Beketov, The spectrum of nonself-adjoint differential operators with periodic coef ficients, Sov. Math. Dokl., 4 (1963), 1563-1564.   Google Scholar

[8]

E. C. Titchmarsh, Eigenfunction Expansion (Part II), Oxford Univ. Press, 1958. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

O. A. Veliev, The spectral resolution of the nonself-adjoint differential operators with periodic coefficients, Differential Cprime Nye Uravneniya, 22 (1986), 2052-2059.   Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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