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March  2020, 19(3): 1537-1562. doi: 10.3934/cpaa.2020077

## On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $L_{2}(-\infty, \infty)$

Received  June 2019 Revised  September 2019 Published  November 2019

In this paper we investigate the non-self-adjoint operator$\ H$ generated in $L_{2}(-\infty, \infty)$ by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which $H$ has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator $H$ by investigating the essential spectral singularities.

Citation: O. A. Veliev. On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $L_{2}(-\infty, \infty)$. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1537-1562. doi: 10.3934/cpaa.2020077
##### References:
 [1] L. V. Ahlfors, Complex Analysis, McGRAW-HILL, 1979., [2] P. Djakov and B. S. Mitjagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Mathematische Annalen, 351 (2011), 509-540.  doi: 10.1007/s00208-010-0612-5. [3] M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, New York: Hafner, 1974. [4] M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential operators, Fankts. Anal. Prilozhen, 14 (1980), 14-19. [5] I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120. [6] F. Gesztesy and V. Tkachenko, A criterion for Hill operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287–353. doi: 10.1007/s11854-009-0012-5. [7] N. B. Kerimov, On a Boundary value problem of N. I. Ionkin type, Differential Equations, 49 (2013), 1233–1245. doi: 10.1134/S0012266113100042. [8] D. McGarvey, Operators commuting with translations by one. Part Ⅱ. Differential operators with periodic coefficients in $L_{p}(-\infty, \infty)$, J. Math. Anal. Appl., 11 (1965), 564–596. doi: 10.1016/0022-247X(65)90105-8. [9] D. McGarvey, Operators commuting with translations by one. Part Ⅲ. Perturbation results for periodic differential operators, J. Math. Anal. Appl., 12 (1965), 187–234. doi: 10.1016/0022-247X(65)90033-8. [10] M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967. [11] A. A. Shkalikov and O. A. Veliev, On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 85 (2009), 647–660. doi: 10.1134/S0001434609050058. [12] O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a non-self-adjoint Hill operators with a locally integrable potential, J. Math. Anal. Appl., 265 (2002), 76–90. doi: 10.1006/jmaa.2001.7693. [13] O. A. Veliev, Asymptotic Analysis of Non-self-adjoint Hill Operators, Cent. Eur. J. Math., 11 (2013), 2234–2256. doi: 10.2478/s11533-013-0305-x. [14] O. A. Veliev, On the simplicity of the eigenvalues of the non-self-adjoint Mathieu-Hill operators, Applied and Computational Mathematics, 13 (2014), 122-134. [15] O. A. Veliev, Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators, Journal of Mathematical Analysis and Applications, 422 (2015), 1390–1401. doi: 10.1016/j.jmaa.2014.09.074. [16] O. A. Veliev, On the spectral singularities and spectrality of the Hill's Operator, Operators and Matrices, 10 (2016), 57–71. doi: 10.7153/oam-10-05. [17] 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. [18] O. A. Veliev, Spectral expansion series with parenthesis for the non-self-adjoint periodic differential operators, Communication on Pure and Applied Analysis, 18 (2019), 397–424. doi: 10.3934/cpaa.2019020.

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##### References:
 [1] L. V. Ahlfors, Complex Analysis, McGRAW-HILL, 1979., [2] P. Djakov and B. S. Mitjagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Mathematische Annalen, 351 (2011), 509-540.  doi: 10.1007/s00208-010-0612-5. [3] M. S. P. Eastham, The Spectral Theory of Periodic Differential Operators, New York: Hafner, 1974. [4] M. G. Gasymov, Spectral analysis of a class of second-order nonself-adjoint differential operators, Fankts. Anal. Prilozhen, 14 (1980), 14-19. [5] I. M. Gelfand, Expansion in series of eigenfunctions of an equation with periodic coefficients, Sov. Math. Dokl., 73 (1950), 1117-1120. [6] F. Gesztesy and V. Tkachenko, A criterion for Hill operators to be spectral operators of scalar type, J. Analyse Math., 107 (2009), 287–353. doi: 10.1007/s11854-009-0012-5. [7] N. B. Kerimov, On a Boundary value problem of N. I. Ionkin type, Differential Equations, 49 (2013), 1233–1245. doi: 10.1134/S0012266113100042. [8] D. McGarvey, Operators commuting with translations by one. Part Ⅱ. Differential operators with periodic coefficients in $L_{p}(-\infty, \infty)$, J. Math. Anal. Appl., 11 (1965), 564–596. doi: 10.1016/0022-247X(65)90105-8. [9] D. McGarvey, Operators commuting with translations by one. Part Ⅲ. Perturbation results for periodic differential operators, J. Math. Anal. Appl., 12 (1965), 187–234. doi: 10.1016/0022-247X(65)90033-8. [10] M. A. Naimark, Linear Differential Operators, George G. Harrap, London, 1967. [11] A. A. Shkalikov and O. A. Veliev, On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 85 (2009), 647–660. doi: 10.1134/S0001434609050058. [12] O. A. Veliev and M. Toppamuk Duman, The spectral expansion for a non-self-adjoint Hill operators with a locally integrable potential, J. Math. Anal. Appl., 265 (2002), 76–90. doi: 10.1006/jmaa.2001.7693. [13] O. A. Veliev, Asymptotic Analysis of Non-self-adjoint Hill Operators, Cent. Eur. J. Math., 11 (2013), 2234–2256. doi: 10.2478/s11533-013-0305-x. [14] O. A. Veliev, On the simplicity of the eigenvalues of the non-self-adjoint Mathieu-Hill operators, Applied and Computational Mathematics, 13 (2014), 122-134. [15] O. A. Veliev, Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators, Journal of Mathematical Analysis and Applications, 422 (2015), 1390–1401. doi: 10.1016/j.jmaa.2014.09.074. [16] O. A. Veliev, On the spectral singularities and spectrality of the Hill's Operator, Operators and Matrices, 10 (2016), 57–71. doi: 10.7153/oam-10-05. [17] 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. [18] O. A. Veliev, Spectral expansion series with parenthesis for the non-self-adjoint periodic differential operators, Communication on Pure and Applied Analysis, 18 (2019), 397–424. doi: 10.3934/cpaa.2019020.
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