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Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators
Department of Mathematics, Dogus University, Acıbadem, 34722, Kadiköy, Istanbul, Turkey |
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.
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. |
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.
|
| [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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